The Hilbert-Huang Transform in Engineering

DK342X Half Title pg 5/18/05 9:51 PM Page 1

TheHilbert-Huang

Transformin

Engineering

© 2005 by Taylor & Francis Group, LLC

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Boca Raton London New York Singapore

A CRC title, part of the Taylor & Francis imprint, a member of theTaylor & Francis Group, the academic division of T&F Informa plc.

TheHilbert-Huang

Transformin

EngineeringEdited by

Norden HuangNii O. Attoh-Okine


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Preface

Data analysis serves two purposes: to determine the parameters needed to constructa model and to confirm that the model constructed actually represents the phenom-enon. Unfortunately, the data — whether from physical measurements or numericalmodeling — most likely will have one or more of the following problems:

• The total data are too limited.• The data are nonstationary.• The data represent nonlinear processes.

These problems dictate how the data can be analyzed and interpreted.This book describes the formulation and application of the Hilbert-Huang trans-

form (HHT) in various areas of engineering, including structural, seismic, and oceanengineering.

The primary objective of this book is to present the theory of the Hilbert-HuangTransform (HHT) and its application to engineering. The presentation of the bookis such that it can be used as both a reference and a teaching text. The authors ofthe individual chapters provide a strong theoretical background and some newdevelopments before addressing their specific application. This approach demon-strates the versatility of the HHT.

The book comprises 13 chapters, covering more than 300 pages. These chapterswere written by 30 invited experts from 6 different countries.

The book begins with an introduction and some recent developments in HHT.Chapter 2 uses HHT to interpret nonlinear wave systems and provides a compre-hensive analysis on the assessment of rogue waves. Chapter 3 discusses HHTapplications in oceanography and ocean-atmosphere remote sensing data and pre-sents some examples of these applications. Chapter 4 presents a comparison of theenergy flux computation for shooting waves of HHT and wavelet analysis techniques.In Chapter 5, HHT is applied to nearshore waves, and the results are compared tofield data. In Chapter 6, the author uses HHT to characterize the underwater elec-tromagnetic environment and to identify transient manmade electromagnetic distur-bances, where the HHT was able to act as a filter effectively discriminating differentdipole components. Chapter 7 presents a comparative analysis of HHT and wavelettransforms applied to turbulent open channel flow data.

In Chapter 8, nonlinear soil amplification is quantified by using HHT. Chapter9 extends the application of HHT to nonstationary random processes. Chapter 10presents a comparative analysis of HHT wavelet and Fourier transforms in somestructural health-monitoring applications. In Chapter 11, HHT is applied to molec-ular dynamics simulations. Chapter 12 presents a straightforward application of HHTto decomposition of wave jumps. Chapter 13, the concluding chapter, presents


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perspectives on the theory and practices of HHT. It attempts to review HHT appli-cations in biomedical engineering, chemistry and chemical engineering, financialengineering, meteorological and atmospheric studies, ocean engineering, seismicstudies, structural analysis, health monitoring, and system identification. It alsoindicates some directions for future research.

One important feature of the book is the inclusion of a variety of modern topics.The examples presented are real-life engineering problems, as well as problems thatcan be useful for benchmarking new techniques.

The studies reported in this book clearly indicate an increasing interest in HHTand analysis for diversified applications. These studies are expected to stimulate theinterest of other researchers around the world who are facing new challenges in newtheoretical studies and innovative applications.

Norden HuangNASA

Nii O. Attoh-OkineUniversity of Delaware


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Acknowledgments

The editors are grateful to the contributing authors. We also wish to express ourthanks to Tao Woolfe, B. J. Clark, and Michael Masiello of Taylor & Francis forproviding useful feedback and guiding the editors throughout the complex editorialphase.

Norden E. Huang would like to thank Professors Theodore T. Y. Wu of theCalifornia Institute of Technology, and Owen M. Phillips of the Johns HopkinsUniversity for their guidance and encouragement throughout the years, withoutwhich the Hilbert-Huang Transform would not be what it is today.

Nii O. Attoh-Okine, the co-editor of the book, wishes to express his gratitudeto his parents, Madam Charkor Quaynor and Richard Attoh-Okine, for their supportand encouragement through the years.


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Contributors

Nii O. Attoh-OkineCivil Engineering DepartmentUniversity of DelawareNewark, Delaware

Brad BattistaInformation Systems Laboratories, Inc.San Diego, California

Rodney R. BuntzenInformation Systems Laboratories, Inc.San Diego, California

Marcus DätigCivil Engineering DepartmentBergische University WuppertalWuppertal, Germany

Brian DzwonkowskiGraduate College of Marine StudiesUniversity of DelawareNewark, Delaware

Colin M. EdgeGlaxoSmithKline PharmaceuticalsHarlow, United Kingdom

Jonathan W. EssexSchool of ChemistryUniversity of SouthamptonSouthampton, United Kingdom

Michael GabbayInformation Systems Laboratories, Inc.San Diego, California

Robert J. GledhillSchool of ChemistryUniversity of SouthamptonSouthampton, United Kingdom

Ping GuDepartment of Civil EngineeringUniversity of Illinois at Urbana-

ChampaignUrbana, Illinois

Norden E. HuangGoddard Institute for Data AnalysisNASA Goddard Space Flight CenterGreenbelt, Maryland

Paul A. HwangOceanography DivisionNaval Research LaboratoryStennis Space Center, Mississippi

Lide JiangGraduate College of Marine

StudiesUniversity of DelawareNewark, Delaware

Young-Heon JoGraduate College of Marine StudiesUniversity of DelawareNewark, Delaware

James M. KaihatuOceanography DivisionNaval Research LaboratoryStennis Space Center, Mississippi

Michael L. LarsenInformation Systems Laboratories, Inc.San Diego, California

Stephen C. PhillipsSchool of ChemistryUniversity of SouthamptonSouthampton, United Kingdom


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Panayotis PrinosDepartment of Civil EngineeringAristotle University of ThessalonikiThessaloniki, Greece

Ser-Tong QuekDepartment of Civil EngineeringNational University of SingaporeSingapore

Jeffrey RidgwayInformation Systems Laboratories, Inc.San Diego, California

Torsten SchlurmannCivil Engineering DepartmentBergische University WuppertalWuppertal, Germany

Martin T. SwainSchool of ChemistryUniversity of SouthamptonSouthampton, United Kingdom

Puat-Siong TuaDepartment of Civil EngineeringNational University of Singapore Singapore

Albena Dimitrova VeltchevaPort and Airport Research InstituteYokosuka, Japan

Cye H. WaldmanInformation Systems Laboratories, Inc.San Diego, California

David W. WangOceanography DivisionNaval Research LaboratoryStennis Space Center, Mississippi

Quan WangDepartment of Civil EngineeringNational University of Singapore Singapore

Wei WangPhysical Oceanography LaboratoryOcean University of ChinaShandong, P. R. China

Yi-Kwei WenDepartment of Civil EngineeringUniversity of Illinois at Urbana-

ChampaignUrbana, Illinois

Adrian P. WileySchool of ChemistryUniversity of SouthamptonSouthampton, United Kingdom

Xiao-Hai YanGraduate College of Marine

StudiesUniversity of DelawareNewark, Delaware

Athanasios ZerisDepartment of Civil EngineeringAristotle University of ThessalonikiThessaloniki, Greece

Ray Ruichong ZhangDivision of EngineeringColorado School of MinesGolden, Colorado


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Contents

Chapter 1 Introduction to Hilbert-Huang Transform and Some Recent Developments...........................................................................1

Norden E. Huang

Chapter 2 Carrier and Riding Wave Structure of Rogue Waves........................25

Torsten Schlurmann and Marcus Dätig

Chapter 3 Applications of Hilbert-Huang Transform to Ocean-Atmosphere Remote Sensing Research..................................................................59

Xiao-Hai Yan, Young-Heon Jo, Brian Dzwonkowski, and Lide Jiang

Chapter 4 A Comparison of the Energy Flux Computation of Shoaling Waves Using Hilbert and Wavelet Spectral Analysis Techniques ....83

Paul A. Hwang, David W. Wang, and James M. Kaihatu

Chapter 5 An Application of HHT Method to Nearshore Sea Waves...............97

Albena Dimitrova Veltcheva

Chapter 6 Transient Signal Detection Using the Empirical Mode Decomposition .................................................................................121

Michael L. Larsen, Jeffrey Ridgway, Cye H. Waldman, Michael Gabbay, Rodney R. Buntzen, and Brad Battista

Chapter 7 Coherent Structures Analysis in Turbulent Open Channel Flow Using Hilbert-Huang and Wavelets Transforms..............................141

Athanasios Zeris and Panayotis Prinos

Chapter 8 An HHT-Based Approach to Quantify Nonlinear Soil Amplification and Damping.............................................................159

Ray Ruichong Zhang


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Chapter 9 Simulation of Nonstationary Random Processes Using Instantaneous Frequency and Amplitude from Hilbert-Huang Transform .........................................................................................191

Ping Gu and Y. Kwei Wen

Chapter 10 Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms for Selected Applications .............................................213

Ser-Tong Quek, Puat-Siong Tua, and Quan Wang

Chapter 11 The Analysis of Molecular Dynamics Simulations by the Hilbert-Huang Transform .....................................................245

Adrian P. Wiley, Robert J. Gledhill, Stephen C. Phillips, Martin T. Swain, Colin M. Edge, and Jonathan W. Essex

Chapter 12 Decomposition of Wave Groups with EMD Method......................267

Wei Wang

Chapter 13 Perspectives on the Theory and Practices of the Hilbert-Huang Transform .........................................................................................281

Nii O. Attoh-Okine


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1
Introduction to Hilbert-Huang Transform and Some Recent Developments
Norden E. Huang

CONTENTS

1.1 Introduction ......................................................................................................11.2 The Hilbert-Huang Transform .........................................................................91.3 The Recent Developments .............................................................................16

1.3.1 The Normalized Hilbert Transform ...................................................161.3.2 The Confidence Limit ........................................................................191.3.3 The Statistical Significance of IMFs .................................................21

1.4 Conclusion......................................................................................................22References................................................................................................................22

1.1 INTRODUCTION

Hilbert-Huang transform (HHT) is the designated name for the result of empiricalmode decomposition (EMD) and the Hilbert spectral analysis (HSA) methods, whichwere both introduced recently by Huang et al. (1996, 1998, 1999, and 2003),specifically for analyzing data from nonlinear and nonstationary processes. Dataanalysis is an indispensable step in understanding the physical processes, but tradi-tionally the data analysis methods were dominated by Fourier-based analysis. Theproblems of such an approach were discussed in detail by Huang et al. (1998). Asdata analysis is important for both theoretical and experimental studies (for data isthe only real link between theory and reality), we desperately need new methods inorder to gain a deeper insight into the underlying processes that actually generatethe data. The method we really need should not be limited to linear and stationaryprocesses, and it should yield physically meaningful results.

1


2

The Hilbert-Huang Transform in Engineering


The development of the HHT is motivated precisely by such needs: first, becausethe natural physical processes are mostly nonlinear and nonstationary, there are verylimited options in data analysis methods that can correctly handle data from suchprocesses. The available methods are either for linear but nonstationary processes(such as the wavelet analysis, Wagner-Ville, and various short-time Fourier spectro-grams as summarized by Priestley [1988], Cohen [1995], Daubechies [1992], andFlandrin [1999]) or for nonlinear but stationary and statistically deterministic pro-cesses (such as the various phase plane representations and time-delayed imbeddedmethods as summarized by Tong [1990], Diks [1997], and Kantz and Schreiber[1997]). To examine data from real-world nonlinear, nonstationary, and stochasticprocesses, we urgently need new approaches.

Second, the nonlinear processes need special treatment. Other than periodicity,we want to learn the detailed dynamics in the processes from the data. One of thetypical characteristics of nonlinear processes, proposed by Huang et al. (1998), isthe intra-wave frequency modulation, which indicates that the instantaneous fre-quency changes within one oscillation cycle. Let us examine a very simple nonlinearsystem given by the nondissipative Duffing equation as

(1.1)

where ε is a parameter not necessarily small and γ is the amplitude of a periodicforcing function with a frequency ω.

In Equation 1.1, if the parameter ε were zero, the system is linear, and thesolution would be easy. For a small, however, the system is nonlinear, but it couldbe solved easily with perturbation methods. If ε is not small compared to unity, thenthe system is highly nonlinear. No known general analytic method is available forthis condition; we have to resort to numerical solutions, where all kinds of compli-cations such as bifurcations and chaos can result.

Even with these complications, let us examine the qualitative nature of thesolution for Equation 1.1 by rewriting it in a slightly different way, as

(1.2)

where the symbols are defined as in Equation 1.1. Then the quantity within theparentheses can be regarded as a variable spring constant or a variable pendulumlength. With this view, we can see that the frequency should be changing fromlocation to location and from time to time, even within one oscillation cycle.

As Huang et al. (1998) pointed out, this intra-frequency frequency variation isthe hallmark of nonlinear systems. In the past, there has been no clear way to depictthis intra-wave frequency variation by using Fourier-based analysis methods, exceptto resort to the harmonics. Even by the classical Hamiltonian approach, in whichthe frequency is defined as the rate of change of the Hamiltonian with respect to

∂∂

+ + =2

23x

tx x tε γ ωcos ,

∂∂

+ + =2

221

x

tx x t( ) cos ,ε γ ω



3


the action, we still cannot gain any more insight, for the definition of the action isan integration of generalized momentum along the generalized coordinates; there-fore, there is no instantaneous value. Thus, the best we could do for any nonlineardistorted waveform in the earlier approaches was to refer to harmonic distortions.Harmonic distortion is, in fact, a rather poor alternative, for it is the result obtainedby imposing a linear structure on a nonlinear system. Consequently, the results maymake perfect mathematical sense, but at the same time they have absolutely nophysical meaning. The physically meaningful way to describe the system should bein terms of the instantaneous frequency.

The easiest way to compute the instantaneous frequency is by the Hilbert trans-form, through which we can find the complex conjugate, y(t), of any real valuedfunction x(t) of Lp class,

(1.3)

in which the P indicates the principal value of the singular integral. With the Hilberttransform, we have

(1.4)

where

(1.5)

Here a is the instantaneous amplitude and θ is the phase function; thus theinstantaneous frequency is simply

(1.6)

As the instantaneous frequency is defined through a derivative, it is very localand can be used to describe the detailed variation of frequency, including theintra-wave frequency variation. As simple as this principle is, the implementation isnot at all trivial. To represent the function in terms of a meaningful amplitude andphase, however, requires that the function satisfy certain conditions. Let us take thelength-of-day (LOD) data, given in Figure 1.1, as an example to explain this require-ment. The daily data covers from 1962 to 2002, for roughly 40 years. After per-forming the Hilbert transform, the polar representation of the data is given in Figure1.2, which is a random collection of intertwined looping curves. The correspondingphase function is given in Figure 1.3, where random finite jumps intersperse within

y t Pxt

d( )( )

,=−

−∞

∞

∫1π

ττ

τ

z t x t j y t a t e j t( ) ( ) ( ) ( ) ,( )= + = θ

a t x y tyx

( ) ; ( ) tan ./

= +( ) = −2 21 2

1θ

ω θ= − ddt

.


4



FIGURE 1.1 (See color insert following page 20). Data of Length-of-Day measure thedeviation from the mean 24-hour-day.

FIGURE 1.2 (See color insert following page 20). Analytic function in complex phaseplane formed by the real data and it Hilbert Transform. It shows not apparent order. After theEMD, the annual cycle is extract and plotted also.



5


the data span. If we follow through the definition of the instantaneous frequency asgiven in Equation 1.6 literally, we would have a totally nonsensical result, as givenin Figure 1.4, where the instantaneous frequency is equally likely to be positive ornegative. Unfortunately, this is exactly the procedure recommended by Hahn (1996).

To show how this should not be the case, let us consider the three curves givenby the following three expressions

(1.7)

shown in Figure 1.5. All three curves are perfect sine functions, but with the meandisplaced: for x1, its mean is exactly zero; for x2, its mean is moved up by half ofits amplitude; for x3, its mean is moved up by 1.5 times its amplitude. As a result,the curve represented by x3 is totally above the zero reference axis. If we performthe Hilbert transform to all three functions given in Equation 1.7, we would getthree different circles in the phase plane, with the centers of two of the circlesdisplaced by the amount of the added constants, as shown in Figure 1.6. Conse-quently, the phase functions from the three circles will be different, as shown inFigure 1.7: for x1, the phase function is a straight line; for x2, the phase function isa wavy line, but the general trend still agrees with the straight line; for x3, the phasefunction is also wavy, but the variation is always within ± π.

FIGURE 1.3 The phase function of the analytic function based on the Length-of-Day data.

x t

x t

x t

1

2

3

0 5

1 5

=

= +

= +

sin ;

. sin ;

. sin ;

ω

ω

ω


6



FIGURE 1.4 Instantaneous frequency obtained form derivative of the phase function withoutdecomposition first. The values are equally like to be positive as negative.

FIGURE 1.5 (See color insert following page 20). Model data to illustrate the fallacy ofthe instantaneous frequency without decomposition.



7


FIGURE 1.6 (See color insert following page 20). The analytic function in complexphase plane of the data given in Figure 1.5.

FIGURE 1.7 (See color insert following page 20). Phase function of the model functiongiven in Figure 1.5.

Ph

ase

An

gle

: rad

ian

s

20Phase Angles for Test Data

15

sin x0.5 + sin x1.5 + sin x

10

5

0

−5800700600500400

Time: (100) second

3002001000


8



Based on these phase functions, the instantaneous frequency values are verydifferent for the three expressions, as shown in Figure 1.8: for x1, the instantaneousfrequency is a constant value, which is exactly what we expected; for x2, theinstantaneous frequency is a variable curve with all positive values; for x3, theinstantaneous frequency is a highly variable curve fluctuating from positive tonegative values. Even if we are prepared to accept negative frequency, the result ofthree different values for the same sine wave with only a displaced mean is veryunsettling: some of the results are, of course, nonsensical. The only meaningfulresult is from the sine curve with a zero mean.

What went wrong was the fact that two of the curves do not have a zero mean,or the envelopes of the curves are not symmetric with respect to the zero axis. Thus,before performing the Hilbert transform, we have to preprocess the data. In the past,any preprocessing usually consisted of band-pass filtering. For some of the datafrom linear and stationary processes, this band-pass filtering method will give thecorrect results. For data from nonlinear and nonstationary processes, however, theband-pass filter will alter the characteristics of the filtered curve. The problem withthe filtering approach is that all the frequency domain filters are Fourier-based, whichmeans they are established under linear and stationary assumptions. When the dataare from nonlinear and nonstationary processes, such Fourier-based analysis willsurely generate spurious harmonics, which are mathematically necessary but phys-ically meaningless, as discussed by Huang et al. (1998, 1999).

Considering these points leads us to this conclusion: The correct preprocessingfor data from nonlinear and nonstationary processes will have to be adaptive and

FIGURE 1.8 (See color insert following page 20). Instantaneous frequency computedfrom the cosine model functions consist of the identical cosine function with different dis-placements.



9


implemented in the time domain. The only method presently known to achieve thisis based on the Hilbert-Huang transform proposed by Huang et al. (1996, 1998,1999, and 2003). This method is the subject of the next section.

1.2 THE HILBERT-HUANG TRANSFORM

The Hilbert-Huang transform is the result of the empirical mode decomposition andthe Hilbert spectral analysis. As the EMD method is more fundamental, and it is anecessary step to reduce any given data into a collection of intrinsic mode functions(IMF) to which the Hilbert analysis can be applied (see, for example, Huang et al.,1996, 1998, and 1999), we will discuss it first. An IMF represents a simple oscillatorymode as a counterpart to the simple harmonic function, but it is much more general:by definition, an IMF is any function with the same number of extrema and zerocrossings, with its envelopes, as defined by all the local maxima and minima, beingsymmetric with respect to zero. Obtaining the EMD consists of the following steps:

For any data as given in Figure 1.9, we first identify all the local extrema andthen connect all the local maxima by a cubic spline line as the upper envelope. Werepeat the procedure for the local minima to produce the lower envelope. The upperand lower envelopes should cover all the data between them. Their mean is desig-nated as m1, as shown in Figure 1.10, and the difference between the data and m1

is the first proto-IMF (PIMF) component, h1:

(1.8)

FIGURE 1.9 Test data to illustrate the procedures of Empirical Mode Decomposition alsoknown as sifting.

x t m h( ) .− =1 1


10



This result is shown in Figure 1.11. The procedure of extracting an IMF is calledsifting. By construction, this PIMF, h1, should satisfy the definition of an IMF, butthe change of its reference frame from rectangular coordinate to a curvilinear onecan cause anomalies, as shown in Figure 1.11, where multi-extrema between suc-cessive zero-crossings still existed. To eliminate such anomalies, the sifting processhas to be repeated as many times as necessary to eliminate all the riding waves. Inthe subsequent sifting process steps, h1 is treated as the data. Then

, (1.9)

where m11 is the mean of the upper and lower envelopes of h1. This process can berepeated up to k times; then, h1k is given by

. (1.10)

Each time the procedure is repeated, the mean moves closer to zero, as shownin Figures 1.12a, b, and c. Theoretically, this step can go on for many iterations, buteach time, as the effects of the iterations make the mean approach zero, they alsomake amplitude variations of the individual waves more even. Yet the variation ofthe amplitude should represent the physical meaning of the processes. Thus thisiteration procedure, though serving the useful purpose of making the mean to bezero, also drains the physical meaning out of the resulting components if carriedtoo far. Theoretically, if one insists on achieving a strictly zero mean, one would

FIGURE 1.10 (See color insert following page 20). The cubic spline upper and the lowerenvelopes and their mean, m1.

h m h1 11 11− =

h m hk k k1 1 1 1( )− − =



11


FIGURE 1.11 (See color insert following page 20). Comparison between data and h1 , asgiven by Equation (1.9). Note most, but not all, riding waves are eliminated in h1.

FIGURE 1.12 (A) (See color insert following page 20). Repeat the sifting using h1 asdata.

Data and h1

Am

pli

tud

e

−10

−8

−6

−4

−2

0

2

4

6

8

10

600550500450400

Time: second

350300250200

h1data

Am

pli

tud

e

200 250 300 350

Time: second

Envelopes and the Mean: h1

Data: h1EnvelopeEnvelopeMean: m2

400 450 500 550−10

−8

−6

−4

−2

0

2

4

6

8

10

600


12



FIGURE 1.12 (B) (See color insert following page 20). Repeat the sifting using h2 as data.

FIGURE 1.12 (C) (See color insert following page 20). After 12 iterations, the firstIntrinsic Mode Function is found.

Am

pli

tud

e

200 250 300 350

Time: second

Envelopes and the Mean: h2

Data: h2EnvelopeEnvelopeMean: m3

400 450 500 550−10

−8

−6

−4

−2

0

2

4

6

8

10

600



13


probably have to make the resulting components purely frequency-modulated functions,with the amplitude becoming constant. Then the resulting component would notretain any physically meaningful information. Thus to attain the delicate balance ofachieving a reasonably small mean and also retaining enough physical meaning inthe resulting component, we have proposed two stoppage criteria. The “stoppagecriterion” actually determines the number of sifting steps to produce an IMF; it isthus of critical importance in a successful implementation of the EMD method.

The first stoppage criterion is similar to the Cauchy convergence test, where wefirst define a sum of the difference, SD, as

(1.11)

then the sifting will stop when SD is smaller than a preassigned value. This definitionis a slight modification from the original one proposed by Huang et al. (1998), wherethe SD was defined simply as

(1.12)

The shortcoming of this old definition as given in Equation 1.12 is that the valueof SD can be dominated by local small values of hk–1, while the definition given inEquation 1.11 sums up all the contributions over the whole duration of the data.Even with this modification, there is still a problem with this seemingly mathemat-ically sound approach: in this definition, the important criterion that the number ofextrema has to equal the number of zero-crossings has not been checked. To over-come this practical difficulty, Huang et al. (1999, 2003) proposed an alternative ina second stoppage criterion.

The second stoppage criterion is based on a number called the S-number, whichis defined as the number of consecutive siftings when the numbers of zero-crossingsand extrema are equal or at most differing by one; it requires that number shallremain unchanged. Through exhaustive testing, Huang et al. (2003) used this S-num-ber method of defining a stoppage criterion to establish a confidence limit for theEMD, to be discussed later.

When the resulting function satisfies either of the criteria given above, thiscomponent is designated as the first IMF, c1, as shown in Figure 1.12c. We can thenseparate c1 from the rest of the data by

(1.13)

SD

h t h t

h t

k k

t

T

k

t

T=

−−=

−=

∑∑

1

2

0

12

0

( ) ( )

( )

;

SDh t h t

h tk k

kt

T

=−−

−=∑ 1

2

12

0

( ) ( )

( ).

X t c r( ) .− =1 1


14



This resulting residue is shown in Figure 1.13. Since the residue, r1, still containsinformation with longer periods, it is treated as the new data and subjected to thesame process as described above. This procedure can be repeated to all the subse-quent rj’s, and the result is

. (1.14)

By summing up Equation 1.13 and Equation 1.14, we finally obtain

(1.15)

Thus, we achieve a decomposition of the data into n IMF modes, and a residue,rn, which can be either a constant, a monotonic mean trend, or a curve having onlyone extremum. Recent studies by Flandrin et al. (2004) and Wu and Huang (2004)established that the EMD is a dyadic filter, and it is equivalent to an adaptive wavelet.Since it is adaptive, we avoid the shortcomings of using an a priori–defined waveletbasis, and we also avoid the spurious harmonics that would have resulted. The

FIGURE 1.13 (See color insert following page 20). Comparison between data and theresidue, r1, after the first IMF, c1, is removed. Notice the residue behaves like a moving meanto the data that bisect all the waves.

Am

pli

tud

e

200 250 300 350

Time: second

Data and residue: r1

DataResidue: r1

400 450 500 550−10

−8

−6

−4

−2

0

2

4

6

8

10

600

r c r

r c rn n n

1 2 2

1

− =

− =−

,

...

X t c rj n

j

n

( ) .= +=

∑1


Introduction to Hilbert-Huang Transform and Some Recent Developments 15


components of the EMD are usually physically meaningful, for the characteristicscales are defined by the physical data. The sifting process is, in fact, a Reynolds-typedecomposition: separating variations from the mean, except that the mean is a localinstantaneous mean, so that the different modes are almost orthogonal to each other,except for the nonlinearity in the data.

Having obtained the intrinsic mode function components, we can apply theHilbert transform to each IMF component and compute the instantaneous frequencyas the derivative of the phase function. After performing the Hilbert transform toeach IMF component, we can express the original data as the real part, RP, in thefollowing form:

(1.16)

Equation 1.16 gives both amplitude and frequency of each component as afunction of time. The same data, if expanded in a Fourier representation, would havea constant amplitude and frequency for each component. The contrast between EMDand Fourier decomposition is clear: the IMF represents a generalized Fourier expan-sion with a time-varying function for amplitude and frequency. This frequency–timedistribution of the amplitude is designated as the Hilbert amplitude spectrum, H(,t), or simply the Hilbert spectrum.

With the Hilbert spectrum defined, we can also define the marginal spectrum,h(), as

(1.17)

The marginal spectrum offers a measure of total amplitude (or energy) contri-bution from each frequency value. It represents the cumulated amplitude over theentire data span in a probabilistic sense.

The combination of the EMD and the HSA is known as the Hilbert-Huangtransform for short. Empirically, all tests indicate that HHT is a superior tool fortime–frequency analysis of nonlinear and nonstationary data. It has an adaptive basis,and the frequency is defined through the Hilbert transform. Consequently, there isno need for the spurious harmonics to represent nonlinear waveform deformationsas in any of the a priori basis methods, and there is no uncertainty principle limitationon time or frequency resolution resulting from the convolution pairs possessing apriori bases. Table 1.1 compares Fourier, wavelet, and HHT analyses.

From this table, we can see that the HHT approach is indeed a powerful methodfor the analysis of data from nonlinear and nonstationary processes: it has an adaptivebasis; the frequency is derived by differentiation rather than convolution — therefore,it is not limited by the uncertainty principle; it is applicable to nonlinear andnonstationary data; and it presents the results in time–frequency–energy space forfeature extraction. This basic development of the HHT method has been followed

X(t) = RP a (t) e .j=1

n

ji (t) dtj∑ ∫ ω

h( ) = H( ,t)dt.0

T

ω ω∫


16 The Hilbert-Huang Transform in Engineering


by recent developments that have either added insight to the results or enhancedtheir statistical significance. Some of the recent developments are summarized inthe following section.

1.3 THE RECENT DEVELOPMENTS

We will discuss in some detail recent developments in the areas of the normalizedHilbert transform, the confidence limit, and the statistical significance of IMFs.

1.3.1 THE NORMALIZED HILBERT TRANSFORM

It is well known that, although the Hilbert transform exists for any function of Lp

class, the phase function of the transformed function will not always yield physicallymeaningful instantaneous frequencies, as discussed above. In addition to the require-ment of being an IMF, which is only a necessary condition, additional limitationshave been summarized succinctly in two theorems.

First, the Bedrosian theorem (1963) states that the Hilbert transform for theproduct of two functions, f(t) and h(t), can be written as

(1.18)

only if the Fourier spectra for f(t) and h(t) are totally disjoint in frequency space,and if the frequency content of the spectrum for h(t) is higher than that of f(t). Thislimitation is critical, for we need to have

(1.19)

otherwise, we cannot use Equation 1.5 to define the phase function. According tothe Bedrosian theorem, Equation 1.19 is true only if the amplitude is varying so

TABLE 1.1Comparisons between Fourier, Wavelet, and Hilbert-Huang Transformin Data Analysis

Fourier Wavelet Hilbert

Basis A priori A priori AdaptiveFrequency Convolution: global,

uncertaintyConvolution: regional, uncertainty

Differentiation: local, certainty

Presentation Energy–frequency Energy–time–frequency Energy–time–frequencyNonlinear No No YesNonstationary No Yes YesFeature Extraction No Discrete: no

Continuous: yesYes

Theoretical Base Theory complete Theory complete Empirical

H f t h t f t H h t[ ( ) ( )] ( ) [ ( )] ,=

H a t t a t H t[ ( )cos ( ) ] ( ) [cos ( )] ;θ θ=




slowly that the frequency spectra of the envelope and the carrier waves are disjoint.This has made the application of the Hilbert transform even to IMFs problematic.To satisfy this requirement, Huang and Long (2003) have proposed the normalizationof the IMFs in the following steps: starting from an IMF, we first find all the maximaof the IMFs, defining the envelope by spline through all the maxima and designatingthe envelope as E(t). Now, we normalize the IMF by dividing the IMF by E(t). Thus,we have the normalized function with amplitude always equal to unity.

Even with this normalization, we have not resolved all the limitations on theHilbert transform. The new restriction is given by the Nuttall theorem (1966). Thistheorem states that the Hilbert transform of cosine is not necessarily the sine withthe same phase function for a cosine with an arbitrary phase function. Nuttall gavean error bound, ∆E, defined as the difference between y(t), the Hilbert transform ofthe data, and Q(t), the quadrature (with phase shift of exactly 90°) of the function:

(1.20)

where Sq is Fourier spectrum of the quadrature function. The proof of this theoremis rigorous, but the result is hardly useful, for it gives a constant error bound overthe whole data range. For a nonstationary time series, such a constant bound willnot reveal the location of the error on the time axis.

With the normalized IMF, Huang and Long (2003) have proposed a variableerror bound based on a simple argument, which goes as follows: let us compute thedifference between the squared amplitude of the normalized IMF and unity. If theHilbert transform is exactly the quadrature, then the squared amplitude of thenormalized IMF should be unity; therefore, the difference between it and unityshould be zero. If the squared amplitude is not exactly unity, then the Hilberttransform cannot be exactly the quadrature. Consequently, the error can be measuredsimply by the difference between the squared normalized IMF and unity, which isa function of time. Huang and Long (2003) and Huang et al. (2005) have conducteddetailed comparisons and found the result quite satisfactory.

Even with the error indicator, we can only know that the Hilbert transform isnot exactly the quadrature; we still do not have the correct answer. This promptsthe suggestion of a drastic alternative, eschewing the Hilbert transform totally. Tothis end, Huang et al. (2005) suggest that the phase function can be found bycomputing the arc-cosine of the normalized function. A checking of the results soobtained has also proved to be satisfactory. The only problem is that the imperfectnormalization will give some values greater than unity. Under that condition, thearc-cosine will break down.

An example of the normalized and regular Hilbert transforms is given in Figure1.14, from the data given in Figure 1.12c. There are three different instantaneousfrequency values: the instantaneous frequency from regular Hilbert transform, thenormalized Hilbert transform, and the generalized zero-crossing, which can serveas the standard in the mean. It is easy to see that the normalized instantaneous

∆E y t Q t dt S dq

t

T

= − =−∞=∫∫ ( ) ( ) ( ) ,

20

0

ω ω




frequency is very close to the zero-crossing values, while the regular Hilbert trans-form result gives large undulations that will never result in the mean as given bythe zero-crossing method. The high undulation results from the large changes ofamplitude and some nonlinear distortions of the waveform, both of which will causethe envelope to fluctuate as shown in Figure 1.15. In the normalization scheme, thesmooth spline helps to eliminate many of the undulations in the resulting instanta-neous frequency. One can also see that the problem of the regular Hilbert transformoccurs always at the location where either the amplitudes change drastically or theamplitude is very low, as predicted by the Nuttall theorem. The normalized Hilberttransform alleviates the problems substantially.

Finally, the error index is given in Figure 1.16; here we can see that the erroris also small in general, unless the waveform is locally distorted. Even over the largeerror location, the index values are smaller than 10%, except for the end region,where the end effect of the Hilbert transform causes additional problems. Thus thenormalized Hilbert transform has helped to overcome many of the difficulties of theregular Hilbert transform, and it should be used all the time.

FIGURE 1.14 (See color insert following page 20). Comparison of the instantaneous fre-quency values derived from different methods. Note that the Instantaneous from IMF is stillnot correct when the amplitude fluctuates too much. The normalized Hilbert Transform,however, gives a much better instantaneous frequency when compared with the values derivedfrom the generalized-zero-crossing method.

Comparison of Instantaneous frequency from Different Methods

0 50 100 150 200 250 300 350 400

Time : second

Am

pli

tud

e

DataIF-HIF-NHIF-Z

0.2

0.15

0.1

0.05

0

–0.05




1.3.2 THE CONFIDENCE LIMIT

The confidence limit for the Fourier spectral analysis is routinely computed. Thecomputation, however, is based on the practice of cutting the data into N sectionsand computing spectra from each section. The confidence limit is defined as thestatistical spread of the N different spectra. This practice is based on the ergodictheory, where the temporal average is treated as the ensemble average. The ergodiccondition is satisfied only if the processes are stationary; otherwise, averaging themwill not make sense. Huang et al. (2003) have proposed a different approach, usingthe fact that there are infinitely many ways to decompose one given function intodifferent components. Even using EMD, we can still obtain many different sets ofIMFs by changing the stoppage criteria. For example, Huang et al. (2003) exploredthe stoppage criterion by changing the S-number. Using the length-of-day (LOD)data, they varied the S-number from 1 to 20 and found the mean and the standarddeviation for the Hilbert spectrum given in Figure 1.15. The confidence limit soderived does not depend on the ergodic theory. By using the same data length, thereis also no downgrading of the spectral resolution in frequency space through sub-dividing of the data into sections.

FIGURE 1.15 (See color insert following page 20). The instantaneous amplitude (or theenvelope) of the test data. Note the improvement in adopting the spline envelope over thesimple analytic function.

Comparison of Amplitude from Different Methods

0 50 100 150 200 250 300 350 400

Time : second

Am

pli

tud

e : c

m

DataA-HA-NHA-Z

10

9

8

7

6

5

4

3

2

1

0




Additionally, Huang et al. (2003) invoked the intermittence criterion and forcedthe number of IMFs to be the same for different S-numbers. As a result, they wereable to find the mean for specific IMFs. Figure 1.17 shows the IMF representingvariations of the annual cycle of the length of day. The peak and valley of theenvelope represent the El Niño events. Of particular interest are the periods of highstandard deviations, from 1965 to 1970 and from 1990 to 1995. These periods turnout to be the anomaly periods of the El Niño phenomena, when the sea surfacetemperature readings in the equatorial region were consistently high based on obser-vations, indicating a prolonged heating of the ocean, rather than the changes fromwarm to cool during the El Niño to La Niña changes.

Finally, from the confidence limit study, an unexpected result was the determi-nation of the optimal S-number. Huang et al. (2003) computed the difference betweenthe individual cases and the overall mean and found that there is always a rangewhere the differences reach a local minimum. Based on their limited experiencefrom different data sets, they concluded that an S-number in the range of 4 to 8performed well. Logic also dictates that the S-number should not be too high (whichwould drain all the physical meaning out of the IMF) nor too low (which wouldleave some riding waves remaining in the resulting IMFs).

FIGURE 1.16 (See color insert following page 20). The Error Index of the normalizedHilbert transform; it has large value whenever the wave form deviated from a smooth sinu-soidal form. But their values are, in general, small except near the ends.

Error Index for Normalized Hilbert Transform

0 50 100 150 200 250 300 350 400

Time : second

Am

pli

tud

e

0.1

0.08

0.06

0.04

0.02

0

–0.02

–0.04

–0.06

–0.08

–0.1




1.3.3 THE STATISTICAL SIGNIFICANCE OF IMFS

The EMD is a method of separating data into different components by their scales.There is always the question: on what is the statistical significance of the IMFsbased? In data that contains noise, how can we separate the noise from informationwith confidence? This question was addressed by both Flandrin et al. (2004) andWu and Huang (2004) through the study of signals consisting of noise only.

Flandrin et al. (2004) studied the fractal Gaussian noises and found that theEMD is a dyadic filter. They also found that when one plotted the mean period androot-mean-square (RMS) values of the IMFs derived from the fractal Gaussian noiseon log–log scale, the results formed a straight line. The slope of the straight line forwhite noise is –1; however, the values change regularly with the different Hurstindices. Based on these results, Flandrin et al. (2004) suggested that the EMD resultscould be used to discriminate what kind of noise one was encountering.

FIGURE 1.17 (See color insert following page 20). The mean envelope of the annualcycle IMF component from LOD data. The peaks of the envelope are all aligned with El Nioevents, when the additional angular momentum imparted to the atmosphere from the overheated Equatorial ocean water. The large scatter of the envelope periods in 1065-70 and 1990-95 represent periods of El Nio anomalies.

Error Index for Normalized Hilbert Transform

1965

MeanAnnual Envelope

M+std

M–std

std

1970 1975 1980 1985 1990 1995 2000

Time : second

Am

pli

tud

e

0.1

0.08

0.06

0.04

0.02

0

–0.02

–0.04

–0.06

–0.08

–0.1




Instead of fractal Gaussian noise, Wu and Huang (2004) studied the Gaussianwhite noise only. They also found the relationship between the mean period andRMS values of the IMFs. Additionally, they have also studied the statistical prop-erties of the scattering of the data and found the bounds of the data distributionanalytically. From the scattering, they deduced a 95% bound for the white noise.Therefore, they concluded that when a data set is analyzed with EMD, if the meanperiod-RMS values exist within the noise bounds, the components most likelyrepresent noise. On the other hand, if the mean period-RMS values exceed the noisebounds, then those IMFs must represent statistically significant information.

1.4 CONCLUSION

HHT is a relatively new method in data analysis. Its power is in the totally adaptiveapproach that it takes, which results in the adaptive basis, the IMFs, from which theinstantaneous frequency can be defined. This offers a totally new and valuable viewof nonstationary and nonlinear data analysis methods. With the recent developmentson the normalized Hilbert transform, the confidence limit, and the statistical signif-icance test for the IMFs, the HHT has become a more robust tool for data analysis,and it is now ready for a wide variety of applications. The development of HHT,however, is not over yet. We still need a more rigorous mathematical foundation forthe general adaptive methods for data analysis, and the end effects must be improvedas well.

REFERENCES

Bedrosian, E. (1963). On the quadrature approximation to the Hilbert transform of modulatedsignals. Proc. IEEE, 51, 868–869.

Cohen, L. (1995). Time-Frequency Analysis. Prentice Hall, Englewood Cliffs, NJ.Diks, C. (1997). Nonlinear Time Series Analysis. World Scientific Press, Singapore.Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia.Flandrin, P. (1999). Time-Frequency/Time-Scale Analysis. Academic Press, San Diego, CA.Flandrin, P., Rilling, G., and Gonçalves, P. (2004). Empirical mode decomposition as a

filterbank. IEEE Signal Proc. Lett. 11 (2): 112–114.Hahn, S. L. (1996). Hilbert Transforms in Signal Processing. Artech House, Boston.Huang, N. E., and Long, S. R. (2003). A generalized zero-crossing for local frequency

determination. U.S. Patent pending. Huang N. E., Long, S. R., and Shen, Z. (1996). Frequency downshift in nonlinear water wave

evolution. Advances in Appl. Mech. 32, 59–117.Huang, N. E., Shen, Z., Long, S. R. (1999). A new view of nonlinear water waves — the

Hilbert spectrum. Ann. Rev. Fluid Mech. 31, 417–457.Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, S. H., Zheng, Q., Tung, C. C., and

Liu, H. H. (1998). The empirical mode decomposition method and the Hilbert spec-trum for non-stationary time series analysis. Proc. Roy. Soc. London, A454, 903–995.

Huang, N. E., Wu, Z., Long, S. R., Arnold, K. C., Blank, K., Liu, T. W. (2005). On instan-taneous frequency. Proc. Roy. Soc. London (submitted).




Huang, N. E., Wu, M. L., Long, S. R., Shen, S. S. P., Qu, W. D., Gloersen, P., and Fan, K.L. (2003). A confidence limit for the empirical mode decomposition and the Hilbertspectral analysis. Proc. Roy. Soc. London, A459, 2317–2345.

Kantz, H., and Schreiber, T. (1997). Nonlinear Time Series Analysis. Cambridge UniversityPress, Cambridge.

Nuttall, A. H. (1966). On the quadrature approximation to the Hilbert transform of modulatedsignals. Proc. IEEE, 54, 1458–1459.

Priestley, M. B. (1988). Nonlinear and nonstationary time series analysis. Academic Press,London.

Tong, H. (1990). Nonlinear Time Series Analysis. Oxford University Press, Oxford.Wu, Z., and Huang, N. E. (2004). A study of the characteristics of white noise using the

empirical mode decomposition method. Proc. Roy. Soc. London, A460, 1597–1611.


DK342X_book.fm copy Page 59 Thursday, May 19, 2005 3:42 PM

3
Applications of Hilbert-Huang Transform to Ocean-Atmosphere Remote Sensing Research Xiao-Hai Yan, Young-Heon Jo, Brian Dzwonkowski, and Lide Jiang
CONTENTS

3.1 Introduction ....................................................................................................603.2 Analyses of TOPEX/Poseidon Sea Level Anomaly Interannual

Variation Using HHT and EOF .....................................................................623.3 Application of HHT to Ocean Color Remote Sensing

of the Delaware Bay ......................................................................................643.4 Mediterranean Outflow and Meddies (O & M)from Satellite

Multisensor Remote Sensing .........................................................................713.5 Conclusion......................................................................................................78Acknowledgments....................................................................................................79References................................................................................................................79

ABSTRACT

The Hilbert-Huang transform (HHT) is a newly developed method for analyzing non-linear and nonstationary processes. Its application in oceanography and oceanatmo-sphere remote sensing research is still in its infancy. In this chapter, we briefly introducethe application of this method in oceanatmosphere remote sensing data analyses andpresent a few examples of such applications.

59


60



3.1 INTRODUCTION

Spectral analysis is a very useful tool to analyze a time series signal. However, thismethod does not fully describe a data set that changes with time. The spectrum givesus the frequencies that exist over the entire duration of the data set. On the otherhand, time–frequency analysis allows us to determine the frequencies at a particulartime. Hence, the fundamental idea of time–frequency analysis is to understand anddescribe phenomena where the frequency content of a signal is changing in time.

Scientists traditionally use short Fourier transform by sliding the window alongthe time axis to get a time–frequency distribution. Since it relies on the traditionalFourier spectral analysis, one has to assume the data to be piecewise stationary.Currently, the most famous time–frequency analysis method is wavelet transform.The most common method used is Morlet wavelet, defined as Gaussian envelopedsine and cosine wave groups with 5.5 waves (1). The problem with Morlet waveletis the leakage generated by the limited length of the basic wavelet function, whichmakes the quantitative definition of the energy–frequency–time distribution difficult.Once the basic wavelet is selected, one has to apply it to analysis of all the data (2).

Recently Huang et al. (3) introduced a new and potentially more robust methodfor time–frequency analysis. This method, the empirical mode decomposition–Hil-bert-Huang transform (EMD-HHT), is applicable to both nonstationary and nonlin-ear signals. In real ocean and ocean atmosphere coupling, most processes are non-linear and nonstationary. One example is that at the onset of El Nino: nonlinearKelvin waves carry warm water from the western Pacific to the east (4). This processis exhibited as a nonlinear pattern in altimeter data. For this reason, we use theEMD-HHT technique in our El Nino study and in many of our other studies.

Since the EMD-HHT is relatively new to the ocean remote sensing community,a brief summary of the technique based on Huang et al. (3) is given in this section.Basically, the EMD-HHT method requires two steps in analyzing the data. The firststep is to decompose time series data into a number of intrinsic mode functions(IMFs). These functions must satisfy the following two conditions: (a) within theentire data set, the total number of extrema (as a function of time) and the totalnumber of zero-crossings must either be equal or differ at most by one, and (b) atany point, the mean value of the envelope defined by the local minima (as a functionof time) and the envelope defined by the local maxima (as a function of time) mustbe zero. The second step is to apply the Hilbert transform to the decomposed IMFsand construct the energy–frequency–time distribution, designated as the Hilbertspectrum. The presentation of the final results of the time–frequency analysis issimilar to the wavelet transform method, which is a spectrogram (time–fre-quency–energy plot). For clarity, a spectral analysis of the corresponding signal isshown next to the spectrogram when we present our results in the next sections.

The decomposition of the time series data (i.e. H(t)) into IMFs uses separatelydefined envelopes of local maxima and minima. Once the extrema are identified, allthe local maxima are connected by a cubic spline to form the upper envelope. Theprocedure is repeated for the local minima to produce the lower envelope. Theirmean is designated as m1(t), and the difference between the time series data andm1(t) is the first component, h1(t). One can repeat this procedure k times, until hk(t)


Applications of Hilbert-Huang Transform

61


is an IMF. Then, hk(t) = c1(t) is the first IMF component of the data. c1(t) shouldcontain the finest scale or the shortest period component of the signal. Then, c1(t)is separated from the data, and the process is repeated until either the componentcn(t) or the residue rn(t) becomes so small that it is less than a predetermined value,or when the residue rn(t) becomes a monotonic function from which no IMF can beextracted.

After all IMFs have been determined, one can check the original data with thesum of the IMF components

. (3.1)

Thus, a decomposition of the data into n empirical modes and a residue rn(t) isachieved. The rn(t) can be either the mean trend or a constant.

After IMFs of the data have been generated, the next step is to apply the Hilberttransform to each IMF time series. For an arbitrary time series X(t), one can defineits Hilbert transform, Y(t), as:

. (3.2)

With this definition, X(t) and Y(t) form a complex conjugate pair, so one has ananalytic signal Z(t) as

, (3.3)

where its amplitude function a(t) is

, (3.4)

and its phase function (t) is

. (3.5)

The instantaneous frequency is defined as

. (3.6)

H t c t r tn

i

n

( ) = ( ) + ( )=∑ 1

1

Y tX

td( ) =

( )−

−∞

∞

∫1π

ττ

τ

Z t X t iY t( ) = ( ) + ( )

a t X t Y t( ) = ( ) + ( ) 2 2

12

θ tY t

X t( ) =

( )( )

arctan

ωθ

=( )d t

dt


62



Once the instantaneous frequency (as function of time) of each IMF has beengenerated, the final result of the time–frequency analysis is similar to the wavelettransform method, time–frequency–energy plot, or spectrogram. The energy densitysimilar to Fourier transform can be generated by summing the spectrogram for thewhole time series along a constant frequency for each frequency.

From the definition of the phase function and instantaneous frequency, we canclearly see that for a simple function such as a = sin(t), the Hilbert spectrum issimply cos(t), and the phase function is a monotonic straight line. Hence, thespectrogram is simply a horizontal line for a whole time along fixed frequency, andits spectrum is simply a single peak at that particular frequency.

In the next sections, we describe a few examples of applications of the HHT inocean-atmosphere remote sensing data processing and research. These examplesinclude an analysis of TOPEX/Poseidon sea level anomaly interannual variationusing HHT and empirical orthogonal function (EOF), application of HHT to oceancolor remote sensing of the Delaware Bay, and Mediterranean outflow and Meddiesdetermined from satellite multisensor remote sensing.

3.2 ANALYSES OF TOPEX/POSEIDON SEA LEVEL ANOMALY INTERANNUAL VARIATION USING HHT AND EOF

The measurement of sea surface height provides a unique opportunity to makesignificant contributions to many of the science objectives of oceanatmosphereremote sensing scientists working in the area of climate variability, oceanatmosphereinteractions, and upper ocean dynamics. Satellite altimeter data can be employed togain vital new understanding of the nature of the ocean’s circulation and oceanatmosphere coupling, both in terms of mean state and variability and interrelation-ships between this circulation, its variability, and global climate change.

We used TOPEX/Poseidon (T/P) monthly sea level anomaly (SLA) data fromJanuary 1993 to August 2002, obtained from the University of Texas’ Center forSpace Research (UT/CSR). The deviation of the sea surface was removed from amean surface and was preceded by instrument corrections (ionosphere, wet and drytroposphere, and electromagnetic bias) and geophysical corrections (tides andinverted barometer). The availability of accurate altimeter data, such as that fromT/P, with the root-mean-square (RMS) error as small as 2 cm (5), allows one toexamine the annual and interannual sea surface variability due to El Niño SouthernOscillation (ENSO) and global climate change. The original spatial range was 65°Sto 65°N, 180°E to 180°W; however, we limited the latitude range to 60°S to 60°N.The resolution of the data set was 1° by 1°.

To isolate the signal of the interannual variation of SLA, the annual fluctuationwas removed by subtracting the overall monthly mean for a given month from theindividual monthly data for that respective month. According to calculation, theinterannual energy accounted for about 60% of the total energy. Furthermore, toanalyze the resulting time series, EOF method was applied. The first three EOFs,EOF-1, EOF-2, and EOF-3, accounted for 35.4%, 12.6%, and 9.8% of the totalinterannual energy, respectively.



63


Due to its important role in interannual variation, EOF1 was further analyzedusing the HHT. Empirical mode decomposition (EMD) was applied, and EOF1 wasdecomposed into three IMFs plus a residue (Figure 3.1). The temporal mode ofEOF1 shows a peak in late ‘97, which correspond to the strong 1997–1998 El Niño.To examine the effect of ENSO events on EOF1, the Southern Oscillation Index(SOI) was also used as a reference by decomposing it using EMD. The IMF4 of theSOI appeared to be a smoothed curve of the SOI, and Salisbury and Wimbush (6)used this to predict future ENSO events.

Here a comparison was made between the EOF-1 and the SOI by calculatingthe correlation coefficient between the IMF-3 of the EOF-1 and IMF-4 of the SOI.The correlation coefficient was found to be as high as 0.85 (Figure 3.2). Consideringthis, the EOF-1 can be considered to behave under the impact of the ENSO, and wecan further infer that the ENSO contributes to about one third of the interannualSLA variation.

Moreover, we can reexamine the spatial mode of EOF1 (Figure 3.1) to see theglobal impact of the ENSO events on human activities. Besides the coastal areas ofAmerica, Australia, and Indonesia, which are most strongly and directly affected,those of east Africa and Japan also are undergoing remarkable ENSO variation. The

FIGURE 3.1 The spatial (upper) and temporal (lower) interannual EOF-1 (left) and EMDof the temporal mode (right) of the T/P-SLA.

−0.02 −0.015 −0.01 −0.005 0

180

Interannual EOF-1 (35.4%)

120W 60W120E60E60S

40S

20S

EQ

20N

40N

60N

0.005 0.01 0.015 0.02

−0.1593 94 95 96 97 98 99 00 01 02

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

93−0.02

−0.01

0

0.01

0.02

94 95 96 97 98

IMF-1 of EOF-1

99 00 01 02

93 94 95 96 97 98 99 00 01 02−0.1

−0.05

0

0.05

0.1IMF-2 of EOF-1

93 94 95 96 97 98 99 00 01 02−0.2

−0.1

0

0.1

0.2IMF-3 of EOF-1

93 94 95 96 97 98 99 00 01 02−0.02

0

0.02

0.04

0.06

0.08Residual of EOF-1


64



impact even spread as far as Antarctica. On the other hand, coastal areas of theAtlantic Ocean are less affected by the ENSO events.

In addition, the HHT spectrum of the EOF1 was investigated to extract thefrequency information (Figure 3.3). The frequency distribution of the EOF1 isbetween 0 and 4 cycle/yr. The 2.5 to 4 cycle/yr range corresponds to the spectrumof the IMF1, which behaves in a relatively random pattern. The 0.5 to 2 cycle/yrrange corresponds to the spectrum of IMF2, which has more energy within thefrequency range 0.5 to 0.8 cycle/yr. The 0.2 to 0.4 cycle/yr range has much higherenergy, which is shown as a dark red curve at the bottom part of the spectrum. Thiscurve corresponds to the frequency feature of IMF3, i.e., a period of 2.5 to 5 yr,which can be considered as the typical frequency of ENSO events. The lowestfrequency part of the 0 to 0.1 cycle/yr range corresponds to the frequency of theresidue, which is the longterm trend. It has a period of tens of years or longer and,therefore, requires much longer time series to analyze.

3.3 APPLICATION OF HHT TO OCEAN COLOR REMOTE SENSING OF THE DELAWARE BAY

The use of satellite-based ocean color data has become an integral part of oceano-graphic studies, aiding in the exploration of a number of important topics. The abilityto collect ocean color data at high temporal and spatial resolutions, which onlysatellite sensors can provide, has given the oceanographic community a rich data

FIGURE 3.2 Correlation between negative IMF-4 of SOI (red) and IMF-3 of EOF-1 (blue)is 0.85. It suggests that the primary interannual EOF is impacted by ENSO.

−0.15

−0.1

−0.05

0

0.05

0.1

030201009998

Correlation = 0.85

9796959493



65


source. However, use of this data source is dependent on the premise that there arediscernable relationships between reflectance exiting the water column and theconstituents in it. A primary constituent of study has been chlorophyll-a on accountof its connection to phytoplankton and thus to primary production and biomass (7).

As coastal management strategies begin to focus on large-scale ecosystem basedprograms, the relationship between chlorophyll-a and primary production and bio-mass has the potential to provide a cost-effective alternative to traditional ship-basedor point source sampling. Monitoring and assessing the health of ecosystem-sizeregions will be much more feasible with satellite-based parameters due to thefrequent repeat periods and large spatial areas covered by satellite sensors. Thus,the quantitative assessment of water constituents from satellite-based ocean colordata holds great promise for implementing dynamic management strategies. How-ever, to interpret ocean color data appropriately, a full understanding of the period-icity of chlorophyll concentrations in a given area is essential. The purpose of thisstudy is to examine the seasonal and interannual chlorophyll-a cycles from satellitedata of the coastal region at the mouth of the Delaware Bay.

Although quantifying chlorophyll concentration from satellite ocean color datahas been successful in the open ocean, coastal areas (case II water) can still presentproblems. Water constituents associated with coastal regions, such as excessivechlorophyll-a concentrations, suspended sediments, and colored dissolved organic

FIGURE 3.3 The HHT spectrum of EOF-1. The dark red curve indicates high energy atfrequencies between 0.2 and 0.4 cycle/yr, which corresponds to the Hilbert spectrum of IMF-3of interannual EOF-1. It indicates that ENSO events have a typical period of 2.5 to 5 yr.

Hil

ber

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material, create an optically complex area in which traditional empirical modelstypically perform poorly (8). However, several studies have used an innovative neuralnetwork (NN) methodology successfully to develop ocean color algorithms in CaseII water that outperform empirical methods (9, 10, 11, 12). NN based algorithmsare advantageous on account of their ability to model complex, nonlinear geophysicaltransfer functions like those required to relate chlorophyll-a concentrations to thereflectance measured by a satellite sensor (13). The NN requires training dataconsisting of input data paired with the desired output, from which it “learns” thegeophysical relationship. The ability of a NN to generalize (i.e. accurately predictthe geophysical parameter) is assessed through a validation set of pair data that theNN has never encountered. Given the success of NNs in coastal water algorithmdevelopment, the chlorophyll-a data used in this study was produced from a NNtrained and validated using the Sea-Viewing Wide Field-of-View Sensor (SeaWiFS)remotely sensed reflectance data paired with in situ chlorophyll data in the DelawareBay and its adjacent coastal water. These paired data points were collected duringa number of different days and seasons. The neural network showed significantimprovement over the current SeaWiFS processing algorithm (Ocean Color 4 [OC4])and was used to reprocess the satellite data used in this study (14).

Over the course of the SeaWiFS project, daily high-resolution picture transmis-sion (HRPT) images of the Delaware Bay and the adjacent coastal zone weresubjectively viewed to identify all the cloud-free images of the Delaware Bay andthe adjacent coastal zone from September 1997 to July 2003. This period coversalmost six years of data, from which 468 images were collected, with an averageof 6.6 images per month. The daily images were obtained at the L1A processinglevel and converted to a remapped L2 remote sensing reflectance (Rrs) product. Theremapping used a standard Mercator projection with a fixed grid of 890 × 890 pixels.Each image ranges from 34°N to 42°N latitude and from 68°W to 78°W longitude,which covers approximately 9.58 × 105 km2. The Rrs product was then used as inputfor the optimal NN model mentioned earlier to produce daily chlorophyll-a mapsof the Delaware Bay and the adjacent coastal zone. Since NNs are extremely goodat interpolation but poor at extrapolation, only a limited subset (the 121 × 121 pixelDelaware Bay and adjacent coastal zone) of the entire SeaWiFS image was processedusing the NN. This ensured that the model would only be applied to the regionwhere it was specifically trained with in situ data. The SeaWiFS imagery was binnedby month and geometrically averaged to produce an image of the mean monthlyvalues of chlorophyll-a concentrations for each month. This organizational processcreated spatially coherent patterns of chlorophyll-a values at evenly spaced timescales that help reduce the effects of sensor and algorithm errors and minimize theinfluence of high frequency variability on seasonal distributions of chlorophyll-aconcentrations (15). Furthermore, due to the fact that chlorophyll pigments tend tobe lognormally distributed, the geometric means were used in the various statisticalprocedures applied to chlorophyll-a values in this study (15). A station at the mouthof the Delaware Bay (38.916°N, –75.100°W) was used as the geographic point fromwhich a monthly times series of chlorophyll-a concentrations was produced. Thismonthly time series is shown in Figure 3.4.



67


Hilbert-Huang transform (HHT) analysis provides an innovative way to studythe time–frequency characteristics of time series, due to its ability to handle nonlinearand nonstationary data and to account for changes in frequency over time. This noveltechnique has been applied to the time series of chlorophyll-a concentrations at themouth of the Delaware Bay, in order to better understand the forcing mechanismsassociated with seasonal and interannual variations. The initial chlorophyll-a timeseries was decomposed into IMFs, to which the Hilbert transform can be applied.The resulting Hilbert spectrum and its associated marginal spectrum are shown inFigure 3.5.

The marginal spectrum is conceptually similar to a spectral density graph pro-duced by Fourier analysis; however, it is produced by summing the instantaneousenergy associated with a constant frequency over the entire time period. Thus, themarginal spectrum provides a measurement of the frequencies that contain the largestamounts of energy. As expected, the dominant energy peak occurs at about 0.09,which is the approximate annual forcing frequency and corresponds to period of11.1 months. This result is supported by the fact that several annual physical forcingfactors, such as sea surface temperature, solar insolation, and stream flow, have beenshown to strongly influence chlorophyll-a concentrations in the Delaware Bay andits adjacent coastal zone (8, 16). The fact that the chlorophyll-a periodicity, typically

FIGURE 3.4 The time series of the monthly means of the chlorophyll-a concentrations (toppanel) and the North Atlantic Oscillation index (bottom panel) from September 1997 to July2003.

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presumed to be annual, is not precisely 12 months is not unexpected, since thebiological processes are not simply controlled by physical processes. Examining theHilbert spectrum along the 0.09 frequency shows that there is generally high energyalong this signal, with the exception of two time periods. From January 1999 toOctober 1999, the annual signal completely degenerates, and in the summer of 2001there is a weakening of this signal. Without the ability of the HHT to ascertaininstantaneous frequency, these breakdowns in the annual periodicity would be dif-ficult to determine.

In addition to the strong annual signal, there is a significant energy peak in thelower frequencies. The marginal spectrum indicates an energy peak at the 0.02frequency, which corresponds to a 4.2-yr period. However, looking at the Hilbertspectrum shows that the energy at the 0.02 frequency is not constant; instead, itappears to be roughly oscillating around the 0.02 frequency. This energy begins toincrease in frequency from mid-1999 to late 2001, when the signal briefly weakensand starts to decrease in frequency. This interannual signal is primarily captured inIMF 4 of the empirical mode decomposition (Figure 3.6). The relatively long periodof the mode suggests that a slowly varying basin-scale process could be a forcingcandidate.

One such basin-scale process is the North Atlantic Oscillation (NAO). NAO isa measure of atmospheric conditions focused around the North Atlantic. The NAOcan have a positive or negative phase, as determined by the NAO index, which isthe surface sea-level pressure difference between the Subtropical (Azores) High and

FIGURE 3.5 The Hilbert spectrum (left panel) and marginal spectrum (right panel) for theIMFs of the chlorophyll-a times series.

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the Subpolar Low. Both positive and negative phases affect basin-wide variations inthe North Atlantic jet stream and storm tracks, as well as in large-scale modulationsof the typical patterns of zonal and meridional heat and moisture transport. Thesealterations will consequently affect temperature and precipitation patterns and canextend from eastern North America to western and central Europe (17). Furthermore,recent studies have shown links between the NAO and hydrographic properties alongthe northeast Atlantic shelf slope and the Gulf of Maine, and zooplankton andchlorophyll-a concentrations within the Gulf of Maine (18). Thus, the interannualfrequency of the NAO was examined.

Monthly mean data of the NAO index from September 1997 to July 2003 wereobtained from the NOAA Climate Prediction Center. The NAO index data are shownin Figure 3.4. This time series was also decomposed into its component IMFs inorder to isolate the NAO interannual mode. After decomposing the NAO index timeseries, the fourth IMF contained an interannual signal that exhibited features verysimilar to the fourth chlorophyll-a IMF (Figure 3.6).

To quantify this relationship, cross-correlation analysis was performed on theIMF 4 of both the chlorophyll-a concentrations and the NAO index. The results areshown in Figure 3.7. This study focused only on positive lag time (chlorophyll-atime series following the NAO time series), because it would not be sensible toassume that negative lags (chlorophyll-a time series leading the NAO time series)were meaningful. The correlation analysis reveals that the NAO and chlorophyll-a

FIGURE 3.6 Interannual variation (IMF 4) for both the NAO index (blue line) and thechlorophyll-a (green line) time series.

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concentrations have their highest correlation (correlation coefficient of –0.67) whenchlorophyll-a concentration lags (follows) the NAO by 10 months. The result issignificant at the 95% confidence level, as shown in Figure 3.7. Although a highcorrelation coefficient does not guarantee a relationship between the two times series,the fact that previous studies (mentioned earlier) have shown that the NAO can affectthe interannual time scales of biological and physical processes in oceanic regionsas distant as the Gulf of Maine suggests that this correlation is important.

This study examined the seasonal and interannual variability in satellite derivedchlorophyll-a data from the mouth of the Delaware Bay for a 6-yr period. Byapplying the HHT to the chlorophyll-a time series, a more complete understandingof its time–frequency characteristics was obtained. The resulting Hilbert spectrumdisplayed a strong annual signal, with an unexpected weakening at two brief inter-vals. In addition, the energy at interannual frequencies was compared to the NAO.The fourth IMF of the NAO index and the fourth IMF of the chlorophyll-a concen-trations displayed strong similarities and had a cross-correlation coefficient of –0.67

FIGURE 3.7 Correlation coefficient (blue line) and the related 95 % confidence interval (reddashed line). The maximum correlation coefficient with a positive lag (i.e. chlorophyll-ainterannual variation following NAO index interannual variation) is –0.67, corresponding toa lag of 10 months.

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71


that was significant at the 95% confidence level. This relatively high correlationcoefficient occurred at a lag of 10 months, which suggests that the possible inter-annual link between chlorophyll-a concentrations at the mouth of the Delaware Bayand the NAO index may result from chlorophyll-a concentrations responding tovariations in the NAO index. Finally, the observations of both the weakening of theseasonal signal and the suggested teleconnection of the interannual variation betweenchlorophyll-a concentrations and the NAO are topics worthy of future investigationthat would have been difficult to identify with traditional time series analysis tech-niques.

3.4 MEDITERRANEAN OUTFLOW AND MEDDIES (O & M)FROM SATELLITE MULTISENSOR REMOTE SENSING

One of the most interesting and prominent features of the North Atlantic Ocean isthe salt tongues originating from an exchange flow between the Mediterranean Seaand the Atlantic through the Strait of Gibraltar. The Mediterranean outflow throughthe strait is heavier than Atlantic water due to its higher salt content. Evaporationin the Mediterranean Sea raises water salinity to approximately 38.4 ppt, comparedto 36.4 ppt in the eastern North Atlantic. After leaving the strait under the incominglighter North Atlantic water, the Mediterranean outflow sinks and turns to the right,due to the Coriolis force, following the continental slope of Spain and Portugal.Eventually, the Mediterranean water leaves the coast and spreads out into the middleNorth Atlantic, forming a tongue of salty water and generating clockwise eddies inthe process. These Mediterranean eddies, or Meddies, are rapidly rotating doubleconvex lenses that contain a warm, highly saline core of Mediterranean water. Theyare typically 100 km in diameter, extend over 800 m vertically, and are located ata depth of 1000 m.

Generally speaking, most of the remotely sensed oceanographic observationsare confined to either the sea surface or the top part of the upper mixed layer. Thislimitation is not always true. Yan et al. (19, 20, 21) and Yan and Okubo (22)developed methods to infer the upper ocean mixed layer depths from multisensorsatellite data. Stammer et al. (23) applied Geosat Exact Repeat Mission altimeterdata to investigate the possible surface signal of the Meddies. In this section, wereport a new method to study the O&M by satellite integration analyses of altim-eter, scatterometer, SST, and XBT data with help of HHT. We computed the sealevel difference, namely is the devi-ation of the sea surface topography of the whole vertical water column, which canbe measured by TOPEX/Poseidon (T/P) altimetry. is the sea surface variationin the upper layer (400 m) due to thermal expansion ( ), calculated from XBT.Salt expansion ( ) and wind stress ( ) are computed from satellite scatterometerdata. The HHT was applied to help interpret ∆η′.

To estimate the sea level variation due to thermal expansion in the upper layer,sea surface height anomaly was calculated using monthly mean XBT ( ) data fromthe Joint Environmental Data Analysis (JEDA) Center by the integration of temper-ature down to the 400 m depth from January 1993 to December 1999, i.e.,

∆ ′ = ′ − ′ = ′ − ′ + ′ + ′η η η η η η ηTotal UL Total T S W( ). ′ηTotal

′ηUL

′ηT

′ηS ′ηW

′ηT


72



∆T is the temperature difference between two different depths (dz), and α is thethermal expansion coefficient, which is calculated as function of salinity (S) andwater pressure (P) based on empirical measurements (24);

The spatial resolution of XBT data is 73° × 61° longitude/latitude, which is inter-polated to 1° × 1° longitude/latitude data for this study. It is possible that errorsoccurred due to interpolation of the time rates of change of the integrated upperocean heat storage anomaly (5 W/m2, on average); this is discussed by White andTai (25). We also checked interpolation error using 1° × 1° longitude/latitude opti-mum interpolation sea surface temperature (OISST). We compared interpolated seasurface temperature (SST) from XBT data and OISST by calculating correlationcoefficients. Correlation coefficients between OISST and SST from XBT were over95% with RMS 0.2°C, except for two small areas, and correlation coefficient 80%with RMS 1.2°C. We conclude that interpolation error of XBTs is less than 1°C atthe sea surface.

Because of scarcity of temporal and spatial salinity data, we estimated the effectof in the upper layer indirectly. Two different correlation and RMS calculationswere made: the first correlation is between and OISST, and the second correlationis between and OISST. Good temporal and spatial agreement between SSTand or suggests that a robust regression between fields may have somephysical significance with respect to thermal expansion, but a low correlation withhigh RMS may have some other variability in the mixed layer or below the mixedlayer due to the salinity. It appears that the difference is due to a change in oceansalinity, which is reflected in the T/P sea level measurements but not in the measurements. The first correlation coefficients were all over 90% with smaller RMSthan the second ones, but the second ones were also over 60% to 80%, with largerRMS than the first ones. This result is caused by a warm and salty core below themixed layer. Using Levitus 94 (26), the vertical structure of the salinity was examinedto see the salinity distribution. The whole upper layer in our study domain has auniform salinity distribution above 1000 m. We conclude that the anomaly of isspatially uniform with only small variability.

Wind effect on the signal was considered using scatterometer data fromEuropean Remote Sensing Satellite-1/2 (ERS-1/2) from January 1993 to December1999. First we calculated the correlations between wind stress curl and the forthe temporal variation. Negative correlation would be expected; rising sea level isassociated with negative wind stress curl. Correlation coefficients showed a relationof about –30% in our study area, with southern areas having larger RMS than

′ =−∫η αT Tdz∆ .400

0

αρ

ρ= − 1

0

∆∆T s p( , ).

′ηS

′ηT

′ηTotal

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′ηT

′ηS

′ηTotal

′ηTotal


Applications of Hilbert-Huang Transform 73


northern areas. Second, we considered the magnitude of sea level variation due towind stress based on a linear barotropic vorticity equation, i.e.,

where η is the sea level height, ρ is the water density, g′ is the reduced gravity, f0

is the Coriolis parameter, and τ is the wind stress. We also estimated by usingNASA’s scatterometer: NSCAT (August 1996 to June 1997) and Quikscat (June1999 to present). The mean sea level variation due to wind stress was around 1 cm.

The result of is apparently some variation of the vertical watercolumn, which is a response according to the different incoming or outgoing watermass under the upper layer. Sea level variation due to salinity (temperature) changeusing mass conservation with 800 m thick is around 60 cm (32 cm), with 1-pptdifferences from background fluid (2°C), derived from the salt expansion coefficient7.5 × 10–4/°C and thermal expansion coefficient 2.0 × 10–4/°C, respectively. Thebottom pressure (deep current) variation is negligible at this region (26, 27). How-ever, since the Meddy was a weakly stratified result of extensive salt fingering (28),there were regions of high stratification above and below the lens, where the back-ground isopycnal surface becomes increasingly broader as it moves above the Meddytoward the sea surface. Because of this isopycnal compensation, the O&M are notrevealed in the signal. This is why we cannot detect a Meddy with the altimeterobservation alone.

We computed the absolute differences of the sea surface height anomaly of and to examine the trajectories of the O&M from January 1993 to December1999. Figure 3.8 is an example of such trajectories from July 1998 to June 1999.One can see a strong signature (|∆η′|) toward west and south from July to November.Generally, southward Meddies are formed near 36°N by separation of the frictionalboundary layer at sharp corners (29) and in the Canary Basin (30, 31, 32). Thesouthward travelling Meddies can be explained by the strongest low frequency zonalmotions driven by baroclinic instability (33) and the influence of the neighboringmesoscale features (cyclonic vortices or Azores Current meanders) in the regions(34). The northward O&M over 36° to 40°N are also shown from August to October.Furthermore, a strong signature over 40°N in February is considered to be theinfluence of the returning Gulf Stream. From March to June the weak O&M signa-tures are found because of the weak salinity deviation in 1000 m depth examinedfrom climatological data.

To investigate the reasons for the change in the direction of propagation ofMeddies, the stream function (ψ) was computed by using the T/P altimeter. Thisrepresentation of the stream function permits observations of the interactionsbetween the sea surface gradient and the Meddies’ propagation. The computationof the sea surface height anomaly η′ in terms of the usual dynamic variables isstraightforward, if the flow is assumed to be quasigeostrophic: The stream function(ψ) is defined as

ddt

gf g

xη

ρτ≈ − ′ ∇

0

,

′ηW

∆ ′ = ′ − ′η η ηT P UL

′ηTotal

′ηUL

′ηTotal




FIGURE 3.8 Calculation of the from July 1998 to June 1999.

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, (3.7)

where ψ is the surface quasigeostrophic stream function (35). This is illustrated inFigure 3.9. Due to the seasonal migration of the tropical water toward the north andpolar water toward the south, we can see that the seasonal signals divide into anorthern part and a southern part at 37°N. We compared the Meddies measured bythe floats and the stream functions in all the months in the experimental periods andfound that the Meddies preferred to travel toward the low streamlines and that theystayed within the northmost and southmost streamlines in April and October, respec-tively.

FIGURE 3.9 Comparisons of Meddy trajectories with stream functions (Equation 3.7) forJanuary, April, July, and October 1994.

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FIGURE 3.10 A comparison of the third EMD modes using HHT are shown at givenlocations. The solid curve is for location A (refer to Figure 3.11 for the location) in all panelsfor comparison. The dotted curve is the individual EMD modes for those locations.

3rd

EMD mode using HHT at A(30W, 30N) with other places

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To find out the dominant signal of the sea surface interaction with the Meddies,the HHT was applied to their EMD modes. We chose eight places to investigate thedominant signal in our study area. Figure 3.10 showed that Place Group 1 (B, C,and D, shown in Figure 3.11) and Group 2 (H and G, shown in Figure 3.11) had asimilar signal. However, Group 3 (E and F, shown in Figure 3.11) had slightlydifferent signals from Groups 1 and 2. Consequently, there were seasonal fluctuationsbetween location A and Group 3. To compute the dominant signal of the power forlocations A and E, HHT was also employed; this is shown in Figure 3.12. In both,the dominant frequency was around f = 0.082 (1 year). However, location A had alower frequency, f = 0.03 (33.3 months), which seems to indicate that wind stresswith 33.3-month period produces sea surface forcing. The surface variations producethe baroclinic instability on the Meddies, which is related to southward translation.The southward translation of the Meddies due to the baroclinic effects were consis-tent with that discussed by Müler and Siedler (36) and by Käse and Zenk (37). Ifthe current is vertically sheared in a stratified fluid, baroclinic instability can occur.Figure 3.12 also demonstrates that the comparison between field observations fromtwo float experiments in 1994 and our computation was excellent.

FIGURE 3.11 Comparisons of annual mean |∆η′| signal from two float experiments in 1994.The gray scale and contours show the annual mean of the |∆η′| signal estimated from ourmethod. All Meddies were discovered during the AMUSE experiment (stars) and SEMA-PHORE experiment (circles) in 1994.

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3.5 CONCLUSION

Three examples of applications of the HHT method in ocean-atmosphere remotesensing research were illustrated in this chapter. These examples show that the HHTmethod is indeed a potentially very useful and powerful tool for ocean engineeringand science studies.

FIGURE 3.12 Time–frequency–energy spectrum of the stream functions at locations A andE. Refer to Figure 3.11 for location A and E. One can see the high energy at the 33-monthperiod at location A, but not at location E.

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ACKNOWLEDGMENTS

This research was supported partially by the National Aeronautics and Space Admin-istration (NASA) through Grant NAG5-12745 and NGT5-40024, by the Office ofNaval Research (ONR) through Grant N00014-03-1-0337, and by the NationalOceanic and Atmospheric Administration (NOAA) through Grants NA17EC2449and NA96RG0029.

REFERENCES

1. Chan, Y. T. (1995). Wavelet Basics. Boston, MA: Academic.2. Huang, N., Long, S., Shen, Z. (1996). The mechanism for frequency downshift in

nonlinear wave evolution. Adv. Appl. Mech. 32:59–111.3. Huang, N., Shen, Z., Long, S., Wu, M., Shin, H., Zheng, Q., Yen, N.-C, Tung, C.,

Liu H. (1998). The empirical mode decomposition and Hilbert spectrum for nonlinearand nonstationary time seriesanalysis. Proc. Royal Soc. Mathematical, Phys. Eng.Science 454:903–995.

4. Yan, X.-H., Zheng, Q., Ho, C.-R., Tai, C.-K., Cheney, R. (1994). Development of thepattern recognition and the spatial integration filtering methods for analyzing satellitealtimeter data. Remote Sens. Environ. 48:147–158.

5. Cheney, R., Miller, L., Agreen, R., Doyle, N., Lillibridge, J. (1994). TOPEX/POSEI-DON: 2 cm solution. J. Geophys. Res. 99:24555–24563.

6. Salisbury, J. I., Wimbush, M. (2002). Using modern time series analysis techniquesto predict ENSO events from the SOI times series. Nonlinear Proc. Geophys.9:341–345.

7. Kiddon, J. A., Paul, J. F., Buffum, H. W., Strobel, C. S., Hale, S. S., Cobb, D., Brown,B. S. (2003). Ecological condition of U.S. mid-Atlantic estuaries. 1997–1998. MarinePollution Bulletin 46(10):1224–1244.

8. Keiner, L. E., Yan, X-H. (1998). A neural network model for estimating sea surfacechlorophyll and sediments from thematic mapper imagery. Remote Sens. Environ.66(2):153–165.

9. Zhang, Y., Pulliainen, J., Koponen, S., Hallikainen, M. (2002). Application of anempirical neural network to surface water quality estimation in the Gulf of Finlandusing combined optical data and microwave data. Remote Sens. of Environ.81:327–336.

10. Baruah, P. J., Oki., K., Nishimura, H. (2000). A neural network model for estimatingsurface chlorophyll and sediment content at Lake Kasumi Gaura of Japan. In: Con-ference Processing of GIS Development. Taipei, Taiwan, December.

11. Buckton, D., O’Mongain, E., Danaher, S. (1999). The use of neural networks for theestimation of oceanic constituents based on the MERIS instrument. Int. J. RemoteSensing 20(9):1841–1851.

12. Keiner, L. E., Yan, X.-H. (1997). Empirical orthogonal function analysis of sea surfacetemperature patterns in Delaware Bay. IEEE Trans. Geoscience Remote Sensing25(5):1299–1306.

13. Zhang, T., Fell, F., Liu Z.-S., Preusker, R., Fischer, J., He, M.-X. (2003). Evalatingthe performance of artificial neural network techniques for pigment retrieval fromocean color in Case I Water. J. Geophys. Res. 108(C9):3286–3298.




14. Dzwonkowski, B. (2003). Development and application of a neural network basedocean color algorithm in coastal waters. Masters thesis, University of Delaware,Newark, DE.

15. Barnard, A. H., Stegmann, P. M., Yoder, J. A. (1997). Seasonal surface ocean vari-ability in the South Atlantic Bight derived from CZCS and AVHRR imagery. Conti-nental Shelf Res. 17:1181–1206.

16. Pennock, J. R. (1985). Chlorophyll distributions in the Delaware estuary: Regulationby light-limitation. Estuarine, Coastal and Shelf Science 21:711–725.

17. Hurrell, J. W. (1995). Decadel trends in the North Atlantic Oscillation regionaltemperatures and precipitation. www.cgd.ucar.edu/cas/papers/science1995/sci.htm.

18. Thomas, A. C., Townsend, D. W., Weatherbee, R. (2003). Satellite-measured phy-toplankton variability in the Gulf of Marine. Continential Shelf Res. 23:971–989.

19. Yan, X.-H., Schubel, J. R., Pritchard, D. W. (1990). Oceanic upper mixed layer depthdetermination by the use of satellite data. Remote Sens. Environ. 32 (1):55–74.

20. Yan, X.-H., Okubo, A., Shubel, J. R., Pritchard, D. W. (1991). An analytical mixedlayer remote sensing model. Deep Sea Res. 38:267–287.

21. Yan, X.-H., Niller, P. P, Stewart, R. H. (1991). Construction and accuracy analysisof images of the daily-mean mixed layer depth. Int. J. Remote Sensing12(12):2573–2584.

22. Yan, X.-H., Okubo, A. (1992). Three-dimensional analytical model for the mixedlayer depth. J. Geophy. Res., 97(C12):20201–20226.

23. Stammer D.,. Hinrichsen, H. H, Käse, R. H. (1991). Can Meddies be detected bysatellite altimetry? J. Geophys. Res. 96:7005–7014.

24. Wilson, W., Bradley, D. (1996). Technical Report NOLTR. 66–103.25. White, W. B., Tai, C.-K. (1995). Inferring interannual changes in global upper ocean

heat storage from TOPEX altimetry J. Geophys. Res. 15:24943–24954.26. Levitus, S., Boyer, T. P. (1994). World Ocean Atlas, Vol. 4, Temperature, NOAA Atlas

NESDIS 4, U.S. Govt. Print. Off., Washington, D.C., 117.27. Iorga, M. C., Lozier, M. S. (1999). Signature of the Mediterranean outflow from a

North Atlantic climatology, 1. Salinity and density fields. J. Geophys. Res.104:25985–26009.

28. Hebert, D. L. (1988). A Mediterranean salt lens. Ph.D. dissertation, Dalhousie Uni-versity, Halifax, Nova Scotia.

29. D’Asaro, E. A. (1993). Generation of submesoscale vortices: A new mechanism. J.Geophys. Res. 93:6685–6693.

30. Armi, L., Stommel, H. (1983). Four views of a portion of the North Atlantic sub-tropical gyre. J. Phys. Oceanogr. 13:828–857.

31. Armi, L., Zenk, W. (1984). Large lenses of highly saline Mediterranean salt lens. J.Phys. Oceanogr. 14:1560–1576.

32. Richardson, P. L., Tychensky, A. (1998). Meddy trajectories in the Canary Basinmeasured during the SEMAPHORE experiment, 1993–1995. J. Geophys. Res.103:25029–25045.

33. Spall, M. A., Richardson, P. L., Price, J. (1993). Advection and eddy mixing in theMediterranean salt tongue. J. Mar. Res. 51:797–818.

34. Tychensky, A., Carton, X. (1998). Hydrological and dynamical characterization ofMeddies in the Azores region: A paradigm for baroclinic vortex dynamics. J. Geophys.Res. 103:25061–25079.

35. Douglas, B. C., Cheney, R. E, Agreen, R. W. (1983). Eddy energy of the northwestAtlantic and Gulf of Mexico determined from GEOS 3 altimetry. J. Geophys. Res.88:9595–9603.


http://www.cgd.ucar.edu/



36. Müler, T. J., Siedler, G. (1992). Multi-year current time series in the eastern NorthAtlantic Ocean, J. Mar. Res. 50:63–98.

37. Käse, R. H., Zenk W. (1996). Structure of the Mediterranean water and Meddycharacteristics in the northeastern Atlantic. In: Ocean Circulation and Climate:Observation and Modeling the Global Ocean, W. Krauss, Ed. Berlin: GebrüderBornträger, Berlin, pp. 365–395.



4
A Comparison of the Energy Flux Computation of Shoaling Waves Using Hilbert and Wavelet Spectral Analysis Techniques
Paul A. Hwang, David W. Wang, and James M. Kaihatu

CONTENTS

4.1 Introduction ....................................................................................................844.2 The Huang-Hilbert Spectral Analysis............................................................844.3 Shoaling Waves and Energy Flux Computation............................................85

4.3.1 Field Measurement.............................................................................854.3.2 Numerical Simulations.......................................................................88

4.4 Discussions.....................................................................................................914.4.1 Resolution and Nonlinearity ..............................................................914.4.2 Edge Effect.........................................................................................93

4.5 Summary ........................................................................................................94Acknowledgments....................................................................................................94References................................................................................................................94

83


84



ABSTRACT

The HHT analysis interprets wave nonlinearity in terms of frequency modulationinstead of harmonic generation. The resulting spectrum energy concentrates in theneighborhood of the fundamental frequency in comparison with the FFT or waveletspectrum. The energy flux computation from the HHT spectrum is considerably largerthan with the FFT or wavelet spectrum.

4.1 INTRODUCTION

Fourier transform decomposes a nonlinear waveform into a fundamental frequencycomponent and its harmonics. As a result, the energy of a nonlinear wave is spreadinto higher frequencies. This may produce difficulties in the interpretation of wavepropagation and quantification of wave dynamic properties such as the energy fluxof the wave field. Recently, Norden Huang and his colleagues developed a newanalysis technique, the Hilbert-Huang transformation (HHT). Through analyticalexamples, they demonstrated the excellent frequency and temporal resolution ofHHT for analyzing nonstationary and nonlinear signals [1, 2].

With the HHT analysis, the physical interpretation of nonlinearity is frequencymodulation, which is fundamentally different from harmonic generation. The HHTspectrum therefore retains its energy density near the fundamental frequency of thewave motion in comparison with the FFT or wavelet spectrum [3]. In this article,we briefly describe the HHT technique and investigate its use in the calculation ofthe energy flux of ocean waves. The resulting HHT spectrum is compared with thecounterpart obtained by the wavelet method. The wavelet technique is based onFourier spectral analysis but uses adjustable frequency-dependent window functions— the mother wavelets — to provide temporal/spatial resolution for nonstationarysignals [4–7]. As expected, the Fourier-based analysis interprets wave nonlinearityin terms of harmonic generation; thus the spectral energy leaks to higher frequencycomponents. The HHT interprets wave nonlinearity as frequency modulation, andthe spectral energy remains near the base frequency. The computed energy flux fromthe HHT method is much higher than that from the wavelet method.

4.2 THE HUANG-HILBERT SPECTRAL ANALYSIS

Hilbert transformation was introduced to water wave analysis in the 1980s. Appli-cations include the study of wave modulation leading to wave breaking [8], localproperties of sea waves such as the group and pulse structure, fluctuation of waveenergy and energy flux [9], and the measurement and quantification of breakingevents of wind-generated surface waves [10]. A key function of the Hilbert analysisis the derivation of local wave number in a spatial series or instantaneous frequencyin a time series. To use the Hilbert transformation, proper preprocessing of the signalsis critical. Large errors in the computed local frequency or wave number can occurwhen small waves are riding on longer waves or when sharp changes of the oscil-lation frequencies occur in the wave signal. Quantitative discussions on the riding


A Comparison of the Energy Flux Computation of Shoaling Waves

85


wave problem [1] and on the rapid change of the oscillation frequencies [11] havebeen published and will not be repeated here.

The key ingredient in the HHT is empirical mode decomposition (EMD),designed to reposition the riding waves at the mean water level. This is achievedthrough a sifting process that repeatedly subtracts the local mean from the originalsignals. The procedure decomposes the signal into many modes with differentfrequency characteristics, and thus also alleviates the problem of sharp frequencychange in the original signal. Huang and his coworkers have provided extensivediscussions on the EMD [1, 2]. From experience, even for very complicated randomsignals, a time series can usually be decomposed into a relatively small number ofmodes, M < log2 N. Each mode is free of riding waves and is suitable for Hilberttransformation to yield accurate local frequency of the mode. The spectrum of theoriginal signal can be obtained from the sum of the Hilbert spectra of all modes.Extensive tests have been carried out, and the HHT technique proves to deliver veryhigh frequency and temporal/spatial resolutions in analyzing nonstationary signals.It is also shown to be able to handle the task of analyzing nonlinear signals producedby exact solution with modulating oscillation frequencies, whereas Fourier-basedtechniques interpret the signals as superposition of harmonics [1–3].

4.3 SHOALING WAVES AND ENERGY FLUX COMPUTATION

The Hilbert view of nonlinear waves is significantly different from the conventionalFourier view. In particular, intra-wave modulation is a key signature of nonlinearitybased on the Hilbert spectral analysis, while harmonic generation is typical of theFourier spectrum. The difference in the interpretations of nonlinearity will lead toquantitative differences in the energy flux computation.

4.3.1 FIELD MEASUREMENT

An example of intra-wave modulation highlighted by the HHT analysis is illustratedin the analysis of shoaling swell shown in Figure 4.1. The three-dimensional (3D)surface wave topography was acquired by an airborne topographic mapper (ATM,an airborne scanning laser system) near Duck, N.C., three days after an extratropicalstorm passed through the area. The wind condition at the time of data acquisitionwas low, resulting in a relatively simple two-dimensional (2D) swell system prop-agating on mild slope bathymetry. More discussions of the environmental conditions,measurement techniques, and the bathymetry of the region are reported by Hwanget al. [12]. Figure 4.1 illustrates the spatial evolution of the airborne measuredshoaling swell (Plot a), the Hilbert spectrum (Plot b), and the wavelet spectrum (Plotc) in the near coast region. For reference, the water depth based on the bathymetrydatabase is shown in Plot d. The spectral energy is distributed narrowly about thedominant frequency. The distinctive intra-wave modulation in the neighborhood ofthe peak frequency of the wave spectrum can be clearly identified in the HHTspectrum (Figure 4.1b). In contrast, the wavelet spectrum distributes the energy intoharmonics, as a result of interpreting nonlinearity as harmonic generation in the


86



Fourier-based spectral processing (Figure 4.1c), as discussed earlier. It is noticeablethat the second harmonic of the wavelet spectrum displays a spatial modulationcharacteristic similar to the intra-wave modulation of the HHT spectrum at the peakfrequency.

The energy flux, F, of the wave field can be computed from the wave spectrum,S, by

, (4.1)

where x is the spatial coordinate in the propagation direction, ω is angular frequency,and Cg is the wave group velocity. The computed results based on HHT and waveletanalyses are shown in Figure 4.2. The magnitude derived from the Hilbert spectrumis in general larger than that obtained by the wavelet spectrum. As explained in thelast section, this is partly caused by the much higher spectral level in the lowerfrequency of the Hilbert spectrum compared to the wavelet spectrum. Because phasevelocity and group velocity of gravity waves increase monotonically with wave-length, the increased low frequency spectral density translates to a higher level ofthe energy flux in the wave field.

The source function, Q, of the wave system can be derived from the spatial rateof change of the energy flux:

FIGURE 4.1 (a) A 3D surface wave topography of shoaling swell in the coastal region. (b)The HHT spectrum of the waves shown in (a). (c) The wavelet spectrum of the waves shownin (a). (d) The water depth from bathymetry database.

F x S x C x dg( ) = ( ) ( )∫ ω ω ω; ;



87


. (4.2)

Figure 4.2b and c shows the derived source function based on the Hilbert and waveletspectra, respectively. The magnitude of the Hilbert source term is considerably largerthan the wavelet source term (notice the difference in scale of the two ordinates).The oscillatory nature of the source function highlights the complex nature of thewave conditions in the field. The group structure can be introduced by many pro-cesses, such as wave nonlinearity, interaction among different wave components,and bathymetry-induced wave scattering. The group structure causes the sourcefunction to fluctuate between positive and negative values, as can be visualized fromEquation 4.1 and Equation 4.2. Further discussion of the group structure will bepresented later.

In appearance, the source functions obtained by the HHT and wavelet methods,as shown in Figure 4.2b and c, are considerably different. In reality, the apparentdifference reflects the much higher temporal/spatial resolution of the HHT method.Figure 4.3 shows the results of the running average of HHT computation, with thewavelet result superimposed. With an averaging bin width of 60 elements, therunning average of the HHT source function is almost identical to the wavelet sourcefunction. The spatial average (excluding the leading and trailing 90 m of the spatialcoverage shown) of the source term is –2.49 × 10–4 for the wavelet processing, –5.64× 10–4 for the HHT processing, and –4.78 × 10–4 for the running average of the HHT

FIGURE 4.2 (a) Energy flux computed from the shoaling swell shown in Figure 4.1. (b) Thesource function computed from the HHT spectrum. (b) The source function computed fromthe wavelet spectrum.

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200

1

2

3F

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200

−0.2

0

0.2

<EF(HHT)>=−6.373e−004, <EF(wavelet)>=−3.083e−004, offset=30

d(F)

/dx H

HT

(b)

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200−0.02

0

0.02

d(F)

/dx W

avel

et

x (m)

(c)

∂∂

Fx

Q=


88



processing. From this limited investigation, we conclude that the energy flux com-putation may be off by a factor of two using FFT based processing techniques.

As mentioned earlier, harmonic generation has created major difficulties for theanalysis of wave dynamics due to the dispersive nature of water waves. For example,the phase and group velocities of each free wave component are frequency depen-dent, yet the harmonics (of nonlinear waves) are not dispersive. Therefore, the energyflux (the product of group velocity and spectral density) of the harmonics associatedwith the nonlinear waves cannot be distinguished from the energy flux of the freewaves at the same frequency, and FFT based processing will always underestimatethe energy flux of the wave field. The underestimation will reflect on the magnitudeof parameters such as the dissipation rate or growth rate of a wave field.

4.3.2 NUMERICAL SIMULATIONS

To investigate further the influence of processing methods and wave nonlinearity onthe quantitative results of energy flux computation, numerical simulations werecarried out using the refraction–diffraction (REFDEF) model [13]. The case simu-lated is a train of monochromatic waves (10 sec wave period and 0.25 m initial waveamplitude) propagating from offshore along a plane sloping beach. In the firstscenario, the nonlinear terms are turned off, and the waveform remains sinusoidalalong the full computational domain (Figure 4.4Aa). In the second scenario, thenonlinear terms are turned on, and wave asymmetry becomes obvious in the near-shore region (Figure 4.4Ba). The spatial evolution of the wave spectra for the two

FIGURE 4.3 A comparison of the wavelet source function, the HHT source function, andthe ensemble average of HHT source function with a bin width of 60 spatial elements.

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1<EF(HHT)>=−6.373e−004, <EF(HHT)>

60=−5.522e−004, <EF(wavelet)>=−3.083e−004, offset=30

d(F)

/dx

x (m)

HHT<HHT>

60wavelet



89


(a)

(b)

FIGURE 4.4 Numerical simulations of shoaling waves. Left panels: linear simulation. Rightpanels: nonlinear simulation. (a) Waveform and water depth, (b) HHT spectrum, (c) waveletspectrum, and (d) characteristic wave numbers kp and k2.


90



scenarios by the HHT processing method is shown in Figure 4.4b. The spectraldensity is concentrated in a very narrow frequency band, as expected from a mono-chromatic wave train. The lack of frequency modulation in the absence of nonlin-earity is clear from a comparison of Figure 4.4Ab with Figure 4.4Bb. The resultbased on the wavelet method is shown in Figure 4.4c. The spectral energy is morespread out because of the finite window size, and the effect of nonlinearity isharmonic generation.

The characteristic wave number is typically defined as the peak wave number,kp, or weighted wave number computed from the spectral moment, kn = (∫knS(k)dk/∫S(k)dk)1/n. With the HHT analysis, kp and kn are almost identical for both linear andnonlinear scenarios because the spectral energy is confined in a narrow frequencyrange. With the wavelet analysis, kp and kn are also very similar in the linear scenario,but they differ considerably in the nonlinear scenario as a result of harmonic gen-eration. The results of phase-average kp and k2 for the two scenarios are shown inFigure 4.4d. Intra-wave frequency modulation as reflected by the oscillation of thecharacteristic wave number is clearly seen in the HHT analysis of the nonlinearsimulation; it is also somewhat noticeable but much weaker in the wavelet analysis.

For the linear case, the computed energy flux and the gradient of energy fluxderived from the two processing methods are very similar (Figure 4.5a,b,c). TheHHT results show small intra-wave modulation due to finite amplitude of the wave-form. For the nonlinear case, the intra-wave modulation is greatly amplified by the

FIGURE 4.5 Same as Figure 4.4 but for the energy flux computation. (a) Energy flux, (b)gradient of energy flux, and (c) same as (b) but phase averaged. A: linear simulation. B:nonlinear simulation.

0 500 1000 15000

0.2

0.4

linear

E C

g

(a)

HHTwavelet

0 500 1000 1500−2

0

2x 10

−3

d(E

F)/d

x

x (m)

(b)

0 500 1000 1500−1

0

1x 10

−3

<d(

EF)

/dx>

60

x (m)

(c)

0 500 1000 15000

0.2

0.4

nonnolear

E C

g

(a)

0 500 1000 1500−2

0

2x 10

−3

d(E

F)/d

x

x (m)

(b)

0 500 1000 1500−1

0

1x 10

−3

<d(

EF)

/dx>

60

x (m)

(c)



91


nonlinearity of the wave system. The differentiation process in calculating the gra-dient of energy flux sometimes produces large spikes in the computation. The resultsbecome very sensitive to small modifications in the data processing procedure. Figure4.5 shows an example of the processed results.

4.4 DISCUSSIONS

4.4.1 RESOLUTION AND NONLINEARITY

Fourier-based spectral analysis methods have been widely used for studying randomwaves. One major weakness of the Fourier-based spectral analysis methods is theassumption of linear superposition of wave components. As a result, the energy ofa nonlinear wave is spread into many harmonics, which are phase-coupled via thenonlinear dynamics inherent in ocean waves. In addition to the nonlinearity issue,strictly speaking Fourier spectral analysis should be used for periodic and stationaryprocesses only. Wave propagation in the ocean is certainly neither stationary norperiodic. Using the HHT analysis, the physical interpretation of nonlinearity isfrequency modulation, which is fundamentally different from the commonlyaccepted concept associating nonlinearity with harmonic generation. Huang et al.[1, 2] argued that harmonic generation results from the perturbation method used insolving the nonlinear equation governing the physical processes, thus the harmonicsare produced by the mathematical tools used for the solution rather than being atrue physical phenomenon. Through analytical examples, they demonstrated theexcellent frequency and temporal resolutions of HHT for analyzing nonstationaryand nonlinear signals. Here we give a few computational examples to illustrate thepoints discussed earlier.

Figure 4.6 presents a comparison of HHT and wavelet analysis of three differentcases [3]. Case 1 is an example of an ideal time (t) or space (x) series of sinusoidaloscillations of constant amplitude; the frequency (f) or wave number (k) of the firsthalf of the signal is twice that of the second half (Figure 4.6a). The spectra computedby the HHT and wavelet techniques are displayed in Figure 6b and c, respectively.The HHT spectrum yields very precise frequency resolution as well as high temporalresolution in identifying the sudden change of signal frequency at about the halfpoint of the time series. In comparison, the wavelet spectrum has only a mediocretemporal resolution of the frequency change. There is also a serious leakage problem,and the spectral energy of the simple oscillations spreads over a broad frequencyrange (the contour interval is 3 dB in the spectral plots).

Case 2 is a single cycle sinusoidal oscillation occurring at the middle of theotherwise quiescent signal stream (Figure 4.6d). The precise temporal resolution ofthe HHT method is clearly demonstrated by the sharp rise and fall of the HHTspectrum coincident with the transient signal, as illustrated in Figure 4.6e. In com-parison, the wavelet spectrum is much more smeared, both in the frequency andtemporal resolutions (Figure 4.6f).

Case 3 is a sinusoidal function, y, with its oscillating frequencies subject toperiodic modulation (Figure 4.6g)


92



, (4.3)

where a is the amplitude, ω is the angular frequency, and ε is a small perturbationparameter. This is the exact solution for the nonlinear differential equation [1]

(4.4)

If the perturbation method is used to solve Equation 4.2, the solution to the firstorder of ε is

(4.5)

The HHT spectrum (Figure 4.6h) correctly reveals the nature of oscillatory frequen-cies of the exact solution (Equation 4.3). In contrast, the wavelet spectrum (Figure4.6i) shows a dominant component at the base frequency and periodic oscillationsof the second harmonic component.

FIGURE 4.6 Examples comparing HHT (middle row) and wavelet (bottom row) analysis ofnonlinear and nonstationary signals. (a,b,c) Time series of the harmonic motion with theoscillation period doubled in the second half, and its HHT and wavelet spectra. (d,e,f) Transientsinusoidal function of one cycle. (g,h,i) Oscillatory motion with modulated frequencies.

500 1000 1500−1

0

1

y

(a)

500 1000 15000

0.1

0.2

0.3

k|f

(b)

500 1000 15000

0.1

0.2

0.3

x|t

k|f

(c)

500 1000 1500−1

0

1

y

(d)

500 1000 15000

0.1

0.2

k|f

(e)

500 1000 15000

0.1

0.2

x|t

k|f

(f)

500 550 600−1

0

1

y

(g)

500 550 6000

0.1

0.2

0.33dB contours

k|f

(h)

500 550 6000

0.1

0.2

0.3

x|t

k|f

(i)

y t a t t( ) = +( )cos sinω ε ω

d y

dtt y x t

2

2

2 20 5

21+ +( ) − −( ) =ω εω ω εω ωcos sin.

00.

y t t t t112

1( ) = − = − −( )cos sin cos cos 2 t2ω ε ω ω ε ω

.



93


These three examples illustrate the excellent temporal (spatial) and frequency(wave number) resolution of the HHT method for processing nonlinear and nonsta-tionary signals. The result from Case 3 is especially interesting, as it illustrates theunique ability of the HHT technique to reveal more accurately the nature of nonlinearprocesses with nonstationary oscillation frequencies. Solutions obtained through theperturbation method and analysis results obtained through Fourier-based techniquesrepresent those processes as superpositions of harmonic motions.

4.4.2 EDGE EFFECT

One problem frequently encountered in data processing involves the treatment ofthe two edges of the data segment. This problem is especially serious when the datasegment is short, as in many transient processes. Figure 4.7a shows an example ofone sinusoidal cycle (cos t) between 100 and 300 time marks, shown by the solidcurve. The Hilbert transformation of the data segment (shown by x) deviates fromits exact solution, sin t, at the two edges. Padding zeros to expand the data segmentdoes not alleviate the problem (Figure 4.7a), but repeating the signal sometimeshelps, as shown in Figure 4.7b. However, if the expanded data sequence containsdiscontinuities (Figure 4.7c), the edge problem persists. A technique to avoid dis-continuities at the two edges when expanding the data segment is to mirror imagethe data, as illustrated in Figure 4.7d. Our experience indicates that the edge problemcan be significantly alleviated with about 30% expansion of the original data segmentwith the mirror-imaging method. More elaborate mirror-imaging techniques mayinclude applying windowing (e.g., exponential decay) to the imaged segments (pri-vate communication, N. E. Huang, 2004).

FIGURE 4.7 An example illustrating the mirror-imaging approach to alleviate edge problems.

0 50 100 150 200 250 300−1

0

1

t

f(t)

(a)cosθ (expanded)Hilbert transformsinθ (expanded)

0 50 100 150 200 250 300−1

0

1

t

f(t)

(b)

0 50 100 150 200 250 300−1

0

1

t

f(t)

(c)exp(−0.01t)cosθ (expanded)Hilbert transformexp(−0.01t)sinθ (expanded)

0 50 100 150 200 250 300−1

0

1

t

f(t)

(d)




4.5 SUMMARY

Analyzing nonlinear and nonstationary signals remains a very challenging task.Many methods developed to deal with nonstationarity are based on the concept ofFourier decomposition; therefore, all the shortcomings associated with Fourier trans-formation are also inherent in those methods. The recent introduction of empiricalmode decomposition [1, 2] represents a fundamentally different approach for pro-cessing nonlinear and nonstationary signals. The associated spectral analysis (HHT)provides excellent spatial (temporal) and wave number (frequency) resolution forhandling nonstationarity and nonlinearity [1–3]. The HHT spectrum also results ina considerably different interpretation of nonlinearity (frequency modulation, ascompared to the traditional view of harmonic generation). Applying the techniqueto the problems of ocean waves, we found that the spectral function derived fromHHT is markedly different from those obtained by the Fourier-based techniques.The difference in the resulting spectral functions is attributed to the interpretationof nonlinearity. The Fourier techniques decompose a nonlinear signal into sinusoidalharmonics; therefore, some of the spectral energy at the base frequency is distributedto the higher frequency components. The HHT interprets nonlinearity in terms offrequency modulation, and the spectral energy remains in the neighborhood of thebase frequency. This results in a considerably higher spectral energy at lower fre-quencies and a sharper dropoff at higher frequencies in the HHT spectrum incomparison with the Fourier-based spectra [3]. The energy flux computed by usingthe HHT spectrum is much higher than that obtained from the FFT or waveletmethods.

ACKNOWLEDGMENTS

This work is supported by the Office of Naval Research (Naval Research LaboratoryProgram Elements N61153 and N62435). This is NRL contribution BC/7330.

REFERENCES

1. Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H. H., Zheng, Q., Yuen, N.C., Tung, C. C., and Liu, H. H. (1998). The empirical mode decomposition and theHilbert spectrum for nonlinear and nonstationary time series analysis. Proc. R. Soc.Lond. 454A: 903–995.

2. Huang, N. E., Shen, Z., and Long, S. R. (1999). A new view of nonlinear waterwaves: The Hilbert spectrum. Annu. Rev. Fluid Mech. 31: 417–457.

3. Hwang, P. A., Huang, N. E., and Wang, D. W. (2003). A note on analyzing nonlinearand nonstationary ocean wave data. Appl. Ocean Res. 25: 187–193.

4. Shen, Z., and Mei, L. (1993). Equilibrium spectra of water waves forced by inter-mittent wind turbulence. J. Phys. Oceanogr. 23(9): 2019–2026.

5. Shen, Z., Wang, W., and Mei, L. (1994). Finestructure of wind waves analyzed withwavelet transform. J. Phys. Oceanogr. 24(5): 1085–1094.

6. Liu, P. C. (2000). Is the wind wave frequency spectrum outdated? Ocean Eng. 27:577–588.


A Comparison of the Energy Flux Computation of Shoaling Waves 95


7. Massel, S. R. (2001). Wavelet analysis for processing of ocean surface wave records.Ocean Eng. 28: 957–987.

8. Melville, W. K. (1983). Wave modulation and breakdown. J. Fluid Mech. 128:489–506.

9. Bitner-Gregersen, E. M., and Gran S. (1983). Local properties of sea waves derivedfrom a wave record. Appl. Ocean Res. 5: 210–214.

10. Hwang, P. A., Xu, D., and Wu, J. (1989). Breaking of wind-generated waves: mea-surements and characteristics. J. Fluid Mech. 202: 177–200.

11. Guillaume, D. W. (2002). A comparison of peak frequency-time plots produced withHilbert and wavelet transforms. Rev. Scientific Inst. 73(1): 98–101.

12. Hwang, P. A., Walsh, E. J., Krabill, W. B., Swift, R. N., Manizade, S. S., Scott, J. F.,and Earle, M. D. (1998). Airborne remote sensing applications to coastal waveresearch. J. Geophys. Res. 103(C9): 18791–18800.

13. Kaihatu, J. M., and Kirby, J. T. (1995). Nonlinear transformation of waves in finitewater depth. Phys. Fluids 7: 1903–1914.


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5
An Application of HHT Method to Nearshore Sea Waves
Albena Dimitrova Veltcheva

CONTENTS

5.1 Introduction ....................................................................................................985.2 Field Data.......................................................................................................995.3 Conventional Methods for Wave Data Analysis..........................................1005.4 Hilbert-Huang Transform Method ...............................................................1065.5 Application of the HHT Method .................................................................108

5.5.1 Offshore Waves During Different Sea Stages .................................1125.5.2 Cross-Shore Transformation of Sea Waves .....................................114

5.6 Conclusions ..................................................................................................117References..............................................................................................................118

ABSTRACT

Real sea waves are nonlinear by nature, and this nonlinearity increases considerablyin the shoreward direction. The presence of wave groups in the sea wave records showsthat the wave process is also not stationary, as it is usually considered in conventionalwave data analysis. In this work, the Hilbert-Huang transform (HHT) method fornonlinear and nonstationary time series analysis is applied to wave data from thenearshore area. The field data were collected at Hazaki Oceanographical ResearchStation (HORS), Port and Airport Research Institute, Japan, and covered different seastates. The importance of a proper choice of an analyzing technique for the correctunderstanding of the examined phenomenon is discussed. The ability of EMD, a keypart of the HHT method, to extract into IMFs the different oscillation modes embeddedin the sea surface elevation records is demonstrated. The variation of IMFs along thebeach profile is examined in an attempt to investigate cross-shore transformation ofsea waves. The dominant oscillations for each data record are determined, and theirvariations during different sea stages are investigated.

97


98



5.1 INTRODUCTION

Real sea waves are nonlinear and nonstationary by nature, and their profile isasymmetrical with respect to the zero level, with sharp crests and flat troughs,especially well pronounced in shallow water. The existence of small ripples ridinglonger waves, as well as an appearance of many small waves due to wave breaking,complicates the wave profile considerably. Real sea waves also appear in groups,which are time localized transient events. The wave data records always register amixture of many wave systems with different periods and energy, generated bydifferent sources. Through analysis of sea surface elevation record, we seek todetermine characteristic time–frequency scales of the energy. The widely used meth-ods of wave analysis are burdened with the traditional concept of sea waves: as asine wave with definition domain from minus infinity to plus infinity. It is alsoassumed, according to linear wave theory, that within the framework of one wavethe period is constant and the wave height is equal to twice the wave amplitude. Inaddition to the positive maxima and negative minima of the wave profile, positiveminima and negative maxima are also often observed in the profile of real sea waves.The neglect of these actual features of sea waves leads to incompleteness andmisinterpretation of the recorded sea state, and thus the analyzing method can distortthe investigated phenomenon.

The dominant approach in the existing methods of wave analysis is decompo-sition of the time series into component basis functions. Global sinusoidal compo-nents of fixed amplitude are the basis for data expansion in the commonly usedFourier spectral analysis methods. The parameters of water waves, determined byFourier spectra, are integral characteristics of the sea state. Despite its being wide-spread and useful, the Fourier spectral technique has an important drawback: theFourier spectrum contains no information about the timing of the analyzed process.Consequently, the local time variations of determined wave characteristics are impos-sible to track.

To understand the sea wave process correctly, we need an adequate method foranalysis. In the last decade the attention of scientists started to turn from globallyestimated wave parameters to local properties of a wave system. The attention hasbeen addressed to the time distribution of the wave energy, complimentary to thefrequency distribution information provided by usual Fourier spectral analysis. Liu(1) suggested that the wave frequency spectrum approach is outdated, presentingthe advantages of the wavelet spectrum in the investigation of wave growth. Thedifferences between the Fourier and the wavelet spectrum are widely discussed byMassel (2) on the basis of the application of the wavelet transform for the processingof sea wave data. Huang et al. (3) also point out the necessity to break down theearlier paradigm of wave analysis, emphasizing that Fourier analysis is not a goodmethod for studying nonlinear and nonstationary water waves.

Several Fourier-based methods for frequency–time distribution of the energy arereviewed by Huang et al. (4): time window-width Fourier transform, evolutionaryspectrum, principal component analysis, and wavelet analysis. These methods sufferfrom the demerits of Fourier spectral analysis — global definition of harmoniccomponents and the use of linear superposition for decomposition of the data



99


(assuming the stationarity and linearity of data). There is also a resolution probleminherent in all Fourier-based analysis due to the Heisenberg Uncertainty Principle.Wavelet analysis provides some improvement in the resolution by using adjustablewindows, but the problems of the nonadaptive nature of these windows are still notsolved completely. It has to be mentioned in addition that in these methods forfrequency–time distribution of the energy, the basis for expansion of the data mustbe chosen in advance. An exception is the principal component analysis, in whichthe basis is derived empirically from the data.

In this paper, we discuss the importance of the analyzing technique for the correctunderstanding of the examined phenomenon and pay special attention to the non-linear and nonstationary behavior of real sea waves. We propose a new method fornonlinear and nonstationary time series analysis, the Hilbert-Huang transform (HHT)method, introduced by Huang et al. (4), as an alternative one for investigation ofreal sea waves. The empirical mode decomposition (EMD), a key part of the HHTmethod, extracts the energy associated with various intrinsic time scales into a setof intrinsic mode functions (IMFs). By virtue of their derivation, the IMFs havewell-behaved Hilbert transform results, and the instantaneous frequencies can becalculated. The frequency–time distribution of energy is designated as a Hilbertspectrum. Any event in the data series can be localized in time as well as in frequency.

We apply the HHT to the wave data from the nearshore area and demonstratethe ability of EMD to extract into IMFs the different oscillation modes embeddedin the sea surface elevation records. The variation of IMFs along the beach profileis examined in an attempt to investigate cross-shore transformation of sea waves.The dominant oscillations for each data record are determined, and their variationsduring different sea stages are investigated.

5.2 FIELD DATA

The wave data used in this work were collected at the Hazaki OceanographicalResearch Station (HORS), Port and Airport Research Institute, Japan, during theperiod from 25 February to 1 March 1989. The HORS is located on the Pacific coastof Japan. The measurements were performed by six wave gauges mounted on a 427m long pier and by three deepwater bottom wave gauges in an extension line of thepier, as shown schematically in Figure 5.1. The offshore distance and water depthof the wave gauges are listed in Table 5.1. The sea surface elevation was recordedevery 6 h, and the records are 2 h long at a sampling data rate of 0.5 sec.

Different sea stages were observed during the measurement period. The varia-tions of offshore significant wave height (Hs) and mean wave period (Tmean) duringthe observation are presented in Figure 5.2. The sea conditions are classified intofour different sea stages: calm, wave growth, wave decay, and post-storm stage, afterKatoh et al. (5). The initial three terms of measurements were in a comparativelycalm sea stage, when sea waves had a period of less than 6 sec and significant waveheight was less than 2 m. Under the direct influence of a passing atmosphericdepression with a strong north wind, growth of the sea waves was observed in thenext three terms. The waves became greater than 3 m in significant wave height, buttheir periods remained short. In the wave decay stage, when the wind speed dropped,


100



the wave periods became longer than 9 sec. At the end of the observation, notedhere as the post-storm stage, the wind changed its direction and the registered waveswere small in amplitude, but with long periods. The wave gauges 7, 8, and 9 werelocated inside the surf zone during the calm sea stage; in other sea stages, the surfzone became wider and also included wave gauges 4, 5, and 6.

5.3 CONVENTIONAL METHODS FOR WAVE DATA ANALYSIS

An example of sea surface elevation, recorded in the surf zone at 5.05 m waterdepth, is shown in Figure 5.3. The wave profile is irregular and considerably asym-metrical. There are small waves, resulting from wave breaking as well as ripples,riding the longer waves. The question is how to analyze properly such considerablynonlinear data time series, in view of the fact that the right choice of method is veryimportant for the correct understanding of the investigated phenomenon.

The conventional methods for wave data analysis propose estimation of wavecharacteristics by means of Fourier spectra or by analysis of time series of individualwaves, determined by some criteria. The stochastic properties of sea waves, derivedby different discretization methods, are presented and discussed in detail by Ochi (6).

The identification of discrete waves from the record of sea surface elevation isnontrivial work, especially for the shallow water wave data. The three criteria most

FIGURE 5.1 Location of wave gauges during the observations (HORS, PARI, Japan).

TABLE 5.1Location of Wave Gauges

Number of Wave Gauge 1 2 3 4 5 6 7 8

Offshore distance (m) 3000 2000 1000 380 320 215 145 80Water depth (m) 24.00 14.00 9.00 5.65 5.05 2.33 1.81 0.30



101


FIGURE 5.2 Offshore characteristics of sea waves during the observation period.

FIGURE 5.3 Example of a wave record from surf zone

Sea

su

rfac

e el

evat

iom

(m

)

Time (s)

1401301201101009080

2

1

0

-1

-2


102



often used for determination of individual waves from the sea surface elevationrecords are briefly described in Table 5.2. The estimation of mean frequency dependson the discretization criterion and is determined as where mk isa kth spectral moment

,

and S( f ) is a spectral density function.The application of the zero-crossing (ZC) method for determination of individual

waves often meets difficulties in the treatment of small amplitude waves. They wereconsidered as “non real” or false waves by Gimenez et al. (7), Kitano et al. (8), andVeltcheva and Nakamura (9), and were eliminated by some of the criteria proposedby Mizuguchi (10), Hamm and Peronnard (11), and Gimenez et al. (12). The maingoal of removing the small waves, especially from the surf zone data, is to ensurethe constancy of wave period from deep to shallow water. It must be stressed thatthe procedure of removing the small waves affects the estimation of wave charac-teristics, assuming a priori the linearity of sea waves.

The orbital criterion, introduced by Gimenez et al. (12), uses the complexpresentation of wave process. The analytical function

, (5.1)

corresponds to the vertical displacement of sea surface elevation X(t). Here X (t) isthe Hilbert transform of the sea surface elevation,

, (5.2)

where P indicates the Cauchy principal values. The Hilbert transform is a convolutionof the sea surface elevation X(t) with 1/t and thus emphasizes the local properties of X(t).

TABLE 5.2Discretization Criteria

Criterion Definition of Individual WavesMean Wave Frequency

Zero-crossing criterion A wave is determined by two consecutive (up–up or down–down) crossings of mean water level.

ω02

Crest to crest criterion A wave is determined by two consecutive maxima of surface elevation.

ω24

Orbital criterion A wave is determined as corresponding to a 2 advance of the phase angle in the complex plane.

ω01

ω πijj i

j im m= −2 ,

m f S f dfkk=

∞

∫0

( )

ξ( ) ( ) ˆ ( )t X t jX t= +

X t PX t

t tdt( ) =

′( )− ′

′∫1π



103


The wave envelope, phase, and frequency are defined as

, (5.3)

, (5.4)

and

, (5.5)

respectively. The envelope A(t), defined by Equation 5.3, provides a direct measureof the wave amplitude and is widely used in the statistical analysis of the sea waves(13, 14, 15, 16, 17), while the phase information is usually not considered. Beingan important physical quantity, the phase φ(t) associated with the sea wave processcan also bear fundamental information about the dynamics of sea waves. But unfor-tunately, the phase is totally disregarded when conventional methods for waveanalysis are employed to study the wave system.

The phase φ(t) is a wrapped phase φ(t) ∈ [– π, π] or an unwrapped phase (t) ∈[–∞, ∞], depending on its interval of definition. An example of the unwrapped andwrapped phase of sea surface elevation X(t) is shown in Figure 5.4. The unwrappedphase in Figure 5.4a increases smoothly with time, in contrast to the wrapped phase,which varies widely in the range [– π, π].

The frequency ω(t), defined by Equation 5.5 as the rate of phase change, iscalled the instantaneous frequency, and its value changes with time. This definitionof frequency was adopted before by Huang et al. (18) and used by Cherneva andVeltcheva (19) for investigation of the local properties of sea waves and their groupstructure. Later, Huang et al. (4) emphasized the considerable controversy in thisdefinition of instantaneous frequency, even with utilization of the Hilbert transform.The instantaneous frequency is, as mentioned by Cohen (20), “one of the mostintuitive concepts, since we are surrounded by light of changing color, by soundsof varying pitch, and by many other phenomena whose periodicity changes. Theexact mathematical description and understanding of the concept of changing fre-quency is far from obvious.…” Blindly applying the definition Equation 5.5 ofinstantaneous frequency to any analytic function may lead to one of the five para-doxes listed by Cohen (20). One of these paradoxes concerns the appearance ofnegative frequency: “Although the spectrum of the analytic signal is zero for negativefrequencies, the instantaneous frequency may be negative.” The enlarged portion ofthe unwrapped phase shown in the upper left-hand corner of Figure 5.4a reveals thejumps up and down in the time variation of the unwrapped phase. The unwrappedphase function ϕ(t) is not always increasing function of time, and consequently theinstantaneous frequency ω(t) will have negative values, which have no physicalmeaning.

A t X t X t( ) ˆ= ( ) + ( )2 2

ϕ( )ˆ ( )( )

t arctgX tX t

=

ω ϕ( )t

ddt

=


104



The negative advances of the phase ϕ(t) produce loops or phase reversals in apolar diagram of the wave process (Figure 5.5), where the analytical function ξ(t)is presented as a radius-vector rotating in the complex plane. The magnitude of thevector is equal to the wave envelope A(t), estimated by Huang et al. (3), and theangle of rotation is equal to the unwrapped phase function ϕ(t). The arrows showthe direction of rotation of the radius-vector. The rotation of the radius-vector withpositive instantaneous frequency is called a proper rotation by Yalcinkaya and Lai(21) in their investigation of the phase dynamics of continuous chaotic flow.

FIGURE 5.4 The phase φ(t) of the analytical process. (a) Unwrapped phase; (b) wrappedphase.

0

200

400

60

80

1000

0 200 400 600 800 1000 1200

dar[esahp

depparwn

U

50

60

70

80

90

100

140

dar[esahp

depparwn

U]

80 100 120Time [s]

] 0

0

Time [s]

-4

-2

0

2

4

0 200 400 600 800 1000 1200

Wr

]dar[esah p

deppa

Time [s]



105


For a clearer illustration, Figure 5.5 shows the polar diagram of the part of thewave record (thick, solid line) of Figure 5.3 between 85 sec and 112 sec. Thetrajectory of ξ(t) in the complex plane generally may have multiple centers ofrotation. The majority of the individual zero-crossing waves in Figure 5.3 correspondto a completely closed trajectory of rotation around the origin of the coordinatesystem in Figure 5.5. The small ZC waves, around 90 sec and between 105 sec and110 sec in Figure 5.3, are represented by loops in Figure 5.5. The centers of rotationof these loops differ from the origin of the coordinate system. The small ridingwaves in the wave profile also produce loops in the polar diagram, like the ridingwave between 91 sec and 95 sec in Figure 5.3. These phase reversals, as seen inEquation 5.5, produce negative frequencies, which have no physical meaning.According Gimenez et al.’s (12) orbital criterion, any discrete waves that do notcorrespond to a 2π advance of the phase in the complex plane are considered asfalse waves and are simply removed. In this way, the wave process is again assumedto be a linear one, since only the completely closed orbit of water particles with 2πadvance of phase angle is accepted as an individual wave.

The small zero-crossing waves and ripples are indeed distinctive features of realsea waves from the surf zone, and thus it is incorrect to eliminate them by the orbitalcriterion for the purpose of achieving the positive frequency. Band-pass filtering,

FIGURE 5.5 Presentation of sea surface elevation in complex plane.

Sea

su

rfac

e el

evat

ion

(m

)

Hilbert transform of sea surface elevation (m)

1.50-1.5

1.5

0

.5-1


106



proposed by Melville (22), also does not resolve the problem with negative frequen-cies. Huang et al. (4) especially stressed the fact that the filtering operation itselfcontaminated the data with spurious harmonics. These unsuccessful attempts toresolve the problem of negative frequencies are probably attributable to an inappro-priately chosen tool for investigation of nonlinear and nonstationary sea waves.

5.4 HILBERT-HUANG TRANSFORM METHOD

Generally, the method of analysis affects the investigated phenomenon. The choiceof an appropriate analyzing technique is very important for a correct understandingof the examined phenomenon, since the functions used in the analysis can bemisinterpreted as characteristics of the investigated phenomenon (23). To reducethis risk, the analyzing function has to be chosen in accordance with the intrinsicstructure of the field to be analyzed. The HHT method, introduced by Huang et al.(4), completely covers this requirement, since the expansion of the data is based onthe local characteristics of the particular data. This method offers a unique anddifferent approach for data processing. The HHT method was motivated “from thesimple assumption that any data consists of different intrinsic mode oscillations” (3).

The main idea of EMD, a key part of the HHT method, is to identify the timescale that will reveal the physical characteristics of the studied process, recorded intime series, and to extract these time scales into IMFs. A time lapse betweensuccessive extrema in the time series is defined as a time scale, and this is the essenceof EMD. The EMD is a data sifting process to eliminate locally riding waves aswell as to eliminate local asymmetry of the time series profile. Huang et al. (3, 4)present a procedure of sifting and several applications of the HHT method. On thebasis of the results of EMD of the data, Veltcheva (24, 25) investigated the groupstructure of sea waves in the coastal zone.

The time series X(t) is first decomposed by EMD into a finite number n IMFs(denoted Cj), which extract the energy associated with various intrinsic time scales,and residual Rn. The superposition of the IMF and residue reconstruct the data:

. (5.6)

By definition, the IMF has to satisfy two conditions. First, the number of extremamust be equal or differ at most by one from the number of zero-crossings; this issimilar to the traditional narrow band requirements for the stationary Gaussianprocess. Practically, this condition corresponds to elimination of riding waves orsmall waves in the time series with multiple centers of rotation. The derived IMFcorresponds to a proper rotation in the complex plane, as the center of rotation isthe origin of the coordinate system. Second, the IMF has to have symmetric envelopes,defined by the local maxima and minima respectively. These upper and lower envelopesare determined by using cubic splines, as suggested by Huang et al. (4). This secondcondition ensures that the instantaneous frequency will not have fluctuations arising

X t C t R tj

j

n

n( ) = ( ) + ( )=

∑1



107


from an asymmetric wave profile. The important condition for the IMF is that onlyone maximum or minimum exists between successive zeros. Each IMF is determinedby a sifting procedure, which is repeated several times in order to ensure that allthe requirements of IMFs are satisfied.

The set of IMFs obtained in this way is unique and specific to the particulartime series, since it is based on and derived from the local characteristics of thesedata. IMFs could be considered as a more general case of the simple harmonicfunctions, but IMFs are claimed to have a physical meaning in additional to amathematical one due to their specific derivation.

In the second step, the Hilbert transform is applied to these IMFs:

(5.7)

where P indicates the Cauchy principal value. The amplitude aj, the phase ϕj, andthe instantaneous frequency ωj are calculated by

, (5.8)

, (5.9)

and

. (5.10)

By using Equation 5.8, Equation 5.9, and Equation 5.10, we can express the originaldata X(t) as a real part (Re) of the complex expansion

, (5.11)

which is considered a generalized form of the Fourier expansion. Here, both ampli-tude aj(t) and instantaneous frequency ωj(t) are functions of time t. Whereas theFourier expansion is made in a global sense, the expansion by HHT is made in alocal sense.

It must be mentioned that, by definition, IMFs always have positive frequencies,because the oscillations in IMFs are symmetric with respect to the local mean. Thus,the HHT method solves the problem of negative frequencies, and there is no need

C t PC t

t tdtj

j( ) =′( )

− ′′

−∞

∞

∫1π

a t C t C tj j j( ) ˆ= ( ) + ( )2 2

ϕ j t arctgC tC t

( )ˆ( )( )

=

ωϕ

jj

td t

dt( ) =

( )

X t a t ej

j

ni t dtj( ) = ( ) ∫

=

( )∑Re1

ω




for any additional elimination of small waves, which are indeed observed peculiar-ities of real sea waves. All the distinguishing features of sea surface elevation aretaken into account, and the subjectivity in estimation of wave characteristics probablywill be reduced. This is a direct result of the use of a more appropriate tool forexamining nonlinear and nonstationary real sea waves.

The frequency–time distribution of the amplitude or squared amplitude is des-ignated as a Hilbert amplitude spectrum or Hilbert energy spectrum. In the presentwork, we use the Hilbert energy spectrum, determined as the frequency–time dis-tribution of the squared amplitude. For simplicity, the Hilbert energy spectrum isdenoted as the Hilbert spectrum H(ω,t). The frequency–time distribution of theenergy allows us to determine which frequency exists at a particular time, whereasthe Fourier frequency spectrum provides information as to which frequency existsgenerally in given data series.

The time resolution of the Hilbert spectrum can be as precise as the samplingdata rate. Since the Hilbert spectrum gives the best fit of a local sine or cosinefunction to the data, the frequency is uniformly defined by a local derivative of thephase function. The lowest extractable frequency is 1/T, where T is the duration ofthe record, and the highest frequency is 1/(l∆t), where l = 5 is the minimum numberof points necessary to define frequency accurately, and ∆t is the sampling rate.

Based on the Hilbert spectrum, the marginal Hilbert spectrum h(ω) can bedefined as

. (5.12)

The marginal Hilbert spectrum represents the cumulated squared amplitude overthe entire data span and offers a measure of total energy contribution from eachfrequency.

5.5 APPLICATION OF THE HHT METHOD

For each wave record, a set of IMFs is obtained by EMD. Figure 5.6 presents detailedresults of decomposition of sea surface elevation data, recorded at an offshore pointwith 24 m water depth at wave decay stage, with significant wave height of 3.18 mand mean period 9.96 sec. Figure 5.6a shows the data time series. The EMD methodyields eight IMFs and residue, shown in Figure 5.6b through 5.6j. Since time scalesare identified as intervals between the successive alternations of local maxima andminima, IMFs represent oscillation modes embedded in the data. The different IMFscorrespond to the different physical time scales that characterize the various dynam-ical oscillations in the time series. A nearly constant (as to time scale) wave com-ponent dominates in each IMF, representing the carrier wave constituent at thespecific time scale.

The sifting procedure of EMD generates modes Cj(t), j = 1, …, n with degree oftime variations in decreasing order. By construction, each mode Cj(t) generates a proper

h H t dt

T

( ) ( , )ω ω= ∫0


An Application of HHT Method to Nearshore Sea Waves 109


FIGURE 5.6 EMD of wave record. (a) Data of sea surface elevation; (b) through (i) eightIMFs (C1 through C8); (j) residue R.

0−3

3

0

X(t

)

1000 2000 3000 4000 5000 6000 7000

0−3

3

0

C1

1000 2000 3000 4000 5000 6000 7000

0−3

3

0C2

1000 2000 3000 4000 5000 6000 7000

0−1.5

1.5

0C3

1000 2000 3000 4000 5000 6000 7000

C4

0−0.5

0.5

0

1000 2000 3000 4000 5000 6000 7000

Time (s)

0−0.2

0.2

0C5

1000 2000 3000 4000 5000 6000 7000

0−0.1

0.1

0C6

1000 2000 3000 4000 5000 6000 7000

0−0.1

0.1

0C7

1000 2000 3000 4000 5000 6000 7000

0−0.1

0.1

0C8

1000 2000 3000 4000 5000 6000 7000

R

0−0.2

0.2

0

1000 2000 3000 4000 5000 6000 7000

Time (s)

(a) (f )

(g)

(h)

(i)

(j)

(b)

(c)

(d)

(e)




rotation in the complex plane of its own analytic function ψj(t) = aj(t) exp [iφj(t)]. Theaverage rotation frequencies obey the order becausethe sifting procedure picks the components with the fastest variations embedded in theoriginal data first and those with the slowest variations last. The mean periods of theIMFs, shown in Figure 5.6, are correspondingly 3.4 sec, 8.4 sec, 16.6 sec, 35.3 sec,89.2 sec, 235 sec, 572 sec, and 1540 sec.

These oscillations not only have different time scales, but they also have adifferent range of energy. The limits of the vertical axes in Figure 5.6b through 5.6iare different for different IMFs. The most energetic in the decomposition is thesecond IMF, C2. The shortest oscillations in decomposition, presented by the firstIMF, C1, have smaller amplitudes than those of C2. Physically, these high frequencyoscillations, extracted in C1, may represent short period waves, such as wind wavesin the growing stage, as well as the small ripples that ride longer waves. The thirdIMF, C3, takes second place in the energy hierarchy. The first three constituents inthe decomposition, C1, C2, and C3, probably represent the contribution of the windwaves oscillations, while the lower frequency oscillations are characterized by theIMFs with higher index: C4, C5, and so forth. The complete presentation of the seawave state recorded in this data can be considered as a composition of dominantwave oscillations, extracted in the second IMF and several (but finite number)components with smaller amplitudes.

The contribution of different Intrinsic Mode Functions to the energy and fre-quency contents of wave data is investigated. The spectrum of wave record iscompared with spectra of its Intrinsic Mode Functions in Figure 5.7 as estimatedFourier spectrum is shown in Figure 5.7a, while a marginal Hilbert spectrum h(ω)is presented in Figure 5.7b. Under a sampling rate of these data, the highest frequencythat can be extracted by Empirical Mode Decomposition is 0.4 Hz. The highestfrequency of Fourier spectrum is 1 Hz, but for simplicity and easy comparison withmarginal spectrum, only until 0.35 Hz is shown in Figure 5.7a. The Fourier spectrumis estimated as an average of the raw spectra, calculated in the overlapped segments,by which the wave record is divided. The spectral estimations are also smoothedwith a Hamming window.

The Intrinsic Mode Functions represent different as period and energy oscilla-tions, extracted by EMD from the wave data. The spectrum of the first IMF C1 (starsline) covers very well the tail of wave spectrum (thick solid line) in the both ñFourier and marginal spectrum. The peak of second IMF C2 (open circle line)coincides with the major peak of the spectrum of sea surface elevation, both Fourierand marginal spectrum.

The marginal Hilbert spectra are projections of real frequency-time distributionof the energy, provided by Hilbert spectrum. There is basic difference in the meaningof frequency in Fourier and Hilbert spectrum. The presence of energy at a givenfrequency in Fourier spectrum means that a trigonometric component with thisfrequency and amplitude exists through the whole time span of the data. In themarginal Hilbert spectrum the existence of energy at a given frequency means thatin the whole time span of the data, there is a higher probability for local appearanceof such a wave.

ω φj jt d t dt( ) ( )= ω ω ω1 2≥ … ≥ n




FIGURE 5.7 Spectra of wave data and its IMFs. (a) Fourier spectra; (b) marginal Hilbertspectra.

100

10–1

10–2

10–3

101

10–4

Fo

uri

er s

pectr

um

(a) Wave dataC1C2C3C4C5C6C7C8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Frequency (Hz)

100

10–1

10–2

10–3

101

10–4

Mar

gin

al s

pectr

um

(b) Wave dataC1C2C3C4C5C6C7C8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Frequency (Hz)




5.5.1 OFFSHORE WAVES DURING DIFFERENT SEA STAGES

The peculiarities of the decomposition of offshore waves during different sea stageswere examined. The energy and time characteristics of individual IMFs were esti-mated and compared. The zero-th moment m0

Cj of the marginal Hilbert spectrumhj(ω) is proposed as a measure of the integrally determined energy of the jth IMF:

. (5.13)

The mean value of the instantaneous frequency , estimated by Equation 5.10, isused as a measure of the frequency of the jth IMF.

Four cases, representative of each of the four different sea stages, were chosen.The characteristics of their decomposition are shown in Table 5.3. The energy ofthe IMFs with index higher than 4 is significantly lower than the energy of the firstfour IMFs. This tendency is confirmed by the analysis of all offshore wave records.From an energetic point of view, the first four IMFs contribute mainly to the energycontents of the wave data, measured at 24 m depth. For a complete description ofthe analyzed data, a complete set of all IMFs is necessary.

In the wave growing stage, C2 has the highest energy, followed by C1, inagreement with the prevailing short period waves in the wave growing stage. Theswell type of sea waves starts to dominate in the wave decay stage; consequently,C3 has energy comparable with or higher than the energy of C1. A similar tendencyin the energy distribution of the IMFs is also observed in the post-storm stage, whenthe majority of the registered sea waves have a longer period. The energy hierarchyof the IMFs in decomposition of a wave record reflects clearly the specific pecu-liarities of the particular wave data.

The different IMFs have different significance in view of the purpose of thespecific study. If the signal of interest is captured in a single IMF, the investigationand eventual prediction of the examined phenomenon is facilitated, since the dynam-ical system is simpler. Salisbury and Wimbush (26) demonstrated this advantage ofthe EMD, analyzing only one IMF, which had the same mean frequency as aphenomenon of interest, instead of considering the original data series.

An understanding of the characteristics of large sea waves is essential for manycoastal engineering problems; therefore, special attention is paid here to the mostenergetic components in the decomposition. In this connection, the IMF most dom-inant in energy among the whole set of IMFs is determined and denoted Cm. InFigure 5.8, the energy of m0

Cm is investigated as a function of the energy of theoriginal wave data m0; the data from all observation terms and all water depths areused. The zero-th moments Equation 5.13 of the marginal Hilbert spectrum are againused as a measure of energy. The dominant IMF, Cm, contained about 55% of theenergy of the wave data.

The specific peculiarities of the wave process are well captured by EMD andreproduced by IMFs. Easy separation of the different time and energy associatedwith the oscillations in the data is well achieved by EMD.

m h dCj

j0 = ∫ ( )ω ω

ω j


An

Ap

plicatio

n o

f HH

T Meth

od

to N

earsho

re Sea Waves

113

TABLE 5.Characteduring C ditions

(Hs

Decay Stage (Hs = 2.86 m, Tmean = 10sec)

Post-Storm Stage (Hs = 2.21 m, Tmean = 8.81 sec)

IMF fp (Hz) m0 (m2) fp (Hz) m0 (m2)

C1 0.1053 0.0693 0.1947 0.0473C2 0.0787 0.5382 0.0747 0.2456C3 0.0693 0.1580 0.0800 0.1232C4 0.0240 0.0130 0.0400 0.0138C5 0.0107 0.0030 0.0187 0.0050C6 0.0053 0.0015 0.0053 0.0021C7 0.0027 0.0007 0.0027 0.0004C8 0.0027 0.0007 0.0027 0.0004

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3ristics of Offshore Sea Waves and Their IMFs alm, Wave Growing, Decay, and Post-Storm Con

Calm Stage = 1.53 m, Tmean = 5.2 sec)

Growing Stage (Hs = 3.25 m, Tmean = 7.4 sec)

fp (Hz) m0 (m2) fp (Hz) m0 (m2)

0.1920 0.0797 0.1120 0.32570.1627 0.0912 0.1093 0.52880.0987 0.0309 0.0907 0.08650.0373 0.0027 0.0373 0.01330.0187 0.0007 0.0160 0.00440.0107 0.0005 0.0080 0.0018 0.0053 0.0002 0.0053 0.00130.0027 0.0003 0.0027 0.0005



5.5.2 CROSS-SHORE TRANSFORMATION OF SEA WAVES

After the wave data, measured at different points of the nearshore, were disintegratedinto IMFs, the cross-shore variations of the energy and frequency of the IMFs weretraced in order to examine the transformation of sea waves. The energy of the IMFs,defined by the zero-th moment of the marginal Hilbert spectrum, is presented inFigure 5.9a, while the frequency of the IMFs, defined as a spectral peak frequency,is shown in Figure 5.9b. The horizontal axes in Figure 5.9 are an index of the firsteight decomposition components Cj (j = 1, …, 8). The data from 24 m and 14 mwater depth were collected in the region before breaking, while the other observationpoints, at 5.65 m water depth and less, were located inside the surf zone. The waveprocess in the surf zone is commonly considered to be a complicated one andassociated with a broadband Fourier spectrum. Here, by virtue of the EMD method,these significantly nonlinear and complicated wave data series are practically decom-posed into a small number of IMFs with a properly defined phase ϕ(t) respectivelyfrequency ω(t).

The energy of the first IMF, C1, characterizing the high frequency oscillationsin the data, is significantly smaller than that of the second component for the caseof offshore waves. However, in the shoreward region of the surf zone (2.33 m, 1.81m, and 0.30 m water depths), the energy of C1 becomes comparable to the energyof C2, C3, and C4. The second IMF, C2, has the highest energy in the decompositionof the offshore waves. After breaking, the energy of the second IMF decreases

FIGURE 5.8 Comparison of energy of the dominant IMF (m0Cm) and wave data (m0).

1.41.210.8

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-1)

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m(m

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)

↑




considerably. At the same time, the oscillations extracted in C2 become shorter. Theenergy and frequency of the lower frequency IMFs, C4, C5, C6, and C7, remainapproximately constant in the process of cross-shore transformation.

These peculiarities in the cross-shore variations of IMFs are reflected in thefrequency–time distribution of the energy. A section of the Hilbert spectrum of seawaves in three different zones — offshore with water depth 24 m, in the seawardsurf zone at 5.05 m depth, and in the shoreward surf zone at 1.81 m — is shown in

FIGURE 5.9 Decomposition of wave data measured in the cross-shore. (a) Energy of IMFs;(b) frequency of IMFs.

0

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(b)




Figure 5.10. The Hilbert spectrum is presented as a three-dimensional plot in theleft side panels. The color scale maps of the same sections of H(ω,t) in a smoothedform are shown in the right side panels. The Hilbert spectrum is smoothed with 9× 9 weighted Gaussian filter. The smoothing operation degrades both the frequencyand time resolution of H(ω,t). The color bars next to the diagrams denote the energyscale.

The frequency–time distribution of the wave energy changes considerably dueto wave transformation in a cross-shore direction. The Hilbert spectrum in Figure

FIGURE 5.10 A section of the Hilbert spectrum of sea waves, as a 3D plot in the left panelsand as a 9 × 9 Gaussian filtered color coded map in the right panels. (a) Offshore zone; (b)seaward surf zone; (c) shoreward surf zone.

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5.10a shows an energy concentration around 0.092 Hz, corresponding with thesecond component, C2, which has the highest energy in the decomposition in Figure5.6. The magnitude of energy decreases, as revealed by changes in the energy scaleof the color bars. A wide variation in the local frequency in the surf zone is evidentin Figure 10b and c, in contrast with the offshore region. This broadening of thewave frequency range is partially due to wave breaking. The strong nonlinearity ofthe sea waves in shallow water, shown by the intra-wave frequency modulation orthe variation of the local frequency within one wave, also contributes to a largevariation in the local frequency in the Hilbert spectrum.

On the basis of the results of EMD, real sea waves could be considered as asystem of oscillations with different periods and amplitudes. It has to be stressedthat the amplitudes and periods of these oscillations are not constant but instead arefunctions of time according to Equation 5.11. The wave system, as a whole, changesits content in cross-shore transformation. The process of wave breaking mainlyaffects the IMFs with the highest energy in the decomposition of offshore waves.Inside the surf zone, the frequency of each IMF tends to increase. These peculiaritiesin the spatial distribution of different IMFs agree with general principles of sea wavetransformation in the cross-shore direction. The analysis of IMFs and the Hilbertspectrum provides information on the cross-shore transformation of sea waves.

The EMD method, with its great ability to extract specific oscillation dataembedded in the data, can facilitate the investigation of sea waves. Attention can beconcentrated on a finite number of IMFs, each characterizing a dynamical systemthat is much simpler than the original one. The application of the HHT method tonearshore waves is a fertile area of research, as future studies will shed more lighton the complicated dynamics of the coastal zone.

5.6 CONCLUSIONS

We have discussed the importance of the analyzing technique for the correct under-standing of the examined phenomenon and paid special attention to the nonlinearand nonstationary behavior of real sea waves. A review of widely used conventionalmethods for wave analysis shows that all of them assume a priori linearity andstationarity of sea waves; these assumptions consequently affect the estimation ofwave characteristics.

The Hilbert-Huang transform method for the analysis of nonlinear and nonsta-tionary time series is proposed as an alternative for the investigation of the nonlinearand nonstationary nature of real sea waves. In this method, the oscillations embeddedin the data are extracted by EMD into a set of IMFs without placing any subjectivepreliminary limitations on the nature of the investigated phenomenon. Easy separa-tion of the different, as time and as energy associated with the oscillation in the datais well achieved by EMD.

The dominant oscillations for each data record were determined and their vari-ations during different sea stages were investigated. The energy hierarchies of theIMFs in decomposition of wave records observed during different sea stages reflectthe specific peculiarities of the particular wave data. The process of wave breaking




mainly affects the IMFs with the highest energy in the decomposition of offshorewaves. Inside the surf zone, the frequency of each IMF tends to increase.

The EMD method, with its great ability to extract oscillation modes embeddedin the data, can facilitate the investigation of sea waves. Attention can be concentratedon a finite number of IMFs, each characterizing a dynamical system that is muchsimpler than the original one. The application of the HHT method to nearshorewaves is a fertile area of research, as future studies will shed more light on thecomplicated dynamics of the coastal zone.

REFERENCES

1. Liu, P. C. (2000). Is the wind wave frequency spectrum outdated? Ocean Eng. 27:577–588.

2. Massel, S. R. (2001). Wavelet analysis for processing of ocean surface wave records.Ocean Eng. 28: 957–987.

3. Huang, N. E., Shen, Z., and Long, S. R. (1999). A new view of nonlinear waterwaves: The Hilbert spectrum. Annu. Rev. Fluid Mech. 31: 417–457.

4. Huang, N. E., Shen, Z., Long, S..R., Wu, M. C., Shin, H. S., Zheng, Q., Yuen, Y.,Tung, C. C., and Liu, H. H. (1998). The EMD and Hilbert spectrum for nonlinearand non-stationary time series analysis. Proc. R. Soc. London. Vol. 454: 903–995.

5. Katoh, K., Nakamura, S., and Ikeda, N. (1991). Estimation of infragravity waves inconsideration of wave groups: An examination on basis of field observation at HORF.Rep. Port Harbour Res. Inst. Vol. 30, 1: 137–163.

6. Ochi, M. K. (1998). Ocean Waves: The Stochastic Approach. Cambridge OceanTechnology Series 6, 319 p. New York: Cambridge University Press.

7. Gimenez, M. H., Sanchez-Carratala, C. R., and Medina, J. R. (1998). Influence offalse waves on wave records statistics. Proc. ICCE 1998, pp. 920–933.

8. Kitano, T., Mase, H., and Nakano, S. (1998): Statistical properties of random waveperiods in shallow water: Analysis by utilizing false waves. Proc. Coast. Eng, JSCE,Vol. 45, 221–225.

9. Veltcheva, A., and Nakamura, S. (2000). Wave groups and low frequency waves inthe coastal zone. Rep. Port Harbour Res. Inst. Vol. 39, 4: 75–94.

10. Mizuguchi, M. (1982). Individual wave analysis of irregular wave deformation in thenearshore zone. Proc. ICCE 82, pp. 485–504.

11. Hamm, L., and Peronnard, C. (1997). Wave parameters in the nearshore: A clarifica-tion. Coastal Eng. 32: 119–135.

12. Gimenez, M. H., Sanchez-Carratala, C. R., and Medina, J. R. (1994). Analysis offalse waves in numerical sea simulations. Ocean Eng. Vol. 21, 8: 751–764.

13. Longuet-Higgins, M. S. (1958). On the intervals between successive zeros of arandom function. Proc. R. Soc. Ser. A, 246: 99–118.

14. Tayfun, M. A. (1983). Frequency analysis of wave heights based on wave envelope.J. Geophys. Res. Vol. 88, C12: 7573–7587.

15. Bitner-Gregersen, E. M., and Gran, S. (1983). Local properties of sea waves derivedfrom a wave record. Appl. Ocean Res. Vol. 5, 4: 210–214.

16. Hudspeth, R. T., and Medina, J. R. (1988). Wave group analysis by Hilbert transform.Coastal Eng. Vol. 1: 885–898.

17. Tayfun, M. A., and Lo, J. M. (1989). Envelope, phase, and narrow-band models ofsea waves. J. Waterway Port Coastal Ocean Eng. ASCE, Vol. 115, 5: 594–613.




18. Huang, N. E., Long, S. R., Tung, C. C., Donelan, M. A., Yuan, Y., and Lai, R. J.(1992). The local properties of ocean surface waves by the phase-time method.Geophysical Res. Lett. Vol. 19, 7: 685–688.

19. Cherneva, Z., and Veltcheva, A. (1993). Wave group analysis based on phase prop-erties. Proc. 1st Int. Conf. Mediterranean Coastal Environ. (MEDCOAST ’93), Tur-key, pp. 1213–1220.

20. Cohen, L. (1995). Time-Frequency Analysis. Prentice Hall Signal Processing Series,Alan V. Oppenheim, series editor, 299 p.

21. Yalcinkaya, T., and Lai, Ying-Cheng (1997). Phase characterization of chaos. Phys.Review Lett. Vol. 79, 20: 3885–3888.

22. Melville, K. (1983). Wave modulation and breakdown. J. Fluid Mech. 128: 489–506.23. Farge, M. (1992). Wavelet transforms and their applications to turbulence. Annu. Rev.

Fluid Mech. 24: 395–457.24. Veltcheva, A. D. (2001). Wave groupiness in the nearshore by Hilbert spectrum. Proc.

4th Int. Symp. Ocean Wave Meas. Anal. (WAVES 2001), San Francisco, pp. 367–376.25. Veltcheva, A. D. (2002). Wave and group transformation by a Hilbert spectrum.

Coastal Eng. J. Vol. 44, 4: 283–300.26. Salisbury, J. I., and Wimbush, M. (2002). Using modern time series analysis tech-

niques to predict ENSO events from the SOI time series. Nonlinear Processes Geo-physics, Vol. 9: 341–345.



6
Transient Signal Detection Using the Empirical Mode Decomposition
Michael L. Larsen, Jeffrey Ridgway, Cye H. Waldman, Michael Gabbay, Rodney R. Buntzen, and Brad Battista

CONTENTS

6.1 Introduction ..................................................................................................1226.2 Empirical Mode Decomposition..................................................................1256.3 EMD-Based Signal Processing....................................................................1266.4 Experimental Validation Using Towed-Source Signals...............................1356.5 Conclusion....................................................................................................137Acknowledgment ...................................................................................................138References..............................................................................................................138

ABSTRACT

In this paper, we report on efforts to develop signal processing methods appropriatefor the detection of manmade electromagnetic signals in the nonlinear and nonstationaryunderwater electromagnetic noise environment of the littoral. Using recent advancesin time series analysis methods [Huang et al., 1998], we present new techniques forsignal detection and compare their effectiveness with conventional signal processingmethods, using experimental data from recent field experiments. These techniques arebased on an empirical mode decomposition which is used to isolate signals to bedetected from noise without a priori assumptions. The decomposition generates aphysically motivated basis for the data.

121


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6.1 INTRODUCTION

In this paper, we discuss application of nonlinear signal processing techniques tounderwater electromagnetic data. Electromagnetic measurements provide a poten-tially useful complement to acoustic detection methods in littoral zones where highnoise levels and reverberation in the littoral limit acoustic detection ranges. Inshallow waters, for example, the ambient noise levels from wind-generated breakingwaves are typically 10 to 100 times greater than those in deep water. Electromagneticmeasurements are affected by noise as well. One source of electromagnetic fieldsin seawater is electrical currents induced by hydrodynamic flow in the Earth’smagnetic field. The ocean is a constantly moving conducting fluid [26], and oceanflows caused by gravity waves (wind waves and far-field swell), internal waves,turbulence, tides, and currents all result in electromagnetic field fluctuations.

While the standard way of viewing the complexity of wave phenomena is toconsider the ocean to be a random, Gaussian process, the importance of nonlinearinteractions in water wave dynamics has been conclusively demonstrated [28]. Fur-thermore, turbulence is also often present at the bottom boundary layer due tononlinear interactions between flows arising from tidal and wind-driven currents,wind waves, swell, and internal waves [4]. In the random sea assumption (usedprimarily in the deep ocean), the evolution of the sea surface is obtained as asuperposition of waves of different frequencies and directions with random phases.The wave amplitudes are determined according to a measured power spectrum, thesea surface is modeled as a linear system, and energy transfer between wave modesis not accounted for. In littoral regimes, however, nonlinear wave interactions aresignificant and cannot be neglected. Higher-order statistics show that wave behaviorin shallow water zones displays a distinctly non-Gaussian regime, where wave–waveinteraction plays an important role and energy transfer between wave componentsoccurs [18, 12].

A typical power spectrum of naturally occurring ocean-bottom magnetic andelectric field data is shown in Figure 6.1. The most obvious peaks in both electricand magnetic spectra in Figure 6.1 are visible at 62 MHz, the dominant oceanicswell frequency. Swells have characteristic frequencies in the 50 to 100 MHz range,and surface waves driven by local winds are found at frequencies above the swellfrequency, up to about 0.5 Hz. The effect of ocean swell on the magnetic field issignificantly more pronounced than its effect on the electric field. An additional peakis also seen in the spectrum of the magnetic field data at twice the swell frequency(125 MHz). Frequency doubling effects due to the nonlinear interference of the swellwith wave fields reflecting off local land masses have been previously observed [27,24]. Spectral peaks above these frequencies are likely due to local turbulence, withthe exception of the Schumann resonance discussed below.

Another major component of underwater electromagnetic noise is actually dueto sources of non-oceanic origin — geomagnetic, atmospheric, and ionosphericelectromagnetic radiation. The Schumann resonance at 8 Hz is caused by worldwidelightning, which resonates in the earth-ionosphere cavity. Geomagnetic noise atlower frequencies originates in the ionosphere, is nonstationary, and increases in



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power at lower frequencies, imparting the strong slope to the spectrum seen in Figure6.1. These effects were observed in our experiments and in nearshore tests done byother researchers [16]. Geomagnetic noise is typically coherent over large distancesand can be mitigated by using a remote reference station [17]. A transfer functioncan be calculated to account for the difference in the conductivity of the underlyinggeology and used to cancel out the effect of magnetospheric sources observed inthe local sensor data. Figure 6.1 shows the spectrum before and after such noisereduction was performed.

Characterizing the nonstationary and nonlinear character of ocean-bottom elec-tromagnetic noise is important for detection of manmade electromagnetic (EM)anomalies. Ships and submarines generate electric fields in the surrounding waterdue to corrosion currents caused by the dissimilar metals used in their constructionand also by onboard cathodic protection systems designed to mitigate such currents.The bronze propeller acts as one node of the vessel’s galvanic cell, while thesacrificial zincs on the hull act as nodes of opposite polarity. A current flows fromthe propeller through the drive shaft, bearings, and hull into the water, returning tothe propeller. This current flow in the surrounding water can be well modeled as anelectric dipole. The horizontal motion of the ship or submarine imparts characteristicultra-low frequency signals to the field components, which change with the speedand range of the submarine. This signature is described as the “quasi-static” horizontalelectric dipole (HED) signature. Impressed on the quasi-static dipole field is an ACmodulation, which is caused by variations in the electrical resistance in the shaft as itrotates. A vessel passing a stationary electric field sensor generates a transient signalin the nonstationary background ambient electric field. Figure 6.2 shows an exampleof simulated horizontal electric field (E-field) components (Ex and Ey).

For a stationary sensor, the measured dipole signal is a function of the vesselspeed and its lateral distance from the sensor at the point of closest approach. Thespectra of moving dipole signatures are generally confined to ultra-low frequencies(0.5 to 5 MHz) and thus fall into the underwater electromagnetic frequency regime

FIGURE 6.1 Power spectra for ocean-bottom electromagnetic fields before and after noisemitigation.

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wer

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dominated by ionospheric noise. When available, ionospheric noise removal cangreatly aid detection of the moving dipole signal, as any reduction in spurious signalpower translates directly into extended ranges [5, 1]. While remote reference miti-gation techniques can remove much of this noise, specialized detection and extractiontechniques are still necessary. A number of techniques for processing underwaterelectromagnetic data exist in the literature. A generalized likelihood ratio test wasdeveloped for HED detection using a 3-axis E-field sensor [6], but under simplifyingGaussian white noise assumptions. The use of higher-order statistics in detectionalgorithms has been investigated [3,20], and the wavelet transform has also proveduseful for detection purposes [21,22].

In this paper, we investigate the use of signal processing techniques for under-water electromagnetic data which use the empirical mode decomposition (EMD)reported by Huang et al. [13]. The EMD was developed specifically to deal with thenonstationary and nonlinear characteristics of ocean wave data. In Section 6.2, wereview the properties of the EMD. In Section 6.3, we describe a detection algorithmbased on the conventional matched filter that employs the EMD for electromagneticanomaly detection. Experimental results are presented in Section 6.4.

FIGURE 6.2 Simulated time series of the two horizontal components of the electric fieldfrom a moving vessel. The dipole appears as a slow transient. The inset is an enlarged viewof the cyclic modulation of the dipole.

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6.2 EMPIRICAL MODE DECOMPOSITION

The EMD facilitates calculation of physically meaningful instantaneous frequencies[13] by using the Hilbert transform. The Hilbert transform can be used to define theinstantaneous amplitude and phase of an arbitrary time series x(t). The Hilberttransform of x(t) yields a time series y(t) given by

(6.1)

where PV denotes the Cauchy principal value. Hence, the Hilbert transform is theconvolution of x(t) with 1/t, which consequently stresses the local nature of thesignal. From the complex analytic signal formed via z(t) = x(t) + iy(t), an instanta-neous amplitude and phase can be calculated. The instantaneous frequency corre-sponds to the time derivative of the phase of z(t).

Direct application of the Hilbert transform to data may yield negative frequen-cies. Only positive instantaneous frequencies, however, correspond to physicallymeaningful oscillatory behavior. In order to avoid nonphysical instantaneous fre-quencies, the empirical mode decomposition is used to separate the data intowell-behaved intrinsic mode functions (IMFs) from which the Hilbert transform willyield positive instantaneous frequencies. An IMF is a function satisfying two con-ditions: i) the number of extrema and the number of zero-crossings of the functionmust either be equal or differ at most by one, and ii) at any instant in time, the meanvalue of the envelopes defined by the function’s local maxima and minima must bezero. Each IMF has a characteristic frequency, although IMFs can overlap in fre-quency content. IMFs are found by using a recursive sifting procedure that generatesthe highest-frequency IMF first. This IMF is subtracted from the time series, andthe process is iteratively applied to the difference until only a non-oscillatory resid-ual, r, which represents the trend in the data, remains. Each IMF xn(t) has a variableinstantaneous amplitude, an(t), and frequency, ωn(t), calculated via the Hilbert trans-form. The time–frequency distribution H(ωn,t) of the amplitude is known as the Hilbertspectrum. The original time series can be written as a sum of a finite number of IMFs:

(6.2)

Completeness of the EMD is assured in principle and depends only on thenumerical accuracy of the sifting process. Orthogonality of the IMFs, while notguaranteed theoretically, is satisfied in practice. The Hilbert spectrum can be inte-grated over time to yield the Hilbert marginal spectrum,

y t PVxt

d( )( )=−

,−∞

∞

∫1π

ττ

τ

x t Re a t e rn

N

ni t dtn( ) ( ) ( )= +

.=

∑1

ω

HMS H t dtT

( ) ( )ω ω= , ,∫0


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which represents the accumulated energy at each frequency over the entire data spanand is related to the fraction of time that a given frequency can be observed in thesystem. The marginal spectrum is analogous to the Fourier power spectral density.

The EMD leads to compact representation of nonlinear or nonstationary signalswithout higher-order harmonics or a proliferation of global modes needed to accountfor rapid transients. It is often observed that individual IMFs often correspond toidentifiable physical processes in the data [13,14].

Recent work is yielding greater understanding of the behavior of the EMD andits potential for signal processing applications [8,7]. It appears to operate on frac-tional Gaussian noise as an almost dyadic filterbank, resembling a wavelet decom-position [10,9]. Furthermore, numerical experiments indicate that its impulseresponse is similar to a cubic spline wavelet [25].

6.3 EMD-BASED SIGNAL PROCESSING

Electromagnetic noise in a littoral, oceanic environment is generally nonstationaryand can also be nonlinear. The spectrum of such noise is colored, insofar as it riseswith lower frequencies according to a power law. Hence, traditional detection meth-ods based upon the assumption of the noise being white, stationary, and Gaussianmay not be optimal. The empirical mode decomposition appears to distinguishsignals originating from different physical processes by grouping them into a limitednumber of intrinsic mode functions [24]. This behavior suggests the use of empiricalmode decomposition for increasing the effectiveness of detection of horizontalelectric dipole in a littoral oceanic EM noise background. Such anomalies generatedby moving sources are transient, with time scales lasting between 10 and 30 min,depending upon the ratio of the closest point of approach (CPA) to the velocity.

The traditional method for detecting a deterministic signal in a noise backgroundis the matched filter, which uses an estimate of the expected signal to filter the data.The matched filter is the optimal filter for detection of a deterministic signal in whiteGaussian noise [15]. It has been used for HED detection by Blampain [2]. Othermethods in the literature for detecting transient HEDs include a generalized likeli-hood ratio test [6] and a method based on the eigen-decomposition of the samplecovariance matrix [23]. All three of these methods assume that the noise is spatiallyand temporally both white and Gaussian, an assumption that is not representativeof the littoral oceanic environment. A method of transient magnetic signal detectionin geomagnetic noise [22], based upon the application of wavelet packets, yieldedmixed results. As a starting point for our analysis, we have used the matched filtermethod as our standard of comparison, with a priori knowledge of the target (i.e.,bearing, CPA, velocity, and time of CPA).

Electromagnetic signal anomalies generated by moving sea vessels are transientin nature. Results from simulated and actual data show that the transient quasi-staticHED signature is often captured in one or two IMFs. This behavior is demonstratedwith a simulated example without noise of a HED passing by at two different ranges,with a simulated double-frequency cyclic component whose propagation is modeledwith the algorithm found in Orr [19]. The EMD of Ex generates five IMFs (seeFigure 6.3).



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Similar results are obtained for noisy signals. A typical track time series fromthe experiment, containing both the HED and cyclic signal generated by a towedsource, is shown in the top graph of Figure 6.4. The cyclic signal (in the second)and the HED (in the seventh panel), highlighted in red, are easily identified in thisclose-pass track, which still contains noise.

We investigated the performance of the matched filter for detecting a movingHED in the ocean using synthetic marine signals, which simulate a single dipolemoving in a straight path past a single receiver. The expected signal is a functionof the dipole amplitude and target range, bearing, and speed. In this study we electedto simplify the situation by assuming a priori knowledge of speed and range, inorder to focus on the comparison between the matched filter and the empirical mode(EMD)-assisted matched filter (EMMF) to be described later. In practice, one wouldsweep through the set of realizations of speed, range, and HED dipole strength, anddetermine the parameters that maximize the matched filter output. The two compo-nents of the electric field were processed together in quadrature as a complex signal,with Ex as the real component and Ey as the imaginary component. A correlationfilter was used with the magnitude of the output being the detection criterion,normalized to a maximum unity output. Through simulation it was determined that

FIGURE 6.3 Simulated, combined dipole and cyclic signals at different ranges and theirderived IMF components. The lower five graphs display the individual IMFs, showing theability of the EMD procedure to separate signals with different characteristics.

Time Series and IMF Components

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the expected signal magnitude depends on the CPA range and target speed but notthe target bearing. All bearings will have the same magnitude response, with differingphase responses. Thus the matched filter output depends only upon using the correctrange and speed of the vessel.

Building on the matched filter, an improved algorithm, the empirical modematched filter, was developed to exploit the EMD’s ability to capture the signal injust a few of the IMFs. This method applies the matched filter to a signal generatedusing the IMFs as basis functions, rather than to the target signal x(t) itself. Thisimproved detector performance. A block diagram of the processing steps of thealgorithm is shown in Figure 6.5. Empirical mode decomposition was first applied

FIGURE 6.4 EMD analysis of underwater electromagnetic data.

FIGURE 6.5 Block diagram of EMD-assisted matched filter, where the matched filter processis preceded by a least-squares fit of the IMFs (generated from the time series data).

−2

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to the time series to generate the IMFs αi(t), i = 1, …, N, where N is the numberof IMFs. In the next step, the linear combination of the IMFs containing the targetsignal s(t) which best approximates the known target signature in the least-squaresense was determined:

The least-squares optimization was constrained so that the weights ci are positive.A matched filter based on the target signature s(t) was then applied to the signalx(t), which is composed of the weighted linear combination of the IMFs,

to generate a detection statistic λ(t).The underlying motivation for this technique is the tendency of the EMD to

generate a physically motivated basis for the data. Individual IMFs can be associatedwith the dipole signal. Often the signal is captured in a single IMF. In such a case,it would be advantageous to apply a matched filter to this IMF, rather than to thewhole time series, which may contain corrupting background signals. In general, itis not known a priori which IMF the dipole will be in; it may even be spread acrossseveral IMFs. Thus we employ the least-squares criterion above. The positive weightconstraint stems from the fact that the IMFs contain no sign ambiguity — their sumis the complete time series. This constraint also reduces false alarms. A sample timeseries of this process from the experiment is shown in Figure 6.6. The process ofweighting the IMFs results in an improved correlation with the target signal ascompared to the full time series.

We examined the performance of the EMMF using a statistical Monte Carloanalysis with real electromagnetic background data from the experiment and simu-lated target signals (which are shown above to match the actual towed source signalsvery well). This procedure validated that the EMMF robustly improves the proba-bility of detection of the HED dipole signal in realistic oceanic/geomagnetic noise.A number of signal-conditioning and pre-processing steps were used as well. Thesimulation analysis was confirmed with detection results for real signals generatedby the DC electric towed source.

The first type of noise background tested for comparison between the linearmatched filter (LMF) and the EMMF was white Gaussian noise generated by arandom signal generator. The noise was band-limited by using a high-pass filter witha frequency cutoff of 0.5 MHz (2000-sec period), because this low-frequency cutoffdoes not change the moving dipole signals we wish to detect. A dipole signal wascreated with a source strength equal to that used in the experiment and numericallyembedded in the white Gaussian noise background, which usually resulted in about

min ( )c

i

N

i ii

s t c≥

=

|| − || .∑01

α

ˆ( )x t ci

N

i i= ,=∑

1

α


130



3000 overlapping samples in the simulation. The window size was set to τ = 2.7(τ = V/RCPA), which from experience yields an effective template for the matchedfilter. The desired Ex and Ey signals are shown in Figure 6.7.

A τ parameter of 2.7 yields signals that are 9 min long for this geometry. Longerτ values include the tails of the signals, which contain little information, and shorterτ values may cut out valuable information about a signal’s shape that makes itdifferent from the background. For this τ and constant source speed V, the signalslengthen in time proportional to their CPA offset RCPA.

Receiver operating curves (ROC) curves were generated for the white Gaussiannoise and signal by calculating distributions of the matched filter output (normalizedto 0–1) for inputs of noise only and the noise plus signal. Once the distributionswere created, ROC curves were extracted, utilizing a sliding threshold, to determinethe probability of detection (PD) at different probabilities of false alarm (PFAs). Theresult, shown in Figure 6.8, confirms the expected theoretical behavior: that thestandard LMF technique yields the best detection statistics for this type of noise.The LMF procedure outperforms the EMMF (also labeled as the “LSQ” in thefigures that follow because the method involves a least-squares fit of the IMFs tothe desired matched filter). As the source strength and signal-to-noise ratio decrease,both the LMF and EMMF ROC performances deteriorate, but the LMF maintainsa slight advantage, as predicted by theory. Signal-to-noise ratio (SNR) is defined

FIGURE 6.6 Comparison of matched filter and EMMF techniques.

0 100 200 300 400 500 600 7002.5

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time (sec)

µV/m

raw signalLSQ wt signalmodel



131


here as the peak excursion of the target signal versus the standard deviation of thebackground noise [11], normalized to a logarithmic dB number:

,

where P is the peak excursion of the target signal (of a specified component), andσ is the standard deviation of the background noise. This broadband definition is

FIGURE 6.7 Ex (solid) and Ey (dotted) signals for a north–south track, where τ = 2.7, whichyields a 9-min signal length at this CPA and speed.

FIGURE 6.8 Comparative ROC curves for HED moving dipole detection in Gaussian whitenoise, for LMF method (lower curve) and EMMF/LSQ method (upper curve). The originalsource strength has been reduced to correspond to SNR values of –14 and –16.4. The LMFmethod outperforms the EMMF in Gaussian white noise, in accordance with signal-processingtheory.

600500400300

Time (seconds)

E-field X/Y components, Tau = 2.7

2001000

E-fi

eld

(uV

/m)

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1

LSQLMF

LSQLMF

SNR P= /20 10log ( )σ




calculated independently of frequency. The SNR for the ROC curves in Figure 6.8is negative (printed on curves) and means that the signal amplitude is lower thanthe standard deviation of the noise. A SNR of –14, for example, corresponds to thepeak signal value being one-fifth the noise standard deviation.

The advantage of the LMF over the EMMF method is lost when an actual noisebackground is used. The background used in the following analysis consists of 52h of experimentally measured underwater electromagnetic data, which is non-Gaus-sian with a colored spectrum and a nonstationary variance. In actual oceanic andgeomagnetic noise, the EMMF method is superior. This is demonstrated in Figure6.9, where we have displayed three representative curves together on one graph,utilizing SNRs between –5 and +14, much higher than the SNRs used in the whiteGaussian noise process. A major characteristic of the real-world noise is its nonsta-tionarity and high power at low frequencies, which can lead to anomalies that mimicthe transient HED.

The ROC performance curves in Figure 6.9 have been enhanced by two sig-nal-conditioning procedures: high-pass filtering of the data to produce first-orderstationarity (see above), and removal of a linear trend within each window beingexamined. The window length is determined by the τ value being sought. (For the

FIGURE 6.9 ROC curves for LMF process (light) and EMMF/LSQ process (dark), forseveral different SNRs. When the SNR is strong, the two processes achieve nearly identicaldetection results, but when the SNR is weaker, the EMMF process has a definite advantage.

0 0.2 0.4 0.6 0.8 10

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Source strength effect on ROC curves, LMF vs. EMMF

SNR 10

SNR 7

SNR 5

EMMF/LSQ

LMF


Transient Signal Detection Using the Empirical Mode Decomposition 133


simulations performed, we assume knowledge of such parameters; for detection ina field situation, one would sweep through a predetermined set of moving dipoleparameters and choose the set that yields the highest matched-filter output.) Othersignal-conditioning techniques were also explored and found to be less effective.For example, when we remove a mean value from the window rather than de-trend-ing, the overall ROC curve degrades, but the EMMF displays a relative advantageover the linear matched filter.

Figure 6.10 shows the effects of source strength and SNR on ROC performancecurves. The source strength is represented in dB relative to 1 A-m. For low-to-medium source strengths (up to 35 dB), the EMMF method yields superior PDs ata given strength. Conversely, for a given PD, the necessary source strength to achievethat PD is reduced by 2 to 3 dB in the PD range of 0.4 to 0.8. This detection gainvanishes at high source strengths (high SNRs), where the two methods yield identicaldetection statistics. It must be emphasized that the analysis here has been done on“unreduced” data, which has not been differenced versus a remote reference and isa high-noise environment. Removing coherent geomagnetic noise by using a remotereference sensor yields a greater range.

FIGURE 6.10 Relative gain for HED detection using EMMF at set PFA (= 0.1) for twodifferent ranges. PD is plotted as a function of source strength in dB. The EMMF method isin blue. For a given PD, the necessary source strength is lowered by 2 to 3 dB, for moderate PD.

0 10 20 30 40 500

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)

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Black: LSQ Red: LMF Blue: LSQ, 2x range Green: LMF, 2x range




We repeated this procedure for a greater range for unreduced data, which yieldsa nearly identical curve, but shifted to the right about 15 dB. Thus the EMMF methodmaintains its increase in effectiveness across range. The detection versus rangebehavior is summarized in Figure 6.11, where we plot PD versus CPA range for aset PFA = 0.1. Up to a certain range, the two methods again yield an identical highprobability of detection, but as the range is further increased (and SNR decreases),the EMMF method (dark curve) is more effective, extending the detection range by10 to 15% for a set PD, or increasing the PD by 0.1 to 0.2 for a given range. Wesee that the EMMF method flattens the falloff curve for low SNR regimes, therebyextending detection ranges. The inclusion of the EMD into the matched filter processadds detection gain in high-amplitude, colored, and nonstationary noise such asobserved in the underwater electromagnetic environment. This effectivenessenhancement is seen generally in low to medium SNRs, whether caused by weakersource strength or a farther CPA.

We also investigated the effectiveness of the EMMF method in a noise regimewhere the common-mode geomagnetic noise has been partially mitigated by usingremote-reference techniques. This yields improved noise reduction so that an algorithm

FIGURE 6.11 Probability of detection versus CPA, for LMF (light) and EMMF (dark)methods. As the range is further increased and SNR decreases, the EMMF method excels,extending the detection range by 10 to 15% for a set PD, or increasing the PD by 10 to 25%for a given range.

0.2

0.4

0.6

0.8

1

Range

PD

(01

) PD versus CPA; unreduced data

EMMFLMF




comparison can be made in a background where oceanic and other processes dom-inate, rather than ionospheric sources. Figure 6.12A shows a plot of the unreducedand reduced time series. Visually, the reduced time series is more uniform acrossits entire length and has both lower variance and fewer outliers than the originaltime series, which is simply high-pass filtered. Histogram analysis at various posi-tions in the time series also indicates that it has more a stationary variance, andspectral analysis shows it to have a flatter power spectrum at low frequencies andthus to be closer to white Gaussian noise.

We repeated the Monte Carlo procedure for generating ROC curves as a functionof range, with the results shown in Figure 6.12B. Detection ranges are noticeablyextended. Also, the LMF method is actually slightly superior to the EMMF, at rangeswhere the SNR is high. However, as the range increases and the SNR decreases, theEMMF method becomes more effective than the linear matched filter, providing aflatter rolloff of PD versus range.

Figure 6.13 shows PD performance versus range for the EMMF and LMFmethods for reduced data. Here we have set the PFA to a constant (= 0.1), andcalculated the PD as a function of range for both methods. The linear method hasa slight advantage at lower ranges and higher SNR, whereas the HHT-enhancedmethod is more effective at long ranges and a small SNR. Overall, the EMMFmethod provides superior performance with flatter rolloff versus range.

6.4 EXPERIMENTAL VALIDATION USING TOWED-SOURCE SIGNALS

We utilized a database of 17 tracks of a towed electromagnetic source collected over6 sensors with various CPAs to test the relative effectiveness of the EMMF method

FIGURE 6.12 Effect of remote-reference noise reduction on matched filter methods. (A)Unreduced (top) and reduced (bottom) time series for background, X component, over 40 h.The geomagnetic noise removal not only lowers the variance, but also makes the resultingseries more stationary. (B) ROC curves generated from reduced data, for EMMF (dark) andLMF (light) methods. The LMF method is more effective at low ranges/high SNR, whereasthe EMMF method increasingly dominates at greater ranges.

PD

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2.5sigma = 0.23

sigma = 0.090

3Unreduced and reduced background noise ROC curves, REDUCED data

Unreducedreduced




versus the LMF method of detection for the moving HED signal on unreduced data.A total of 36 track/sensor combinations existed in this CPA range. For each individualtrack, assuming knowledge of the track bearing and CPA, we performed the algo-rithms discussed above and measured the correlation signal. Such signals generallyproduced a peak very near the CPA. To determine whether a detection occurred, weapplied the matched filter to data measured at a remote sensor which contains onlybackground noise, and calculated a threshold for a given false alarm level (a PFAlevel of 5% was used).

The results are displayed in Figure 6.14A for the LMF method and Figure 6.14Bfor the EMMF method. As expected from the ROC analysis, the EMMF method issuperior. EMMF correlations are uniformly higher with a much flatter falloff versusCPA than for the LMF method. Additionally, the threshold (actually a mean valueof the thresholds over all 36 cases) for the EMMF method raises only slightly, whichresults in an overall increase in detectability using the EMMF. As displayed in Figure6.15, in the 36 cases tested, the EMD-based method results in 24 detections, versusonly 14 for the linear matched filter (points very near to the detection threshold werecounted as detections), using a PFA of 5%.

FIGURE 6.13 Summary graph of behavior of PD versus range for EMMF (dark) and LMF(light), after coherent geomagnetic noise removal. The LMF method is slightly improved atlower ranges, but overall the EMMF method provides a superior curve with flatter rolloffversus range.

0

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(01

) PD versus CPA, REDUCED data




6.5 CONCLUSION

In summary, we have exploited the empirical mode decomposition to isolate physicalsignals into a discrete number of intrinsic mode functions in order to enhance thedetection of transient HED signals in a background of geomagnetic and hydrody-namic noise. We have numerically compared this method to the standard matchedfilter by deriving ROC curves via a Monte Carlo simulation for experimental data.In white Gaussian noise, the LMF method yields superior ROC curves, in accordance

FIGURE 6.14 Comparison of (A) the LMF method with (B) the EMMF method, utilizingsignals created by the towed electric source. The EMMF has a flatter falloff with range andmore detections (24/36 versus only 14 for the LMF).

FIGURE 6.15 Comparison of number of detections of the LMF method (light) with theEMMF method (medium), utilizing signals created by the towed electric source.

Co

rrel

atio

n (

0–

1)

0.1

0.2

0.3

0.4

0.5

0.6

0.7 threshold for PFA = 0.05

A): 14/36 detections

threshold for PFA = 0.05

A): 24/36 detections

0.8

0.9

1

Co

rrel

atio

n (

0–

1)

0.1

0.2

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1(a) (b)

LMF-based correlation vs. threshold EMMF-based correlation vs. threshold

Range

Num

ber

of d

etec

tions

, ch

ance

s

Detections vs. range: LMF vs EMMF for data with coherent noise reduction

Possible chances

Detections EMD

Detections LMF




with signal-processing theory. In a realistic, non-Gaussian, high-noise environment,the EMMF method outperforms its linear counterpart by enhancing detection rangesand minimum detectable source strengths. When it is possible to mitigate much ofthe geomagnetic noise by using remote reference methods, then the traditional LMFprovides slightly superior detection at high SNRs, but the EMMF method yields aflatter PD versus range curve and significantly extends detection ranges for a givenPD. The results for unreduced data were confirmed by applying the matched filtermethods to experimental data, where the EMMF method increases detection rangesand flattens the detection curve with range. Further investigation of the optimal useof such methods is warranted.

ACKNOWLEDGMENT

This work was funded by the Office of Naval Research and others. The authors aregrateful for the experimental assistance of the Swedish Defense Ministry and theFOI, and for the assistance of Dave Rees of SPAWARSYSCEN San Diego.

REFERENCES1. J. Berthier, F. Robach, B. Flament, and R. Blanpain. Geomagnetic noise reduction

from sea surface, high sensitivity magnetic signals using horizontal and verticalgradients. In Proc. 1998 Oceans Conf., July 2001.

2. R. Blampain. Traitement en temps réel du signal issu d'une sonde magnetometrique.Doctoral thesis of engineering, INPG, Grenoble, 1979.

3. C. Chichereau, J.L. Lacoume, R. Blanpain, and G. Dassot. Short delay detection ofa transient in additive Gaussian noise via higher order statistic test. In IEEE DigitalSignal Proc. Workshop, pp. 299–302, 1996.

4. C.S. Cox, X. Zhang, S.C. Webb, and D.C. Jacobs. Electro-magnetic fluctuationsinduced by geomagnetic induction in turbulent flow of seawater in a tidal channel.In MARELEC, 3rd Int. Conf. Marine Electromagnetics, July 2001.

5. B. Deniel. Undersea magnetic noise reduction. In Proc. 1997 MTS/IEEE Conf., vol.2, pp. 784–788, 1997.

6. R. Donati. Detection of the electric field due to corrosion phenomena. In Proc. 2ndInt. Conf. Marine Electromagnetics (MARELEC), Vol. 2, pp. 322–325, 1999.

7. P. Flandrin, P. Gonclaves, and G. Rilling. Detrending and denoising with empiricalmode decompositions. In Proc. IEEE-EURASIP Workshop Nonlinear Signal ImageProcess. NSIP-04, to appear in 2004, available at http://www.inrialpes.fr/is2/peo-ple/pgoncalv/pub/emd-eusipco04.pdf.

8. P. Flandrin and P. Gonclaves. Sur la decomposition modale empirique. In Proc.GRETSI-03, 2003.

9. P. Flandrin, G. Rilling, and P. Gonclaves. Empirical mode decomposition as a filterbank. IEEE Sig. Proc. Lett., 11(2):112–114, 2004.

10. P. Flandrin. Empirical mode decompositions as data-driven wavelet-like expansionsfor stochastic processes. In Proc. Conf. Wavelets Statistics, 2003.

11. B.C. Fowler, H.W. Smith, and F.W. Bostick Jr. Magnetic anomaly detection utilizingcomponent differencing techniques. Technical Report 154, Electrical GeophysicsResearch Lab., Univerisity of Austin Texas, 1973.


http://www.inrialpes.fr/

http://www.inrialpes.fr/



12. K. Hasselmann. On the nonlinear energy transfer in a gravity wave spectrum. Jour.Fluid Mech., 12:481–500, 1962.

13. N.E. Huang, Z. Shen, S.R. Long, M.L. Wu, H.H. Shih, Q. Zheng, N. Yen, C.C. Tung,and H.H. Liu. The empirical mode decomposition and Hilbert spectrum for nonlinearand non-stationary time series analysis. Proc. R. Soc. London A, 454: 903–995, 1998.

14. N.E. Huang, Z. Shen, and S.R. Long. A new view of nonlinear water waves: theHilbert spectrum. Annu. Rev. Fluid Mech., 31:417–57, 1999.

15. S.M. Kay. Fundamentals of Statistical Signal Processing, Volume 2: Detection Theory.Prentice Hall, Englewood Cliffs, N.J., 1998.

16. M.J. Morey, T. Bentson, and C. Osborn. TET E-field data analysis, summer 1991experiment. Tech. Document 2482, NCCOSC RDT & E Divsion, 1993.

17. E.A. Nichols, J. Clarke, and H.F. Morrison. Signals and noise in measurements oflow-frequency geomagnetic fields. J. Geophys. Res., 93:13743–13754, 1988.

18. M.K. Ochi. Ocean Waves: The Stochastic Approach. Cambridge Ocean Technology,Series 6. University Press, Cambridge, 1998.

19. A.M. Orr. Computational techniques for evaluating extremely low frequency electro-magnetic fields produced by a horizontal electrical dipole in seawater. Dept. ofPhysics, King's College London, 2001.

20. A. Quinquis and D. Declerq. Magnetic noise subtraction with wavelet packets. InProc. OCEANS '95. MTS/IEEE. Challenges of Our Changing Global Environment,Vol. 1, pp. 227–232, 1995.

21. A. Quinquis and C. Dugal. Using the wavelet transform for the detection of magneticunderwater transient signals. In Proc. OCEANS '94: Oceans Engineering for Today'sTechnology and Tomorrow's Preservation, Vol. 2, pp. 548–543, 1992.

22. A. Quinquis and S. Rossignol. Detection of underwater magnetic transient signalsby higher order analysis. In Proc. Int. Symp. Signal Process. Appl., pp. 741–749, 1996.

23. J. Rantakokko. Localization of static electric and magnetic underwater sources. Tech-nical Report FOA report No. FOA-R–99-01265-409–SE, Swedish Defense ResearchEstablishment (F.O.A.), Stockholm, Sweden, 1999.

24. J. Ridgway, M.L. Larsen, C.H. Waldman, M. Gabbay, R.R. Buntzen, and C.D. Rees.Analysis of ocean electromagnetic data using a Hilbert spectrum approach. In Proc.7th Exper. Chaos Conf., pp. 333–338, August, 2003.

25. G. Rilling, P. Flandrin, and P. Gonclaves. On empirical mode decomposition and itsalgorithms. In Proc. IEEE-EURASIP Workshop Nonlinear Signal Image Process.NSIP-03, 2003.

26. T.B. Sanford. Motionally induced electric and magnetic fields in the sea. J. Geophys.Res., 76:3476–3492, 1971.

27. S. Webb and C.S. Cox. Pressure and electric fluctuations on the deep seafloor:background noise for seismic detection. Geophys. Res. Lett., 11:967–970, 1984.

28. H.C. Yuen and B.M. Lake. Nonlinear dynamics of deep-water gravity waves. Adv.Appl. Mech., 22:67–229, 1982.



7
Coherent Structures Analysis in Turbulent Open Channel Flow Using Hilbert-Huang and Wavelets Transforms
Athanasios Zeris and Panayotis Prinos

CONTENTS

7.1 Introduction ..................................................................................................1427.2 Results ..........................................................................................................143

7.2.1 Academic Signals.............................................................................1437.2.2 Turbulent Signals .............................................................................1437.2.3 Coherent Structures.........................................................................143

7.3 Conclusions ..................................................................................................150References..............................................................................................................156

ABSTRACT

In this study the Hilbert-Huang transform with empirical mode decomposition isapplied for analyzing and quantifing turbulent velocity data from open channel flow.This novel technique, proposed by Huang et al. [1,2], is applied using experimentalvelocity signals in turbulent, fully developed open channel flow, with and without theimposition of bed suction. Signals were obtained with laser Doppler and hot filmtechniques [3,4]. Velocity data for the above flow were analyzed in previous studies,with traditional methods, for various suction rates for the description of the flow fieldand for the identification of the intense events related with coherent structures thatcontribute to the organized motion. Here, an attempt is made to exploit the noveldecomposition method in the domain of joint frequency–time analysis of the turbulencedata.

141


142



7.1 INTRODUCTION

Details for the instrumentation and experimental results from measurements of theapplication of suction from the bed in open channel have been presented in previousarticles [3,4,5]. We mention for completeness of the present study that measurementswere conducted with LDA (tracker) and hot film methods, the suction rate is basedon discharge ratio (Qs/Qtot, Qs = suction discharge, Qtot = total discharge), the lengthof suction region L = 0.035 m, and the Re number based on upstream velocity Um

and flow depth was Re = Umh/ = 59,000. The empirical mode decomposition (EMD) method proposed by Huang [1] and

the Hilbert transform of the decomposed signal are summarized as follows:All the extrema of the signal x(t) are identified, and an interpolation for the

maxima and the minima is attempted with cubic splines curves, following recom-mendations and proposals from previous work [1,2]. The local mean m(t) is com-puted, and the difference between the local mean m(t) and the initial signal isextracted. Until a stopping criterion [2] is achieved, an iteration process takes place.In this study the criterion that we adopt is that the number of extrema equals (ordiffers by one from) the number of zero-crossings. When this criterion has beenachieved, the remainder is considered as an intrinsic mode function (IMF) anddenoted c1; this IMF contains the shortest periods of the signal. The residual

,

is considered as a new signal and subjected to the iterative process repeatedly, until thelast residue is smaller than a predetermined small value or is a monotonic function.

Preliminarily, the problem of extrema interpolation is checked from the view-point of (a) it is time consuming especially when the signal belongs to the multi-frequency ones as turbulent signals that needs a large number of iterations for acorrect decomposition and (b) because of end effects propagation of the splines intothe interior of the signal. We used different approaches as linear, linear with empiricalmodifications, and iterative process with preselected iteration numbers. With thecriterion of the most perfect composition of the original signal we chose, for thepresent article, to extract IMFs with the originally proposed method [2] cubic splinewith the addition of sine waveforms at the beginning and at the end of the analyzeddata.

With the aid of the Hilbert transform, Huang et al. [1,2] proposed the constructionof the analytic signal for every IMF component. From them we get the time distri-butions of amplitude and phase

,

x t c r t( ) − = ( )1

h t c t H c t( ) = ( ) + ( ) 2

φ tH c t

c t( ) =

( )( )( )

−tan 1


Coherent Structures Analysis in Turbulent Open Channel Flow

143


where h(t) and φ(t) are the instantaneous amplitude and phase respectively. Fromthis quantity, values for the instantaneous frequency are obtained as

With the extraction of IMFs as a time series, the tools for the HHT ampli-tude–frequency–time representation of a signal — the instantaneous frequency andthe amplitude — are available, and, depending on the signal’s nature, their qualityis better than or equal to the results of other well documented methods, such asshort FFT, proper orthogonal decomposition, Wigner-Ville, and the wavelet decom-position method.

7.2 RESULTS

7.2.1 ACADEMIC SIGNALS

Some academic signals are used for the estimation of the quality of decompositionapplied in the present study. The traditional frequency shift sine signal (Figure 7.1a)is examined through frequency time variation of the first IMF with the point offrequency shift well localized. Figure 7.1c also gives the Hilbert spectrum andFigure 7.1d presents the FFD distribution.

In Figure 7.2a the sawtooth signal is presented along with the distribution ofthe instantaneous frequency (Figure 7.1c) because this kind of variation is a candidate(and as the edge detection problem) as part of an average representative patternindicating the presence of organised structures.

7.2.2 TURBULENT SIGNALS

Figure 7.3 shows a typical velocity signal, and Figure 7.4 illustrates its analysis ineight IMF components. Figure 7.5 shows the distribution of these components inthe frequency domain. Figure 7.5b presents the filtering approach of the Ci compo-nents divided by the root-mean-square (RMS) value of every component.

Figure 7.6a and b gives the variation of frequency with time. For clarificationonly, Figure 7.6a shows the frequency–time distribution with a median filter of highorder for a better visual observation of the different components. In addition, Figure7.6c gives a representative energy–frequency–time distribution. Figure 7.7 presentsthe time variations in instantaneous energy IE(t) for a short time period for a velocitysignal at y+ = 100 from hot film measurements.

7.2.3 COHERENT STRUCTURES

In turbulent wall flows, a great portion of the momentum transport is connected withthe presence of organized structures in the form of burst structures in the regionnear the wall. According to well-documented models, streaks of low velocity startto oscillate and to lift up from the wall. The ejections from the decomposition of

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these streaks, appropriately grouped, are characterized as burst structures. In thesearch for quantitative information, many techniques have been introduced duringthe past decades toward the identification of patterns and levels in the signal velocityconnected with these organized structures. These techniques include the variable

FIGURE 7.1 (a) Frequency shift signal; (b) instantaneous frequency variation; (c) Hilbertspectrum; (d) FFT distribution.

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interval time average (VITA) technique [6,7], quadrant splitting [8] of Re stresses,the temporal pattern average (TPAV) [10], linear stochastic estimation [9], and theproper orthogonal decomposition (POD) and wavelet decomposition–based tech-niques [11].

Traditional techniques, such as VITA (focusing on local variance that exceedspart of the whole signal variance) and quadrant splitting for velocity and Re stress,with empirical threshold criteria k = 1.0, predetermined time scale T* = 10, andH = 1 for quadrant splitting, respectively, have been applied in previous studies [3,4]to extract intense events in the flow field inside the porous suction region (x/L =0.5) and beyond the suction region (x/L = 1.2) from hot film and LDA results. Figure7.8 gives an example of results from the VITA technique. Here the effect of suctionis evident at the exit of the suction region from the frequency appearance of theintense events.

With the tool of signal decomposition, it is possible to overcome the shortcom-ings resulting from the single scale limitations of the VITA method. The VITAmethod does not permit the identification of events separated by a time smaller thanthat imposed by the VITA treatment. Also, a main disadvantage for VITA is the factthat it requires the selection of two rather subjective criteria (time scale and thresh-old).

Wavelets decomposition allows intense events to be identified based on the valueof the wavelet coefficients in the representation of the translation — scale planeusing a nonsubjective threshold criterion.

Analysis in the scale time plane of turbulent statistical measures gives scaledependent behavior for quantities such skewness and flatness factors. Strongnon-Gaussian behavior of the smaller scales is an indication of the intermittency.For intermittency estimation at each scale, Farge [11] proposed the local intermit-tency index Im,n , the ratio of local energy to the mean energy at the respective scale.As this index takes an extreme value at a time instant, a strong percentage of energyand intermittency is found in the corresponding time and scale.

With the above in mind, it is possible to use EMD to determine the ratio of localenergy (amplitude squared) to mean energy for every IMF in the energy–fre-quency–time distribution. Figure 7.9 shows the above ratios with the 4th order Daub4wavelet and for the HHT method.

The next step in the identification of coherent events is to apply the appropriatethresholds. Universal proposed relations from wavelets such as Donoho andJohnstone [12] is considered that cannot contribute to these turbulent data becauseof the Gaussian consideration of the incoherent part.

In this study, we applied different values for thresholding the ratio of instanta-neous energy for every IMF mode. We propose the selection of a threshold value inthe region where the number of the detected events are not independent from the

FIGURE 7.3 Typical velocity signal.

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FIGURE 7.4 (a) IMFs; (b) residual.

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threshold. Instead of using arbitrary predefined thresholds, we used relative thresh-olds (local value as percentage of the maximum, predefining only the existence ofstructure). Figure 7.10b presents instantaneous frequencies (that localize the corre-sponding amplitude squared) for a range of IMFs and for values 2 < IMF < 5, andFigure 7.10c shows Hilbert-Huang transforms of local intermittency thresholds basedon these relative criteria.

Having defined the points in time and IMF mode where strong events areidentified, and having checked the decomposition process for the quality of recon-struction of the original time series and the degree of orthogonality, it is possible to

FIGURE 7.5 Frequency distribution of Ci.

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select the corresponding points at every IMF that allow the coherent signal to bedistinguished from the noncoherent signal in the reconstruction formula. As anexample, Figure 7.11 presents the reconstruction of the organized and disorganizedpart of the signal applying a hard thresholding method for the Hilbert-Huang trans-form.

With well-known strategies [14,15], it is possible to average the coherent pattern[6,7,13,14] and correct the phase jitter (through an iterative process with cross-cor-relation and a time shift between the ensemble average pattern and each individualpattern), in order to extract the averaged patterns for every turbulent quantity asstreamwise, vertical velocities, and Re stresses.

Figure 7.12 illustrates the shortcomings of the corresponding wavelets recon-struction in the case of the Daub4 wavelet, where the signature of the wavelet itselfis observed in the reconstruction formula (N = 1024, m = 2, scale reconstruction).Figure 7.13 presents an IMF analysis (sum from coarser to finer scales Ri) of theintense events for two regions of the flow field where suction is applied in openchannel flow. A stronger effect of suction is observed near the wall and at the exitof the suction region.

Because of the indication of organized activity of the non-Gaussian behaviour ofthe turbulent data, we propose the analysis of the skewness and kirtosis coefficients,

FIGURE 7.6 (a,b) Frequency–time distributions of velocity signal; (c) energy–fre-quency–time distribution.

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from the reconstruction formula (starting the composition of the initial signal fromresidual and the larger scales). Figure 7.14 a presents the skewness coefficient Skfor a short signal of N = 8192. The value of the whole signal is –0.23. Figure 7.14bshows a scale analysis of the same signal with Daub4 wavelet.

Figure 7.15 shows, analysis with distributions for every IMF, of the turbulentquantities with the same length of signal for comparative reasons. Figure 7.15apresents the energy content of streamwise component Eiu and the vertical componentEiv, as percentage of the whole energy resulting from hot film measurements thatlead to conclusions for the principal time scales. Figure 7.15b shows the ratio Eiu/Eiv.Figure 7.15c gives the percentage of energy for every IMF, from LDA measurementswith and without suction application near the wall at the exit of the suction region.

7.3 CONCLUSIONS

Having established an approach using empirical mode decomposition to study openchannel flow, we must mention the need for a systematic application in differentflow situations. Nonstationary flows, vortex shedding over obstacle, and wake flowsmay comprise an ideal domain for application of this new technique. Further doc-umentation of the sifting process is necessary, including the stopping criteria andextrema interpolation, especially for difficult environments such as turbulent flowwhere the appearance of all the frequency bands makes the overall process timeconsuming. Applications that demand real-time processing, such as active control,must be investigated under this new method with concepts such as the modulated

FIGURE 7.7 (a) Velocity signal; (b) instantaneous energy distribution.

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intra-wave frequency. Nonstationary flows with short time variations would be thefield for comparison of HHT with other older techniques, such as those of entropy(MEM)–based methods. However, the final general impression remains the abilityof this novel technique to exploit the Hilbert transform and the analytic signal formultifrequency signals to achieve better localization in the frequency time domainand avoid the shortcomings of wavelets coefficients, such as the strong smear effect.

FIGURE 7.8 Frequency of events with VITA analysis (y+ = 8, x/L = 1.2).

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FIGURE 7.9 Local intermittency measure with IMF and Daub4.

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FIGURE 7.10 (a) Velocity signal; (b) instantaneous frequency variations; (c) HHTLIM withthresholds.

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FIGURE 7.11 (a) Disorganized part; (b) organized part.

FIGURE 7.12 Reconstruction with Daub4 (N = 1024, m = 2).

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Coherent Structures Analysis in Turbulent Open Channel Flow 155


FIGURE 7.13 Ratio of intense events for suction and no-suction cases.

FIGURE 7.14 (a) Sk distributions with residuals; (b) Sk distribution with scales Daub4.

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Coherent Structures Analysis in Turbulent Open Channel Flow 157


REFERENCES

1. Huang N., Shen Z., Long S. (1999). A new view of nonlinear water waves: TheHilbert Spectrum. Annu. Rev. Fluid Mech. 31: 417–457.

2. Huang N. E., Shen Z., Long S., Wu M., Shih H., Zheng Q. et al. (1998). The EmpiricalMode Decomposition and the Hilbert Spectrum for non linear and non stationarytime series analysis. Proc. R. Soc. London A 454, 903–995.

3. Zeris A. (2001). Turbulent flow in open channel with suction from the bed. Ph.D.thesis, Aristotle University, Thessaloniki, Greece, 2001.

4. Zeris A., Prinos P. (2001). Measurements of bed suction effects on an open channelflow with Laser–Doppler/Hot-Film Anemometry. EFHT Congr., pp. 1099–1103.

5. Zeris A., Prinos P. Open channel flow with suction from the bed. J. Hydraulic Eng.,submitted 2004.

6. Blackwelder R., Haritonidis J. Scaling of the bursting frequency in turbulent boundarylayers. J. Fluid Mech. Vol. 132, 87–103, 1983.

7. Bogard, D. G., Tiederman W. G. (1986). Burst detection with single point velocimetrymeasurements. J. Fluid Mech. Vol. 162, 389–413.

8. Lu, S. S., Wilmarth, W. W. (1973). Measurements of structure of Reynolds stress ina turbulent boundary layer. J. Fluid Mech. Vol. 60, 481–511.

9. Adrian R. J. On the role of conditional averages in turbulence theory. In Turbulencein Liquids, Zakin and Patterson, Eds., Princeton Science Press, 1977.

10. Wallace, J. M., Eckelmann, H., Brodkey, R. S. (1972). The wall region in turbulentshear flow. J. Fluid Mech. Vol. 54, 39–48.

11. Farge M. (1992). Wavelet transforms and their applications to turbulence. Annu. Rev.Fluid Mech. Vol. 24, 395–497.

12. Donoho D., Johnstone I. M. (1994). Ideal spatial adaptation by wavelet shrinkage.Biometrika, Vol. 81, 425–455.

13. Camussi R., Gui G. (1997). Orthonormal wavelet decomposition of turbulent flows:intermittency and coherent structure. J. Fluid Mech. Vol. 348, 177–199.

14. Raupach, M. (1981). Conditional statistics of Reynolds stress in rough wall andsmooth wall turbulent flow. J. Fluid Mech. Vol. 108, 363–382.

15. Subramanian C. S., Rajagopalan S., Antonia R., Chambers A. (1982). Comparisonof conditional sampling and averaging techniques in a turbulent boundary layer. J.Fluid Mech. Vol. 23, 335–362.



8
An HHT-Based Approach to Quantify Nonlinear Soil Amplification and Damping
Ray Ruichong Zhang

CONTENTS

8.1 Introduction ..................................................................................................1608.2 Symptoms of Soil Nonlinearity ...................................................................1618.3 Fourier-Based Approach for Characterizing Nonlinearity ..........................1628.4 HHT-Based Approach for Characterizing Nonlinearity ..............................1668.5 Applications to 2001 Nisqually Earthquake Data.......................................172

8.5.1 Detection of Nonlinear Soil Sites....................................................1738.5.2 HHT-Based Factor of Site Amplification ........................................1768.5.3 Influences of Window Length of Data ............................................1788.5.4 Comparison of HHT- and Fourier-Based Factors for Site

Amplification....................................................................................1808.5.5 HHT-Based Factor for Site Damping ..............................................184

8.6 Concluding Remarks and Discussion ..........................................................185Acknowledgments..................................................................................................187References..............................................................................................................187

ABSTRACT

This study proposes to use a method of nonlinear, nonstationary data processing andanalysis, i.e., the Hilbert-Huang transform (HHT), to quantify influences of soil non-linearity in earthquake recordings. The paper first summarizes symptoms of soil non-linearity shown in earthquake ground motion recordings. It also reviews the Fou-rier-based approach to characterizing the nonlinearity in the recordings anddemonstrates the deficiencies. It then offers the justifications of the HHT in addressingthe nonlinearity issues. With the use of the 2001 Nisqually earthquake recordings andresults of the Fourier-based approach as a reference, this study shows that the

159


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HHT-based approach is effective in characterizing soil nonlinearity and quantifyingthe influences of nonlinearity in seismic site amplification. Primary results are asfollows:

The first HHT-based component and the Hilbert amplitude spectra can identifyabnormal high-frequency spikes in the recording at sites where strong soilnonlinearity occurs; this can help to detect the nonlinear sites at a glance.

The HHT-based factor for site amplification is defined as the ratio of marginalHilbert amplitude spectra, similar to the Fourier-based one that is the ratio ofFourier amplitude spectra. The HHT-based factor is effective in quantifying soilnonlinearity in terms of frequency downshift in the low-frequency range andamplitude downshift in the intermediate-frequency range.

Hilbert and marginal damping spectra are identified in ways similar to Hilbertand marginal amplitude spectra. Consequently, the HHT-based factor for sitedamping is found as the difference of marginal Hilbert damping spectra, whichcan be extracted from the HHT-based factor for site amplification and used asan alternative index to measure the influences of soil nonlinearity in seismicground responses.

8.1 INTRODUCTION

Site amplification is the phenomenon in which the amplitude of seismic wavesincreases significantly when they pass through soil layers near the earth’s surface.It can be illustrated by considering the seismic energy flux along a tube of seismicrays, which is proportional to the impedance (density × wave speed) and squaredshaking velocity. Since the energy should be constant in the absence of damping,any reduction in the impedance is compensated by an increase in the shaking velocity,thus yielding site amplification for seismic waves in soil layers. Site amplificationis a key factor in mapping seismic hazard in urban areas (e.g., [1]) and designinggeotechnical and structural engineering systems on soils (e.g., [2]).

In general, site amplification is not linearly proportional to the intensity of inputseismic motion at bedrock because of soil nonlinearity under large-amplitude earth-quakes. The extent of soil nonlinearity can be characterized by the change of twodynamic features of soil layer, i.e., soil resonant frequency and damping, in thefrequency-dependent site amplification. Consensus has been building that thesite-amplification factors in the current codes overemphasize the extent of soilnonlinearity and thus potentially underestimate the level of site amplification. It hasbeen demonstrated [3] that the recording-based amplification factors are larger thanthose in codes for a certain range of base acceleration intensity. In addition, somefeatures of site-amplification factors used in codes and guidance for structural designcontradict recent findings from the 1994 Northridge ground motion data set [4].

The aforementioned problem might exist partly because seismologists and engi-neers lack sufficient understanding of the underlying causes in nonlinear soil. Forexample, the influence of soil heterogeneity does not scale linearly, even when thesoil is perfectly linear [5]. In other words, a linear elastic medium with randomheterogeneity can change ground motion in a way similar to that caused by medium


An HHT-Based Approach to Quantify Nonlinear Soil Amplification

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nonlinearity. Consequently, it is possible to interpret the motion influenced by ran-dom heterogeneous media as soil nonlinearity (e.g., in the form of damping [6]),which can distort the quantification of site amplification. The problem may also existpartly because there is lack of an effective approach to properly characterize non-linear features of ground motion in recordings and then to quantify them.

The objective of this study is to propose the use of a method for nonlinear,nonstationary data processing [7], referred to as the Hilbert-Huang transform (HHT)method, to characterize the soil nonlinearity from earthquake recordings. In partic-ular, this paper first summarizes symptoms of soil nonlinearity shown in earthquakeground motion recordings from previous studies and literature. It will also reviewFourier-based approaches to characterizing nonlinearity from earthquake motion anddemonstrate their deficiencies. It then proposes to use the HHT method for charac-terizing soil nonlinearity in the motion. For illustration, the study analyzes a hypo-thetical recording to show the HHT-based characterization of nonlinearity. Finally,it examines the mainshock and aftershock of the 2001 Nisqually earthquake record-ings to demonstrate the validity and effectiveness of the proposed approach forcharacterizing soil nonlinearity and the influences.

8.2 SYMPTOMS OF SOIL NONLINEARITY

In general, the stress–strain relationship of a soil becomes nonlinear and hystereticfor a large-amplitude input excitation. Such nonlinearity and hysteresis correspondto a reduction of soil strength, increased soil damping, and deformed waveform ofresponse in comparison with those of the linear case (e.g., [8–12]).

With a reduction of soil strength such as shear modulus (G) for nonlinear soil,the shear-wave velocity (v = (G/ρ)½, where ρ is the soil density) and thus thefundamental resonant frequency (f = v/4h) of the soil layer with thickness h decrease.Therefore, seismic motion recordings over a nonlinear soil layer could show strongwave response at a lower resonant frequency than for the same layer with linearexcitation. Accordingly, increased site amplification at the downshifted soil resonantfrequency can be regarded as a signature of soil nonlinearity observable in groundmotion records (e.g., [13, 14]).

On the other hand, increased damping for nonlinear soil will decrease groundmotion, thus moderating the site amplification. Since soil damping is typicallyfrequency dependent, so is the change of damping for nonlinear soil. The increaseddamping of nonlinear soil is likely lower at higher frequencies [15]. The increasedsite amplification at the downshifted soil resonant frequency and the fre-quency-dependent increased damping imply that the change in site amplificationdue to soil nonlinearity should be strongly dependent upon frequency.

Soil nonlinearity can also sometimes be inferred from abnormal high-frequencyspikes and recorded waveforms that appear only in recordings that are close to thelocations where strong nonlinearity occurred (e.g., [16]). The cusped waveforms andhigh-frequency spikes observed in the 2001 Nisqually earthquake recordings, forexample, are symptomatic of a nonlinear response at some soil sites (e.g., [17]).


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8.3 FOURIER-BASED APPROACH FOR CHARACTERIZING NONLINEARITY

In practice, the Fourier series expansion is frequently used for representing andanalyzing recorded digital data of earthquake acceleration X(t), i.e.,

, (8.1)

where ℜ denotes the real part of the value to be calculated, i = (–1)∫ is an imaginaryunit, amplitudes Aj are a function of time-independent frequency Ωj that is definedover the window in which the data is analyzed, and the Fourier amplitude spectrumis defined as

. (8.2)

To apply this Fourier spectral analysis to estimate the influences of soil nonlinearity in the seismic wave responses at soil site or simply site amplification, two sets of recordings are typically needed [18], one at a soil site and the other at a referenced site such as bedrock or outcrop. For a frequency, the Fourier-based factor of site amplification (FF) for an earthquake event (either mainshock or aftershock) can then be found by

, (8.3)

where subscripts s and r denote respectively the soil and referenced sites, andsubscripts h1 and h2 denote the two horizontal components. Note that Equation 8.3is one of many representatives for site-amplification factor that can be the ratio ofcharacteristics of seismic waves or spectral responses at a site versus referenced site.

Since the wave paths and earth structures except the soil layer are almost thesame for the soil and referenced sites, the factor for site amplification in Equation8.3 eliminates approximately the influences of source from the earthquake event andthus provides essentially the dynamic characteristics of the soil. In addition, therecordings at the referenced site are generally believed to be the results of linearwave responses and the recordings at the soil site subject to the large-amplitudemainshock to be the results of nonlinear wave responses. Accordingly, comparingthe factors from the mainshock and the aftershock could help us explore and quantifythe influences of soil nonlinearity in site amplification.

While the Fourier-based approach given here and similar methods are widelyused, they have the following deficiencies in characterizing the nonstationarity of

X t A e A t iA tj

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the earthquake motion that is caused by source, different types of propagating waves,and soil nonlinearity if the earthquake magnitude is large enough.

A Fourier-based approach defines harmonic components globally and thus yieldsaverage characteristics over the entire duration of the data. However, some charac-teristics of data, such as the downshift of soil resonant frequency at a nonlinear site,may occur only over a short portion of a record. This is particularly true when theintensity of the seismic input to a soil layer is not strong, such that the soil becomesnonlinear over only a portion of the entire duration of motion and in only a certainfrequency band. As a result, the averaging characteristic in Fourier spectral analysismakes it insensitive for identifying time-dependent frequency content. While awindowed (or short-time) Fourier-based approach can be used to improve the aboveanalysis to a certain extent, it also reduces frequency resolution as the length of thewindow shortens. Thus, one is faced with a tradeoff. The shorter the window, thebetter the temporal localization of the Fourier amplitude spectrum, but the poorerthe frequency resolution, which directly influences the measurement of downshiftof soil resonance that typically arises in a low to intermediate frequency band.

More important, a Fourier-based approach explains data in terms of a linearsuperposition of harmonic functions. Therefore, it is an appropriate, effective methodfor characterizing linear phenomena such as waves with time-independent frequency,rather than nonlinear phenomena with time-dependent frequency. An example oftime-independent and time-dependent frequency waves is a hypothetical wave recordy(t) = y1(t) + y2(t), where decaying waves y1(t) = cos[2πt + εsin(2πt)]e–0.2t havetime-dependent frequency of 1 + εcos(2πt) Hz, with ε denoting a constant factor,and noise y2(t) = 0.05sin(30t) has time-independent frequency of 15 Hz. Note thatthe waves shown in Figure 8.1 with ε = 0.5 are physically related to one type of

FIGURE 8.1 A hypothetical wave recording, consisting of nonlinear waves and noise withfrequencies 1 + 0.5cos(2πt) Hz and 15 Hz, respectively.

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water waves that result from a nonlinear dynamic process and are also representativeof seismic responses at a nonlinear soil site (to be elaborated).

The time-dependent frequency waves can be expanded into and thus interpretedby a series of time-independent frequency waves, as done by the Fourier spectralanalysis in which y(t), or y1(t) in particular, can be interpreted as to contain Fouriercomponents at all frequencies (see Equation 8.1, Figure 8.2, and Figure 8.3).

Alternatively, the expansion of y1(t), i.e.,

(8.4)

suggests that the Fourier transform of y1(t) consists primarily of two harmonicfunctions centered respectively at 1 Hz and 2 Hz for ε << 1, and the widths of theseharmonic functions are proportional to the exponential parameter 0.2, which isrelated to the damping factor. Note that Figure 8.1 and Figure 8.2 use ε = 0.5, whichis not a small number in comparison to unity, and thus they have the third observableharmonic function at 3 Hz in Figure 8.2. Therefore, one can equally well describey1(t) by saying that it consists of just two frequency components, each componenthaving a time-varying amplitude that is proportional to e–0.2t. Indeed, if one were toexamine the local behavior of y1(t) in the neighborhood of a given time instant, say

FIGURE 8.2 Fourier and marginal Hilbert amplitude spectra of the recording in Figure 8.1.

y t t t e ft1

0 20 5 2 0 5 4( ) [ . cos( ) . cos( )] ,.≈ − + + −ε π ε π oor ε << 1



165


t0, this is precisely what one would observe. The Fourier-based analysis or interpre-tation given here can also be seen in Priestley [19] and Zhang et al. [20], amongothers.

Because the true frequency content of the waves y1(t) is bounded between 1 –ε and 1 + ε, much less than 2 Hz, analysis of the above example suggests that Fourierspectral analysis typically needs higher-frequency harmonics (at least 2 Hz for theexample) to simulate the nonlinear waveform of the data. Stated differently, Fourierspectral analysis distorts the nonlinear data. Consequently, the Fourier-basedapproach in Equation 8.3 twists the influences of soil nonlinearity in site amplifica-tion. The above assertions are confirmed in Huang et al. [21] and Worden andTomlinson [22], among others, with the aid of solutions to classic nonlinear systemssuch as the Duffing equation in general, and in Zhang et al. [23] with the nonlinearsite amplification in particular.

In theory, Fourier spectral analysis in general and Fourier-based approaches forsite amplification in particular can be further used for evaluating damping factor.For example, the resonant amplification method or half-power method uses theamplitude change or width of the peaks at a certain frequency in the Fourier ampli-tude spectrum to find the damping factor of dynamic systems such as a soil layer(e.g., [24]). However, the distorted Fourier amplitude spectrum for nonlinear data

FIGURE 8.3 Fourier components (fj, j = 1, 2, 3, 4, 5) of the recording in Figure 8.1 at selectedfrequencies (i.e., 10 Hz, 5 Hz, 2 Hz, 1 Hz, and 0.5 Hz).


166



will mislead the subsequent use for damping evaluation with nonlinear soil. Forexample, the damping factor evaluated at the first and second peaks in the Fourieramplitude spectrum in Figure 8.2 suggests that the damping is associated withfrequency at 1 Hz and 2 Hz. In fact, the damping of the hypothetical record isdependent only on the true frequency content of the waves y1(t) bounded between1 – ε and 1 + ε, or 0.5 Hz and 1.5 Hz with ε = 0.5 in Figure 8.1. Accordingly, aFourier-based approach would misrepresent the influences of damping factor as itrelates to soil nonlinearity.

8.4 HHT-BASED APPROACH FOR CHARACTERIZING NONLINEARITY

A method for nonlinear, nonstationary data processing [7] can be used as an alter-native to the Fourier-based approach for characterizing soil nonlinearity. Thismethod, referred to as the Hilbert-Huang transform, consists of empirical modedecomposition (EMD) and Hilbert spectral analysis (HSA). Any complicated timedomain record can be decomposed via EMD into a finite, often small, number ofintrinsic mode functions (IMFs) that admit a well-behaved Hilbert transform. TheIMF is defined by the following conditions: (1) over the entire time series, the numberof extrema and the number of zero-crossings must be equal or differ at most by one,and (2) the mean value of the envelope defined by the local maxima and the envelopedefined by the local minima is zero at any point. An IMF represents a simpleoscillatory mode similar to a component in the Fourier-based harmonic function,but more general.

The EMD explores temporal variation in the characteristic time scale of the dataand thus is adaptive to nonlinear, nonstationary data processes. The HSA defines aninstantaneous or time-dependent frequency of the data via Hilbert transformation ofeach IMF component. These two unique features endow the HHT with a possiblyenhanced interpretive value, making it a useful alternative to Fourier componentsand amplitude spectra.

The HHT representation of data X(t) is

, (8.5)

where Cj(t) and Yj(t) are respectively the jth IMF component of X(t) and its Hilberttransform,

,

where P denotes the Cauchy principal value, and the time-dependent amplitudesaj(t) and phases θj(t) are the polar-coordinate expression of Cartesian-coordinateexpression of Cj(t) and Yj(t), from which the instantaneous frequency is defined as

X t a t e C t iY tj

j

ni t

j j

j

n

j( ) ( ) [ ( ) ( )( )= ℜ = ℜ += =

∑ ∑1 1

θ ]]

Y t PC t

t tdtj

j( )( )

=′

− ′′∫1

π


An HHT-Based Approach to Quantify Nonlinear Soil Amplification 167


. (8.6)

It should be emphasized here that the instantaneous frequency has physicalmeaning only through its definition on each IMF component as in Equation 8.5 andEquation 8.6; by contrast, the instantaneous frequency defined through the Hilberttransformation of the original data is generally less directly related to frequencycontent [7].

Equation 8.5 and Equation 8.6 indicate that the amplitudes aj(t) are associatedwith ωj(t) at time t, or, in general, functions of ω and t. Similar to the Fourieramplitude spectrum, the Hilbert amplitude spectrum is defined as

, (8.7)

and its square gives the temporal evolution of the energy distribution. The marginalHilbert amplitude spectrum, h(ω), defined as

, (8.8)

provides a measure of the total amplitude or energy contribution from each frequencyvalue, in which T denotes the time duration of the data.

In comparison with Equation 8.2, the Hilbert amplitude spectrum H(ω,t) providesan extra dimension by including time t in motion frequency and is thus more generalthan the Fourier amplitude spectrum F(Ω). While the marginal amplitude spectrumh(ω) provides information similar to the Fourier amplitude spectrum, its frequencyterm is different. The Fourier-based frequency (Ω) is constant over the sinusoidalharmonics persisting through the data window, as seen in Equation 8.1, while theHHT-based frequency ω varies with time based on Equation 8.6. As the Fouriertransformation window length reduces to zero, the Fourier-based frequency (Ω)approaches the HHT-based frequency (ω). The Fourier-based frequency is locallyaveraged and not truly instantaneous, for it depends on window length, which iscontrolled by the uncertainty principle and the sampling rate of data.

Recordings that are stationary and linear can typically be decomposed or rep-resented by a series of time-independent frequency waves through the Fourier-basedapproach in Equation 8.1. If the jth IMF component, i.e., Cj(t) in Equation 8.5,corresponds to a Fourier component with a sine function at a time-independentfrequency, the Hilbert transform of the sine function, i.e., Yj(t) in Equation 8.5, canbe found to equal the cosine function at the same frequency in opposite sign. Becausethe sign can be changed with adding a constant phase, the above analysis essentiallyleads to the consistence between Fourier- and HHT-based approaches in general,

ωθ

jjt

d t

dt( )

( )=

H t a tj

j

n

( , ) ( )ω ==

∑1

h H t dt

T

( ) ( , )ω ω= ∫0




and Fourier and marginal Hilbert amplitude spectra in particular, in characterizinglinear phenomena with time-independent frequency.

For recordings that are nonstationary and nonlinear, such as large-magnitudeearthquakes, many studies have showed [21] that the marginal Hilbert amplitudespectra can truthfully represent the nonlinear data in comparison with Fourier ampli-tude spectra. As a result, an HHT-based approach for characterizing nonlinear siteamplification can be proposed that is similar to a Fourier-based approach. TheHHT-based factor of site amplification (FH) is defined

. (8.9)

While this HHT-based factor could provide an alternative insight in character-izing and quantifying the influences of nonlinear soil in site amplification, the roleof damping in nonlinear site responses is not discerned from the general features ofsoil nonlinearity, which should be implicitly involved in the factor.

To single out the damping factor from the HHT-based factor in Equation 8.9,which is associated with amplitude aj(t) in Equation 8.7 and Equation 8.8, thephysical meanings of the jth IMF component that forms the amplitude aj(t) inEquation 8.5 is first examined below. Since all the IMF components are extractedfrom acceleration records that are the result of seismic waves generated by a seismicsource and propagating in the earth, they should reflect the wave characteristicsinherent to the rupture process and the earth medium properties. Indeed, with theaid of a finite-fault inversion method, Zhang et al. [25] examined signatures of theseismic source of the 1994 Northridge earthquake in the ground acceleration record-ings. That study looks over only the second to fifth IMF components because theyare much larger in amplitude than the remaining higher-order, low-frequency IMFcomponents. The first IMF component was not investigated in that study because itcontains information that is not simply or easily related to the seismic source (e.g.,wave scattering in the heterogeneous media). That study shows that the second IMFcomponent is predominantly wave motion generated near the hypocenter, withhigh-frequency content that might be related to a large stress drop associated withthe initiation of the earthquake. As one progresses from the second to the fifth IMFcomponent, there is a general migration of the source region away from the hypo-center with associated longer-period signals as the rupture propagates. In addition,that study shows that some IMF components (e.g., the fifth IMF) can exhibit motionfeatures reflecting the influences of nonlinear site condition.

Because of the significance of IMF components that directly relate amplitudesaj(t), Equation 8.5 can be rewritten as

(8.10)

FHh h

h hs

s h s h

r h r h

ω( ) =+

+, ,

, ,

12

22

12

22

X t a t e t ej

j

ni t

j

j

ntj j( ) ( ) ( )( ) ( )= ℜ = ℜ

= =

− +∑ ∑1 1

θ ϕΛ ii tjθ ( )




where time-dependent amplitudes Λ j(t) can be interpreted as the source-relatedintensity, ϕj(t) are the exponential factors characterizing the time-dependent decayof the waves in the jth IMF component due to damping, and

. (8.11)

Similar to the description of the instantaneous frequency in Equation 8.6, theinstantaneous damping factor can be defined as

. (8.12)

With the aid of Equation 8.11, the Hilbert damping spectrum can be found as

. (8.13)

The marginal Hilbert damping spectrum is

. (8.14)

Equation 8.14 indicates that the marginal Hilbert damping spectrum consists oftwo terms: one is from the time-dependent amplitudes aj(t) that are related tomarginal and Hilbert amplitude spectra, and the other is from source-related intensity,i.e., time-dependent amplitudes Λ j(t).

It is of interest to note that the definition of instantaneous damping factor inEquation 8.12 and subsequent spectra in Equation 8.13 and Equation 8.14 aredifferent from those in Salvino [26] and Loh et al. [27]. For recordings ofimpulse-induced or ambient linear vibration responses, some IMF components canbe extracted from the data that are related to certain vibration modes [28–30].Consequently, Λj(t) are constant and ηj(t) are proportional to the damping ratio anddamped frequency. The modal damping ratio can then be found. This is essentiallythe same as those in Salvino [26] and Loh et al. [27], if the latter can judiciallyrelate the IMF components to the vibration/wave modes.

For recordings to an earthquake, Λj are functions of time and unknown, whichare dependent upon the seismic source. Their influences in the site amplification,however, can be removed if two recordings at soil and referenced sites are used.Similar to the HHT-based factor of site amplification, the difference of marginalHilbert damping spectra at soil and referenced sites, or HHT-based factor of sitedamping, could approximately eliminate the influences of the source that is associated

a t t ej jtj( ) ( ) ( )= −Λ ϕ

ηϕ

jjt

d t

dt( )

( )=

D t ta t

a t

t

tj

j

nj

j

j

j

( , ) ( )( )

( )

( )

( )ω η= = − +

=∑

1

ΛΛ

=

∑j

n

1

d D t dta t

a t

t

t

T

j

j

j

j

( ) ( , )( )

( )

( )

( )ω ω= = − +

∫0

ΛΛ

= +∫∑=

dt d d

T

j

n

a

01

( ) ( )ω ωΛ




with Λj and thus provide essentially the characterization of the damping in the soilsite. The HHT-based factor of site damping can be found as

(8.15)

where use has been made in the last approximation of the fact that the source-relateddamping terms at the soil and referenced sites are approximately equal, i.e., ds

Λ (ω)dr

Λ (ω). Finally, comparing the HHT-based factors of site damping from the mainshock

and the aftershock could help quantify the influences of nonlinear soil damping insite responses.

To illustrate the HHT-based characterization of nonlinearity, the hypotheticalrecord in Figure 8.1 is analyzed again. Figure 8.4 shows the five IMF componentsdecomposed from the data by EMD. The first and second components (c1 and c2)capture the noise and primary waveform, while the other three (c3 to c5), withnegligible amplitudes, represent the numerical error in the EMD process. ComparingFigure 8.3 and Figure 8.4 suggests that some IMF components can be not only morephysically meaningful than the Fourier components, they can also be in principleused to explore the damping factor with the use of, e.g., the consecutive peak values

FIGURE 8.4 The five IMF components of the recording in Figure 8.1.

c5c4

−0.02

0

0.02

c3

−0.02

0

0.02

c2

−0.50

0.5

c1

−0.5

0

0.5

0

5

10

×10−3

0 1 2 3 4

Time (sec)

5 6 7 8 9 10

d d d d ds h r h s h r h∆ ( ) [ ( ) ( )] [ ( ) (, , , ,ω ω ω ω= − + −1 12

2 2 ωω

ω ω ω

)]

[ ( ) ( )] [ ( ), , , ,

2

1 12

2≈ − + −d d d ds ha

r ha

s ha

r hha

22( )]ω




and corresponding time elapse in the second IMF component, if that component isrelated to a free-vibration response or forced response at a certain mode.

The Hilbert amplitude spectrum in Figure 8.5 shows a clear picture of tempo-ral–frequency energy distribution of the data, i.e., primary waves with frequencydependence modulated around 1 Hz and bounded by 0.5 Hz and 1.5 Hz, noise at15 Hz, and the decaying energy of the primary waves with the color changing fromthe red/yellow at the beginning to the dark blue at the end of the record. In contrast,the Fourier amplitude spectrum in Figure 8.2 not only loses the information pertain-ing to temporal characteristics of the motion, but, more important, it also distortsthe information of the record by introducing higher-order harmonics, notably at 2Hz and 3 Hz. For comparison, the marginal amplitude spectrum of the recording isalso plotted in Figure 8.2, showing truthfully the energy distribution of the motionin frequency.

Figure 8.6 shows the marginal Hilbert damping spectrum, calculated withoutusing any smooth function, which was not so done in the calculation of the abovemarginal Hilbert amplitude spectrum with the use of Hilbert-Huang TransformationToolbox [31]. While oscillation of the curve in Figure 8.6 is originally due to thenumerical calculation of ·aj(t) and ·aj(t)/aj(t) that subsequently influences the compu-tation of the spectra in Equation 8.13 and Equation 8.14, the estimated mean dampingfactor at frequency 0.5 to 1 Hz is around the true value of 0.2. Compared with no

FIGURE 8.5 Hilbert amplitude spectra of the recording in Figure 8.1.

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Fre

qu

enc

y (H

z)

20

18

16

14

12

10

8

6

4

2

00 1 2 3 4 5

Time (sec)

Hilbert Spectrum (Hypothetical Water Wave)

6 7 8 9 10




damping in the record at frequency larger than 1.5 Hz, the very small mean dampingfactor in Figure 8.6 (about 0.04, lower at a factor of five than the damping factor ofwaves) due to the aforementioned unavoidable, cumulated numerical error is stillacceptable. It is believed that the oscillated curve in Figure 8.6 can be improved byreplacing it with the instantaneous mean curve. The mean curve can be found bymany methods, one of which is the use of the summation of all the IMF components,excluding the first IMF component, that are extracted from the data of the oscillatedcurve with one sifting process. The large damping factors at around 1.5 Hz are dueto the numerical error caused by the transition of two damping factors from 0.2 to0.04, although such abrupt damping change is unlikely in practice. The large damp-ing factor below 0.5 Hz is caused by the high-order, low-frequency IMF components(primarily from the third to fifth IMFs). The error in overestimating damping factorat very low frequency can be theoretically minimized if high-order, low-frequencyIMF components with very small amplitudes are judicially not used in the dampingcalculation.

8.5 APPLICATIONS TO 2001 NISQUALLY EARTHQUAKE DATA

In this section, the HHT-based approach is used to analyze the recordings of theM6.8 mainshock and the ML3.4 aftershock of the 2001 Nisqually earthquake at SDSand LAP. The SDS is a station on artificial fill with nearby liquefaction, and theaverage shear-wave velocity in the top 30 m for the soft soil is estimated as Vs30 =

FIGURE 8.6 Marginal Hilbert damping spectrum of the recording in Figure 8.1.

Mar

gin

al D

amp

ing

Sp

ectr

um

10−2

10−1

100

100

Frequency (Hz)

Wave Data–Marginal Damping Spectrum

101




148 m/s. The LAP is located over a stiff soil with Vs30 = 367 m/s. To examine thesoil nonlinearity from recordings at the two stations, recordings at SEW are usedas referenced ones, because Vs30 = 433 m/s at SEW is within the range of Vs30values for typical rock sites in the western United States. Previous Fourier-basedstudies (e.g., [17]) suggest that SDS experienced strong soil nonlinearity during themainshock while the LAP did not, with recordings at SEW as a reference.

8.5.1 DETECTION OF NONLINEAR SOIL SITES

Figure 8.7a shows the NS-acceleration recording of the 2001 Nisqually earthquakeat SDS. The one-sided cusped waves and high-frequency spikes in the S-coda waves,notably between 20 sec and 25 sec, are similar to the waveform with sharper crestsand rounded-off troughs in Figure 8.1, and have been concluded [17] to be symp-tomatic of nonlinear response at the soft soil site.

Indeed, the high-frequency spikes are believed to appear only in recordings thatare close to the locations where strong nonlinearity occurred (e.g., [16]). Therefore,singling out the spikes will help detect strongly nonlinear sites. Frankel et al. [17]used Fourier-based high-frequency band-pass filtering (10 to 20 Hz) to identify the

FIGURE 8.7 (a, top) NS-acceleration recording and (b, bottom) its first IMF of the Nisquallymainshock at SDS (soft soil).

Acc

eler

atio

n (

g)

−0.05

0

0.05

0.1

−0.14035302520

Time (sec)

1st Component Sds1

151050

Acc

eler

atio

n (

g)

−0.1

0

0.2

0.1

0.3

−0.2

Mainshock Time History Sds1

4035302520151050




spikes. It should be noted that there exist other tools to identify the spikes. Forexample, Hou et al. [32] used a wavelet-based approach to characterize the spikesfrom nonlinear vibration recordings in the vicinity of a damaged-structure locationsubject to a severe earthquake. The disadvantages of these approaches, however,reside with the subjective selection of frequency band in a Fourier-based approachand subjective selection of another wavelet in a wavelet-based approach, amongothers.

This study presents the effectiveness of the HHT-based approach in identifyingthe spikes. Figure 8.7b depicts the first IMF component of the NS-component ofmotion, clearly showing the two largest spikes between 20 sec and 25 sec. Whilethe spikes in the recording of Figure 8.7a can be visualized without using any tools,the first IMF component in Figure 8.7b is shown simply for validation of theHHT-based approach in identifying the spikes. Indeed, the EW-component of thesame recording in Figure 8.8a does not clearly show the spikes between 20 sec and25 sec. The corresponding first IMF component in Figure 8.8b is, however, able toreveal them explicitly, suggesting that the first IMF component is effective at detect-ing the high-frequency spikes. This study also analyzed the horizontal recordingsof the ML3.4 aftershock at the same location in Figure 8.9 and Figure 8.10, in whichone does not observe the spikes in the S-coda waves in the corresponding first IMF

FIGURE 8.8 (a, top) EW-acceleration recording and (b, bottom) its first IMF of the Nisquallymainshock at SDS (soft soil).

Acc

eler

atio

n (

g)

−0.04

−0.02

0

0.04

0.02

0.06

−0.064035302520

Time (sec)

1st Component Sds1

151050

Acc

eler

atio

n (

g)

−0.2

0

0.2

0.4

−0.4

Mainshock Time History Sds2

4035302520151050




components. These observations partially confirm that the nonlinearity-related spikescan be detected via the first IMF component.

It should be pointed out here that the high-frequency spikes before the S-codawaves in the first IMF component (such as those before 20 sec in Figure 8.7b, Figure8.8b, Figure 8.9b, and Figure 8.10b), which also show up in the Fourier-basedhigh-frequency band-pass (10 to 20 Hz) approach, may not be simply, directly relatedto the soil nonlinearity.

The aforementioned nonlinearity-related spikes can also be identified via Hilbertamplitude spectra. Specifically, the Hilbert amplitude spectra of the mainshock inFigure 8.11 and Figure 8.12 show two beams of high-frequency energy (5 Hz andabove) between 20 sec and 25 sec, when the dominant energy of the motion isprimarily below 5 Hz. In contrast, those of the aftershock in Figure 8.13 and Figure8.14 do not show the distinguishing high-frequency energy beams in the same timeperiod. This suggests that the first IMF component and Hilbert amplitude spectracan be used effectively to single out nonlinearity-related high-frequency spikes.Accordingly, the results will help us, at the first glance, to detect strong nonlinearsoil sites.

FIGURE 8.9 (a, top) NS-acceleration recording and (b, bottom) its first IMF of the Nisquallyaftershock at SDS (soft soil).

Acc

eler

atio

n (

g)

−1

0

1

2×10

−4

−24035302520

Time (sec)

1st Component Sds1

151050

Acc

eler

atio

n (

g)

−0.5

0

0.5

1×10

−3

−1

Aftershock Time History Sds1

4035302520151050




8.5.2 HHT-BASED FACTOR OF SITE AMPLIFICATION

Figure 8.15a shows the HHT-based factors of site amplification of the mainshockand aftershock at SDS. In calculating the factors in Equation 8.9, the correction for1/R geometrical spreading in the recordings at SDS and SEW is not carried out sincethe hypocentral distances for the sites under investigation are similar. In addition,the marginal Hilbert amplitude spectra are not smoothed in the calculation, for thenonsmoothed spectra more clearly show the characteristics of the HHT-basedapproach. Examining Figure 8.15a shows the following:

• The profile of the HHT-based factor in the frequency band up to 2.5 Hz(referred to as the low-frequency range) is generally downshifted in fre-quency from the aftershock to the mainshock, with an average shift ofapproximately 0.7 Hz. This downshift is exemplified by the peaks of theaftershock at 0.95 Hz and 2.1 Hz downshifted to those of the mainshockat 0.25 Hz and 1.4 Hz, respectively. The degree of soil nonlinearity underthe mainshock motion can be quantified by the frequency downshift.

FIGURE 8.10 (a, top) EW-acceleration recording and (b, bottom) its first IMF of theNisqually aftershock at SDS (soft soil).

Acc

eler

atio

n (

g)

−2

0

4

2

6

8×10

−4

−44035302520

Time (sec)

1st Component Sds2

151050

Acc

eler

atio

n (

g)

−0.5

0

0.5

1×10

−3

−1

Aftershock Time History Sds2

4035302520151050




• The profile of the HHT-based factor in the frequency band 2.5 to 7 Hz(intermediate-frequency range) is generally reduced in amplitude fromthe aftershock to the mainshock, with an average difference of approxi-mately a factor of 0.43. Note that the measure is carried out in the waythat the averaged factor of 7 for the aftershock is reduced to 3 for themainshock in frequency range 3 to 4 Hz.

• There is no evidence to support a difference in HHT-based factor fromabout 7 Hz up (high-frequency range) between the mainshock and after-shock.

To facilitate understanding of these observations, this study analyzes the HHT-basedfactor of the mainshock and aftershock at LAP in Figure 8.16a, which shows thefollowing:

• In the low-frequency range (below 2.5 Hz), Figure 8.16a shows a down-shift profile in both frequency and amplitude from the aftershock tomainshock that is similar to Figure 8.15a in the low to intermediatefrequency range, but the former shows a smaller shift (about 0.1 Hz in

FIGURE 8.11 Hilbert amplitude spectra of NS-acceleration recording of the 2001 Nisquallyearthquake mainshock at SDS (soft soil).

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

Nisqually Mainshock – 2-D Hilbert Spectrum - SDSns

Time (s)

0 5 10 15 20 25 30 35 40

Fre

qu

enc

y (H

z)

10−1

100

101




frequency and a factor of 0.2) than the latter (about 0.7 Hz and a factorof 0.43).

• In the intermediate to high frequency range, there is almost no differencein the two factors between the mainshock and the aftershock.

Comparison of the HHT-based factors at SDS and LAP suggests that SDS hadsevere soil nonlinearity during the mainshock and LAP had slight soil nonlinearity.Site LAP can also be regarded as having no soil nonlinearity under the mainshockif the aforementioned small downshift in both frequency and amplitude in thelow-frequency range is the result of variation of data collection and sampling.

8.5.3 INFLUENCES OF WINDOW LENGTH OF DATA

The above results were obtained on the basis of 40-sec record lengths. It can beseen from Figure 8.7 through Figure 8.14 that the characteristics of the motionduring the 40 sec change from one time interval to another. The noticeable featuresin different sections of the 40 sec (see Figure 8.7a and Figure 8.8a) are the dominantP-wave signals with high frequencies and low amplitude from 0 to 10 sec, followedby the dominant S-wave signals with intermediate frequencies and high amplitudefrom 10 to 20 sec, and finally the surface and coda waves with low frequencies and

FIGURE 8.12 Hilbert amplitude spectra of EW-acceleration recording of the 2001 Nisquallyearthquake mainshock at SDS (soft soil).

0.11

0.1

0.09

0.08

0.07

0.06

0.05

0.03

0.04

0.02

0.01

Nisqually Mainshock – 2-D Hilbert Spectrum - SDSnew

Time (s)

0 5 10 15 20 25 30 35 40

Fre

qu

enc

y (H

z)

10−1

100

101




intermediate amplitude from 20 to 40 sec. Accordingly, the HHT-based factor cal-culated over 40 sec is a characteristic averaged in frequency and amplitude, i.e.,mixing low to high frequencies and amplitudes. It is also noticed that the soilnonlinearity is likely most prevalent in the strong S-wave motion. To characterizethe soil nonlinearity precisely, one should select an appropriate window of motionfor analysis in which the soil nonlinearity shows up most strongly. However, theselection of an appropriate window is subjective.

Nevertheless, this study next examines the influence of window length on theHHT-based factor by selecting two windows, 0 to 10 sec and 10 to 20 sec. In thefirst window, 0 to 10 sec, where the P-wave signals are dominant with relativelysmall amplitudes and high frequencies, the soil is likely linear under both mainshockand aftershock motions, which should lead to the same factor in a broad spectrumof frequency for the mainshock and aftershock. This is verified in Figure 8.17a,except for a large difference in amplitude between 3 Hz and 6 Hz. The differencemight be caused by unknown factors such as a lack of recorded frequency contentin the referenced site or indeed by a change of physical properties in the soil. Furtherresearch is needed along this line.

In the second window, 10 to 20 sec, where the S-wave signals are dominant withlarge amplitudes and intermediate frequencies, the soil is likely strongly nonlinear under

FIGURE 8.13 Hilbert amplitude spectra of NS-acceleration recording of the 2001 Nisquallyearthquake aftershock at SDS (soft soil).

6

×10−4

5

3

4

2

1

Nisqually Aftershock – 2-D Hilbert Spectrum - SDSns

Time (s)

0 5 10 15 20 25 30 35 40

Fre

qu

enc

y (H

z)

10−1

100

101




the mainshock. The downshift in both amplitude and frequency from the HHT-basedfactors between mainshock and aftershock is clearly seen in Figure 8.17b.

8.5.4 COMPARISON OF HHT- AND FOURIER-BASED FACTORS FOR SITE AMPLIFICATION

To further illustrate the characteristics of the HHT approach, this study comparesthe HHT-based factors of site amplification at SDS shown in Figure 8.15a with theFourier-based ones shown in Figure 8.15b (i.e., Figure 8.7 in Frankel et al. [17]).

In the low-frequency range, Figure 8.15b shows a frequency-downshift profilefrom the aftershock to mainshock that is similar to Figure 8.15a, but the formershows a smaller shift (about 0.2 Hz) than the latter (about 0.7 Hz). Because of theaveraging characteristic in Fourier spectral analysis, as indicated in Section 8.3, thefrequency downshift measured from the HHT-based factors in Figure 8.15a maygive a more truthful indication of the soil nonlinearity than that measured fromFourier-based factors in Figure 8.15b. In addition, the factor in the low-frequencyrange in Figure 8.15a is generally somewhat larger than the factors in Figure 8.15b.

In the intermediate-frequency range, Figure 8.15b shows an amplitude-reductionprofile from the aftershock to mainshock that is similar to Figure 8.15a, but the

FIGURE 8.14 Hilbert amplitude spectra of EW-acceleration recording of the 2001 Nisquallyearthquake aftershock at SDS (soft soil).

5.5

×10−4

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5

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FIGURE 8.15 (A) HHT-based factor for site amplification at SDS (soft soil) for mainshockand aftershock of the 2001 Nisqually earthquake. (B) Fourier-based factor for site amplifica-tion at SDS (soft soil) for mainshock and aftershock of the 2001 Nisqually earthquake.

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ectr

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FIGURE 8.16 (A) HHT-based factor for site amplification at LAP (stiff soil) for mainshockand aftershock of the 2001 Nisqually earthquake. (B) Fourier-ased factor for site amplificationat LAP (stiff soil) for mainshock and aftershock of the 2001 Nisqually earthquake.

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ectr

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FIGURE 8.17 (A) HHT-based factor for site amplification with the use of recordings inwindow 0 to 10 sec at SDS (soft soil) for mainshock and aftershock of the 2001 Nisquallyearthquake. (B) HHT-based factor for site amplification with the use of recordings in window10 to 20 sec at SDS (soft soil) for mainshock and aftershock of the 2001 Nisqually earthquake.

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ectr

al R

atio

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former shows a smaller reduction (about a factor of 0.2) than the latter (about afactor of 0.43).

In the high-frequency range, Figure 8.15b shows a significant increase in theFourier-based factor from the aftershock to mainshock, while Figure 8.15a does not.Following the discussion in Section 8.3 and numerical illustration in Figure 8.1 andFigure 8.2, the high-frequency content in the Fourier amplitude spectra may beinfluenced by higher-order harmonics used to represent a nonlinear waveform, whichconsequently increases the Fourier-based factor of the mainshock in the high-fre-quency range.

This study also compares HHT- and Fourier-based factors at LAP in Figure 8.16aand b. Almost no fundamental difference in the factors is observed in terms of overallprofile, amplitude value, frequency downshift, and amplitude reduction between themainshock and aftershock, indicating that the two approaches are essentially consistentwith each other in estimating linear or approximately linear site amplification.

8.5.5 HHT-BASED FACTOR FOR SITE DAMPING

Figure 8.18 shows the HHT-based factors for site damping at SDS calculated byEquation 8.15, an alternative perspective of soil nonlinearity from recordings. It

FIGURE 8.18 Difference of marginal Hilbert damping spectra at SDS and SEW for main-shock and aftershock of the 2001 Nisqually earthquake.

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Nisqually Earthquake - Difference of Damping Factor (SDS-SEW)

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reveals that the site damping during the mainshock is much larger than that in theaftershock at frequency 0.5 to 5 Hz, suggesting that strong soil nonlinearity occurredduring the mainshock in this frequency band. The increased damping will decreasethe amplified magnitude of seismic wave responses through the nonlinear soil andthus reduce the site amplification factor. This can be confirmed from Figure 8.15a,which shows that the HHT-based factor for site amplification is observably reducedfor the mainshock from the aftershock in the similar frequency band of 0.5 to 7 Hz.

Figure 8.18 also shows that the second largest increased site damping for themainshock is in the frequency band 8 to 16 Hz. However, the HHT-based factorsfor site amplification in Figure 8.15a do not appear to change between the mainshockand aftershock. This can be explained as follows: On the one hand, the increasedsite damping for the mainshock over the frequencies 8 to 16 Hz will reduce theseismic wave responses in the same frequency band. Note that the soil nonlinearitytypically occurs under large-amplitude seismic waves incident to the soil layer, andthat the amplitude of the motion changes with time. This suggests that the soilnonlinearity is likely most prevalent in the strong S-wave motion or following surfaceor S-coda waves, but not in the P-wave motion during the mainshock. Figure 8.7aand Figure 8.8a reveal that except for the abnormal high-frequency spikes between20 sec and 25 sec, the mainshock accelerations after the S-wave arrival at about 10sec contain less high-frequency motion than does the aftershock in general, and inthe high-frequency band of about 8 to 16 Hz in particular. This fact is consistentwith the increased damping of the nonlinear soil during the mainshock in thehigh-frequency band of 8 to 16 Hz, which damped out the high-frequency waveresponses in the recordings.

On the other hand, the soil nonlinearity indeed introduces large-amplitudehigh-frequency spikes from 20 to 25 sec in the mainshock recordings in Figure 8.7aand Figure 8.8a.

Together with the fact that the HHT-based factors in Figure 8.15a show theaveraged characteristics of soil linearity and nonlinearity, these observations suggestthat the overall high-frequency (around 8 to 16 Hz) content of the mainshock maybe equivalent to that of the aftershock. This yields the same factors in Figure 8.15ain the frequency range 8 to 16 Hz.

For further clarification, the HHT-based factor for site damping at LAP isexamined. Figure 8.19 shows that the profiles of soil damping are essentially nodifferent between the mainshock and aftershock, suggesting that site LAP is linearduring the mainshock. This can be further verified from the HHT-based site-ampli-fication factors shown in Figure 8.16a.

8.6 CONCLUDING REMARKS AND DISCUSSION

This study investigates the role of the HHT method in effectively detecting soilnonlinearity and precisely quantifying soil nonlinearity in terms of site amplificationand damping. In particular, it reveals that the first IMF component can help identifythe high-frequency spikes related to soil nonlinearity, which can also be furtherconfirmed by Hilbert amplitude spectra.




The HHT-based factor of site amplification is defined as the ratio of marginalHilbert amplitude spectra, similar to the Fourier-based factor that is the ratio ofFourier amplitude spectra. The HHT-based factor has the following relationshipswith Fourier-based one:

• The HHT-based factor is essentially equivalent to the Fourier-based factorin quantifying linear site amplification.

• The HHT-based factor is more effective than the Fourier-based factor inquantifying soil nonlinearity in terms of frequency downshift in thelow-frequency range and amplitude reduction in the intermediate-fre-quency range.

• In the low to intermediate frequency range, the HHT-based factor isgenerally larger than the Fourier-based factor, which is also dependentupon the window length in which the recordings are used.

• For nonlinear site amplification, the Fourier-based factor is increased withrespect to the linear one at high frequencies; this phenomenon is not seenin the HHT-based factor. This difference may be due to the introductionof high frequencies in Fourier spectral analysis in the characterization ofa nonlinear waveform.

FIGURE 8.19 Difference of marginal Hilbert damping spectra at LAP and SEW for main-shock and aftershock of the 2001 Nisqually earthquake.

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Frequency (Hz)

Nisqually Earthquake - Difference of Damping Factor (LAP-SEW)

Mainshock

Aftershock

86420

Diff

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ce o

f D

amp

ing

Fac

tor

10−1

10−2




The HHT-based factor for site damping can be extracted from the HHT-basedfactor for site amplification and used as an alternative index to measure the influencesof soil nonlinearity in seismic wave responses at sites.

It should be pointed out that the results from this study rely mainly on the useof advanced signal processing techniques to explore and then quantify the signatureof soil nonlinearity from recordings. They must, therefore, be validated bymodel-based simulation. Recently, significant advances have been made in boreholedata collection and simulation techniques. These include data from 17 boreholearrays in Southern California [33] and from the Port Island vertical array for the1995 Hyogoken Nanbu earthquake, geophysical data including the S-wave velocityprofile in the top layer(s) at key strong motion station sites (Rosrine 2002 athttp://geoinfo.usc.edu/rosrine, USGS open file reports), and simulated broadbandground motion for scenario earthquakes including nonlinear soil effects [34]. Theabove information should allow the further validation of the observations and resultsfrom this study.

ACKNOWLEDGMENTS

The author would like to express sincere gratitude to Norden E. Huang at NASA;Stephen Hartzell, Authur Frankel, and Erdal Safak at USGS; Lance VanDemarkfrom Colorado School of Mines; and Yuxian Hu from China Seismological Bureauand Jianwen Liang from Tianjin University of China for providing data, Fourieranalysis and calculations, and more important, constructive suggestions. This workwas supported by the National Science Foundation with Grant Nos. 0085272 and0414363, and by the US-PRC Researcher Exchange Program administered by Mul-tidisciplinary Center for Earthquake Engineering Research. The opinions, findings,and conclusions expressed herein are those of the author and do not necessarilyreflect the views of the sponsors.

REFERENCES1. Frankel, A., C. Mueller, T. Barnhard, D. Perkins, E. Leyendecker, N. Dickman, S.

Hanson, and M. Hopper (2000), “USGS national seismic hazard maps,” EarthquakeSpectra, v. 16, pp. 1–19.

2. Kramer, S.L. (1996) Geotechnical Earthquake Engineering, Prentice-Hall, Inc. UpperSaddle River, NJ.

3. Borcherdt, R.D. (2002) “Empirical evidence for site coefficients in building codeprovisions,” Earthquake Spectra, Vol. 18, pp. 189–217.

4. Hartzell, S. (1998) “Variability in nonlinear sediment response during the 1994Northridge, California, earthquake,” Bull. Seism. Soc. Am. Vol. 88, 1426–1437.

5. O’Connell, D.R.H. (1999) “Influence of random-correlated crustal velocity fluctua-tions on the scaling and dispersion of near-source peak ground motions, Science, Vol.283, pp. 2045–2050, March 26.

6. Yoshida, N. and S. Iai (1998) “Nonlinear site response and its evaluation and predic-tion,” The Effects of Surface Geology on Seismic Motion, (Irikura, Kudo, Okada andSasatani, eds.), Balkema, Rotterdam, pp. 71–90.


http://geoinfo.usc.edu/



7. Huang, N.E., S. Zheng, S.R. Long, M.C. Wu, H,H. Shih, Q. Zheng, N-C. Yen, C.C.Tung, and M.H. Liu (1998). “The empirical mode decomposition and Hilbert spec-trum for nonlinear and nonstationary time series analysis." Proc. R. Soc Lond., A 454903–995.

8. Hardin, B.O. and V.P. Drnevich (1972a) “Shear modulus and damping in soils:measurement and parameter effects,” J. Soil. Mech. Foundations Div. ASCE, 98,603–624.

9. Hardin, B.O. and V.P. Drnevich (1972b) “Shear modulus and damping in soils: designequation and curves measurement and parameter effects,” J. Soil. Mech. FoundationsDiv. ASCE, 98, 667–692.

10. Erdik, M. (1987) Site response analysis. In Strong Ground Motion Seismology (Erdik,M. and Toksoz, M., eds.), D. Reidel Publishing Company, Dordrecht, 479–543.

11. Vucetic, M. and R. Dobry (1991) “Effect of soil plasticity on cyclic response, J.Geotech. Eng., 117, 89–107.

12. Silva, W., S. Li, B. Darragh, and N. Gregor (1999). “Surface geology based motionamplification factors for the San Francisco Bay and Los Angeles areas,” A PearlReport.

13. Field, E.H., P.A. Johnson, I.A. Beresnev, and Y. Zeng (1997) “Nonlinearground-motion amplification by sediments during the 1994 Northridge earthquake,Nature, 390, 599–602.

14. Beresnev, I.A., E.H. Field, P.A. Johnson, and K.E.A. Van Den Abeele (1998) “Mag-nitude of nonlinear sediment response in Los Angeles basin during the 1994Northridge, California, earthquake, Bull. Seis. Soc. Am., Vol. 88, pp. 1097–1084.

15. Joyner, W.B. (1999). “Equivalent-linear ground-response calculations with fre-quency-dependent damping,” Proceedings of the 31st Joint Meeting of the US-JapanPanel on Wind and Seismic Effects (UJNR), Tsukuba, Japan, May 11–14, 1999, pp.258–264.

16. Bonilla, L.F., D. Lavallee, and R.J. Archuleta (1998). “Nonlinear site response:laboratory modeling as a constraint for modeling accelerograms,” The Effects ofSurface Geology on Seismic Motion, (Irikura, Kudo, Okada and Sasatani, eds.),Balkema, Rotterdam.

17. Frankel, A.D., D.L. Carver, and R.A., Williams (2002) “Nonlinear and linear siteresponse and basin effects in Seattle for the M6.8 Nisqually, Washington, Earthquake,”Bull. Seism. Soc. Am., v. 92, pp. 2090–2109.

18. Safak, E. (1997). “Models and methods to characterize site amplification from a pairof records,” Earthquake Spectra, Vol. 13, pp. 97–129.

19. Priestley, M.B. (1981). Spectral Analysis and Time Series, Vol. 2, Multivariate Series,Prediction and Control, Academic Press, London, 1981.

20. Zhang, R., S. Ma, E. Safak, and S. Hartzell (2003b) “Hilbert-Huang transformanalysis of dynamic and earthquake motion recordings,” ASCE Journal of Engineer-ing Mechanics, Vol. 129, No. 8, pp 861–875.

21. Huang, N.E., Z. Shen, and R.S. Long, (1999). “A new view of nonlinear water waves:Hilbert Spectrum.” Annu. Rev. Fluid Mech. 31,417–457.

22. Worden, K. and Tomlinson, G.R. (2001) Nonlinearity in Structural Dynamics: Detec-tion, Identification and Modeling, Institute of Physics Publishing, Bristol and Phila-delphia.

23. Zhang, R., L. VanDemark, J. Liang, and Y.X. Hu (2003) “Detecting and quantifyingnonlinear soil amplification from earthquake recordings,” Advances in StochasticStructural Dynamics (Zhu, Cai and Zhang, eds.), pp. 543–550.




24. Clough, R.W. and J. Penzien (1993). Dynamics of Structures, Second Edition,McGraw-Hill, Inc., New York.

25. Zhang, R., S. Ma, and S. Hartzell (2003) “Signatures of the seismic source inEMD-based characterization of the 1994 Northridge, California, earthquake record-ings,” Bulletin of the Seismological Society of America, Vol. 93, No. 1, pp 501–518.

26. Salvino, L.W. (2001) “Evaluation of structural response and damping using empiricalmode analysis and HHT,” in CD ROM, 5th World Multiconference on Systemics,Cybernetics and Informatics, July 22–25, 2001, Orlando, Florida.

27. Loh, C.H., T.C. Wu and N.E. Huang (2001). Application of the empirical modedecomposition-Hilbert spectrum method to identify near-fault ground-motion char-acteristics and structural responses, Bull. Seism. Soc. Am. 91, 1339–1357.

28. Yang, J.N., Y. Lei, S. Pan and N. Huang (2002a). “System identification of linearstructures based on Hilbert-Huang spectral analysis. Part I: Normal modes,” Earth-quake Eng. Structural Dynamics, Vol. 32, pp. 1443–1467.

29. Yang, J.N., Y. Lei, S. Pan and N. Huang (2002b). “System identification of linearstructures based on Hilbert-Huang spectral analysis. Part II: Complex modes,” Earth-quake Eng. Structural Dynamics, Vol. 32, pp. 1533–1554.

30. Xu, Y.L., S.W. Chen, and R. Zhang (2003) “Modal identification of Di Wang buildingunder typhoon York using HHT method,” The Structural Design of Tall and SpecialBuildings, Vol. 12, No. 1, pp. 21–47.

31. Hilbert-Huang Transformation Toolbox (2000). Professional Edition V1.0, PrincetonSatellite Systems, Inc., Princeton, New Jersey.

32. Hou, Z., M. Noori and R. St. Amand (2000) “Wavelet-based approach for structuraldamage detection,” ASCE Journal of Engineering Mechanics, 126(7), pp. 677–683.

33. Archuleta, R.J. and J.H. Steidl (1998). “ESG studies in the United States: resultsfrom borehole arrays,” The Effects of Surface Geology on Seismic Motion, (Irikura,Kudo, Okada and Sasatani, eds.)., Balkema, Rotterdam, pp. 3–14.

34. Hartzell, S., A. Leeds, A. Frankel, R.A. Williams, J. Odum, W. Stephenson and W.Silva (2002). “Simulation of broadband ground motion including nonlinear soileffects for a magnitude 6.5 earthquake on the Seattle fault, Seattle, Washington,” Bull.Seism. Soc. Am. Vol. 92, No. 2, pp. 831–853.



9
Simulation of Nonstationary Random Processes Using Instantaneous Frequency and Amplitude from Hilbert-Huang Transform
Ping Gu and Y. Kwei Wen

CONTENTS

9.1 Introduction ..................................................................................................1919.2 Hilbert-Huang Transform (HHT).................................................................1949.3 IMF Recombination Method .......................................................................1979.4 Improved Wen-Yeh Method.........................................................................2009.5 Conclusions ..................................................................................................203Acknowledgments..................................................................................................207References..............................................................................................................207Appendix: Relation between Hilbert Marginal Spectrum and Fourier Energy Spectrum ......................................................................................209

9.1 INTRODUCTION

With the ever-increasing power of the computer and of the capacity of computationalmechanics, simulation has become an indispensable tool in engineering. It is moreso in earthquake engineering since earthquake ground motions are highly random andstructural behaviors are nonlinear and inelastic, making random vibration solutions

191


192



difficult to obtain. Generally speaking, structural responses are functions of the entireground excitation time history. Current codes use a scalar intensity measure such asspectral acceleration or displacement for the earthquake demand on structures. Thesemeasures do not reflect satisfactorily the effects of near source or effects due tohigher modes and other important structural response behaviors caused by groundmotions. For performance evaluation of complex structural systems, simulation ofearthquake ground motions and time history response analysis have played a moreand more important role.

Many models for simulation based on earthquake records have been proposed.For example, a stationary white noise model was first proposed [1], followed byfiltered white noise models to reflect the site characteristics [2, 3, 4]. A high-passfilter was later added to remove the undesirable zero frequency content of the powerspectral density function [5]. To account for the intensity variation with time, auniformly (amplitude) modulated random process was first used [6, 7]. Nonstationarymodels with frequency content variation with time — in particular, the long-durationacceleration pulses observed in many near-fault earthquake records — have alsobeen developed.

Saragoni and Hart [8] proposed a modulated filtered Gaussian white noise modelwith different intensity and frequency content in three consecutive time intervals.The arbitrary selection of time intervals and the resultant abrupt change of frequencycontent in this model, however, are difficult to justify physically. The oscillatoryprocess described by an evolutionary power spectral density suggested by Priestley[9], allowing both amplitude and frequency content variation with time, has beenwidely used in earthquake ground motion simulation. For example, Lin and Yong[10] formulated an evolutionary Kanai-Tajimi [2, 3] model as a filtered pulse train,where the filter represents the ground medium and the pulse train represents inter-mittent ruptures at the earthquake source.

Deodatis and Shinozuka [11] simulated the earthquake ground motions usingan autoregressive (AR) model based on selected parametric forms of the evolutionarypower spectral density for given earthquake records. Li and Kareem [12, 13] furtherextended the method using the fast Fourier transform (FFT) and digital filteringtechnique to make it more computationally efficient. Der Kiureghian and Crempien[14] proposed an evolutionary earthquake model composed of individually modu-lated band-limited white noise processes and estimated the parameters of evolution-ary power spectral density by matching the moments of each component process.Papadimitriou [15] produced a parsimonious nonstationary model by applying asecond-order filter with slowly varying parameters to time-modulated white noise.Conte and Peng [16] proposed to use Thomson’s spectrum to estimate the evolu-tionary power spectra and proposed a model for simulation based on this. Byextending the energy principle of wavelet transform developed by Iyama and Kuwa-mura [17], Spanos and Failla [18] and Spanos et al. [19] proposed methods usingthe wavelet spectrum to estimate the evolutionary power spectral density.

The main challenge in evolutionary process-based methods lies in parameterestimation. In addition to estimating the power spectral density function, one also


Simulation of Nonstationary Random Processes

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needs to know the modulation functions of frequency and time, which are not knowna priori and are difficult to estimate. Motivated by an amplitude- and frequency-mod-ulated process proposed by Grigoriu et al. [20], Yeh and Wen [21] proposed to modelground motion as I(t)ς(φ(t)), where I(t) is a deterministic intensity envelope thatcontrols the amplitude of the ground motion, ς(φ(t)) is a stationary filtered whitenoise in time scale φ and depicts the spectral density form of the process, and φ(t)is a smooth, strictly increasing function that depicts the frequency modulation. Thestationary white noise was filtered to have a Clough-Penzien spectrum [5] with zeromean and unit variance. By changing the time scale of the stationary filtered whitenoise, the spectral density of the stationary process becomes time varying:

. (9.1)

This method does not require the “slowly time-varying” assumption implied inthe evolutionary spectrum method, and it gives a time-dependent power spectraldensity as a simple modification of the spectral density of the stationary process.However, the frequency modulation function is estimated by a zero-crossing rateprocedure and is independent of the amplitude variation. It is therefore somewhatrestrictive in nature and only suited for a certain special class of random processes.It has been applied to earthquake ground motion simulation with some success butlacks general theoretical basis for wider applications.

Recently, Huang et al. [22] introduced a new method of spectral analysis ofnonstationary time series based on empirical modal decomposition (EMD), Hilberttransform, and the concept of instantaneous frequency, commonly referred to as themethod of Hilbert-Huang transform (HHT). Unlike the Fourier transform, the Hilberttransform emphasizes local behavior and does not decompose the signal into sineand cosine waves of fixed frequencies; thus it is suitable for nonstationary processes.HHT decomposes the signal into orthogonal functions called intrinsic mode func-tions (IMFs) directly based on data. The Hilbert transform of IMFs yields theinstantaneous amplitude and instantaneous frequency. HHT can be used to overcomethe difficulties of the Wen-Yeh simulation method since EMD provides a reasonableand physically meaningful decomposition of the nonstationary signal. The Hilberttransform yields instantaneous amplitude and frequency of each IMF, which can beused directly in the Wen-Yeh method. It bypasses the difficult tasks of estimatingand constructing analytical functions for frequency and amplitude modulation. Also,since these two modulation functions are obtained from Hilbert transform of IMFs,the dependence between the two functions is preserved.

In this study, an IMF recombination method for simulation of nonstationaryprocesses is first proposed, followed by an improved Wen-Yeh method. After a briefintroduction of HHT, we describe the IMF recombination method, followed by theimproved Wen-Yeh Method. We then compare the two. These methods are demon-strated by numerical examples of simulation of earthquake ground motions.

S tt

StCPςς ω

φω

φ( , )

( ) ( )=

′ ′

1


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9.2 HILBERT-HUANG TRANSFORM (HHT)

What follows is a brief description of the background of the HHT. Details can befound in Huang et al. [22]. First, note that a time series can be decomposed into aseries of IMFs. An IMF is defined as a time series that satisfies two requirements:

1. Its number of extrema and number of zero-crossings are either equal ordiffer by one.

2. The mean value of the envelopes defined by the local maxima and minimais zero.

The extraction of IMFs out of a time series requires a repeated “sifting” proce-dure called empirical mode decomposition. EMD is described briefly as follows:Connect all the local maxima by a cubic spline as the upper envelope and connectall the local minima by a cubic spline as the lower envelope. Then calculate themean of the envelopes and subtract the mean from the time series. The result is thefirst component. Ideally it should have satisfied the requirements for an IMF, but inpractice a gentle hump on a slope can be amplified to become a local extreme. Sothe procedure is repeated until an IMF is obtained. This is the first IMF. Subtractthe first IMF from the original time series, then continue the procedure on the residueto obtain the second IMF. Repeat the process to obtain the third, fourth, fifth, andsubsequent IMFs, until the residue is too small to be of any consequence or becomesa monotonic function from which no more IMF can be extracted.

After the EMD, the time series X(t) can be expressed in terms of IMFs as follows:

(9.2)

in which Cj(t) is the jth IMF, and rn(t) is the residue that can be the mean trend ora constant. Taking Hilbert transform of Cj(t),

(9.3)

in which P indicates Cauchy principal value. One can then form an analyticalfunction

Zj(t) = Cj(t) + iDj(t) = aj(t) . (9.4)

The instantaneous frequency of the IMF is given by

. (9.5)

X t C t r tj

j

n

n( ) ( ) ( )= +=

∑1

D t PC

tdj

j( )( )

=−∫1

πττ

τ

ei tjθ ( )

ωθ

jjt

d t

dt( )

( )=



195


One can then express X(t) as follows:

(9.6)

in which Re denotes the real part. Therefore, aj(t), j = 1, …, n, therefore control theamplitude variation; each has an instantaneous frequency at a given time. Collec-tively, the functions aj(t) and j(t) define the Hilbert amplitude spectrum. Similarly,aj

2(t) and j(t) define the energy distribution with respect to time and frequency, calledthe Hilbert energy spectrum or simply Hilbert spectrum, H(,t). The marginal Hilbertenergy spectrum is obtained by integration over the duration:

. (9.7)

Because of the local nature of the Hilbert transform and the concept of instan-taneous frequency, the Hilbert marginal spectrum describes an energy contributioncorresponding to a given frequency at a certain time but not the entire duration ofthe process, as would be the case for a Fourier spectrum. This is a fundamentaldifference between these two approaches.

As an example, the method is applied to the well-known N-S component of theEl Centro earthquake record of 1940 and the Newhall ground motion of the 1994Northridge earthquake in SAC-2 ground motion records [23], representing far-sourceand near-source records respectively. The time histories of the ground accelerations(in cm/sec2) are shown at the top of Figure 9.1 (At the bottom of the figure, foursimulated samples of the ground accelerations are shown based on the IMF recom-bination method described later in this paper.) The IMFs of the Newhall record areshown in Figure 9.2. The Hilbert spectra are calculated and shown in Figure 9.3.

As can be seen, one distinct advantage of the Hilbert spectrum is that the energydistribution over time and frequency is always sharp. For example, a darker shadeof color in the plot indicates higher energy, which can be clearly seen in the Newhallrecord from 4 to 10 seconds and below 20 rad/sec (3 cps), a frequently observedfeature of near-source records; it can also be seen over a longer time period in theEl Centro record. Figure 9.4 compares the marginal Hilbert energy spectrum withthe Fourier energy spectrum obtained from the same record. There are some obviousdifferences due to the two different methods of spectral description. The scale ofthe Hilbert marginal spectra has been adjusted by a factor of π for better comparison.Details of comparison of these two spectra and the conversion factor are shown inthe Appendix. Note that the Hilbert transform can be carried out by first performingthe Fourier transform on the original signal, changing the phase of each positivefrequency component by /2 and the phase of each negative frequency componentby –/2, and finally performing an inverse Fourier transform to obtain the Hilberttransform of the sample [24]. One can use FFT to make the procedure more efficient.

X t a t e r tji t

j

n

nj( ) Re ( ) ( )( )= +

=∑ θ

1

h H t dt

T

( ) ( , )ω ω= ∫0


196



In spite of its numerical difficulties — such as in end swing caused by splinefitting, convergence in the sifting procedure; ambiguity of the local symmetryrequirement of IMFs; and dealing with intermittent signals, negative frequency, andend distortion in Hilbert transform — HHT represents a new approach to thechallenging problem of time–frequency analysis of nonstationary signals, especiallyfor slowly varying signals, that has many advantages over existing methods. Insteadof decomposing the signal into predetermined analytical functions, it obtains theorthogonal functions (IMFs) directly from the data; thus it is more adaptive, and theresulting IMFs usually carry physical meaning. For example, it has been shown thatIMFs of earthquake recordings have physical meanings related to seismic sourcemechanism [25]. Unlike the Fourier transform, the Hilbert transform emphasizeslocal behavior, as the Hilbert transform is the convolution of the signal with a windowfunction of 1/t. Because of these characteristics, HHT represents an attractive alter-native to other methods currently available, such as short-time Fourier transform,wavelet transform, instantaneous spectrum, double-frequency spectrum, Kar-hunen-Loeve decomposition, and so forth.

The emphasis of HHT study so far has been on data analysis and interpretationof a deterministic time series of a real physical phenomenon [22, 26–33]. Much lesshas been done to further development of the method as a simulation tool for non-

FIGURE 9.1 Recorded (top) and simulated El Centro and Newhall ground motions usingIMF recombination method.



197


stationary random processes. Such extension is essential for performance evaluationof engineering systems, since the demands need to be modeled as random processes.This paper proposes to use the new approach for simulation of nonstationary randomprocesses and to apply the simulation methods to earthquake engineering.

9.3 IMF RECOMBINATION METHOD

The Hilbert spectral representation of a signal,

suggests that the underlying random process can be represented by introducingrandom elements as follows:

, (9.8)

FIGURE 9.2 IMFs of the 1994 Northridge Newhall record (SAC LA14).

x t a t e r tji t

j

n

nj( ) Re ( ) ( )( )=

+

=∑ θ

1

X t a t e rji t

j

n

nj j( ) Re ( ) ([ ( ) ]=

++

=∑ θ φ

1

tt)


198



in which φj is an independent random phase angle uniformly distributed between 0and 2π. X(t) is therefore a random process. The process has the following mean,covariance, and variance functions:

(9.9)

FIGURE 9.3 Hilbert spectra of 1940 El Centro and 1994 Northridge Newhall ground motion.

Fre

qu

enc

y (r

ad/s

)

140

120

100

80

60

40

20

0 5 10 15

Time (s)

Hilbert Energy Spectrum for EI Centro Record

20 25

Fre

qu

enc

y (r

ad/s

)

140

120

100

80

60

40

20

0 5 10 15

Time (s)

Hilbert Energy Spectrum for Newhall Record

20 25 30

µ θ φX j

i t i

j

n

t a t e E e rj j( )Re ( ) ( )( )

=∑

+

1

nn nt r t( ) ( )=



199


FIGURE 9.4 Comparison of Hilbert marginal spectrum (solid line) with Fourier energyspectrum (dashed line).

103

102

101

100

rad/s

Hilbert Marginal Spectrum and Fourier Energy Spectrum for Newhall Record

10−1

10−2

10−3

10−2

10−1

100

101

102

103

104

105

106

Fourier Energy Spectrum

Hilbert Marginal Spectrum (adjusted)

102

101

100

rad/s

Hilbert Marginal Spectrum and Fourier Energy Spectrum for EI Centro Record

10−1

10−6

10−7

10−5

10−4

10−3

10−2

10−1

Fourier Energy Spectrum

Hilbert Marginal Spectrum (adjusted)


200



(9.10)

(9.11)

X(t), as given in Equation 9.8, has a Hilbert energy spectrum characterized by aj2(t)

with instantaneous frequency d(j(t))/dt, for j = 1, …, n. Due to the Central LimitTheorem, X(t) approaches a Gaussian process for large n. This Hilbert spectralrepresentation model suggests a simple method for simulation of a nonstationaryprocess as shown in Equation 9.8. One can generate the random phase angles oncomputer and use Equation 9.8 to simulate the process. The name reflects the factthat it is essentially a method of recombination of the IMFs. The Hilbert spectrumof each simulated sample can be shown to be the same as that of the recorded groundmotion.

The physical basis for this method is that the frequency and amplitude of eachIMF represent those of a sub-wave caused by a given physical mechanism of therupture surface of the earthquake, as shown by Zhang [25]. For comparison withthe traditional Fourier-based source inversion solution obtained by Hartzell et al.[34] for the Northridge earthquake based on 70 records, Zhang studied the same 70earthquake records using HHT. The spatial distributions of the slip amplitude overthe finite fault, each of which corresponds respectively to the second to fifth IMFsof those records, were plotted. It was found that the large slip amplitude regions arelocated at almost the same regions as those in Hartzell’s study, indicating that wavemotions in each IMF are generated by a fraction of the source. In addition, therupture with large-slip region scatters out to the northwest from the hypocenter, withthe IMF changing from the second to the fifth, indicating that the seismic waves aregenerated sequentially from the dominant short-period signals to main long-periodsignals as the rupture propagates. Therefore, the IMFs are closely related to thesource mechanism in general and the source heterogeneity and rupture process inparticular. For future earthquakes of similar characteristics, the rupture surface spatialfeatures are random and unpredictable. The IMF recombination method representsan efficient way to capture these variabilities.

Simulations of the N-S component of the El Centro and the Northridge Newhallrecords (SAC LA14) were carried out, and sample time histories are shown in Figure9.1. The simulated samples represent a reasonably statistical image of the intensityand frequency variation with time of the records. The method has also been extendedto vector processes.

9.4 IMPROVED WEN-YEH METHOD

The IMF recombination method has certain shortcomings compared to the Wen-Yehmethod. They are as follows:

K t t a t a t t tXX j j j j( , ) ( ) ( )cos[ ( ) ( )]1 2 1 2 1 212

= −θ θjj

n

=∑

1

σ x X j

j

n

u t a t22

2

1

12

(t) E X t= ( ) − ( ) = ( )=

∑



201


• In the context of the Wen-Yeh method, the underlying stationary processfor each IMF has only one frequency component.

• There is only a small number of random elements. For example, Equation9.8 has only n random variables, n being the number of IMFs, which isusually not very large (usually <15 for earthquake accelerograms).

• No smoothing of aj(t) and j(t) is done to separate noise from the underlyingtrue physical characteristics.

Taking advantage of both methods, an improved Wen-Yeh method is proposed.In this method, the recorded ground motion is first decomposed into IMFs. Thenthe smoothing of aj(t) and j(t) of each IMF is performed to separate noise from theunderlying true physical characteristics, from which the underlying stationary ran-dom process of the IMF is obtained and simulated by using the well-known spectralrepresentation method [35]. The frequency and amplitude of the simulated sampleof the stationary random process are then modulated in time. The frequency mod-ulation is done by changing the time scale as in the Wen-Yeh method. The frequencymodulation function and amplitude modulation function are obtained directly fromthe Hilbert transform. Finally, the sample function of the process is simulated bycombining the simulated IMFs. The procedure is described in detail as follows:

1. Decompose the record into IMFs Cj(t), j = 1, …, n, by EMD, and determinethe aj(t) and j(t) for each IMF by Hilbert transform.

2. Smooth aj(t) and j(t) and denote the smoothed functions respectively asajs(t) and js(t).

3. Obtain the reduced process Cjr(t) from Cj(t) by removing the amplitudemodulation ajs(t) from Cj(t) and changing the time scale by ωjs

–1 (t). Chang-ing time scale is done by making the signal a function of θ instead of t,θ being the integration of js(t). Then resample the signal using an evenlyspaced.

4. Simulate the reduced process Cjr(t) as stationary process and obtain sam-ples of the process Sjr(t).

5. Restore the time scale by using the function ωjs(t) in Sjr(t), and restore theamplitude modulation by using ajs(t). The result is the simulated jth IMFSj(t). Restoring time scale is done by first resampling the underlyingstationary process of θ and then expressing the signal as a function oftime t.

6. Add all Sj(t), j = 1, …, n.

The procedure is entirely numerical and based on sample data. No analyticalforms are assumed for the frequency and amplitude modulation or for the spectrumof the underlying stationary random process. It is therefore conceptually simple andeasy to implement.

Figure 9.5 shows eight simulated samples using this method along with therecord for the Newhall Northridge earthquake. Figure 9.6 shows the response spectraof the samples. As can be seen, the long-duration pulses of the near-fault groundmotion are reproduced well. Both the amplitude and frequency changes with time




are reflected well in the simulated samples. The response spectra generally fluctuatearound that of the record. Unlike samples generated by methods based on analyticalmodels of the spectra, the response spectra of the simulated samples retain the jaggedlook of those of the records. The artificial smoothness of the response spectra of thesimulated ground motions obtained by existing power spectrum-based methods hasbeen criticized by seismologists and practicing engineers.

To further illustrate the procedure, the steps leading to the first sample in Figure9.5, shown below by figures, are as follows:

1. Decompose the record into IMFs (Figure 9.2). 2. Smooth aj(t) and j(t). Figure 9.7 shows a2(t) and a2s(t), and Figure 9.8

shows 2(t) and 2s(t) as an example. Here, 24-point rectangular windowsmoothing with zero-padding is used.

3. Obtain the reduced process Cjr(t). The reduced process C2r(t) is shown inFigure 9.9 as an example. As can be seen, both the amplitude and fre-quency are much more uniform than the original time history of the secondIMF. Figure 9.10 illustrates the effect of changing time scale: it removesthe frequency modulation from the signal and makes it stationary.

4. Simulate the reduced process Cjr(t) as a stationary process. A sample ofS2r(t) is shown in Figure 9.11.

FIGURE 9.5 Simulation of Newhall Northridge 1994 by improved Wen-Yeh method.

Simulated samples of Newhall LA14

Original record of Newhall LA 14

302520151050−1000

0

1000−1000

0

1000−1000

0

1000−1000

0

1000−1000

0

1000−1000

0

1000−1000

0

1000−1000

0

1000−1000

0

1000


Simulation of Nonstationary Random Processes 203


5. Restore the time scale and the amplitude modulation to obtain the simu-lated jth IMF Sj(t). The restored sample in Step 4 of the second IMF isshown in Figure 9.12 and compared with the original second IMF.

6. Add all Sj(t), j =1 to n. The result is Sample 1 in Figure 9.5.

Further investigation is needed, including refinement of the smoothing process,modeling of the possible dependence among the IMFs. This method is being usedin a study of the effects of near-fault ground motions on buildings and structuresand in simulation of suites of uniform hazard ground motions for performanceevaluation.

9.5 CONCLUSIONS

Two new methods of simulation of nonstationary random processes are presentedbased on the Hilbert-Huang transform of sample observations, namely the IMF recom-bination method and the improved Wen-Yeh method. Both take advantage of EMD andthe instantaneous frequency and amplitude of the Hilbert transform, thus overcomingdifficulties in the estimation of the frequency modulation and interdependence of

FIGURE 9.6 Response spectra of Newhall LA14 simulations by improved Wen-Yeh method.

1098765

Period (sec)

43210

Pse

ud

o A

ccel

erat

ion

(g

)

3.5

3

2.5

2

1.5

1

0.5

0

Simulation 1

Simulation 2

Simulation 3

Simulation 4

Simulation 5

Simulation 6

Simulation 7

Simulation 8

Original record

Response Spectra of Newhall LA 14 Simulations using

Improved Wen-Yeh Method (5% damping)




FIGURE 9.7 Original (dashed line) and smoothed (solid line) instantaneous amplitude func-tion of the second IMF.

FIGURE 9.8 Original (dashed line) and smoothed (solid line) instantaneous frequency func-tion of the second IMF.

3025201510

Instantaneous amplitude function of the 2nd IMF

50

500

450

400

350

300

250

200

150

100

50

0

Amplitude function

Smoothed amplitude function

3025201510

Instantaneous frequency function of the 2nd IMF

50

120

100

80

60

40

20

0

Frequency function

Smoothed frequency function




FIGURE 9.9 Reduced (stationary) second IMF.

FIGURE 9.10 Effect of change in time scale to remove frequency modulation: original (top);frequency modulation removed (bottom).

350300250200150

Reduced 2nd IMF

100500

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−1.540 50 60 70 80 90 100 110 120 130

−1

−0.5

0

0.5

1

1.5

−1.53 4 5 6 7 8 9 10

−1

−0.5

0

0.5

1

1.5




FIGURE 9.11 Simulated sample of reduced (stationary) second IMF.

FIGURE 9.12 Comparison of original (dotted line) and simulated (solid line) second IMF.

−1.5

−1

−0.5

0

0.5

1

1.5

2

−20 50 100 150

Simulated sample of reduced 2nd IMF

200 250 300 350

30252015

Simulated (solid) and original (dotted) 2nd IMF

1050−500

−400

−300

−200

−100

0

100

200

300

400

500




frequency and amplitude modulation functions faced by most currently availablemethods. These new methods extract and preserve the true physical features of theprocess. They have advantages over most currently available methods since assump-tions about any functional forms of the spectra or about piecewise stationarity ofthe process are unnecessary; difficulties associated with estimation of these analyticalfunctions from data are therefore bypassed. These methods are largely numericaland therefore suited for computer-aided model-based simulations. Numerical exam-ples on earthquake ground motion simulation were carried out. The sample groundmotions and records are shown to have the same time-varying intensity and spectralcharacteristics as in the samples from the underlying nonstationary process. Theresponse spectra of the simulated ground motions retain the jagged features of thespectra of the real records. The proposed HHT method has been shown to have greatpotential in engineering applications when dealing with nonstationary processes.

ACKNOWLEDGMENTS

This research is supported by the University of Illinois Research Board and the NSFEarthquake Engineering Research Center Program under Award NumberEEC-9701785. Information exchange with C. H. Loh of National Taiwan Universityis greatly appreciated.

REFERENCES1. Bycroft, G. N. (1960). White noise representation of earthquakes. J. Eng. Mech. Div.,

ASCE, Vol. 86, EM2: 1–16.2. Kanai, K. (1957). Semi-empirical formula for the seismic characteristics of the

ground. Bull. Earthquake Res. Inst., Univ. Tokyo, Vol. 35, 309–325.3. Tajimi, H. (1960). A statistical method of determining the maximum responses of a

building structure during an earthquake. Proc. Second World Conf. Earthquake Eng.,Tokyo and Kyoto, Japan, Vol. II.

4. Housner, G. W. and Jenning, P. C. (1964). Generation of Artificial Earthquakes, J.Eng. Mech. Div., ASCE, Vol. 90, EM1: 113–150.

5. Clough, R. W. and Penzien, J. (1975). Dynamics of Structures. McGraw-Hill BookCo., New York.

6. Shinozuka, M. and Sato, Y. (1967). Simulation of nonstationary random processes.J. Eng. Mech. Div., ASCE, Vol. 93, EM1: 11–40.

7. Amin, M. and Ang, A. H.-S. (1968). Nonstationary stochastic model for earthquakemotions. J. Eng. Mech. Div., ASCE, Vol. 94, EM2: 559–583.

8. Saragoni, G. R. and Hart, G. C. (1974). Simulation of artificial earthquakes. Earth-quake Eng. Structural Dynamics, Vol. 2, 249–267.

9. Priestley, M. B. (1967). Power spectral analysis of non-stationary random processes.J. Sound Vibration, Vol. 6, 1: 86–97.

10. Lin Y. K. and Yong, Y. (1987). Evolutionary Kanai-Tajimi earthquake models. J. Eng.Mech. Div., ASCE, Vol. 113, 8: 1119–1137.

11. Deodatis, G. and Shinozuka, M. (1988). Auto-regressive model for nonstationarystochastic processes. J. Eng. Mech., ASCE, Vol. 114, 4: 1995–2012.




12. Li, Y. and Kareem, A. (1991). Simulation of multivariate nonstationary randomprocesses by FFT. J. Eng. Mech., ASCE, Vol. 117, 5: 1037–1058.

13. Li, Y. and Kareem, A. (1997). Simulation of multivariate nonstationary randomprocesses: Hybrid DFT and digital filtering approach. J. Eng. Mech., ASCE, Vol. 123,12: 1302–1310.

14. Der Kiureghian, A. and Crempien, J. (1989). An evolutionary model for earthquakeground motion. Struct. Safety, 6: 235–246.

15. Papadimitriou, C. (1990). Stochastic characterization of strong ground motion andapplication to structural response, Rep. No. EERL 90-03, California Institute ofTechnology, Pasadena, Calif., U.S.A.

16. Conte, J. P. and Peng, B. F. (1997). Fully nonstationary analytical earthquakeground-motion model. J. Eng. Mech., Vol. 123, 1: 15–24.

17. Iyama, J. and Kuwamura, H. (1999). Application of wavelets to analysis and simu-lation of earthquake motions. Earthquake Eng. Structural Dynamics, 28: 255–272.

18. Spanos, P. D. and Failla, G. (2002). Multi-scale modelling via wavelets of evolution-ary spectra in mechanics applications. Proc. Int. Symp. Multiscaling in Mechanics,Messini, Greece.

19. Spanos, P. D., Tratskas, P. and Failla, G. (2002). Wavelets applications in structuraldynamics. In: Structural Dynamics, EURODYN2002, Grundmann & Schueller (eds.).Swets & Zeitinger, Lisse, ISBN 90 5809 510 X.

20. Grigoriu, M., Ruiz, S. E., and Rosenblueth, E. (1988). Nonstationary models ofseismic ground acceleration. Earthquake Spectra, 4: 551–568.

21. Yeh, C. H. and Wen, Y. K. (1990). Modelling of nonstationary ground motion andanalysis of inelastic structural response. Structural Safety, 8: 281–298.

22. Huang, N. E., et al. (1998). The empirical mode decomposition and the Hilbertspectrum for nonlinear, nonstationary time series analysis. Proc. R. Soc. Lond. A,454: 903–995.

23. Somerville, P., Smith, N., Punyamurthula, S., and Sun, J. (1997). Development ofground motion time histories for Phase 2 of the FEMA/SAC steel project. ReportNo. SAC/BD-97/04, SAC Joint Venture, Sacramento, California, U.S.A.

24. Bendat, J. S. and Piersol, A. G. (2000). Random Data, Analysis and MeasurementProcedures, 3rd ed. John Wiley & Sons, Inc., New York.

25. Zhang R. (2001). The role of Hilbert-Huang transform in earthquake engineering.Proc. World Multiconference Systemics, Cybernetics Informatics, Vol. XVII, Orlando,Florida, U.S.A.

26. Huang, et al. (2001). A new spectral representation of earthquake data: Hilbert spectralanalysis of Station TCU129, Chi-Chi, Taiwan, 21 September 1999. Bull. Seismolog-ical Soc. Am., Vol. 91, 5: 1310–1338.

27. Huang, N. E., Shen, Z. and Long, S. R. (1999). New view of nonlinear water waves:The Hilbert spectrum. Annu. Rev. Fluid Mech., Vol. 31, 417–457.

28. Loh, C. H., Wu, T.C. and Huang, N. E. (2001). Application of the empirical modedecomposition-Hilbert spectrum method to identify near-fault ground-motion char-acteristics and structural responses. Bull. Seismological Soc. Am., Vol. 91, 5:1339–1357.

29. Yang, J. N. and Lei, Y. (2000). System identification of linear structures using Hilberttransform and empirical mode decomposition. Proc. Int. Modal Anal. Conf., Vol. 1,213–219.

30. Zhang, R. and Ma, S. (2000). HHT analysis of earthquake motion recordings and itsimplications to simulation of ground motion. In: Monto Carlo Simulation, G. I. Schuellerand P.D. Spanos (eds.), 483–490. A. A. Balkema Publishers, Lisse, the Netherlands.




31. Zhang R. R. (2001). The signature of structural damage: An HHT view. In: StructuralSafety and Reliability, Corotis et al. (eds.), Swets & Zeitlinger, ISBN 90 5809 197 X.

32. Salvino, L. W. (2000). Empirical mode analysis of structural response and damping.Proc. SPIE Int. Soc. Optical Eng., Vol. 4062, 503–509.

33. Gravier, B. M. et al. (2001). An assessment of the application of the Hilbert spectrumto the fatigue analysis of marine risers. Proc. Int. Offshore Polar Eng. Conf., Vol. 2,268–275.

34. Hartzell, S., Liu, P., and Mendoza, C. (1996). The 1994 Northridge, Californiaearthquake: Investigation of rupture velocity, risetime, and high-frequency radiation.J. Geophys. Res., Vol. 101, 20091–20108.

35. Shinozuka, M. and Deodatis, G. (1991). Simulation of stochastic processes by spectralrepresentation. App. Mech. Reviews, Vol. 44, 4: 191–203

APPENDIX: RELATION BETWEEN HILBERT MARGINAL SPECTRUM AND FOURIER ENERGY SPECTRUM

Consider a time series y(t) which allows Fourier transform, defined as follows:

(9A.1)

Its complex conjugate is then

(9A.2)

Now if we define the Fourier energy spectrum by

(9A.3)

then the area under the Fourier energy spectrum for all frequency is given by

y y t e dti t( ) ( )ω ω= −

−∞

∞

∫

y y t e dti t∗

−∞

∞

= ∫( ) ( )ω ω

E yf ( ) ( )ω ω=2

E y y d y t y t e d dt dti t i t= =∗ − +( ) ( ) ( ) ( )ω ω ω ωω ω1 2 1 2

1 2

−−∞

∞

−∞

∞

−∞

∞

−∞

∞

∫∫∫∫

= −−∞

∞

−∞

∞

∫∫ y t y t t t dt dt( ) ( ) ( )1 2 1 2 1 22πδ




(9A.4)

if y(t) is defined for 0 < t < T, and zero otherwise. Therefore, the area under theFourier energy spectrum defined by Equation 9A.3 is equal to 2π times the totalenergy of the time series, defined by the integral of the square of the time series.

The area under the Hilbert energy spectrum can be illustrated by a simple caseof a time series that is a summation of n sine functions defined for a given timeperiod 0 to T,

for 0 < t < T (9A.5)

where T = mπ/2ωj for all j, m an integer.When we apply EMD to y(t), we in effect recover the n sine functions as n

IMFs. From Equation 9.1, Equation 9.2, and Equation 9.3 (in the text), one can seethat if Cj(t) = Ajsinωjt, the Hilbert transform of each of the sine functions producesDj(t) = –Ajcosωjt, and aj

2 (t) = Aj2 . The Hilbert energy spectrum is therefore equal to

Aj2 at ωj for all time. For a finite period of time T, the discrete marginal spectrum is

equal to Aj2Tat ωj. The total area under the Hilbert marginal spectrum is therefore

The total energy over the time period is given by

(9A.6)

Hence, the total area under Hilbert marginal spectrum is

= 2 (9A.7)

More generally, for an arbitrary time series,

(9A.8)

= =−∞

∞

∫ ∫2 22 2

0

π πy t dt y t dt

T

( ) ( )

y t A tj j

j

n

( ) sin( )==

∑ ω1

A Tj

j

n

2

1

.=

∑

y t dt T AjT

j

n

2

0

2

1

12

( )∫ ∑==

A Tj

j

n

2

1=∑ y t dt

T2

0( )∫

y t a t tj j

j

n

( ) ( ) cos[ ( )]= ⋅=

∑ θ1




if we neglect the residual term from Equation 9.5. Its total area under the Hilbertmarginal spectrum is

Its total energy over time is

(9A.9)

Due to the orthogonality of IMFs (see Huang et al. [22]), the cross-terms (i ≠ j) inEquation 9A.9 can be neglected. Equation 9A.9 reduces to

(9A.10)

When the IMFs have relatively smooth and sinusoidal shapes, for integer numberof quarter-waves, we have

(9A.11)

For high-frequency IMFs, the number of quarter-waves is usually large, so theportion that does not contain a complete quarter-wave is small and can be neglected.For low-frequency IMFs, this portion cannot be neglected, but low-frequency IMFsusually have small amplitudes and contribute insignificantly to the total energy.Therefore one arrives at

(9A.12)

The total area under Hilbert marginal spectrum is

(9A.13)

which is the same as in Equation 9A.7.

a t dtj

T

j

n

2

01∫∑

=

( ) .

y t dt a t a t tT T

i j i

j

n

2

0 01

( ) ( ) ( )cos[ ( )]cos[∫ ∫∑==

θ θθ j

i

n

t dt( )]=∑

1

y t dt a t t dtT

j

T

j

j

n

2

0

2

0

2

1

( ) ( )cos [ ( )]∫ ∫∑==

θ

a t t dt a t dtj j

T

j

T2 2

0

2

0

12

( )cos [ ( )] ( )θ∫ ∫≈

y t dt a t dtT

j

T

j

n

2

0

2

01

12

( ) ( )∫ ∫∑≈=

a t dt y t dtj

T

j

n T2

01

2

02∫∑ ∫

=

≈( ) ( )




Comparing Equation 9A.13 with Equation 9A.4, we conclude that the total areaunder the Hilbert marginal spectrum needs to be multiplied by a factor, to becomparable with the area under the Fourier energy spectrum defined by Equation9A.3.



10
Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms for Selected Applications
Ser-Tong Quek, Puat-Siong Tua, and Quan Wang

CONTENTS

10.1 Introduction ..................................................................................................21410.2 Hilbert-Huang Transform.............................................................................21510.3 Applications of HHT, WT, and FT to Experimental Data..........................216

10.3.1 Locating Crack in Aluminum Beam Using HHT ...........................21610.3.2 Detection of Edges in Aluminum Plate...........................................21910.3.3 Determination of Modal Frequencies of Aluminum Beam ............222

10.4 Concluding Remarks....................................................................................236Acknowledgments..................................................................................................239References..............................................................................................................243

ABSTRACT

This paper illustrates the suitability of three different signal-processing techniques,namely, the Hilbert-Huang transform (HHT), the wavelet transform (WT), and the fastFourier transform (FFT), under different practical situations in relation to structuralhealth monitoring. Three experimentally obtained signals are analyzed. First, in thetime-of-flight analysis of flexural wave propagation in an aluminum beam, HHT isfound to be a more direct method compared to WT, as no knowledge of the actuationfrequency is required. However in the case of WT, if prior knowledge of the actuationfrequency is utilized combined with a refined scale search, WT produces more accurateresults. In another experiment involving the time-of-flight analysis of acoustic Lambwave propagation in an aluminum plate, piezoelectric actuators and sensors (PZTs) areused to excite and receive direct and reflected waves at a specific frequency. HHT and

213


214



WT give similar results in terms of precision and accuracy. Last, a modified Fouriertransform (FT) is investigated based on the concept of HHT and applied to the signalscollected from the free vibration of an aluminum beam under three different supportconditions. The proposed method is able to reveal the presence of higher vibrationmodes than can be revealed by conventional FT of the original signal. The resultsproduced are close to that of the more common WT. It is believed that supporters ofFT will be more receptive to this modified HHT method.

Keywords Hilbert-Huang transform, wavelet transform, fast Fourier transform,structural health monitoring, flexural wave, Lamb wave, free vibration

10.1 INTRODUCTION

Signal processing has gradually become an indispensable tool in the field of engi-neering for extraction of important information from raw signals. In civil engineer-ing, many practical and robust nondestructive evaluation (NDE) techniques devel-oped for assessment of structural performance involve two key components, namely:(1) data acquisition and (2) signal processing and interpretation. Although the useof vibration measurements is a simpler and less costly method with respect toinstrumentation system in comparison with infrared thermography, ground-penetrat-ing radar, acoustic emission monitoring, and eddy current detection, a key factor foridentifying damage precisely lies in having an appropriate data analysis method.

The most well known conventional signal processing technique is the Fouriertransform (FT). Fourier spectral analysis provides a general method for examiningthe global energy–frequency distributions of a given signal. FT has high resolutionin the frequency domain but loses the resolution in the time domain. Despite this,FT has been employed in health monitoring. For example, Crema et al. [1] used afast Fourier transform (FFT) analyzer to perform modal analysis to detect damagein composite material structures. A broadband impulse input was applied, and thedamage that was induced by loads well above the working load caused a smalldecrease in the eigen-frequencies for several modes. However, it can be problematicto locate damage by using FT-based modal analysis techniques.

A few methods are available for processing signals of nonlinear and nonstation-ary systems, of which the short-time Fourier transform (STFT) method and wavelettransform (WT) method are widely adopted. Cook and Berthelot [2] used thetime-dependent energy spectrum from STFT to distinguish backscattered signalsfrom a single crack against those from a series of closely spaced cracks on steelsurface. Hurlebaus et al. [3] used the time–frequency spectrum obtained by perform-ing STFT on the Lamb waves generated by laser to obtain the group velocity–fre-quency domain. The notches are located from the autocorrelation plot of a series ofgroup velocity spectra at different assumed propagation distances.

Wavelet analysis is another adjustable window signal-processing technique tohandle nonstationary signals. Although WT appears similar to STFT, the basic wavecomponents are not limited to sinusoidal functions. Rather, classes of functions havebeen proposed, from the simplest Haar wavelet to the more complicated higher orderDebauchies wavelet functions, to address various problems of time–frequency


Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms

215


resolution. WT has been widely used in the field of damage detection, such as inmodal-based detection (Staszewski [4]). An early work using WT to obtain a dis-persive relation of group velocity of flexural waves in beam was presented byKishimoto et al. [5]. Lee and Liew [6] studied the sensitivity of WT in space domainfor identifying the damage location of a beam for small damage not easily detectedvia conventional eigenvalue analysis. Quek et al. [7], on the other hand, used waveletcoefficients over the time domain to locate cracks in beams via the time-of-flightanalysis of the wave propagation along the beam. However, WT has its shortcomings.Experience shows that it can produce many spurious harmonics under different scalesthat makes the analysis difficult or sometimes meaningless.

The most recently introduced signal-processing technique for analyzing nonlin-ear and nonstationary time series data is the Hilbert-Huang transform (HHT) [8].Empirical mode decomposition (EMD) of the data is first performed to segregatethe signals into narrow band components, and the Hilbert transform (HT) is thenapplied on each mode. Its ability to analyze nonlinear and nonstationary time seriesdata [8–10] has attracted many applications. For example, Zhang et al. [11] adoptedHHT to assess structural damage from vibration recordings. Shim et al. [12]exploited HHT’s ability to exhibit nonlinear and nonstationary behavior to locatedamages on bridges by using frequency sweep and controlled excitations. Quek etal. [13] also employed HHT to analyze structural dynamic responses to detect andlocate anomalies in beams and plates. In addition, HHT has been used to analyzeexperimental data collected by oscilloscope to investigate the dominance of thelowest antisymmetric (A0) and symmetric (S0) modes of Lamb wave vary across thefrequency range adopted for excitation and good agreement with, which agrees wellwith theoretical response prediction in terms of their relative dominance [14].

Individual signal-processing techniques have their own strength and weaknesses.Suitability of a particular technique may, therefore, be problem dependent, especiallywith respect to structural health monitoring problems. This paper compares the threesignal-processing techniques, namely, HHT, WT, and FFT, for signals obtainedexperimentally. The first example concerns the time-of-flight analysis of flexuralwave propagation in an aluminum beam, where WT and HHT are compared. Next,we compare the analysis from the two techniques on a two-dimensional probleminvolving the propagation of acoustic Lamb waves in an aluminum plate. Last, weapply a modified FT based on the EMD concept of HHT to the free vibration signalsof an aluminum beam under three different support conditions, we compare theseresults with results from using conventional direct FFT, HHT, and WT to obtain themodal frequencies.

10.2 HILBERT-HUANG TRANSFORM

Hilbert transform was first developed to process nonstationary narrow-band signals [15].To apply it to signals in general, Huang et al. [8] proposed using the empirical modedecomposition technique to decompose any given signal into a set of narrow-bandsignals. Each component (which may be nonstationary) of the set is termed an intrinsicmode function (IMF) of the original signal, on the IMF where HT can be carried out.


216



The distinct features of a narrow-band signal, and hence an IMF, are that (a)over its entire length, the number of extrema and the number of zero-crossings eitherare equal or differ at most by one; and (b) at any point, the mean value of theenvelope of the signal defined by the local maxima and the envelope defined by thelocal minima is zero. This forms the basis on which the EMD technique wasdeveloped. For the EMD to be applicable, it is assumed that (a) the signal has atleast one maximum and one minimum; (b) the characteristic time scale is definedby the time lapse between the extrema; and (c) if the data are totally devoid ofextrema but contain only inflection points, then the signal can be differentiated oneor more times to reveal the extrema. Hilbert spectral analysis can then be performedon each IMF. MATLAB 6.1 is adopted for the implementation of the HHT in thisstudy.

10.3 APPLICATIONS OF HHT, WT, AND FT TO EXPERIMENTAL DATA

An accurate signal-processing technique to obtain the flight times is crucial to locatedamages in structures. Both WT and HHT are potentially applicable. The advantagesof HHT for damage detection in beams using flight times from narrow-band non-stationary signals have been discussed by Quek et al. [13]. Waves in plates are evenmore complex due to the dispersive nature of Lamb waves; hence, a technique thatprovides good representation of localized events in both the frequency and energyis vital for determining the location of the damage precisely. Another practicalproblem of interest is the determination of higher frequencies from free vibrationdata. In this section we examine a proposal to combine FT with HHT, using analuminum beam under three different boundary conditions as illustration.

10.3.1 LOCATING CRACK IN ALUMINUM BEAM USING HHT

Quek et al. [7] presented a wavelet analysis on experimental data to locate a crackin an aluminum beam based on simple wave propagation considerations. The beamconsidered has geometrical and material properties given in Table 10.1. Three sets

TABLE 10.1Geometrical and Material Properties of Aluminum Beam/Plate

Beam Plate

Dimensions (mm) 650 × 32 × 6 600 × 600 × 2Young’s modulus, E (GPa) 73.1 102Shear modulus, G (GPa) — 26Mass density, ρ (kg/m3) 2790 2700



217


of data with different boundary conditions are analyzed, namely, fixed-fixed, simplysupported, and cantilever conditions. The basic concept is shown in Figure 10.1, ina schematic view of wave propagation due to an impact. The piezoelectric sensorcaptures the wave signal, which contains timings of the direct, damage-reflected,and boundary-reflected waves. By estimating these timings with a signal-processingtechnique, the damage location can be deduced. In view of the dispersive nature ofthe wave, only a wave at a particular frequency, with a corresponding wave propa-gation velocity, is used so that accurate results can be obtained.

HHT is applied in this study on the same three sets of signals. One of the signalsobtained (and shown in Figure 10.2a) was decomposed via EMD into its IMFcomponents, shown in Figure 10.2b. It can be observed that the first IMF componenthas higher amplitude than the other IMFs. This IMF component is then subjectedto HT, and the corresponding frequency–time and energy–time distributions plottedin Figure 10.2c and d, respectively. From the energy–time distribution, the timingsof the energy peaks in the signal, which are due to the propagating wave passingthe sensor, are obtained. The frequencies corresponding to the energy peaks are alsonoted so as to establish that they do not deviate significantly from each other. Itshould be pointed out that not all IMFs are used, as the first IMF is distinct fromthe others and contains only the frequency of interest, whereas other IMFs do not.

The results for the estimated damage position are summarized in Table 10.2 andcompared with those obtained with Gabor wavelet analysis with coarse (discreteWT, scale = 12) and fine (“continuous” WT or CWT, scale = 12.2) resolution. Thewavelet results have been discussed in detail by Quek et al. [7], including the 3Dwavelet plot, and will not be repeated here. It can be observed that the HHT methodis able to identify the damage position with reasonable accuracy; the results arebetter than the discrete wavelet results but worse than the continuous wavelet results.However, using CWT for this application requires determining an appropriate scaleto process the final results. From past experience, this requires a fair amount of skill,as the selection can be rather subjective and may lead to varied end results (see, for

FIGURE 10.1 Schematic view of wave propagation for impact at 200 mm position.


218



FIGURE 10.2 (a) Dynamic response of a beam; (b) IMFs of response signal after EMD; (c)frequency; (d) energy spectra of first IMF component.

TABLE 10.2Comparison of Results Based on HHT Method and Wavelet Analysis

End Condition

HHT Analysis

Wavelet Analysis

Scale 12 Scale 12.2

Estimated Damage Position

Error (%)


Error (%)


*Error (%)

Fixed ended 437.5 –2.78 492.0 9.33 450 0.00Simply Supported 471.7 4.83 493.0 9.56 457 2.00Cantilever 403.8 –10.26 542.0 20.00 457 2.00

Note: Actual damage position is 450 mm from left of the beam.

Inp

ut

1086420

−2−4−6−8

−100 0.5 1 1.5 2 2.5

Time, t(s)

3 3.5 4 4.5 5

×10−3

(a)

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qu

enc

y (H

z)

5000

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3000

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3 4 5

×10−3

(c)

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00 1 2

Time, t(s)

3 4 5

×10−3

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c(1

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10

0

c(2

)

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0

c(3

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c(4

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idu

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3 3.5 4 4.5 5

×10−3

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(b)



219


example, Table 3 of Quek et al. [7]). This is due to the shortcomings of WT, whichproduces false harmonics depending on the scale adopted, although WT also hasthe ability to achieve high resolution in both frequency and time domain. Theadvantage of HHT over WT in this case is that the HHT procedure is more directand can be performed to obtain the desired accuracy for the results without priorknowledge of the excitation frequency (which in this case of impact load is broad-band). It should be noted that there are other applications where CWT may be moreappropriate.

10.3.2 DETECTION OF EDGES IN ALUMINUM PLATE

In this example, we adopted the time-of-flight analysis of acoustic Lamb wavepropagation in a 600 mm × 600 mm aluminum plate to detect the plate’s edges, Inthe experiment, a through crack, 1 mm wide and 30 mm long, inclined at 30º to oneside of the plate (side AB in Figure 10.3), was cut with a milling machine. Thelocation of the center of the crack is (270, 290) mm. Piezoelectric sensors andactuators (PZTs) were used to excite and receive direct and reflected waves at aspecific frequency. A Lamb wave with a narrow-band (600 kHz) actuation pulse(Figure 10.4) was activated based on the following equation:

(10.1)

FIGURE 10.3 Schematic view of PZT positions on plate.

D

A B

C600

500

400

300

200

100

06005004003002001000

y

X

Line crack

Actuator

Sensor

50 mm

75 mm

X t t e a t t( ) cos ( )= ( ) − −2 02

πω


220



where ω is the actuation frequency (Hz), and a and t0 are constants. This was donewith a function generator (Yokogawa FG300 15 MHz Synthesized FG) to feed aPZT actuator (8.0 mm × 8.0 mm × 0.5 mm). The responses were captured by a PZTsensor placed at a distance away and channeled to an oscilloscope (Yokogawa DL71616-Channel) for monitoring and analysis. Figure 10.3 shows the positions of theactuator and sensor. By estimating the flight time due to the propagation of eithera S0 or an A0 Lamb wave, the distance traveled by the wave from the actuator tothe sensor can be computed as

, (10.2)

where ti is the time of flight referenced from the actuation time, and cg is the groupvelocity of either the S0 or A0 mode, determined either theoretically or experimen-tally.

The response signal captured by the PZT sensor (Figure 10.5a) was processedby both HHT and Gabor WT. The resulting energy spectra are given in Figure 10.5dand Figure 10.6. In the HHT analysis, the peaks in the energy–time spectrum forthe IMF component containing the highest energy (in this case the first IMF) givethe wave arrival times of interest. Based on these energy peaks, the distances traveledby the wave from the actuator to the sensor can be computed; these are summarizedin Table 10.3. It can also be observed from the frequency spectrum (see Figure10.5b) that the frequency corresponding to the energy peaks is very close to theactuation frequency, which indicates the accurate representation of localizationevents by HHT. On the other hand, Gabor WT analysis on the same signal responsewas performed based on the prior knowledge that the actuation frequency is 600kHz, which corresponds to a scale of 20.2. Similar energy peaks were obtained,which accounted for the wave pulses on the different paths received by the sensor.The results obtained by WT analysis are also given in Table 10.3. These indicate

FIGURE 10.4 Actuation pulse using Equation 10.1 at frequency 600 kHz.

4

×10−5

3.532.5

Time (s)

21.5

Am

pli

tud

e (V

)

−20

−15

−10

−5

0

5

10

15

20

l t ci i g= ∆



221


FIGURE 10.5 (a) Response signal captured by sensor on aluminum plate; (b) IMFs ofresponse signal after EMD; (c) frequency; (d) energy spectra of first IMF component.

FIGURE 10.6 Gabor wavelet analysis of response signal given in Figure 10.5a.

−0.6−0.4−0.2

0

0 1

Time, t(s)

2

×10−4

0.20.40.60.8

1

−0.8−1

Am

pli

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(a)

×10−4

×105

0 1

Time, t(s)

2

15

10

5

0

−5

Fre

qu

enc

y, (

Hz)

(c)

×10−4

0

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0.4

0.6

0.8

1

0 1

Time, t(s)

2

Am

pli

tud

e

(d)A0/S0(incident)

A0(crack) A0

(boundary, CD)

10

0 1 2

0 1 2

0 1 2

0 1 2

0 1 2

0 1 2

0 1 2

0 1 2

0 1 2

0 1 2

0 1 2

−1

c(1

)

0.050

−0.05

c(2

)

0.020

−0.02

c(3

)

0.020

−0.02

c(4

)

0.010

−0.01

c(5

)

0.010

−0.01

c(6

)0.01

0−0.01

c(8

)

0.10

−0.1c(1

1)

0.10

0 1

Time, t(s)

2

×10−4

−0.1

Res

0.20

−0.2

c(9

)

0.020

−0.02c(1

0)

5 ×10−3

0−5

c(7

)

(b)

2.0E–041.5E–041.0E–04

Time (s)

A0/S0

(incident)

5.0E–050.0E+000.0E+00

2.0E–04

4.0E–04

6.0E–04

8.0E–04

1.0E–03

1.2E–03

Am

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A0

(boundary, CD)

A0

(crack)


222



that HHT and WT give similar results in terms of precision and accuracy. HHT doesnot require prior knowledge of the activation frequency.

10.3.3 DETERMINATION OF MODAL FREQUENCIES OF ALUMINUM BEAM

In this part of the study, we consider an aluminum beam under three supportconditions, shown in Figure 10.7, with material properties similar to the plate inSection 10.3.2 (see Table 10.1). The theoretical modal frequencies of the beam werefirst obtained (see Table 10.4) by using the following equation [16]:

(10.3)

where l, E, I, ρ, and A are the length, Young’s modulus, moment of inertia, density,and cross-sectional area of the beam, respectively; n corresponds to the vibrationmodes; and βn is solved from the following equations, respectively, for cantilever,simply supported, and fixed-fixed conditions.

(10.4a)

(10.4b)

(10.4c)

For the experiment, the free vibration responses of the beam under the three differentsupport conditions (Figure 10.7) were monitored and collected by an oscilloscope(Yokogawa DL716 16-Channel; see Figure 10.8). FFT was first performed directly on

TABLE 10.3Results of Wave Propagation in Aluminum Plate

SignalAnalysis Technique

Theoretical Velocity

(m/s)

Distance Traveled by Incident Wave

(mm)a

Distance Traveled by Reflected Wave from Line Crack

(mm)b

Distance Traveled by Reflected Wave

from Boundary (CD; mm)c

A0 A0

Error (%) A0

Error (%) A0

Error (%)

HHT 3134 52.9 5.80 196.1 –1.95 425.0 –1.62WT 52.7 5.40 196.4 –1.80 424.0 –1.85

a Actual distance traveled by incident wave is 50 mm.b Actual distance traveled by crack-reflected wave is 200 mm.c Actual distance traveled by boundary (CD)-reflected wave is 432 mm.

fl

l

EIA

nnn= =( )

, , , , ...βπ ρ

2

221 2 3

cos coshβ βn nl l = −1

sinβn l = 0

cos coshβ βn nl l = 1



223


the response signal. For example, Figure 10.9a and b shows the linear and logarithmicplots of the Fourier spectrum, respectively, for the cantilevered beam. In the linearplot of the Fourier spectrum (Figure 10.9a), the first two modal frequencies of thebeam can be easily identified (5.99 Hz and 34.93 Hz respectively). However, thethird mode may be easily overlooked due to the low energy of the displayed peak.On the other hand, the logarithmic scale plot of the spectrum (Figure 10.9b) exhibitsthe first three modal frequencies clearly. This is also true for the fixed-fixed supportcondition (see Figure 10.10). However, for the simply supported condition, it is noteasy to distinguish the modes from the logarithmic scale plot of the Fourier spectrum.It can be observed in Figure 10.11b that besides the three fundamental modalfrequencies, there are also a number of energy peaks at other frequency rangeshaving amplitudes of order close to the modal frequencies. Hence for beams withclosely spaced frequencies, the Fourier spectrum may not be appropriate.

EMD was performed on the responses given in Figure 10.8. Consider the resultsfor the cantilevered beam: three decomposed IMF components and one residualcomponent are obtained, as depicted in Figure 10.12a. The individual IMFs aresubjected to FT, and the resulting spectra are shown in Figure 10.12b through d.The residual component is not subjected to FT as it represents the mean or residualtrend of the captured response signal. Figure 10.12b through d clearly identifies the

FIGURE 10.7 Schematic setup of aluminum beam under different support conditions: (a)cantilever; (b) simply supported; (c) fixed-fixed for modal frequency analysis.

Piezoelectric

Sensor

150 mm0 mm800 mm

Aluminum

beam

32 mm

6 mm

x

y(b)

Piezoelectric

Sensor

150 mm0 mm800 mm

Aluminum

beam

32 mm

6 mm

x

y

(c)

Piezoelectric

Sensor

150 mm0 mm920 mm

Aluminum

beam

32 mm

6 mm

x

y

(a)


224

The H

ilbert-H

uan

g Transfo

rm in

Engin

eering

TABAna r Different Support Conditions Obtained by F

Supp

EMD + FT

HHT WT

Freq (Hz)

Error (%)

Freq (Hz)

Error (%)

Freq (Hz)

Error (%)

Canti 5.76 -1.72 5.77 1.52 5.64 3.7335.23 –4.09 34.63 5.72 35.86 2.3998.00 –4.73 96.62 6.06 101.56 1.26

Simp 22.95 5.44 22.80 4.75 23.20 6.5982.85 –4.81 77.88 –10.52 81.25 –6.65

167.50 –14.47 159.18 –18.72 162.50 –17.02Fixed 48.00 –2.69 46.96 4.80 49.85 1.06

138.00 1.49 135.34 0.47 135.42 –0.41 — — 263.19 1.26 270.83 1.60

460.00 4.39 — — 451.39 2.44

DK

342X_book.fm

copy Page 224 Sunday, May 8, 2005 2:25 PM

© 2005 by Taylor &

Francis Group, L

LC

LE 10.4lysis of Modal Frequencies of Aluminum Beam undeFT, EMD Followed by FT, HHT, and WT

ort Conditions ModeTheoretical

Frequency (Hz)

FFT

Freq (Hz)

Error (%)

lever 1st 5.86 5.99 2.202nd 36.73 34.93 –4.913rd 102.86 96.80 –5.89

ly supported 1st 22.77 22.89 5.172nd 87.03 81.80 –6.013rd 195.84 165.80 –15.34

-fixed 1st 49.33 48.00 –2.692nd 135.97 138.00 1.493rd 266.56 268.00 0.544th 440.64 446.00 1.22

Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms 225


FIGURE 10.8 Free vibration response of aluminum under (a) cantilever; (b) simply sup-ported; (c) fixed-fixed conditions.

Am

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first three modal frequencies of the beam (5.76 Hz, 35.23 Hz, and 96.8 Hz respec-tively). Even for the third mode, the appearance of the model frequencies is verydistinctive because the EMD process has effectively decomposed the response signalinto its respective components, which are narrow band. Hence, the energy peak ofthe third mode is not obscured by the first and second modes, which contain relativelyhigher energies. Similar analyses are done for the same beam under the two othersupport conditions, and the results (see Figure 10.13 and Figure 10.14) obtained arecomparable to those obtained by the conventional direct FT. However, for the caseof the fixed-fixed condition, the third modal frequency could not be identified bythe FT method (Figure 10.14). Instead, the first and second modal frequencies are

FIGURE 10.9 FFT analysis of response of cantilever beam given in Figure 10.8a, plotted on(a) linear scale and (b) log-scale.

Am

pli

tud

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00 50 100

Frequency (Hz)

150 200

(a)

250

0.5

1

1.5

2

2.5×10

4

1st

Mode

= 5.99 Hz

2nd

Mode

= 34.93 Hz 3rd

Mode

= 96.8 Hz

Am

pli

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st Mode

= 5.99 Hz

2nd

Mode

= 34.93 Hz3

rd Mode

= 96.8 Hz

10−1

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150 200 250

100

101

102

103

104

105




each contained in two IMF components (fourth and fifth IMFs and second and thirdIMFs respectively; see Figure 10.14). On closer observation, the third and fourthmodal frequencies can be spotted in the Fourier spectrum of the second IMF com-ponent, but they are obscured by the stronger presence of the second modal fre-quency. This setback of the proposed method can be explained by the fact that IMFsmay contain mix modes. This may occur when the frequency of interest in an IMFis only contained along a particular segment of the signal and not throughout theentire time domain. This is the so-called intermittency problem discussed by Huanget al [17].

The original HHT was also performed on the decomposed IMF components ofthe response signals for the beam under the different support conditions. However,as mentioned earlier, the frequency of interest in an IMF may be contained only

FIGURE 10.10 FFT analysis of response of fixed-fixed beam given in Figure 10.8c, plottedon (a) linear scale and (b) log-scale.

Am

pli

tud

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00 50 100

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150 200

(a)

250

400

200

600

800

1000

12001

st Mode

= 22.89 Hz

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= 81.8 Hz

3rd

Mode

= 165.8 Hz

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Mode

= 22.89 Hz 2nd

Mode

= 81.8 Hz

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10−4

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Frequency (Hz)

150 200 250

100

102




along a particular segment of the signal; hence, a weighted average using theamplitude is performed on segments of the frequency spectrum from HT for eachIMF to obtain the modal frequencies of the beam. To illustrate, spectra for the beamunder the three support conditions will be presented in detail individually.

First, consider the HT of the IMFs for the cantilever beam (Figure 10.15). Fromthe Hilbert spectra of the three IMF components, it can be observed that the EMDhas effectively decomposed the original response signal into its different compo-nents, which are well contained within a narrow band, as the fluctuation of thefrequency along the time axis is rather small. For this case, weighted averaging wasperformed over the entire time domain for each of the IMF components. The modalfrequencies obtained are 101.56 Hz, 35.86 Hz, and 5.64 Hz from the first, second,and third IMFs, respectively, which corresponds well with the theoretical values.

FIGURE 10.11 FFT analysis of response of simply supported beam given in Figure 10.8b,plotted on (a) linear scale and (b) log-scale.

Am

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Frequency (Hz)

300 400

(a)

500

400

200

600

800

1000

1400

1200 1st

Mode

= 48 Hz

2nd

Mode

= 138 Hz

3rd

Mode

= 268 Hz

Am

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(b)1st

Mode

= 48 Hz2

nd Mode

= 138 Hz

3rd

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= 268 Hz

4th

Mode

= 446 Hz

10−3

10−2

10−1

0 100 200

Frequency (Hz)

300 400 500

100

103

102

101

104




Next, for the simply supported beam, it can be observed in the HT spectra shownin Figure 10.16 that the frequency is stable only over a certain time range for eachIMF. For example, for the first IMF, the frequency has a small fluctuation between

FIGURE 10.12 (a) IMF components of response of cantilever beam given in Figure 10.8a;FT analysis of (b) first; (c) second; (d) third IMF component.

Am

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3.5×10

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1.5

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1st

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(d)

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Time, t(s)

0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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0.1 sec and 0.3 sec; hence, a weighted average on the frequency is done over thistime range. The frequency obtained is 159.18 Hz, which is closer to the thirdvibration mode of the beam. Similar analysis is done for the second IMF (between

FIGURE 10.13 (a) IMF components of response of simply supported beam given in Figure10.8b; FT analysis of (b) first; (c) second; (d) third IMF component.

Am

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40

35

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20

15

10

5

00 50 100

3rd

Mode

= 167.5 Hz

Frequency (Hz)

150 200 250

(b)

Am

pli

tud

e

4.5

4

3.5

3

2.5

2

1.5

1

0.5

00 50 100

2nd

Mode

= 82.85 Hz

Frequency (Hz)

150 200 250

(c)

5

0

c(1

)

−5

1

−1

0 0.1 0.2 0.3 0.4 0.5

Time, t(s)

0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

Res

idu

e

(a)

−2

5

0

c(2

)

−5

10

0

c(3

)

−10

(d)

Am

pli

tud

e

900

800

700

600

500

400

300

200

100

00 50 100

1rd

Mode

= 22.95 Hz

Frequency (Hz)

150 200 250




FIGURE 10.14 (a) IMF components of response of fixed-fixed beam given in Figure 10.8c;and FT analysis of (b) first; (c) second; (d) third; (e) fourth IMF; (f) fifth IMF component.

10

0

c(1

)

−10

0

Time, t(s)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.5

Res

idu

e

(a)

−0.5

5

0

c(2

)

−5

5

0

c(4

)

−5

2

0

c(5

)

−2

10

0

c(3

)

−10

(d)

Am

pli

tud

e

600

500

400

300

200

100

00 200100 300

2nd

Mode

= 138 Hz

Frequency (Hz)

400 500 600

(e)

Am

pli

tud

e

350

250

300

200

150

100

50

00 200100 300

1st

Mode

= 48 Hz

Frequency (Hz)

400 500 600

Am

pli

tud

e

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 200100 300

Frequency (Hz)

400 500 600

(b)

Am

pli

tud

e

90

80

70

60

50

40

30

20

10

00 200100 300

2nd

Mode

= 138 Hz

3rd

Mode

(268 Hz)

4th

Mode

(460 Hz)

Frequency (Hz)

400 500 600

(c)




0.12 sec and 0.28 sec), and the result (77.88 Hz) gives the second vibration mode.For the third IMF, a rather stable fluctuation of the frequency is observed over theentire time domain; hence, a weighted average is taken over the entire data range. Thisobviously gave the result corresponding to the first fundamental frequency (22.80 Hz).

The Hilbert spectra for the IMF components of the fixed-fixed condition areshown in Figure 10.17. Considering the first IMF, a large fluctuation of the frequencyover the entire time domain is observed. This can be explained by the fact that thefiltered component effectively contains the noise in the original response signal;hence, meaningful modal parameters of the beam can be extracted from it. Thesecond IMF reveals a stable frequency region with high amplitude over the timedomain between 0.05 sec and 0.12 sec. Taking the weighted average over this timerange gives the second modal frequency (135.34 Hz). For the third IMF, two stablefrequency fluctuation regions with high amplitude can be identified, namely, from0 to 0.16 sec and from 0.16 to 0.46 sec. By performing a weighted average on thetwo regions, two fundamental frequencies are obtained (138 Hz and 263.19 Hz). Asimilar stable region is observed for the fourth IMF, from 0.04 to 0.15 sec; thiscorresponds to an averaged frequency of 46.96 Hz, which gives the first modalfrequency of the beam. In the fifth IMF, the two stable regions can be pinpointed,from 0 to 0.15 sec and from 0.24 to 0.48 sec. These two regions give a weightedaverage frequency of 42 Hz and 43.3 Hz respectively.

Last, wavelet analysis using the Morlet function was also performed on the threesignals of the beam under the three support conditions. WT was first done on theoriginal response signals for the beam over a large scale range. By observing thedominant energy contents in this overall spectrum, different scales were zoomed into obtain the dominant frequency contents. For example, WT was first performedfor the vibration response of the cantilevered beam given in Figure 10.8a over awide scale range of 1 to 256. This spectrum plot (see Figure 10.18a) shows threeregions with a consistent existence of a corresponding frequency throughout the

FIGURE 10.14 (continued)

600500

(f )

400300

Frequency (Hz)

2001000

Am

pli

tud

e

45

40

35

30

25

20

15

10

5

0

1st

Mode

= 48 Hz




entire time domain, namely, in scales 105 to 183, 14 to 27, and 4 to 12. By focusingtoward these regions, a more refined dominant frequency can be deduced, namelyat scales of 144, 22.66, and 8, which corresponds to frequencies of 5.64 Hz, 35.86

FIGURE 10.15 HT spectrum for (a) first, (b) second, and (c) third IMF component forresponse of cantilever beam as given in Figure 10.12a.

Fre

qu

enc

y, (

Hz)

500

450

400

350

300

250

200

150

100

50

00 0.2 0.4

Time, t(s)

0.6 0.8 1

(a)

2.0

1.0

0

Fre

qu

enc

y, (

Hz)

100

90

80

70

60

50

40

30

20

10

00 0.2 0.4

Time, t(s)

0.6 0.8 1

(b)

8.0

4.0

0

Fre

qu

enc

y, (

Hz)

60

50

40

30

20

10

00 0.2 0.4

Time, t(s)

0.6 0.8 1

(c)

20

10

0




Hz, and 101.56 Hz respectively. These results give the three fundamental frequenciesof the beam. The wavelet analyses for the other two support conditions are presentedin Figure 10.19 and Figure 10.20.

FIGURE 10.16 HT spectrum for (a) first, (b) second, and (c) third IMF component forresponse of simply supported beam as given in Figure 10.13a.

Fre

qu

enc

y, (

Hz)

500

450

400

350

300

250

200

150

100

50

00 0.2 0.4

Time, t(s)

0.6 0.8 1

(a)

4.0

2.0

0

Fre

qu

enc

y, (

Hz)

300

250

200

150

100

50

00 0.2 0.4

Time, t(s)

0.6 0.8 1

(b)

3.0

1.5

0

Fre

qu

enc

y, (

Hz)

350

300

250

200

150

100

50

00 0.2 0.4

Time, t(s)

0.6 0.8 1

(c)

8

4

0




The results obtained by the different signal analysis methods are compared inTable 10.4. It can be observed that all the methods produce results of approximatelythe same order of error. The methods are equally feasible in determining the modal

FIGURE 10.17 HT spectrum for (a) first, (b) second, (c) third, (d) fourth, and (e) fifth IMFcomponent for response of fixed-fixed beam as given in Figure 10.14a.

Fre

qu

enc

y, (

Hz)

5000

4500

4000

3500

3000

2500

2000

1500

1000

500

00 0.1 0.2

Time, t(s)

0.3 0.4 0.5

(a)

10

5

0

Fre

qu

enc

y, (

Hz)

2500

2000

1500

1000

500

00 0.1 0.2

Time, t(s)

0.3 0.4 0.5

(b)

5

2.5

0

Fre

qu

enc

y, (

Hz)

2000

1800

1600

1400

1200

1000

800

600

400

200

00 0.1 0.2

Time, t(s)

0.3 0.4 0.5

(c)

10

5

0




frequencies of the beam. However, these methods do contain imperfections. Forexample, the direct FT method may not be very effective for cases where the modalfrequencies have energy of same order as that of other frequency ranges. Similarly,for the modified FT with EMD process, the identification of certain modal frequen-cies may not be possible when the IMF components contain mixed modes. For HHT,the definition of the stable frequency region over the time domain, although subjec-tive, is least problematic. For WT, identification of the precise scale from a rangeof scales for the dominant frequency can be ambiguous.

10.4 CONCLUDING REMARKS

A comparison of three signal-processing techniques is presented. In the first study offlexural wave propagation in an aluminum beam, HHT is shown to be a more directmethod compared to WT when no knowledge of the actuation frequency is available.In the second experiment, involving acoustic Lamb wave propagation in an aluminumplate, HHT and WT give similar results for the analysis of the propagating Lamb wavesactuated and received by the PZTs. Good WT results hinge on the fact that the actuation


Fre

qu

enc

y, (

Hz)

800

700

600

500

400

300

200

100

00 0.1 0.2

Time, t(s)

0.3 0.4 0.5

(d)

6

3

0

Fre

qu

enc

y, (

Hz)

300

250

200

150

100

50

00 0.1 0.2

Time, t(s)

0.3 0.4 0.5

(e)

2

1

0




frequency is narrowband and known a priori. For the last case, the combination of theEMD process from HHT with FT on the decomposed IMF components has proven tobe capable of revealing modal frequencies of the aluminum beam having relativelylower energy contents. These modal frequencies may be overlooked by the traditionaldirect FFT, unless a logarithmic plot is done for the energy spectrum. However, theproblem of mixed modes within individual IMFs needs to be addressed. The original

FIGURE 10.18 Morlet WT analysis for response of cantilever beam given in Figure 10.8aat scales (a) 1–256, (b) 144, (c) 22.66, and (d) 8 with sampling rate 1000 samples/sec.

(a)

Sca

le

114274053667992

105118131144157170183196209222235248

Scale of colors from MIN to MAX

Time

(b)

Am

pli

tud

e

−300100 200 300 400 500

Time

600 700 800 900 1000

−200

−100

0

100

200

300Coefficients Line - Ca, b for scale a = 144 (frequency = 5.642)

Sca

le

144





HHT with knowledge of the problem does provide good results when it is used withan averaging concept. These examples show that HHT is a suitable tool for processingnonstationary wave propagation signals encountered in damage detection studies. Itprovides a good representation of localized events in both the frequency and energy ofany transient signal collected. Indeed, it is a simple signal-processing tool to use andprovides reasonably good results.


(d)

Am

pli

tud

e

−10100 200 300 400 500

Time

600 700 800 900 1000

−5

0

5

10


Sca

le

8


(c)

Sca

le

22.66


Am

pli

tud

e

−50100 200 300 400 500 600 700 800 900 1000

0

50Coefficients Line - Ca, b for scale a = 22.66 (frequency = 35.856)

Time




ACKNOWLEDGMENTS

The authors wish to thank the National University of Singapore for providing thesupport to perform this study.

FIGURE 10.19 Morlet WT analysis for response of simply supported beam given in Figure10.8b at scales (a) 1–256, (b) 33.5, (c) 10, and (d) 5 with sampling rate 1000 samples/sec.

(a)

Sca

le

114274053667992

105118131144157170183196209222235248


Time

(b)

Am

pli

tud

e

−40100 200 300 400 500

Time

600 700 800 900 1000

−20

0

20


Sca

le

35






(d)

Am

pli

tud

e

−8100 200 300 400 500

Time

600 700 800 900 1000

−2−4−6

02468

Coefficients Line - Ca, b for scale a = 5 (frequency = 162.500)

Sca

le

5


(c)

Sca

le

10


Am

pli

tud

e

−4−3−2−1

100 200 300 400 500 600 700 800 900 1000

0

4321


Time




FIGURE 10.20 Morlet WT analysis for response of fixed-fixed beam given in Figure 10.8cat scales (a) 1–2042, (b) 163, (c) 60, (d) 28, and (e) 18 with sampling rate at 10000 samples/sec.

(a)

Sca

le

1104207310413516619722825928

1031113412371340144315461649175218551958


Time

(b)

Am

pli

tud

e

−80500 1000 1500 2000 2500 3000 3500 4000 4500 5000

−60−40−20

0

4020

6080


Time

Sca

le

163






(c)

Sca

le

60


Am

pli

tud

e

−80−60−40−20

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0

80604020


Time

(d)

Sca

le

30


Am

pli

tud

e

−15

−10

−5

0

5

10


500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Time




REFERENCES

1. Crema L B, Peroni I and Castellani A (1985). Modal Tests on Composite MaterialStructures Application in Damage Detection. Proc. Int. Modal Anal. Conf. ExhibitVol. 2, pp. 708–713.

2. Cook D A and Berthelot Y H (2001). Detection of Small Surface-Breaking FatigueCracks in Steel Using Scattering of Rayleigh Waves. NDT & E Int. 34:483–492.

3. Hurlebaus S, Niethammer, Jacobs L J and Valle C (2001). Automated Methodologyto Locate Notches with Lamb Waves. Acous. Soc. Am. ARLO 2(4):97–102.

4. Staszewski W J (1998) Structural and Mechanical Damage Detection Using Wavelets.Shock Vibr. Digest 30(6):457–472.

5. Kishimoto K, Inoue H, Hamada M and Shibuya T (1995). Time Frequency Analysisof Dispersive Waves by Means of Wavelet Transform. ASME J. Appl. Mech. 62:841–846.

6. Lee Y Y and Liew K M (2001). Detection of Damage Locations in a Beam Usingthe Wavelet Analysis. Int. J. Struct. Stab. Dyn. 1(3):455–465.

7. Quek S T, Wang Q, Zhang L and Ong K H (2001). Practical Issues in Detection ofDamage in Beams Using Wavelets. Smart Mater. Struct. 10(5):1009–1017.

8. Huang H, Norden E, Zheng S, Long S R, Wu M C, Shih H H, Zheng Q, Yen N C,Tung C C and Liu H H (1998). The Empirical Mode Decomposition and HilbertSpectrum for NonLinear and Nonstationary Time Series Analysis. Proc. R. Soc. Lond.A 454:903–995.

9. Huang H, Norden E, Zheng S and Long S R (1999). A New View of Nonlinear WaterWaves: The Hilbert Spectrum. Annu. Rev. Fluid Mech. 31:417–457.


(e)

Sca

le

163


Am

pli

tud

e

−80

−60

−40

−20

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0

80

60

40

20


Time




10. Huang H, Norden E, Chern C C, Huang K, Salvino L W, Long S R and Fan K L(2001). A New Spectral Representation of Earthquake Data: Hilbert Spectral Analysisof Station TCU129, Chi-Chi, Taiwan, 21 September 1999. Bull. Seism. Soc. Am.91(5):1310–1338.

11. Zhang R R, King R, Olson L and Xu Y (2001). A HHT View of Structural Damagefrom Vibration Recordings Proc. ICOSSAR ’01 (San Diego) – CD.

12. Shim S H, Jang S A, Lee J J and Yun C B (2002). Damage Detection Method forBridge Structures Using Hilbert-Huang Transform Technique. 2nd Int. Conf. Struct.Stab. Dynam., Dec. 16–18, 2002, Singapore.

13. Quek S T, Tua P S and Wang Q (2003). Detecting Anomaly in Beams and PlateBased on Hilbert-Huang Transform of Real Signals. Smart Mater. Struct. 12(3):447–460.

14. Tua P S, Jin J, Quek S T and Wang Q (2002). Analysis of Lamb Modes Dominancein Plates via Huang Hilbert Transform for Health Monitoring. Proc. 15 KKCNN Symp.Civil Eng., Ed. S T Quek and D W S Ho, Dec. 19–20, 2002, Singapore.

15. Hahn S L (1996). Hilbert Transform in Signal Processing. London: Artech HouseBoston.

16. Graff K F (1975). Wave Motion in Elastic Solids. Oxford: Clarendon Press. 17. Huang H, Norden E, Zheng S and Long S R (1999). A New View of Nonlinear Water

Waves: The Hilbert spectrum. Annu. Rev. Fluid Mech. 31:417–457.



11
The Analysis of Molecular Dynamics Simulations by the Hilbert-Huang Transform
Adrian P. Wiley, Robert J. Gledhill, Stephen C. Phillips, Martin T. Swain, Colin M. Edge, and Jonathan W. Essex

CONTENTS

11.1 Introduction ..................................................................................................24611.2 The Hilbert-Huang Transform .....................................................................24911.3 Molecular Dynamics ....................................................................................25111.4 Reversible Digitally Filtered Molecular Dynamics.....................................25411.5 Limitations ...................................................................................................26211.6 Conclusions ..................................................................................................262References..............................................................................................................263

ABSTRACT

Proteins are an integral component of all living systems, and obtaining a detailedunderstanding of their structure and dynamics is of considerable importance. Moleculardynamics computer simulations are an important tool in this process. However, thelength of simulation required to probe the full range of motions available to proteins

245


246



is prohibitively expensive. As a solution to this problem, we have developed a technique,Reversible Digitally Filtered Molecular Dynamics (RDFMD), that can enhance orsuppress specific frequencies of motion. We have demonstrated that RDFMD is ableto amplify the low frequency vibrations in the system that are responsible for large-scale changes in structure.

In this paper we describe the use of empirical mode decomposition (EMD) and theHilbert-Huang transform (HHT) to probe the effect of RDFMD on the internal vibra-tions of molecular systems, including a simple protein, in a molecular dynamics (MD)simulation. EMD allows the decomposition of the MD trajectory into intrinsic modefunctions (IMFs) that are shown to be of physical relevance to the input signal. Inparticular, spontaneous changes in molecular shape are shown to be accompanied byincreased energy in low frequency IMFs. Moreover, we show that the HHT enables asuperior analysis of the MD trajectories than Fourier-based techniques and providesessential information that may be used to determine the optimum parameters for theRDFMD method.

11.1 INTRODUCTION

Proteins are polymer chains made up from 20 possible monomer units (amino acids),each sharing a common backbone structure with differing functional side chains (1).YPGDV, a simple protein used in later discussion, consists of five amino acids:tyrosine, proline, glycine, aspartic acid, and valine (Figure 11.1).

It is convenient to consider the amino acid chains (the primary structure) interms of relatively rigid and stable secondary structure units. These are classified aseither α-helices, where a section of coiled amino acids forms a rod-like helix, orβ-sheets, where amino acid strands form layered structures. These units of secondarystructure are connected by flexible loop regions, and secondary structure units canthemselves associate to form compact globular units called domains that move in acorrelated fashion.

Understanding the functions and mechanisms of biological molecules requiresdetailed knowledge of their structure and internal motions. The arrangement, orconformation, of a protein is of considerable importance to its biological function(2, 3). For example, Creutzfeldt-Jakob Disease (CJD) is a fatal neurodegenerative

FIGURE 11.1 Pentapentide YPGDV (Tyr-Pro-Gly-Asp-Val) with backbone dihedral anglesφi and ψi labeled.


The Analysis of Molecular Dynamics Simulations

247


disorder that is generally believed to be caused by conformational change in theprion protein (4). The prion (proteinaceous infectious particle) theory proposes thatthe prion protein (a normal constituent of mammalian cells) undergoes a conforma-tional change to an active form that induces the same change in further prion proteins.Protein aggregation then occurs, which is associated with cell death.

Large-scale conformational motion occurs predominantly by rotation about thesingle bonds in the protein, the so-called dihedral angles, since a small change inthese angles can have a significant effect on the structure of the protein. Theimportant dihedral angles in the main-chain of YPGDV are marked in Figure 11.1.Anharmonic, low frequency motions in dihedral angles are responsible for confor-mational changes (5, 6).

There are several experimental methods for determining three-dimensional struc-tures of molecules, of which X-ray crystallography (7) is undoubtedly the mostcommon. The molecular arrangements seen in X-ray crystallography are, however,“snapshots” of a protein that is not in its natural environment. Often, proteins canbe crystallized in a variety of arrangements that differ significantly (8, 9), and it isbelieved that crystal-packing forces can have a significant influence on the conformerobserved (10). Analytical techniques, such as essential dynamics (11) or normalmode analysis (12), can be applied to experimentally obtained structures in order togive limited information about the flexibility of the protein (13).

Another prominent experimental technique for studying protein structures isnuclear magnetic resonance (NMR) (14). Here data is gathered about the proteinstructure when it is in solution, so that the crystal-packing artifacts associated withX-ray crystallography are avoided. Unfortunately the method is time consumingowing to the need for significant isotopic replacements to allow for the spectra tobe assigned.

Circular dichroism (CD) is a form of light absorption spectroscopy that is verysensitive to the secondary structure of proteins (15). The relative amounts of disor-dered, helical, and sheet secondary structure can be determined, although problemscan be encountered if side chains also contribute to the spectrum.

Experimental methods are generally unable to follow atomic coordinates withsufficient time resolution to describe the nature of many protein motions. For exam-ple, the T4 lysozyme (T4L) enzyme consists of two main domains linked by a hingeregion, and genetic mutants have been crystallized with a range of hinge angles (8).Unfortunately, the natural form of the enzyme can only be crystallized in a closedstate in which there is no space for known substrates to bind. Although NMR datasuggest that the closed state is abundant in solution (16), there is significant evidencethat an opening event must occur. The structure of this open conformer is thoughtto be similar to that crystallized from a mutated form of the protein (17). Experi-mental methods are unable to observe this opening/closing event. To increase ourunderstanding of the mechanisms of conformational change, a technique is thereforerequired that can follow atomistic detail on the time scales over which the relevantmotions occur.

Molecular dynamics (MD) is a computational modeling tool that follows thetrajectories of atoms through time. Improved software and the decreasing cost of


248



computer power now allow routine simulations of large protein systems of around10 ns. Over this time scale, conformational events such as the opening of the twolarge flaps in HIV-1 protease can be observed (18).

A classical description of atoms is generally used in MD; the energy of thesystem is assumed to be independent of the location of electrons (BornOppenheimerapproximation), so atoms are not polarizable and are described by a point locationonly. MD effectively solves Newton’s Second Law,F = ma, for each atom, using afinite difference integrator to overcome the many body problem. One such integrator,the Verlet algorithm (19), is shown in Equation 11.1. The time step, δt, must besufficiently small to keep the system energy stable, yet as large as possible forefficient time evolution of the system (a 2 fs time step is commonly employed).Forces are obtained from the gradient of a potential energy function. This functionhas separate terms for bond stretching, angle deformations, dihedral motion, andnon-bonded interactions. The parameters used for the potential energy function areoptimized with data from vibrational spectra, experimental geometry information,and ab initio quantum mechanical calculations (20).

. (11.1)

Simulations can be run with constant energy, although various thermostats (21,22, 23, 24) and barostats (22, 23) can be applied to reproduce more biologicallyrelevant constant temperature and constant pressure ensembles.

Frequency analysis of molecular dynamics trajectories yields spectra that canbe compared to infra-red experimental data. The majority of intramolecular motionsare wavelike in nature and exist either as localized oscillations in single bonds orangles, or as collective motions (or modes) characterized by coupled vibrations. Forexample, the amide I mode in proteins that originates from the carbonyl stretchingvibration also includes a contribution from the backbone carbon–nitrogen bond (25).

Postsimulation digital filtering has been applied to molecular dynamics simula-tions to remove high frequency motions that are irrelevant to conformational changes.One simple approach Fourier transforms the atomic trajectories, applies a frequency-domain filter, and inverse Fourier transforms the result (26). Alternatively the filteringcan be done in the time-domain by using non-recursive digital filters (27). Thesemethods have proved useful for the visualization of low frequency motions that areimportant for conformational change.

Although MD has been used to study protein conformational change, the timescales over which these motions occur are generally longer than those accessible byconventional simulation routes. Methods to overcome this problem are thereforerequired. Reversible Digitally Filtered Molecular Dynamics (RDFMD) is a methodthat applies digital filters to the velocities in an evolving simulation to promote lowfrequency motions. RDFMD has been successfully applied to a range of systems(28, 29). The analysis of an RDFMD simulation cannot be achieved with sufficienttime resolution by using traditional Fourier-based methods, especially given thediscovery of nonstationary frequency targets (30). The Hilbert-Huang transformprovides the necessary analysis tools, as presented here.

r r r at t t t t t t+( ) = ( ) − −( ) + ( )δ δ δ2 2



249


11.2 THE HILBERT-HUANG TRANSFORM

The Hilbert-Huang transform (HHT) is a combination of empirical mode decompo-sition (EMD) and the Hilbert transform (HT). The method has been applied acrossa broad range of fields, including biology (31, 32), geophysics (33), and solar physics(34). Numerous applications to simple systems and comparisons to Fourier-basedtechniques have also been performed (30, 31).

The Hilbert transform takes a time-domain signal and transforms it into anothertime-domain signal, unlike the Fourier transform, which moves from the time to thefrequency domain. The HT of a real-valued function, x(t), over the range –∞ < t < +∞is another realvalued function, h(t), which is the convolution of x(t) with 1/πt(Equation 11.2). In this equation, P signifies the Cauchy principal value. The rapiddecay of 1/(t–u) biases the transform heavily to points close to t, and thus the HTacts on time-localized data.

(11.2)

In practice, the Hilbert transform can be computed by taking the Fourier trans-form of the data, setting the negative frequency components to zero, doubling thepositive frequency components, and back-transforming (35). This method relies onthe infinite replication of the data set required by the Fourier transform, yieldingunreliable regions at the start and end of the transformed data due to the disconti-nuities introduced. These regions are excluded from our analysis.

It can be shown that the signal magnitude is unchanged by HT, but the phase isadjusted by π/2 (35). The original signal, x(t), and its Hilbert transform, h(t), maybe considered part of a complex signal, z(t):

(11.3)

This can also be written in terms of amplitude, A, and phase, φ:

(11.4)

where

(11.5)

and

. (11.6)

h tx u

t udu( )

( )=−( )

−∞

∞

∫Ρπ

z t x t ih t( ) = ( ) + ( )

z t A t ei t( ) = ( ) ( )φ

A t x t h t( ) = ( ) + ( )2 2

φ th t

x t( ) =

( )( )

−tan 1


250



The rate of change of the phase angle with time is the frequency of motionoccurring. The instantaneous frequency, f(t), is thus defined as:

. (11.7)

The minimum frequency that can be reliably sampled by this method is 1/T (31),where T is the time length of the data set. For an instantaneous frequency to bephysically meaningful, the signal must carry no riding waves and must be locallysymmetrical about its mean point, as defined by the envelope of the local maximaand minima. Real-world phenomena that are described by signals matching thesecriteria are not common.

EMD is a recently developed signal-processing method. It decomposes a signal,which may be nonstationary, into a set of intrinsic mode functions (IMFs) suitablefor the Hilbert transform (31). An IMF is defined as a wave in which the numberof extrema and the number of zero-crossings differ by a maximum of one and where,at any point, the mean of the envelope defined by the local maxima and minima iszero.

To generate an IMF, the local mean is repeatedly determined and subtractedfrom the data set until the number of extrema and zerocrossings in the residual datadiffer by at most one (this process is termed “sifting”). The local mean is found bytaking the mean of a curve through all of the maxima and a curve through all of theminima. These curves are defined, in this work, by cubic spline functions. Thealgorithm proceeds by subtracting each generated IMF from the remaining signaluntil either the recovered IMF or the residual data is small, or the residual hasbecome a trend component with only a single maximum and minimum. The finalresidual of the data is similar in concept to the DC component of a Fourier transform,except that it contains the overall trend of the data.

Once a signal has been decomposed into a set of IMFs, each IMF is separatelyHilbert transformed, giving the amplitude and instantaneous frequency at discretepoints in time. If the IMF is a harmonic oscillator of variable amplitude and fre-quency, then its signal energy at time t can be written in terms of the signal amplitude,A, and frequency, v:

. (11.8)

In the HHT spectra presented in this work, the energy in each IMF is calculatedand presented using Equation 11.8.

This paper presents the application of the HHT method to the analysis of MDand RDFMD simulation trajectories, with a view to obtaining a better understandingof system dynamics, validating the assumptions underlying the RDFMD methods,and deriving some of the parameters associated with the RDFMD approach.

f td t

dt( ) =

( )12π

φ

E t v t A t( ) ∝ ( ) ( )2 2



251


11.3 MOLECULAR DYNAMICS

The YPGDV protein is a common target for conformational studies by moleculardynamics methods (36) and NMR (37), and it has been used as an RDFMD testcase (29). Results included here have been derived from previously published con-stant energy (30) and RDFMD (29) simulations.

The versatility of computer simulation allows the selection of specific internalcoordinates for analysis. For example, it is trivial to extract the progression of anangle or bond during a constant energy YPGDV simulation and to locate the fre-quencies of motion with a windowed Fourier transform. Figure 11.2 illustrates abond-stretching and angle-bending motion in a section of YPGDV, and Figure 11.3shows the Fourier transform of the simulation trajectory for these degrees of freedom.These harmonic motions are stationary throughout the simulation, and the analysisextracts the expected results: the amide I vibration between 1600 cm–1 and 1800 cm–1,the angular motion at around 500 cm–1, and collective backbone motions between800 cm–1 and 1200 cm–1. The angular motion has a large component at the bond-stretching frequency of 1600 to 1800 cm–1, showing that the angles are coupled tothe bond stretches. However, the bond motion has no low frequency component.

The time integral of the HHT spectrum, the marginal spectrum (data not shown),is broadly similar to the Fourier spectrum. However, the HHT analysis can becontinued to greater depth. Figure 11.4 shows the HHT for the angle-bendingCA-C-N vibration between glycine and aspartate, and Figure 11.5 presents thespectrum for C=O bondstretching vibration. The Hilbert spectra show energy flowinginto and out from the high frequency vibrations at the same times, demonstratingthe coupled nature of these vibrations. Figure 11.6 plots the highest frequency IMFsof the glycine–aspartic acid backbone angle (CACN) and of the glycine carbonyllength (C=O); these are clearly in phase. Subtracting the first IMF from the originalsignal and taking the Fourier transform of the result shows that this IMF is respon-sible for all the vibrational motion above approximately 1250 cm–1.

The signals of most relevance to conformational change are the low frequencydihedral motions. These are generally nonstationary, and Fourier analysis providesa poor description of conformational events (30). By looking at the energy in lowfrequency regions as a function of time, Hilbert techniques allow recognition of

FIGURE 11.2 The bond length (C=O) and angle (CA-C-N) used in the frequency analysis.


252



events that occur simultaneously across several dihedrals. Figure 11.7 plots the eightmain-chain dihedral angles of YPGDV that are most important for conformationalchange (see Figure 11.1) as a function of time in a region where a spontaneousconformational transition occurred. The energy in the 0 to 10 cm–1 region obtainedby integrating the HHT spectrum is also shown, and the three highest energy peaksoccur from 15 to 22 ps, corresponding to a rearrangement of dihedrals ψ2, φ3, andψ3. Several dihedrals are often seen moving at once, as this reduces the need forsignificant solvent reorganization.

It is necessary to show that the EMD method produces physically relevant IMFs.Results for the φ3 data of the spontaneous transition are shown in Figure 11.8. Thelow frequency IMFs follow physical motions from the signal, and there is an obviousseparation between the high frequency (IMFs 1 to 5) and low frequency motions(IMFs 6 to 10). The application of EMD as a signal-smoothing technique is also clear.

It is a requirement of the HHT method that frequencies of motion be separable.Without this, artifacts may enter the data (31). This is inherent in the method because,in each sifting iteration, the extracted IMF tries to follow the highest frequencymotion in the remaining signal. Meaningful separation is not always guaranteed fordihedral data due to the flexible nature of the simulation model, and so EMD resultsmust always be interpreted with caution. However, we have shown the physicalrelevance of IMFs and, as Figure 11.8 shows, we find that a single IMF describes

FIGURE 11.3 Fourier transform of backbone degrees of freedom using a Hanning windowand a 50-point running average.



253


conformational events. Sometimes the frequency separation of IMFs is not complete,and the physical relevance of a single IMF is not clear. A blind analysis of a singleIMF is therefore not sensible, but by averaging results over several signals or byintegrating over long time periods, a more robust analysis may be obtained.

In this section the ability of the HHT to yield extra information over and abovethat available by a conventional Fourier analysis has been demonstrated. The couplednature of vibrational motion in molecular systems has been clearly shown, as hasthe ability of EMD to smooth signals of high frequency data that are not requiredand can indeed obscure overall trends. The analysis of conformational change eventshas revealed that these motions are associated with low frequency vibrations, vin-dicating the assumptions underpinning the RDFMD method. The physical relevanceof the IMFs obtained from these signals has been shown, although the separabilityof scales required for HHT does not always apply. Consequently, the EMD resultsmust always be interpreted with caution.

FIGURE 11.4 HHT of bond angle trajectory CA-C-N.

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0 10 20

Time/ps

30


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11.4 REVERSIBLE DIGITALLY FILTERED MOLECULAR DYNAMICS

The basis of the RDFMD method is the application of digital filters to the atomicvelocities in an ongoing molecular dynamics simulation to promote low frequencymotion and hence increase the rate of conformational change (29). The digital filtersused in RDFMD are linear phase, non-recursive filters designed in MATLAB (38).A time series of input velocity data is required, and a mathematical function isapplied to the data to produce modified output velocities. For example, to apply afilter to a single atom in the simulation, the atom’s velocity is stored for every timestep until a buffer of 2m + 1 velocities, v, is obtained. The filter is a list of 2m + 1real coefficients, ci, which is applied to the velocity buffer to produce a single outputvelocity, v ′, as shown in Equation 11.9. The application of a filter to a signal (thevelocity buffer in this case) may affect the frequencies in the signal in different ways— it may enhance the amplitudes of some frequency components and reduce theamplitudes of others. The filter coefficients can be chosen to control the effect of

FIGURE 11.5 HHT of bond length trajectory C=O.

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FIGURE 11.6 The first IMFs of the coupled angle, Gly / Asp CA-C-N, and bond motion,Gly C=0.

FIGURE 11.7 Top: The eight dihedral angle trajectories around a spontaneous conforma-tional transition. Bottom: The energy in the 0 to 10 cm–1 region extracted using HHT.

Sig

nal

Am

pli

tud

e

0 0.1 0.2

Time/ps

1st IMF of Gly C = 0

1st IMF of Gly/Asp CA-C-N

0.3

0 10 20 30Time / ps

0En

erg

y in

0 -

10

cm

-1 r

egio

n

0 10 20 30

100

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An

gle

/ d

egre

es


256



the filter on the input signal. For our purposes, the desired frequency response is anamplification of low frequencies (typically those lower than 100 cm–1) with nochange in the amplitude of higher frequencies. The coefficients of the digital filterare chosen such that the actual frequency response of the filter matches the desiredresponse as closely as possible (see Figure 11.9). The longer the list of coefficients,the closer the actual frequency response is to the desired frequency response. How-ever, short filters are desirable as they require fewer data points and hence lesssimulation. We have found filters of 1001 coefficients (i.e., m = 500) to be sufficient.

. (11.9)

The RDFMD method generally applies a digital filter to a set of atoms in anMD simulation that are of particular interest to the simulator (e.g., just the soluteand not the solvent, or just a region of a protein). A normal MD simulation isperformed to fill a buffer of length 2m + 1 (corresponding to the number of coeffi-cients in the filter) for each atom of interest, and the filter is applied to each of theatoms to enhance or suppress motion in the chosen frequency ranges. A digital filteraffects the phase of each frequency component in the signal as well as the frequency,but, by using a property of linear phase filters, the phase can also be controlled. Theoutput of a linear phase filter is in phase with the midpoint of the filter input buffer.It is for this reason that the system state (coordinates, accelerations, and forces) is

FIGURE 11.8 YPGDV ϕ3 data during a trajectory containing a spontaneous conformationaltransition. Top: Signal (solid line) and residual (dashed line). Upper middle: Sum of highfrequency IMFs (1 to 5). Lower middle: Low frequency IMFs (6 to 10). Bottom: Sum of lowfrequency IMFs (6 to 10).

′ = −=−∑v vt i t i

i m

m

c



257


saved as the midpoint and then restored after the application of the filter — ensuringthat the new velocities are in phase with the atomic positions.

After applying a filter, we allow the system to evolve for a number of steps d.Another filter may then be applied in order to build up gradually the kinetic energyin the low frequency modes. We have been using a value of d = 20 time steps (29),and as a safeguard against unrealistic, high energy events, the filter series is termi-nated early if the kinetic energy of the system exceeds that equivalent to a temper-ature of 2000 K. Since new velocities are applied at the midpoint of a buffer, if dis less than m, molecular dynamics must be run backward in time to fill the buffer,as shown in Figure 11.10. This is possible with the velocity Verlet MD integrator (39).

This flexible method allows many filters to be applied, with subsequent filterapplications positioned anywhere in time relative to the first. It is, however, possiblethat the system is not approaching a conformational transition, and amplifying lowfrequency motions will have little effect. It is for this reason that the number of filterapplications in a series is limited to ten, after which the simulation proceeds asnormal from the end of the final filter buffer. Once filter applications have beenterminated (either by reaching the maximum number of filters allowed or by heatingto the temperature cut-off), 20 ps of MD are performed, giving a new startingstructure from which the filtering process can be repeated.

Prior to the RDFMD simulation, a desired frequency target must be found. Ashort MD simulation of YPGDV has been used for this purpose. Since conforma-tional motions are complex with non-stationary frequencies, Fourier methods yield

FIGURE 11.9 The desired and actual frequency responses for a 1001 coefficient filter.

−0.5

0

0.5

1

1.5

1007550

Frequency/wavenumbers

250

Fre

qu

enc

y R

esp

on

se

Actual frequency response

Desired frequency response




little information. Figure 11.11 presents a spontaneous conformational change andshows the energy present in defined frequency ranges. These data are derived byintegrating the associated HHT spectrum over frequency. It is clear that the sponta-neous conformational change is associated not only with vibrations in the 0 to10 cm–1 region but also with vibrations up to 50 cm–1. Furthermore, there are otherlocations in the trajectory where energy is present in the region 25 to 50 cm–1. Thesedata suggest that although the conformational change itself is associated with verylow (< 10 cm–1) frequency motion, to enhance conformational change, the digitalfilter should be designed to amplify vibrations up to 50 cm–1.

For this paper, a set of RDFMD simulations of YPGDV in water solvent hasbeen performed. The digital filter was applied to all atoms of YPGDV and not tothe solvent. A starting conformation was chosen, and ten sets of random initialvelocities were generated. From each set, seven filter application series were run,using digital filters to amplify the 0 to 25 cm–1 range by two, three, four, five, six,seven, and eight times. The filters used 1001 coefficients, and each filter seriesconsisted of up to ten filter applications using a delay, d, of 20 steps and a temperaturecut-off of 2000 K.

Once a filter has been applied to a buffer, the next buffer should contain anamplification of any low frequency motions present in the system. The HHT methodcan help demonstrate this, as Figure 11.12 shows. The top of the figure shows arepresentative example of a dihedral angle trajectory progressing through two filterbuffers. Only the data corresponding to the middle section of the buffers are shown;the filter is applied at step 500. The bottom graph of the figure is the sum of theEMD trend data and the IMFs with components in the low frequency (0 to 50 cm–1)

FIGURE 11.10 RDFMD sequence showing the delay parameter, d, and the filter length, 2m + 1.


The Analysis of Molecular Dynamics Simulations 259


FIGURE 11.11 YPGDV ψ2 data during a trajectory showing a spontaneous conformationaltransition. The signal and the energy in low frequency bands (cm–1), derived from the HHTspectrum, are shown.

FIGURE 11.12 YPGDV ψ2 data. Top: Signal before and after the application of a digitalfilter. Bottom: Sum of the residual and the low frequency IMFs showing the trend in the data.

200

150

Signal0–100–250–50 100

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edral A

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/deg

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1st buffer2nd buffer




region. The dihedral motion is clearly amplified either side of the central point inthe direction of the low frequency trend.

The amount of additional motion in the backbone dihedral angles caused by theapplication of a filter to the system should correlate with the amount of energy inthe low frequency region at the point of filter application. To test this hypothesis,we define a measure of the additional dihedral motion going from one buffer to thenext as the average of the root-mean-square deviations (RMSDs) of the eight dihedralangles in the central region of the buffer (the midpoint ± 100 steps). The amount ofenergy in the low frequency region is calculated by taking the instantaneous fre-quencies and amplitudes of the dihedral IMFs at the midpoint of the buffer andcalculating the energy at each frequency, as described previously. To estimate theamount of energy in the filter amplification region, these energies are multiplied bythe filter’s frequency response, scaled from unity at 0 cm–1 to zero at 100 cm–1.Finally, the energies are summed to provide our measure of the kinetic energyavailable for amplification.

Figure 11.13 plots the dihedral angle RMSDs against the energy amplified forfive successive applications of a ×5 amplifying filter. The energy and the dihedralmotion are clearly correlated, though the data is quite spread. We also see a steadyincrease in the energy as the digital filter is applied to successive filter buffers. Fewerpoints are displayed for the later buffers due to simulations terminating as theyexceed the temperature cut-off. A trend line has been fitted to the entire data set bya least-squares procedure, plotted to the extremes of the data set only.

FIGURE 11.13 Change in YPGDV backbone dihedral angles as low frequency kinetic energyis successively amplified.

Dih

edra

l an

gle

ch

ang

e (R

MS

D)

30

20

10

0Energy

Filter 1Filter 2Filter 3Filter 4Filter 5




This analysis can be extended to investigate the effect of different amplificationfilters. High amplification filters would intuitively yield greater induced conforma-tional change, although the temperature cutoff will be reached more quickly. InFigure 11.14, the analysis reported in Figure 11.13 has been extended to includedifferent amounts of amplification by the digital filters. Only the trend lines obtainedby least-squares fitting for all the filters are reported. As expected, as the filter’samplification factor increases, so does the induced conformational change. It wasfound that the data for amplification in excess of ×6 were poorly correlated, pre-sumably as a result of rapid simulation termination due to the temperature cut-offbeing exceeded, and consequently these data are not presented.

The controlled increase of energy in low frequency motions is likely to be ofgreater importance to larger protein systems, for which lower temperature cutoffsmay be required. The energy in the low frequency region reached a maximum at anamplification factor of ×3. However, factors of ×4 to ×6 all reach significant levelsof induced conformational change, and higher factors do this in fewer buffers andthus in less computer time. It is believed that the tradeoff between slowly buildingup energy in low frequency motions and efficiently inducing a conformationalresponse will have to be determined for each system to which RDFMD is applied.

In this section, HHT has been applied to examine the hypothesis underpinningthe RDFMD technique, namely that the amplification of low frequency vibrationsin a molecular dynamics simulation of a protein increases the amplitude of dihedralangle vibrations and hence conformational change. Supporting this hypothesis, acorrelation has been observed between the energy available for amplification by the

FIGURE 11.14 Change in YPGDV backbone dihedral angles as low frequency kineticenergy is successively amplified using different digital filters.




digital filter and the resulting RMSD in dihedral motion. As expected, using largeramplifying filters increases the energy in the critical degrees of freedom more rapidly,although there will inevitably be a compromise between the need for efficiency byamplifying as rapidly as possible and the requirement that the conformational tran-sitions be physically sensible without significantly disrupting the protein structure,for which gentle amplification will be required. Finally, more detailed analysis ofspontaneous conformational transitions has confirmed that, although such transitionsare indeed associated with very low frequency motions (< 10 cm–1), they are alsoassociated with higher frequencies (25 to 50 cm–1). Analysis of the remainder of thesimulations trajectory has identified other occasions where significant energy residesin this frequency region, but without associated conformational change. We suspectthat these conditions are associated with possible conformational change events.Amplification of frequencies of up to 50 cm–1 may therefore be more efficient interms of driving conformational change.

11.5 LIMITATIONS

The limitations of the HHT method as applied to molecular dynamics simulationsare essentially threefold. First, analysis of short (1001 time step) RDFMD filterbuffers results in a low frequency limit in the Hilbert transform of 16.7 cm–1. Giventhat this limit lies in the frequency range over which amplification is taking place,it restricts our ability to analyze the simulation behavior over the content of the filterbuffer. Second, conformational transitions often leave large residuals, giving dihedralangle plots similar to step functions. EMD produces physically unrealistic IMFs inan attempt to recreate the unusual signal given by the data less the residual, as shownin Figure 11.15. Although the data could be windowed to reduce the size of theaffected regions, low frequency resolution would be lost. This limitation is due tothe nature of the data and suggests that the HHT must be used with care, withconstant checks of the physical relevance of the IMFs. Third, the HHT requirementof separable scales is not necessarily met by signals from molecular dynamicssimulations. Great care must therefore be taken when performing analyses by, forexample, careful examination of the IMFs to confirm their physical relevance,repeated analysis over many different simulations, averaging over a number ofsimulation signals, and integration of the HHT spectra over frequency and time.

11.6 CONCLUSIONS

In this paper the application of the HHT method to the analysis of moleculardynamics computer simulations has been reported. The non-stationary data associ-ated with these simulations are unsuitable for analysis with Fourier methods. How-ever, it has been shown that the HHT method does yield useful insight into thedynamics. In particular, the assumptions underpinning the RDFMD method ofenhancing conformational change, namely that low frequency vibrations are asso-ciated with conformational change and that non-recursive digital filters may be usedto amplify these vibrations, have been confirmed. HHT analysis has also suggested




that frequencies greater than those earlier identified with spontaneous conformationalchange may in fact be more appropriate targets for RDFMD. However, the presenceof large residuals and an incomplete separation of scales indicate that HHT cannotbe applied to these simulations as a “black box” and that care must be taken.

REFERENCES

1. Creighton, T. E. (1993). Proteins structures and molecular properties. 2nd ed. NewYork: W. H. Freeman and Company, pp. 192–193.

2. Gerstein, M., Lesk, A. M., Chothia, C. (1994). Structural mechanisms for domainmovements in proteins. Biochemistry. 33(22):6739–6749.

3. Källblade, P., Dean, P. M. (2003). Efficient conformational sampling of local sidechainflexibility.J. Mol. Biol. 326:1651–1665.

4. Cardone, F., Liu, Q. G., Petraroli, R., Ladogana, A., D’Alessandro, M., Arpino, C.,Di Bari, M., Macchi, G., Pocchiari, M. (1999). Prion protein glycotype analysis infamilial and sporadic Creutzfeldt-Jakob disease patients. Brain Res. Bull. 49(6):429–433.

5. Caliskan, G., Kisliuk, A., Tsai, A. M., Soles, C. L., Sokolov, A. P. (2003). Proteindynamics in viscous solvents. J. Chem. Phys. 118(9):4230–4236.

6. Rosca, F., Kumar, A. T. N., Ionascu, D., Ye, X., Demidov, A. A., Sjodin, T., Wharton,D., Barrick, D., Sligar, S. G., Yonetani, T., Champion, P. M. (2002). Investigationsof anharmonic low-frequency oscillations in heme proteins. J. Phys. Chem A. 106(14):3540–3552.

7. Rhodes, G. (1993). Crystallography Made Crystal Clear. 2nd ed. San Diego, Calif.:Academic Press.

FIGURE 11.15 YPGDV ψ2 data. Top: Signal and residual (dotted line). Middle: Signal afterthe residual is removed (equivalent to the sum of the IMFs). Bottom: IMFs generated by EMD.




8. Zhang, X., Wozniak, J. A., Matthews, B. W. (1995). Protein flexibility and adaptabilityseen in 25 crystal forms of T4 lysozyme. J. Mol. Biol. 250:527–522.

9. Hayward, S. (1999). Structural principles governing domain motions in proteins.Proteins: Struct. Funct. Genet. 36:425–435.

10. Gogo, N. K., Skrynnikov, N. R., Dahlquist, F. W., Kay, L. E. (2001). What is theaverage conformation of bacteriophage T4 lysozyme in solution? A domain orienta-tion study using dipolar couplings measured by solution NMR. J. Mol. Biol. 308:745–764.

11. Amadei, A., Linssen, A. B. M., Berendsen, H. J. C. (1993). Essential dynamics ofproteins. Proteins: Struct. Funct. Genet. 17(4):412–425.

12. Buchner, M., Ladanyi, B. M., Stratt, R. M. (1992). The short-time dynamics ofmolecular liquids. Instantaneous-normal-mode theory. J. Chem. Phys. 97(11):8522–8535.

13. de Groot, B. L., Hayward, S., van Aalten, D. M. F., Amadei, A., Berendsen, H. J. C.(1998). Domain motions in bacteriophage T4 lysozyme: a comparison betweenmolecular dynamics and crystallographic data. Proteins: Struct. Funct. Genet. 31:116–127.

14. Sanders, J. K. M., Hunter, B. K. (1993). Modern NMR Spectroscopy. 2nd ed. Oxford:Oxford University Press.

15. Johnson, W. C. (1990). Protein secondary structure and circular dichroism — apractical guide. Proteins: Struct. Funct. Genet. 7(3):205–214.

16. Fischer, M. W. F., Majumdar, A., Dahlquist, F. W., Zuiderweg, E. R. P. (1995). 15N,13C, and 1H NMR assignments and secondary structure for T4-lysozyme. J. Magn.Reson., Ser. B 108:143–154

17. Mulder, F. A. A., Mittermaier, A., Hon, B., Dahlquist, F. W., Kay, L. E. (2001).Studying excited states of proteins by NMR spectroscopy. Nat. Struct. Biol. 8(11):932–935.

18. Scott, W. R. P., Schiffer, C. A. (2000). Curling of flap tips in HIV1 protease as amechanism for substrate entry and tolerance of drug resistance. Structure 8:1259–1265.

19. Verlet, L. (1967). Computer experiments on classical fluids. I. Thermodynamicalproperties of Lennard-Jones molecules. Phys. Rev. 159:98–103.

20. Weiner, P.W., Kollman, P.A. (1981). AMBER: assisted model building with energyrefinement. A general program for modelling molecules and their interactions. J.Comput. Chem., 4:287–303.

21. Woodcock, L. V. (1971). Isothermal molecular dynamics calculations for liquid salts.Chem. Phys. Lett. 10:257–261.

22. Berendsen, H. J. C., Postma, J. P. M., van Gunsteren, W. F., Di Nola, A., Haak, J. R.(1984). Molecular dynamics with coupling to an external bath. J. Chem. Phys. 81:3684–3690.

23. Anderson, H. C. (1980). Molecular dynamics simulations at constant pressure and/ortemperature. J. Chem. Phys. 72:2384–2393.

24. Hoover, W. G. (1985). Canonical dynamics: equilibrium phase-space distributions.Phys. Rev. A31:1695–1697.

25. Creighton, T. E. (1993). Conformational properties of polypeptide chains. In: Proteinsstructures and molecular properties. 2nd ed. New York: W. H. Freeman and Company,pp. 192–193.

26. Sessions, R. B., Dauber-Osguthorpe, P., Osguthorpe, D. J. (1989). Filtering molecular-dynamics trajectories to reveal low-frequency collective motions — phospholipase-A2. J. Mol. Biol. 210(3):617–633.




27. Levitt, M. (1991). Real-time interactive frequency filtering of moleculardynamicstrajectories. J. Mol. Biol. 220(1):1–4.

28. Phillips, S. C., Essex, J. W., Edge, C. M. (2000). Digitally filtered molecular dynam-ics: the frequency specific control of molecular dynamics simulations. J. Chem. Phys.112(6):2586–2597.

29. Phillips, S. C., Swain, M. T., Wiley, A. P., Essex, J. W., Edge, C. M. (2003). Reversibledigitally filtered molecule dynamics. J. Phys. Chem. B. 107(9):2098–2110.

30. Phillips, S. C., Gledhill, R. J., Essex, J. W., Edge, C. M. (2003). Application of theHilbert-Huang transform to the analysis of molecular dynamics simulations. J. Phys.Chem. A. 107(24):4869–4876.

31. Huang, N. E., Chen, Z., Long, S. R., Wu, M. L. C., Shih, H. H., Zheng, Q. N., Yen,N. C., Tung, C. C., Liu, H. H. (1971). The empirical mode decomposition and theHilbert spectrum for nonlinear and nonstationary time series analysis. Proc. R. Soc.London Ser. A Math. Phys. Eng. Sci. 454:903–995.

32. Echeverria, J. C., Crowe, J. A., Woolfson, M. S., HayesGill, B. R. (2001). Applicationof empirical mode decomposition to heart rate variability analysis. Med. Biol. Eng.Comput. 39(4):471–479.

33. Chen, K. Y., Yeh, H. C., Su, S. Y., Liu, C. H., Huang, N. E. (2001). Anatomy ofplasma structures in an equatorial spread F event. Geophys. Res. Lett. 28(16):3107–3110.

34. Komm, R. W., Hill, F., Howe, R. (2001). Empirical mode decomposition and Hilbertanalysis applied to rotation residuals of the solar convection zone. Astrophys. J.558(1):428–441.

35. Bendat, J. S. (1985). The Hilbert transform and application to correlation measure-ments. Denmark: Brüel & Kjær.

36. Wu, X. W., Wang, S. M. (2000). Folding studies of a linear pentamer peptide adoptinga reverse turn conformation in aqueous solution through molecular dynamics simu-lations. J. Phys. Chem. B. 104(33):8023–8034.

37. Dyson, H. J., Rance, M., Houghten, R. A., Wright, P. E., Lerner, R. A. (1988). Foldingof immunogenic peptide-fragments of proteins in water solution. 2. The nascent helix.J. Mol. Biol. 201(1):201–217.

38. MATLAB 5.3.0. Natick, MA: The MathWorks Inc. (1999).39. Swope, W. C., Andersen, H. C., Berens, P. H., Wilson, K. R. (1982). A computer-

simulation method for the calculation of equilibrium-constants for the formation ofphysical clusters of molecules — application to small water clusters. J. Chem. Phys.76(1):637–649.



12
Decomposition of Wave Groups with EMD Method
Wei Wang

CONTENTS

12.1 Introduction ..................................................................................................26712.2 Decomposition of Intermittent Small-Scale Fluctuations

from the Large Wave ...................................................................................26912.3 Method for Decomposing Wave Groups .....................................................27212.4 Validations ....................................................................................................27412.5 Conclusions ..................................................................................................280References..............................................................................................................280

ABSTRACT

By adding a proper real temporary time series that can induce a series of additionalextrema, intermittent small fluctuations can be accurately decomposed from the largewaves they ride on. We can thus effectively solve the mode mixing problem that ariseswhen the amplitudes of the components of a time series are very different. Anotherkind of mode mixing arises when the frequencies of the wave components are closeto each other, such as in wave groups. Here a temporary complex time series isintroduced to shift down the frequencies of the components. This will greatly enlargethe ratio of the frequencies in a wave group, and the wave groups can then easily bedecomposed with the empirical mode decomposition method.

Keywords wave groups, EMD method

12.1 INTRODUCTION

Being among the most energetic events in the ocean wave field, wave groups playan important role in ocean wave research. The significant characteristic of a wave

267


268



group is that the group velocity Cg is equal to the energy propagation speed of thewave field; this is widely used in ocean wave models such as WAM. The formationof wave groups is assumed analytically to be the result of the superposition of severalcosine waves with almost the same frequency. In practice, this kind of analysis oftencannot be achieved since no method up to now has been available to decompose awave group efficiently into several simple components. This problem has been themain obstacle in wave group studies.

Feasible methods that might be used to decompose wave groups are Fouriertransform, wavelet transform (Chan 1995), and empirical mode decomposition(EMD) analysis (Huang et al. 1998, 1999). For a continuous uniform wave group,in which the wave group arises from the superposition of continuous constant-ampli-tude cosine waves, Fourier transform can distinguish the frequencies of all thecomponents accurately. For an intermittent wave group, in which the wave grouparises from the superposition of intermittent cosine waves, Fourier transform canyield only a collection of frequencies that will not represent the actual components.The reason is that Fourier transform is a high-resolution frequency detector with noresolution in the time domain; as a result, the intermittency of wave groups willshow as additional frequencies in the spectrum, and the actual frequencies of thewave components will be altered to new locations.

Compared with Fourier transform, wavelet transform is more suitable for dealingwith intermittent signals. It has higher resolution in the time domain and lowerresolution in the frequency domain. Theoretically, intermittent wave groups can bedecomposed by wavelet transform when the frequencies of the wave componentsare widely different, but this method would fail when the frequencies are only slightlydifferent. In practice, the lower resolution of wavelet transform in the frequencydomain will also induce a rich distribution of harmonics around the real componentsand will ultimately result in difficulties in the distinction of the actual wave groupcomponents.

In consideration of the resolution in the time–frequency domain, the most accept-able method is EMD. Compared to the Fourier transform and the wavelet transform,the EMD method has higher resolution in both the time and frequency domains. Asan example, Huang et al. (1998) decomposed a wave group that came from the linearsum of two cosine waves into eight components with up to 3000 sifting processesand an extremely stringent criterion. Although the first two components containedmost of the energy and were similar to the original components, the results are notacceptable for the following reasons:

1. The extra components would not have any physical meaning. 2. Using too many sifting cycles is not suitable for intermittent wave groups

because “too many sifting cycles could reduce all components to a con-stant-amplitude signal with frequency modulation only and the compo-nents would lose all their physical significance” (Huang et al. 1999). Asignal can be decomposed by the EMD method only when the ratio ofthe frequencies is larger than approximately 1.5 and when the number ofsifting cycles is no more than 20. This is a limitation of the EMD methodand reflects a kind of mode mixing.



269


Another kind of mode mixing is introduced by the intermittency of thesmall-scale fluctuations riding on large waves. Huang et al. (1999) gave a methodfor solving this kind of mode mixing, but defining which signal in the intrinsic modefunction (IMF) components is the one we want may be difficult, as shown in Figure12.1.

In the first part of the present paper, we improve the limitation on the ratio ofthe frequencies of the wave components to 1.2 by introducing a high frequencytemporary wave. However, decomposition of a wave group remains impossible withthis method, since the ratio of the frequencies for a typical wave group is alwayssmaller than 1.1.

In the second part of this paper, instead of improving the frequency resolutionof the EMD method, we introduce a temporary complex signal to shift the frequen-cies of the components down. This will greatly increase the ratio of the frequenciesin the wave group, allowing it to be decomposed easily with the EMD method.

12.2 DECOMPOSITION OF INTERMITTENT SMALL-SCALE FLUCTUATIONS FROM THE LARGE WAVE

In general, a fluctuation with high frequency and small amplitude can only changethe large-scale wave profile slightly, but not alter the stationarity of the large scalewave (not introduce additional extreme value). In such a case, the EMD method willtake no account of the existence of the small-scale fluctuations. Moreover, if wecarry out the Hilbert transform, the Hilbert spectrum will treat this small-scale waveas a nonlinear characteristic of the large-scale wave itself, which seriously distortsthe original physical meaning. Only when the small wave is superimposed on thewave crest or trough of the large-scale wave can it affect the stationarity of thelarge-scale wave, and in this case the EMD method can decompose it.

Apparently, if the envelope of this kind of signal is very close to the dominantwave profile, the intermittent signal with high frequency and small amplitude canbe effectively separated. Based on this idea and the characteristics of the EMDmethod, we introduced a temporary signal with high frequency and large amplitude(called the temp-signal hereafter). As a result, the IMF components with highfrequency derived through the EMD method will contain the temp-signal along with

FIGURE 12.1 The original time series and the first two IMF components decomposed withthe EMD method. The solid line in left panel is the original signal, the dashed line in the leftpanel is the envelope, and the lines in the right panel are the IMF components.

−2

−1

0

0 500 1000 1500 2000 2500 3000

1

2

0 500 1000 1500 2000 2500 3000−2

0

2

C1

0 500 1000 1500 2000 2500 3000−2

0

2

C2


270



the high frequency fluctuations in the original signal, and it can easily be separatedby subtracting the temp-signal from the IMFs. But an optionally selected temp-signalwill not match the original signal. Hence, the selection of the temp-signal is veryimportant.

For the case of a large wave with single frequency, containing an intermittentsignal with high frequency and small amplitude, we can write:

(12.1)

where ω2 ω1, a1 > a2. The temp-signal is defined as

(12.2)

The data series with the temp-signal is

(12.3)

The temp-signal should satisfy the following requirements:

• The temp-signal can introduce a series of additional extrema, so its fre-quency should not be near that of the dominant wave; that is, ω3 ω1.

• a3 should be large enough to introduce a series of extrema to conceal thediscontinuity caused by the intermittency.

• The envelope of X(t) calculated by spline function S(t), must satisfy ε a1, where ε = S(t) – a1 cos ω1t.

It is easily found that in order to restrict ε to smaller than 5%, the suitable choiceof the temp-signal satisfies 4T2/5 < T3 T1, in which T1, T2, and T3, are the periodsof the signal of the dominant wave, the fluctuations, and the temp-signal, respectively.In addition, the amplitude of the temp-signal should be large enough to form a seriesof extrema in the original signal.

For the signal of Figure 12.1, the mathematical description is:

(12.4)

f ta t t t t T

a t( )

cos ( , ) ( / , )

cos=

∈ +1 1 1 1 2 1

1 1

0 2ω π ωω

∪++ ∈ +

a t t t t2 2 1 1 22cos [ , / ]ω π ω

X t a t t t Ttmp( ) cos ( , )= ∈3 3 1 1ω

X ta t a t t t t T

( )cos cos ( , ) ( / ,

=+ ∈ +1 1 3 3 1 1 2 10 2ω ω π ω∪ ))

cos cos cos [ , / ]a t a t a t t t t1 1 2 2 3 3 1 1 22ω ω ω π ω+ + ∈ +

X t

t t t

( )

cos ,

co

=

< >2640

1264 1296π

as or

ss . sin ,2640

0 02232

1264π π

t t

−

≤as tt ≤

1296



271


The signal includes a small-scale fluctuation with frequency 50 times larger andamplitude 20 times smaller than those of the dominant wave. Based on the discussionabove, the temp-signal

should be selected in the range 26 < Ttmp < 640. Here we chose atmp = 0.2 and Ttmp

= 60. The composed signal is shown in Figure 12.2.After processing with the EMD method and Deng et al.’s (2001) method of

dealing with the boundary condition, the intermittent small-scale fluctuation isdecomposed successfully, as shown in Figure 12.3a. As a comparison, differentchoices of atmp and Ttmp were used, and the result was shown in Figure 12.3b.

The above example represents a case where the small-scale intermittent fluctu-ation is located on the crest of the dominant wave. For a case where the small-scale

FIGURE 12.2 The shape of the original signal with the superposition of the temp-signal.

FIGURE 12.3 Comparison of the original small-scale fluctuation and the one decomposedwith the EMD method. The solid line represents the decomposed signal, and the dashed linerepresents the original signal. In the left panel, the temp-signal is defined by atmp = 0.2 andTtmp = 60. In the right panel, the temp-signal is defined by atmp = 50 and Ttmp = 120.

X(t

) +

Xtm

p (

t)

−1.50 500 1000 1500

t

2000 2500 3000

−1

−0.5

0

0.5

1

1.5

−0.03

−0.02

−0.01

0

1200 1250 1300 1350 1400

0.01

0.02

0.03(a)

−0.03

−0.02

−0.01

0

1200 1250 1300 1350 1400

0.01

0.02

0.03(b)

X t aT

ttmp tmptmp

( ) cos=

2π


272



intermittent fluctuation is located at an arbitrary location other than crest or trough,the stationarity of the dominant wave will not be changed. As a result, decomposingby the EMD method will give no result. Even under such a condition, the methodwe propose can work. Figure 12.4 shows the original signal and the decomposedresults obtained with our method; the original signal is defined by Equation 12.4,but the position of the intermittent small-scale fluctuation is changed. Signals inwhich singularities exist, such as the one shown in Figure 12.5, can also be decom-posed with this method.

In addition to offering the advantages already mentioned, this method can beused to decompose wave groups when the ratio of the frequencies of the componentsis not less than 1.2 and the distortion of each component is quite small even after alarge number of sifting cycles.

12.3 METHOD FOR DECOMPOSING WAVE GROUPS

As mentioned above, the key point in wave group decomposition is the ratio of thefrequencies of wave group components. Let us consider the wave group as

(12.5)

FIGURE 12.4 Same as Figure 12.3, but the intermittent small-scale fluctuation is changedto the position marked by A in the left panel. The temp-signal is selected as atmp = 50 andTtmp = 120.

FIGURE 12.5 Decomposition of signals with singularities. The temp-signal is selected asatmp = 50 and Ttmp = 120.

−0.03

−0.02

−0.01

0

1950 2000 2050 2100 2150

0.01

0.02

0.03(b)

−1

−0.5

0

0 1000500 1500 25002000 3000

0.5

1(a)

A

−1

−0.5

0

0 1000500 1500 25002000 3000

1.5

1

0.5

2(a)

−0.5

0

1100 13001200 1400 1500 1600

0.5

1(b)

y t t t( ) cos( ) cos( ),= + +ω ω φ1 2



273


in which ω2 > ω1. The ratio of the two frequencies is α = ω2/ω1. Typically α issmaller than 1.1, but it can be greatly enlarged when we are subtracting a temporaryfrequency, ω0, in the numerator and denominator simultaneously. For a real signal,such a downshifting is hard to achieve. Theoretically, Equation 12.5 can be rewrittenin a complex form as

(12.6)

where Y(t) is the complex form of y(t), which can easily be constructed by usingthe Hilbert transform. For an arbitrary time series, y(t), we can always have itsHilbert transform, y(t), as

(12.7)

where P indicates the Cauchy principal value. This transform exists for all functionsof class Lp (Huang et al. 1998). Therefore, the complex function Y(t) can be formed as

(12.8)

The downshifting can easily be achieved by multiplying Y(t) by a function, exp(–iω0t), to yield a new complex signal:

(12.9)

The ratio of the frequencies for signal Z(t) is α = (ω2 – ω0)/(ω1 – ω0).For a well-selected ω0, α can be much larger than 1.5, the limitation in the EMD

method. Zr(t) and Zi(t) can easily be decomposed, either with the EMD method orthe method described in Section 12.2, and represented by their IMF components as:

(12.10)

y t Y t i t i t( ) Re ( ) Re exp( ) exp( ( ))= = + + ω ω φ1 2 ,

y t Pyt

d( )( )

,=−−∞

∞

∫1π

ττ

τ

Y t y t iy t( ) ( ) ( ).= +

Z t Z t iZ t

Y t i t

i t

r i( ) ( ) ( )

( ) exp( )

exp( )

= +

= ⋅ −

=

ω

ω

0

1 ++ + ⋅ −

= −

exp( ( )) exp( )

exp ( )

i t i t

i t

ω φ ω

ω ω

2 0

1 0 + − + exp (( ) ) ,i tω ω φ2 0

Z t C t r

Z t C t r

r rk

k

n

nr

i ik

k

n

ni

( ) ( ) ,

( ) ( )

= +

= +

=

=

∑

∑1

1

..




The combination of the corresponding IMF components of Zr(t) and Zi(t) willreconstruct Z (t) as a sum of complex components Ck(t):

(12.11)

According to Equation 12.9, Y(t) can be written as

(12.12)

Finally, the original signal y(t) may be represented as:

(12.13)

where ck(t) are IMF components of the original signal, and rn is the trend or meanvalue.

12.4 VALIDATIONS

We offer two examples to verify the performance of the present method. The firstis a continuous uniform wave group with a phase shift in one of the components.

For ω1 = 2π/30, ω2 = 2π/32, and φ = π/6, the wave profile represented by Equation12.5 is given in Figure 12.6. In this case α = 1.067. With ω0 = 2π/28, the waveprofile of Z(t), governed by Equation 12.9, is shown in Figure 12.7. Here, a and brepresent the real and imaginary components of Z(t), respectively.

Under such conditions, the ratio of the frequencies of the components is approx-imately 2. Thus, Zr(t) and Zi(t) can easily be decomposed into a series of IMF

Z t Z t iZ t

C t iC t r

r i

r i kk

n

nr

( ) ( ) ( )

( ) ( ) (

= +

= +( ) +=

∑1

++

= +=

∑

ir

C t R

ni

k

k

n

n

)

( ) .1

Y t Z t i t

C t i t Rk

k

n

n

( ) ( ) exp( )

( ) exp( )

= ⋅

= ⋅ +=

∑

ω

ω

0

0

1

⋅⋅exp( ).i tω0

y t Y t

C t i t Rk

k

n

n

( ) Re ( )

Re ( ) exp( ) e

=

= ⋅ + ⋅=

∑ ω0

1

xxp( )

( ) ,

i t

c t rk

k

n

n

ω0

1

= +=

∑


Decomposition of Wave Groups with EMD Method 275


components. Figure 12.8 shows the wave profiles of all the IMF components of thereal and imaginary parts.

Results calculated from Equation 12.11, Equation 12.12, and Equation 12.13are shown in Figure 12.9 and Figure 12.10. The solid lines in Figure 12.9 and Figure12.10 correspond to the first and second modes in Equation 12.5, respectively. Forcomparison, the actual modes are also plotted as dotted lines in the same figures.The agreement is excellent.

FIGURE 12.6 Wave profile represented by Equation 12.9 with ω1 = 2π/30, ω2 = 2π/32, andφ = π/6.

FIGURE 12.7 Wave profile represented by Equation 12.10 with ω0 = 2π/28. (a) The realcomponent of Z(t); (b) the imaginary component of Z(t).

1000800600400

t

2000

y (t

)

−1.5

−2

−1

−0.5

0

0.5

1

1.5

2

Zr

(t)

−2

0

2

Zi (

t)

−2

0

2

1000800600

t

4002000




Figure 12.11 illustrates the differences between the IMF components and theactual modes. It is clear that the differences are negligible.

The second example is an intermittent wave group. The intermittent wave groupis formed as

(12.14)

FIGURE 12.8 The IMF components of Zr(t) and Zi(t). (a,b) IMF components of Zr(t); (c,d)IMF components of Zi(t).

FIGURE 12.9 Comparison of the first IMF component calculated from Equation 12.11, Equa-tion 12.11, and Equation 12.13 (solid line) and the first mode of Equation 12.5 (dotted line).

C2

r

−1

−2

0

1

2

0 200 400

t

600 800 1000

C1

r

−1

−2

0

1

2

0 200 400

t

600 800 1000

C1

i

−1

−2

0

1

2

0 200 400

t

600 800 1000

C2

i

−1

−2

0

1

2

0 200 400

t

600 800 1000

1000800600400

t

2000−1.5

−1

−0.5

0

0.5

1

1.5

y tas t t or t t

t t as t( )

,

cos cos ,=

< >( ) + +( )

0 1 2

1 2 1ω ω φ ≤≤ ≤

t t2




For ω1 = 2π/30, ω2 = 2π/32, φ = π/6, t1 = 1740, and t2 = 3150, the wave profilerepresented by Equation 12.14 is shown in Figure 12.12. With ω0 = 2π/28, the waveprofile of Z(t), governed by Equation 12.10, is shown in Figure 12.13, where a andb represent the real and imaginary components of Z(t), respectively.

In the first example (shown in Figure 12.9, Figure 12.10, and Figure 12.11),only negligible errors exist at the ends because we have confined end effects witha method of adding characteristic waves at the ends. In the second example, however,

FIGURE 12.10 Comparison of the second IMF component calculated from Equation 12.11,Equation 12.11, and Equation 12.13 (solid line) and the second mode of Equation 12.5 (dottedline).

FIGURE 12.11 Differences between IMF components and actual modes. The upper panelshows the differences between the first IMF component and the first mode in Equation 12.5.The lower panel shows the differences between the second IMF component and the secondmode in Equation 12.5.

1000800600400

t

2000−1.5

−1

−0.5

0

0.5

1

1.5

1000800600400

t

2000−1

−0.5

0

0.5

1

10008006004002000−1

−0.5

0

0.5

1




end effects are hardly confined because the ends of the wave groups are not at theends of the data set. If the ends of the wave groups are left unattended, wide swingscaused by spline fitting will exist at the ends and will propagate to the interior andulterior as the number of sifting iterations increases. Figure 12.14 and Figure 12.15show the final results.

FIGURE 12.12 Wave profile represented by Equation 12.14 with ω1 = 2π/30, ω2 = 2π/32,φ = π/6, t1 = 1740, and t2 = 3150.

FIGURE 12.13 Wave profile represented by Equation 12.10 with ω0 = 2π/28. (a) The realcomponent of Z(t); (b) the imaginary component of Z(t).

y (t

)

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2500045004000350030002500

t

2000150010005000

Zi (

t)

−1

0

1

2

−2500045004000350030002500

t

2000150010005000

Zr

(t)

−1

0

1

2

−2

(a)

(b)




FIGURE 12.14 The IMF components of Zr(t) and Zi(t): (a,b) IMF components of Zr(t); (c,d)IMF components of Zi(t).

FIGURE 12.15 Comparison of the IMF components calculated from Equation 12.11, Equa-tion 12.12, and Equation 12.13 (solid line) and the two actual modes of Equation 12.14 (dottedline).

Cr1

−1

−2

0

1

2(a)

Ci1

−1

−2

0

1

2(c)

Cr2

−1

−2

0

1

2

0 1000 2000

t

3000 4000 5000

(b)

Ci2

−1

−2

0

1

2

0 1000 2000

t

3000 4000 5000

(d)

−1

0

1

2

−2500045004000350030002500

t

2000150010005000

−1

0

1

2

−2

(a)

(b)




12.5 CONCLUSIONS

The present paper introduces two methods for decomposing a wave group into aseries of components of relatively single scale. The first method was used not onlyin solving the problem of decomposition of wave groups but also as an improvedmethod of EMD for a variety of uses. The second method can decompose wavegroups of small α, but the sensitivity to the selection of the temporary frequency isstill an open question.

REFERENCES

Chan, Y.T. (1995). Wavelet basics. Boston: Kluwer.Deng, Y.J., W. Wang, C.C. Qian, Z. Wang and D.J. Dai. (2001). Boundary-processing tech-

nique in EMD method and Hilbert transform. Chinese Science Bulletin, 46(11),954–960.

Huang, N.E., Z. Shen, S.R. Long, et al. (1998). The empirical mode decomposition and theHilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc.Lond. A, 454, 899–995.

Huang, N.E., Z. Shen and S.R. Long. (1999). A new view of nonlinear water waves: theHilbert spectrum. Annu. Rev. Fluid Mech., 31, 417–457.



13

281

Perspectives on the Theory and Practices of the Hilbert-Huang TransformNii O. Attoh-Okine

CONTENTS

13.1 Introduction ..................................................................................................28213.2 Basic Ideas ...................................................................................................282

13.2.1 The Hilbert-Huang Transform .........................................................28213.2.2 Starting Point....................................................................................288

13.3 Current Applications ....................................................................................28913.3.1 Biomedical Applications ..................................................................28913.3.2 Chemistry and Chemical Engineering.............................................28913.3.3 Financial Applications......................................................................29013.3.4 Meteorological and Atmospheric Applications ...............................29013.3.5 Ocean Engineering...........................................................................29113.3.6 Seismic Studies ................................................................................29213.3.7 Structural Applications.....................................................................29513.3.8 Health Monitoring............................................................................29613.3.9 System Identification........................................................................296

13.4 Some Limitations .........................................................................................29713.5 Potential Future Research ............................................................................298References..............................................................................................................299Addendum: Perspectives on the Theory and Practices of the Hilbert-Huang Transform............................................................................302

13A.1 Analytical .........................................................................................30213A.2 Hybrid Method.................................................................................30213A.3 Bidimensional EMD ........................................................................30313A.4 Some Observations...........................................................................304

References..............................................................................................................304


282



ABSTRACT

The theory and practice of the Hilbert-Huang transform (HHT) provide another wayof analyzing nonlinear and nonstationary time series. The central idea of HHT is theempirical mode decomposition (EMD), which is then decomposed into a set of intrinsicmode functions (IMFs) and analyzed by using the Hilbert spectrum. This chapter isprimarily a summary of current applications of HHT and suggests areas for futureresearch based on my personal views.

13.1 INTRODUCTION

The Hilbert-Huang transform (HHT) provides a new method of analyzing nonsta-tionary and nonlinear time series data. It allows the exploration of intermittent andamplitude-varying motions. The classical signal analysis theory and practice havebeen dominated by the Fourier transform, which represents the signal system in thefrequency domain. The Fourier works well for strictly periodic or stationary randomfunctions of time. To address the issues of nonstationarity and nonperiodic functions,various methods have been introduced in the literature. These include wavelet anal-ysis, the Wigner-Ville distribution, and the evolutionary spectrum. Unfortunately,these methods failed in one way or another [Huang et al., 1998].

HHT is emerging as a powerful signal-processing tool. During the past fiveyears, a great deal has been learned about the computation, interpretation, andimplementation of the HHT technique. This work has been scattered in a wide varietyof fields, including signal processing, structural health monitoring, earthquake engi-neering, nondestructive testing, transportation engineering, ocean engineering, andfinancial and biomedical applications. No one has drawn the new work together ina comprehensive manner.

This chapter is primarily a summary of my own current views. My failure tomention a particular contribution should not be taken as an indication of disinterestor disagreement. After this general introduction, basic ideas of HHT are presentedin a manner that should be accessible to readers with no previous familiarity withHHT. The paper then discusses various application areas, outlines some perceived“difficulties,” and finally highlights a future research direction of HHT.

13.2 BASIC IDEAS

13.2.1 THE HILBERT-HUANG TRANSFORM

Huang et al. [1998] developed the Hilbert-Huang transform to decompose anytime-dependent data series into individual characteristic oscillations by using theempirical mode decomposition (EMD). The decomposition is developed from theassumption that any data or signal consists of different intrinsic mode functions(IMFs), each representing embedded characteristics of the data/signal. The decom-position can be viewed as an expansion of the data in terms of the IMFs. The HHTmethod is a two-step data analyzing method that consists of EMD and Hilbertspectral analysis (HSA). The first step is the EMD, by which a completed time


Perspectives on the Theory and Practices of the Hilbert-Huang Transform

283


history of the signal is decomposed into a finite and often small number of IMFssuch that they may be Hilbert transformed. The EMD is based on local characteristicsin the time scale of the data, making the approach direct, posterior, and adaptive.The IMFs admit well-behaved Hilbert transforms.

An IMF is defined by the following two criteria:

1. The number of extrema and zero-crossings must either be equal or differby no more than one.

2. At any instant in time, the mean value of the envelope defined by the localmaxima and the envelope of the local minima is zero.

With the above definition, any function can be decomposed as follows:

• Identify all the local extrema, then connect all the local maxima by cubicspline as upper envelope.

• Repeat the procedure for the minima to produce the lower envelope. Theupper and lower envelope should cover all the data. If the mean is desig-nated as m1 and the difference between the data and m1 is the first com-ponent h1, then

(13.1)

where h1 is an IMF.The mean m1 is given by the sum of local extrema connected by cubic spline:

where U is the local maximum and L is the local minimum.The IMF can have both amplitude and frequency modulations. In many cases,

there are overshoots and undershoots after the first round of processing, which istermed sifting. The sifting process serves two purposes [Huang et al., 2001]: elim-inating riding signals/profile and making the signals or profile more symmetric. Thesifting process has to be repeated many times. In the second sifting process, h1 istreated as the data and as the first component. h1 is almost an IMF, except someerror might be introduced by the spline curve-fitting process. To treat h1as new setof data, a new mean is computed. Then,

. (13.2)

After repeating the sifting process up to k times, h1k becomes an IMF. That is:

. (13.3)

x t m h( ) − =1 1

mL U

1 2= +

h m h1 11 11− =

h m hk k k1 1 1 1( )− − =


284



Let h1k = c1, the first IMF from the data. c1 should contain the finest scale or theshortest period component of the data/profile. Now c1 can be separated from the restof the data by

. (13.4)

Since r1 is the residue, it contains information of longer period component; itis now treated as the new data and subjected to the same sifting process. Theprocedure is repeated for all subsequent rjs, and the result is

(13.5)

c2 is now the second IMF of the data.The sifting process can be stopped by predetermined criteria: either when the

component of cn or the residue rn becomes so small that it has a predetermined valueof substantial consequence, or when the residue rn becomes a monotonic functionfrom which no IMF can be extracted.

Summing Equation 13.4 and Equation 13.5 yields the following equation:

, (13.6)

where cj is the jth IMF and n is the number of sifted IMFs. rn can be interpreted asa trend in the signal/profile. The cj has zero mean. Due to the iterative procedure,none of the sifted IMFs derived is closed analytical form [Schlurmann, 2002]. TheIMFs can be linear or nonlinear based on the characteristics of the data. The IMFsare almost orthogonal and form a complete basis. Their sum equals the original data[Salisbury and Wimbush, 2002]. The EMD then picks out the highest-frequencyoscillation that remains in the signal.

Flandrin et al. [2003] established how EMD can be used as a filterbank. Theyrepresented the signal as follows:

(13.7)

where mk(t) stands for a residual “trend,” and the modes dk(t), k = 1, …, K anddk(t) stand for details.

The orthogonality of the IMF components can be checked as follows, if Equation13.6 is expressed as

. (13.8)

x t c r( ) − =1 1

r c r r c rn n n1 2 2 1− = ..., − =; − ;

x t c rj

n

j n( ) = +=

∑1

x t m t d tk

k

K

k( ) ( ) ( )= +=

∑1

x t c tj

n

j( ) ( )==

+

∑1

1



285


That is, one has included the last residual or data trend as an additional element:

. (13.9)

If the decomposition is orthogonal, then the cross terms given in the second parton the right hand side should be zero when they are integrated over time. The overallindex of orthogonality OI can be defined as follows:

. (13.10)

To ensure that the IMF components retain enough physical sense of both ampli-tude and frequency modulations, we can limit the size of the standard deviation(SD), computed from two consecutive sifting results as

A value of SD between 0.2 and 0.3 was used.In the next step, the Hilbert transform is applied to each of the IMFs, subse-

quently providing the Hilbert amplitude spectra a significant instantaneous fre-quency. The Hilbert transform of each IMF is represented by

(13.11)

Equation 13.7 gives both the amplitude and frequency of each component as afunction of time. The Fourier representation would be as follows:

(13.12)

with both aj and j constants.The Fourier transform represents the global rather than any local properties of

the data/signal. The major difference between the conventional Hilbert transformand HHT is the definition of instantaneous frequency. The instantaneous frequencyhas more physical meaning through its definition of each IMF component, while

x t c t c t cj

n

j

j

n

k

n

j k2

1

12

1

1

1

1

2( ) ( ) ( ) (= +=

+

=

+

=

+

∑ ∑∑ tt)

IOc t c t

x tt o

T

j

n

k

n

j k=

=

=

+

=

+

∑ ∑ ∑1

1

1

1

2

( ) ( )

( )

SDh t h t

h tt

T

k k

k

= −

=

−

−∑

0

12

12

| ( ) ( ) |

( )

x t Re a t i t dtj

n

j( ) ( )exp( ( ) )==

∑ ∫1

ω

x t Re aj

ji tj( ) exp=

=

∞

∑1

ω


286



the classical Hilbert transform of the original data might possess unrealistic features[Huang et al., 1998]. This implies that the IMF represents a generalized Fourierexpansion. The variable amplitude and instantaneous frequency enable the expansionto accommodate nonstationary data. Huang et al. [2001] demonstrated that theFourier- and wavelet-based components and spectra may not have a clear physicalmeaning as in HHT. For example, the wavelet-based interpretation of a pavementprofile is meaningful relative to selected mother wavelets [Attoh-Okine, 1999].

Furthermore, Hilbert analysis [Long et al., 1995] is based on almost noncausalsingular information. Therefore, at any given time, the data or signal has only oneamplitude and one frequency, both of which can be determined locally. Theserepresent the best information at that particular time. Physically, the definition ofinstantaneous frequency has true meaning for monocomponent signals, where thereis one frequency, or at least a narrow range of frequencies, varying as a function oftime. Since most data do not show these necessary characteristics, sometimes theHilbert transform makes little physical sense in practical applications. In the presentmethod, the EMD is used to decompose the signal into a series of monocomponentsignals. Furthermore, to extract significant information, the time–frequency–ampli-tude joint distribution [a(t), (t), t] can be developed. This joint distribution in 3Dspace can be replaced by [H(,t)], where x(t) = H(,t). This final representation isreferred to as the Hilbert spectrum.

With the Hilbert spectrum defined, we can define a marginal spectrum:

. (13.13)

The marginal spectrum offers a measure of the total amplitude (or energy) contri-bution from each of the frequency values [Huang et al., 1999a]. The degree ofstationarity is defined as follows:

. (13.14)

The closer the above equation is to zero, the more stationary the system is and theinstantaneous energy:

(13.15)

(13.16)

≅ by definition. In Equation 13.16, is the jth component of the total Hilbertspectrum. The equation

h H t dtt

( ) ( )ω ω= ,∫0

DT

H th T

dtT

( ) (( )

( ))ω ω

ω= − ,

/∫11

0

IE t H t d( ) ( )= ,∫ωω ω2

H t H t A tj

n

j

j

n

j( ) ( ) _ ( )ω ω, = , ≈= =

∑ ∑1 1

H j



287


(13.17)

provides a measure of the total amplitude contribution for each frequency value[Zhang et al., 2003].

By using the Hilbert spectrum, first and second order time-moments and thefrequency-dependent order can be derived. The first and second order of time-depen-dent moments are as follows:

(13.18)

(13.19)

The first and second frequency-dependent distributions are as follows:

(13.20)

(13.21)

The marginal in the time and frequency domains can be calculated as follows:

(13.22)

(13.23)

h hj a t dtj

n

j

n T

j( ) ( ) _ ( )ω ω= ≈= =

∑ ∑ ∫1 1

0

m tH t d

H t dω

ω ω ω

ω ω,

−∞

∞

−∞

∞=,

,

∫∫

1( )( )

( )

m tH t d

H t d

m tω ω

ω ω ω

ω ω,

−∞

∞

−∞

∞ ,=,

,−∫

∫2

2

1( )( )

( )( )

mtH t dt

H t dtt,

−∞

∞

−∞

∞=,

,

∫∫

1( )( )

( )ω

ω

ω

mt H t dt

H t dt

mt t,−∞

∞

−∞

∞ ,=,

,−∫

∫2

2

1( )( )

( )( )ω

ω

ωω

m t H t dω ω ω,∞

∞

∫ ,0( ) ( )

m H t dtt,∞

∞

= ,∫0( ) ( )ω ω




The information content of the marginal mt,0() is much reduced [Huang et al., 2001].The time marginal corresponds to the instantaneous power |x(t)|2 of the signal, andthe frequency marginal corresponds to the spectral energy density |X()|2.

Huang et al. [2003a] developed a confidence limit for HHT. The ensemble meanand standard deviation of IMF sample sets obtained with different stopping criteriaare calculated and form a simple random set. The confidence limit for EMD/HSAis then defined as a range of standard deviations from the assembled mean. Theauthors attempted to establish a confidence limit for the EMD/HSA approach as astatistical measure of the validity of the results. The method of establishing aconfidence limit depends on factors like stopping criteria, the maximum number ofsiftings, intermittence, sifting methods, and the nomenclature needed for the confi-dence limit. The method does not invoke the ergodic assumption. Qu and Jarzynski[2001] did a comparative study of different time–frequency representations of Lambwaves. The authors noted that the time–frequency accuracy of the Hilbert spectrumis dependent on the accuracy of the EMD. That is, if the decomposition into IMFsdoes not capture the signal’s behavior, the resulting Hilbert spectrum will not giveprecise time–frequency results.

13.2.2 STARTING POINT

The starting point of HHT is the Hilbert transform. The Hilbert transforms wereoriginally developed to integrate equations [Duffy 2004]. Instead of expressing afunction of time with its Fourier transform, which depended only on frequency, theHilbert transform yields another temporal function:

(13.24)

Gabor (1946) developed the concept of analytic signal:

(13.25)

or

(13.26)

which is a local time-varying wave with amplitude a(t) and phase (t). The Hilberttransform can be considered to be a filter that simply shifts the phase of its inputby –/2 radians. Unfortunately, most signals are not band limited. One of Huang’snotable contributions was to devise a method — the sifting method — that transformsa wide class of signals into set of band-limited functions. The instantaneous fre-quency may possess negative values. Lai (1998) developed a proper method toimplement positive frequencies. The Hilbert spectrum is similar to the Fourierspectrum in a physical sense, but the Fourier approach is more global. In the Hilbert

ˆ ( )f

ft

d=−−∞

∞

∫1π

ττ

τ

Y t f t i f t( ) ( ) ( )= + ∧

Y t a t ei t( ) ( ) ( )= θ


Perspectives on the Theory and Practices of the Hilbert-Huang Transform 289


spectrum, at any given time, the signal has only one amplitude and one frequency,both determined locally [Long et al., 1995].

13.3 CURRENT APPLICATIONS

13.3.1 BIOMEDICAL APPLICATIONS

Huang et al. [1999b] analyzed the pulmonary arterial pressure on conscious andunrestrained rats. The authors analyzed the signal by using HHT and compared theHilbert spectrum to the Fourier spectrum. The Hilbert spectrum clearly depicts thefluctuations of the frequency with time, whereas the Fourier spectrum gives onlythe distribution of energy over frequencies without allowing the frequency of theoscillations to be variable in the whole time window. Huang et al. [1999c] alsostudied the signals obtained from pulmonary hypertension. Their analysis attemptedto address the linearity and nonlinearity of the dependencies of blood pressure onother variables. Huang et al. [1999c] developed a generalized function connectingthe IMFs and an analytic equation:

(13.27)

(13.28)

where t0 is the instant in time when oxygen concentration drops suddenly, t1 is theinstant when oxygen concentration increases suddenly, A is the mean value of Mk(t)before time t0, and B, C, T1, and T2 are constants.

13.3.2 CHEMISTRY AND CHEMICAL ENGINEERING

Phillips et al. [2003] performed a comparative analysis of HHT and wavelet methods.They investigated a conformational change in Brownian dynamics (BD) and molec-ular dynamics (MD) simulations. Most of the MD trajectories were generated byusing a reversible digitally filtered MD method. The authors used HHT to analyzethe trajectories. The HHT analysis of BD trajectories was able to isolate characteristicfrequencies of motion. The authors compared their results with the Morlet wavelet,and the results yielded similar information. The authors also tried to use Akimaspline, but this did not enhance the algorithm.

Wiley et al. [2004] used HHT to investigate the effect of reversible digitallyfiltered molecular dynamics (RDFMD), which can enhance or suppress specificfrequencies of motion. HHT was applied to conformational motion, which occurspredominantly by rotation about single bonds in a protein, called dihedral angles.A small change in these angles can have significant effects on the structure of theprotein. Unfortunately, there were limitations of the HHT method as applied to theMD simulations. These were:

M t IMF IMF IMFk k k n( ) = + + +−1 …

M t A B t t e C e t tk

t tT

t tT( ) ( ) ( )= + − + − ; ≤

− −− −

0 0

0

1

0

21 tt1




• The analysis of short RDFMD filters buffers results in a low frequencylimit in the HHT.

• EMD produces physically unreliable IMFs. In an attempt to recreate theusual signal given by the data, less the residual.

• The HHT requirement of separable scales is not necessarily met by signalsfrom MD simulations.

Montesinos et al. [2002] applied HHT to signals obtained from BWR neuronstability. The authors used data from a nuclear power plant that were recorded duringstartup of the plant. They show a sharp peak produced by sudden activity insertiondue to the change of the recirculation pump velocity from low to high speed. Theauthors attempted to recognize the IMF that is related to the stability of the BWR.They further used an autoregressive model to analyze the fourth IMF and showedthat the IMFs are mutually orthogonal.

13.3.3 FINANCIAL APPLICATIONS

Huang et al. [2003b] applied HHT to nonstationary financial time series and useda weekly mortgage rate data. The latter authors illustrated the differences amongFourier, wavelet, and Hilbert spectral analysis. The authors relate the slope to theglobal level of the data. By using the HHT, the authors address the question ofvolatility, introducing a new measure of volatility, designated as “variability,” basedon EMD applied to produce IMF components. The variability was defined as theratio of the absolute value of the component(s) to the signal at any time:

(13.29)

where T corresponds to the period at the Hilbert spectrum peak of the high-passedsignal up to h terms, and

. (13.30)

The unit of this variability parameter is the fraction of the market value. This isa simple and direct measure of the market volatility.

13.3.4 METEOROLOGICAL AND ATMOSPHERIC APPLICATIONS

Salisbury and Wimbush [2002], using Southern Oscillation Index (SOI) data, appliedthe HHT technique to determine whether the SOI data are sufficiently noise freethat useful predictions can be made and whether future El Niño southern oscillation(ENSO) events can be predicted from SOI data. The authors were able to identifyan IMF that roughly corresponded to a 3.6-yr mean interval between ENSO warm

V t TS tS t

h( )( )( )

; =

S t c th

j

h

j( ) ( )==

∑1




events. The authors further developed a dynamical model using selected IMFs andSOI data that can make a predictive estimate.

Pan et al. [2002] used HHT to analyze satellite scatterometer wind data over thenorthwestern Pacific and compared the results to vector empirical orthogonal func-tion (VEOF) results. Duffy [2004] used HHT to analyze three different data files:

• Sea level heights• Incoming solar radiation• Barographic observations

The HHT persistently detected periodic features such as tides, snowmelt, andheavy precipitation events in terms of sea level heights. Using the radiation data,Duffy [2004] observed anomalies during ENSO and its effect on the aerosol con-centration in the troposphere. Some of the IMFs of the observation data could bepaired with meteorological events, such as the passage of a cold front, a warm front,and a trough. Generally, the HHT is capable of discerning mesoscale and synopticevents.

With a five-day time series of temperature, Lundquist [2003] demonstrated howthe HHT allows simultaneous identification of the stationary diurnal temperaturecycle as well as intermittent and nonstationary cooling events such as frontal pas-sages and density currents. One of the IMFs obtained by the authors appears to beassociated with inertia gravity waves.

13.3.5 OCEAN ENGINEERING

Schlurmann [2002] introduced the application of HHT to characterize nonlinearwater waves from two different perspectives, using laboratory experiments. Theauthor examined monochromatic and transient (freak) waves and compared theresults with wavelet analysis. The HHT was able to identify superharmonic frequen-cies, which were not evident in the wavelet spectrum, in both cases. Schlurmannand Datig [2005] applied HHT to rogue waves (freak waves). The authors determinedstatistical characteristics of the IMFs and then deduced that the natural data sets donot behave like white noise. The statistical analysis demonstrated that the numberof extrema, the mean periods, and the corresponding standard deviation of the IMFsare strongly related to each other. The authors developed an equation relating theaverage period and mode number. Various coefficients were defined for differentranges of modes.

Veltcheva [2002] applied HHT to wave data from nearshore sea. The IMF wasable to group the waves into calm and wave-decay stages at the offshore. Veltcheva[2004] also compared the HHT and Fourier transforms for nearshore sea waves. Theauthor noted that the process of wave breaking mainly affects the IMF with thehighest energy in the decomposition of offshore waves. Wang [2004] presented animproved EMD for the decomposition of wave groups. Hwang et al. [2004] comparedthe energy flux computation of shoaling waves using HHT, FFT, and wavelet tech-niques. They found that the associated HHT spectral analysis provides superiorspatial (temporal) and wavenumber (frequency) resolution for handling nonstationarity




and nonlinearity. They also found that the energy flux computed using the HHTspectrum is higher than that from wavelet and FFT techniques.

Larsen et al. [2004] used HHT to characterize the underwater electromagneticenvironment and identify transient manmade electromagnetic disturbances. Theauthors collected field data based on in-water magnetic data. The HHT was able toact as a filter, effectively discriminating different dipole components. Therefore, themethod helps in detecting moving electric dipoles in the ocean. The authors furtherdeveloped an empirical mode–matched filter. The main idea is to apply a matchedfilter to a signal generated by using IMFs as basis function, rather than to the targetsignal. In the empirical mode–matched filter, EMD is initially applied to the signal(time series) to generate the IMFs, i(t), i = 1, …, N, where N is the number of IMFs.The second step involves determining the linear combination of the IMFs containingthe target signal s(t) that is best approximated to the known target signature in theleast-square sense:

. (13.31)

A matched filter based on the target signature s(t) is applied to the modeledsignal, which is composed of weighted linear combination of the IMFs. The authorsattempted to use receiver operator characteristics (ROC) curves, HHT, and thematched filter to address the probability of detection. Zeris and Prinos [2004]compared the results of using HHT and wavelet transform for turbulent open channelflow data. Laser Doppler and hot films techniques were used to obtain signals inturbulent open channel flow, with and without the imposed bed suction. The authorswere fairly successful in the application. Unfortunately, it is not clear how the authorsdid the comparative analysis.

13.3.6 SEISMIC STUDIES

Huang et al. [2001] used HHT to develop a spectral representation of earthquakedata. The HHT offers a better representation of the earthquake signals in the lowfrequency range. The authors were able to address the shortcomings of variousmethods typically used in earthquake spectral analysis. For example, the use ofFourier-based methods and Duhanle integral assumes stationarity in the data; inaddition, the selection of an absolute maximum in the responses makes the HHTspectrum less dependent on the record length. Therefore, the design spectrum,defined as the envelope of a collection of response spectra, will inherit the short-comings of the response spectrum: the underrepresentation of the most critical lowfrequency range and its dangerous energy content. In the HHT approach, the fre-quency is instantaneous and is not determined by convolution. Loh et al. [2001]used EMD and HHT to analyze the free field ground motion and to estimate theglobal structural characteristics of building and bridge infrastructures. The processwas used to identify damage from the response data.

min ( )c

i

N

i ii

s t c≥

=

−∑01

α




Chen et al. [2002a] used HHT to determine the dispersion curves of seismicsurface waves and compared their results to Fourier-based time–frequency analysis.The surface-wave dispersion measurement is essential for the study of the crust andupper mantle structures, earthquake source mechanisms, and inelastic properties ofthe earth. The authors applied EMD to the seismogram and obtained the IMFs fromwhich the Hilbert spectrum is calculated. Then, for each frequency, the time of themaximum amplitude corresponds to the group arrival at which the group velocitycan be obtained. The result of group velocity and phase velocity determined by usingHHT is more accurate than by Fourier-based methods.

Shen et al. [2003] applied HHT to ground motion and compared the HHT resultwith the Fourier spectrum. The HHT is capable of identifying the occurrence andproperties of destruction near fault ground motions. The authors used the methodas a demonstration tool in the estimation of seismic loading in bridge design.Magrin-Chagnolleau and Baraniuk [undated] applied HHT to decompose a seismictrace into different oscillatory modes and were able to extract instantaneous frequen-cies of these modes. Chun-xiang and Qi-feng [2003] made a comparative study ofwavelet and HHT using earthquake record. The IMF directly decomposed from theoriginal data, whereas wavelet components are decomposed according to motherwavelets, which means the results can be influenced by the selected mother wavelets.The Hilbert spectrum clearly presents the energy distribution with time and fre-quency.

Zhang et al. [2003] examined the applicability of HHT to dynamic and earth-quake loading motion recordings. The IMF was capable of capturing low and highfrequency components. The authors commented how EMD can act like a filter bygrouping the IMFs. The authors defined EMD-based high frequency motions as thesummation of the first few IMF components, whereas the EMD-based low frequencymotion is the summation of the remaining components. Unfortunately, the authorsfailed to give the exact number of IMFs needed to be summed. This choice, high-lighted by the authors, is very subjective.

Wen and Gu [2004; Gu and Wen, 2004] extended the HHT and applied theirmethods to earthquake ground motions. The authors express x(t) as follows:

(13.32)

and introduced j, an independent random phase angle uniformly distributed between0 and 2:

. (13.33)

This makes x(t) a random process. The process then has the following mean, cova-riance, and variance functions:

x t Re a t r tj

n

ji t

nj( ) ( )exp ( )( )= +

=∑

1

θ

x t Re a t r tj

n

ji t

nj j( ) ( )exp ( )[ ( ) ]= +

=

+∑1

θ φ




, (13.34)

, (13.35)

. (13.36)

The authors extended the approach to a vector process and developed thecross-correlation covariance as follows:

. (13.37)

With the modified approach, the spectra of the simulated samples compare wellwith those of the record.

Zhang [2005] used a modified HHT to estimate the damping factor of nonlinearsoil and its role in seismic wave responses at soil sites, using data from earthquakerecordings. Zhang presented the following equation:

(13.38)

where j(t) can be interpreted as the source-related intensity, j(t) is exponential factorcharacterizing the time-dependent decay of waves in the jth IMF component due todamping, and

. (13.39)

The authors defined the instantaneous damping factor as:

. (13.40)

µ θ φx

j

n

ji it Re a t e t E ej j( ) ( ) ( ) ( )=

+

=∑

1

rr t r tn n( ) ( )=

K t t a t a t tXX

j

n

j j j j( ) ( ) ( )cos[ ( )1 2

1

1 2 112

, = −=

∑ θ θ (( )t2

σ µ2 2

1

212

x t E x t x t a tj

n

j( ) [ ( ) ( )] ( )= − =

=∑

x t Re a t r tk

j

n

jki t

nkjk jk( ) ( )exp ( )[ ( ) ]= +

=

+∑1

θ φ

x t Re a t x t Re tj

n

ji t

j

n

jj( ) ( )exp ( ) (( )= = =

= =∑ ∑

1 1

θ Λ ))exp[ ( ) ( )]− +ϕ θj jt i t

a t Re tj

j

n

jtj( ) ( )exp ( )=

=

−∑1

Λ ϕ

ηϕ

jjt

d t

dt( )

( )=




The Hilbert damping spectrum is defined as follows:

(13.41)

whereas the marginal Hilbert damping spectrum is:

. (13.42)

Equation 13.4 consists of two components, the time-dependent amplitude, whichis related to marginal and Hilbert amplitude spectra, and the source-related intensitytime-dependent amplitude. The new approach can be used to evaluate site dampingand to address the site amplification factor for the influences of soil nonlinearity inseismic analysis. The authors suggested that their approach must be validated bymodel-based simulation.

13.3.7 STRUCTURAL APPLICATIONS

Quek et al. [2003] illustrate the feasibility of the HHT as a signal process tool forlocating an anomaly in the form of a crack, delamination, or stiffness loss in beamsand plates based on physically acquired propagating wave signals. The authorsconsidered four different cases:

1. Experimental data of a damaged aluminum beam to estimate the damageand boundary conditions.

2. Delamination in a sandwiched aluminium plate.3. Healthy and damaged states of reinforced concrete (RC) slab based on

experimental response time history.4. Boundary and damage location of an aluminum square.

It appears that the authors were successful in all cases. Quek et al. [2004] dida comparative analysis of HHT, wavelet transform, and FFT in damage detection.Although the HHT was found to be a more direct method compared to the wavelettransform, the authors provide similar results in terms of precision and accuracy.The authors advocated the combination of EMD with Fourier transform on thedecomposed IMF components. They ascertained that this combination is capable ofrevealing modal frequencies with relatively low energy content.

Using HHT, Li et al. [2003] analyzed the results of a pseudodynamic test of tworectangular reinforced concrete bridge columns. The authors conducted extensivelaboratory investigation to model a near-fault ground acceleration of the Chi-ChiEarthquake. The HHT was used to analyze the signal output from various columnstested. The aim of these analyses was to understand the structural behaviors of the

D ta t

a t

t

tj

nj

j

j

j

( )( )

( )

( )

( )ω, = −

.+

.

=∑

1

ΛΛ

dt

d D t dta t

a t

tT

j

n Tj

j

j( ) ( )( )

( )

(ω ω= , = −

.+

.∫ ∑ ∫=

01

0

Λ ))

( )( ) ( )

ΛΛ

j

a

tdt d d

= +ω ω




columns under near-fault ground acceleration by observing the characteristics of theHilbert spectrum. The authors compared their results with Fourier transform results.The authors drew various conclusions, notably that HHT can be used to obtain theinstantaneous natural frequencies of the bridge columns and to understand therelationship between frequency changes and bridge conditions. Furthermore, theHHT method outperforms the FFT method. Finally, the authors noted during theexperimental investigation that the more wide spread the distribution of the Hilbertspectrum, the more severe the damage of the column.

13.3.8 HEALTH MONITORING

Pines and Salvino [2002] applied HHT in structural health monitoring. The authorsanalyzed a civil building model with and without the presence of damage. Theauthors also used the same model to simulate a seismic loading. The authors inves-tigated two case studies: one looked at ground floor damage and the other at topfloor damage. To correctly identify damage structure, the authors used the conceptof phase dereverberation and applied EMD to dereverberation time series. To addresscharacteristic frequency change as a result of damage, the authors suggest that thereverberated/dereverberated time domain response and the phase angles of bothdereverberated/reverberated signal can be used as indicators of damage.

Yang et al. [2004] used HHT for damage detection, applying EMD to extractdamage spikes due to sudden changes in structural stiffness. This enabled them toidentify damage time and locations. They also used the method to identify naturalfrequencies and damping ratios of the structure before and after damage. The authorsnoted that the capability of detecting damage spikes in a signal is a statistical variabledepending on the severity of damage and level of noise pollution. This implies thatwhen the severity of damage is low or the level of noise pollution is high, the currentmethod cannot detect the damage.

HHT was also used to detect cracks at rivet holes in thin plates by using Lambwaves [Osegueda et al., 2003]. The authors conducted experimental studies on aplate with a hole and with a notch. Area scanning was used to obtain time seriesinformation on the wave propagation characteristics. The location of the notch wasdetected accurately, and the extent of flaw was accurately predicted for all thenotches, except for the 3.2 mm long notch. The extent of flaw was overpredictedfor the shortest notch.

Yu et al. [2003] used HHT for fault diagnosis of roller bearings. The accelerativevibration signal of roller bearings was analyzed. A wavelet was used to decomposethe signal to decrease the influence of lower frequency noise, after EMD was appliedto the signals.

13.3.9 SYSTEM IDENTIFICATION

Chen and Xu [2002] explored the possibility of using HHT to identify the modaldamping ratios of a structure with closely spaced modal frequencies and comparedtheir results to FFT. The damping ratios determined by using the combined HHT




and random decrement technique (RDT) are much better than those resulting fromthe fast Fourier approach. In their studies, to identify instantaneous frequency anddamping ratio, the measured response time history of the MDOF system is decom-posed into IMFs. The authors checked the validity of model by applying it to a realcivil engineering structure. The results show that the FFT-based method fails toidentify modal damping ratios when two modal frequencies are too close.

Xu et al. [2003] compared the modal frequencies and damping ratios in varioustime increments and different winds for one of the tallest composite buildings in theworld. The results were then compared to FFT. Once again, the HHT methodoutperformed the FFT. The FFT-based results significantly overestimate the firstlateral and longitudinal modal damping ratios of the building.

Yang et al. [2002] developed a damage identification–based HHT. The authorsapplied the technique to the ASCE structural health monitoring benchmark structure.The approach was quite successful in identifying damage sections. Yang et al.[2003a] used HHT to identify multi-degree-of-freedom linear systems by usingmeasured free vibration time histories. The authors proposed band filter and EMDcases with polluted signals and high modal frequencies. Failure to introduce theband-pass filtering approach will result in a large number of siftings. The Fourierspectrum of the acceleration signal helps to identify the range of natural frequencies.The acceleration signal is processed through band-pass filters. The time historyobtained from each band-pass proceeds through EMD. A linear least-square fitprocedure is proposed to identify natural frequencies, damping ratios, mode shapes,mass matrix, damping matrix, and stiffness matrix. Yang et al. [2003b] extendedtheir work [Yang et al., 2003a] to identify general linear structures with complexmodes by using free vibration response data polluted by noise. Simulated resultsshow that the approach is capable of identifying the complex mode shapes as wellas the mass, stiffness, and damping matrices. This can help with the completedynamic characteristics of general linear structure.

13.4 SOME LIMITATIONS

Quek et al. [2003] noted the following:

• The approximate envelopes obtained using the local maxima and minimado not always encompass all the data.

• The use of discrete signals instead of continuous introduces numericaluncertainty in the true extrema.

The spline at the beginning and the end of the envelope can result in largevariation, which is very critical in a low frequency signal.

Deng et al. [2001] discussed the use of a cubic spline in calculating the lowerand upper envelopes. The authors mentioned the data extension method as key inaddressing this issue. They used a two-step extension method based on a neuralnetwork approach. The original signal was extended forward and backward, and aneural network was used to learn the envelopes.




Chen and Feng [undated] proposed a technique to improve the HHT procedure.The authors noted that the EMD has its limitation in distinguishing differentcomponents in narrow-band signals. The narrow band may contain either (a) com-ponents that have adjacent frequencies or (b) components where the frequenciesmay not be adjacent, but one of the components has a dominant energy intensitymuch higher than the other components. The improved technique is based on beat-ing-phenomenon waves.

Olhede and Walden [2004] compared HHT to a maximal-overlap discrete wave-let transform (MODWT) and wavelet packet transforms (MODWPT). The authorsreplaced EMD projections by a wavelet-based one and developed a Hilbert spectrumvia wavelet packets. The authors claimed the MODWPT and MODWT to be superiorto the HHT using a particular class of Fejer-Korovkin wavelet filters. It appears thattheir assertion may be true only for nonlinear data. Attoh-Okine [2004] did acomparative analysis of HHT, MODWPT, and MODWT on pavement profile dataand deduced that the HHT outperforms the MODWPT and MODWT in the case ofnonlinear and nonstationary data.

Datig and Schlurmann [2004] did the most comprehensive studies on the per-formance and limitations of HHT with particular applications to irregular waves.The authors did extensive investigation into the spline interpolation. The authorsdiscussed using additional points, both forward and backward, to determine betterenvelopes. They also did a parametric study on the proposed improvement andshowed significant improvement in the overall EMD computations. The authorsnoted that HHT is capable of differentiating between time-variant components fromany given data. Their study also showed that HHT was able to distinguish betweenriding and carrier waves.

13.5 POTENTIAL FUTURE RESEARCH

The HHT is becoming one of the most useful signal-processing techniques. The ideaof cubic spline needs a critical look, although it has been addressed by a few authors.Another area that needs further research is proper interpretation of all of the IMFs.This depends on a variety of factors, especially the proper understanding of thesystem. Researchers have to address what criteria should be used to eliminate IMFswithin a particular signal. Is the statistical analysis and distribution of the IMFs theanswer? What are proper approaches for using the HHT technique as a filter method?Comparative analyses of HHT, FFT, and wavelet techniques have been welladdressed, but more research is needed to investigate the MODWPT and MODWT,and which of these approaches is more applicable than the other. The use of com-putational intelligence has not been addressed in the overall HHT applications. Forexample, how can neural networks be used effectively in constructing the upper andlower envelopes? The use of HHT in image analysis and two-dimensional problemsalso needs to be explored.




REFERENCES

Attoh-Okine, N. O. (1999). Application of Wavelets in Pavement Profile Evaluation andAssessment. Proc. Estonian Acad. Science, 5, 53–63.

Attoh-Okine, N. O. (2004). Comparative Analysis of Hilbert-Huang Transform and MaximalOverlap Discrete Wavelet Packet Transform. Working paper, University of DelawareEngineering Department.

Chen, C. H., Li, C. P., and Teng, T. L. (2002a). Surface-Wave Dispersion MeasurementsUsing Hilbert-Huang Transform. TAO 13, 2:171–184.

Chen, J., and Xu, Y. L. (2002). Identification of Modal Damping Ratios of Structures withClosely Spaced Modal Frequencies. Struct. Eng. Mech. 14, 4:417–434.

Chen, Y., and Feng, M. Q. (Undated). A Technique to Improve the Empirical Mode ofDecomposition in the Hilbert-Huang Transform. p. 24.

Chun-xiang, S., and Qi-feng, L. (2003). Hilbert-Huang Transform and Wavelet Analysis ofTime History Signal. Acta Seismological Sinica 166, 4:422–429.

Datig, M., and Schlurmann, T. (2004). Performance and Limitation of the Hilbert-HuangTransformation with an Application to Irregular Water Waves. Ocean Eng. 31, 14–15:1783–1834.

Deng, Y., et al. (2001). Boundary-Processing Technique in EMD Method and Hilbert Trans-form. Chinese Science Bull. 46, 11:954–961.

Duffy, D. (2004). The Application of Hilbert-Huang Transform to Meteorological Datasets.J. Atmospheric Oceanic Technol. 21, 599–611.

Flandrin, P., Rilling, G., and Goncalves, P. (2003). Empirical Mode of Decomposition as aFilter Bank. IEEE Signal Process. Lett.

Gabor, D. (1946). Theory of Communication. Proc. IEE 93, 429–457.Gu, P., and Wen, Y. K. (2005). Simulation of Nonstationary Random Processes Using Instan-

taneous Frequency and Amplitude from Hilbert-Huang Transform. In this Volume.Huang, N. E., et al. (1999a). A New View of Nonlinear Water Waves: The Hilbert Spectrum.

Annu. Rev. Fluid Mech. 31, 417–57.Huang N. E., et al. (2003a). A Confidence Limit for the Empirical Mode Decomposition and

Hilbert Spectrum Analysis. Proc. R. Soc. London A, 459, 2317–2344.Huang, N. E., Chern, C. C., Huang, K., Salvino, L. W., Long, S. R., and Fan, K. L. (2001).

A New Spectral Representation of Earthquake Data: Hilbert Spectral Analysis ofStation TCU129, Chi-Chi Taiwan, 21 September 1999. Bull. Seismological Soc. Am.91, 5:1310–1338.

Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, E. H., Zheng, Q., Tung, C. C., and Liu,H. H. (1998). The Empirical Mode Decomposition Method and the Hilbert Spectrumfor Non-Stationary Time Series Analysis. Proc. R. Soc. London A 454, 903–995.

Huang, W., Shen, Z., Huang N. E., and Fung, Y. C. (1999b). Engineering Analysis of BiologicalVariables: An Example of Blood Pressure Over 1 Day. Proc. National Acad. Science95, 4816–4821.

Huang, W., Shen, Z., Huang N. E. and Fung, Y. C. (1999c). Nonlinear Indicial Response ofComplex Nonstationary Oscillations as Pulmonary Hypertension Responding to StepHypoxia. Proc. National Acad. Science 96, 1834–1839.

Huang, N. E., Wu, M.-L., Qu, W., Long, S. R., Shen, S. S. P., and Zhang, J. E. (2003b).Application of Hilbert-Huang Transform to Non-Stationary Financial Time SeriesAnalysis. Appl. Stochastic Models Bus. Industry 19, 245–268.

Hwang, P. A., Wang, D. W., and Kaihatu, J. M. (2005). A Comparison of the Energy FluxComputation of Shoaling Waves Using Hilbert and Wavelet Spectral Anlysis Tech-niques. In this Volume.




Lai, Y. (1998). Analytic Signals and the Transition to Chaos Deterministic Flows. Phys. Rev.E 58, 6:R6911–6914.

Larsen, M. L., Ridgway, J., Waldman, C. H., Gabbay, M., Buntzen, R. R., and Battista, B.(2005). Nonlinear Signal Processing of Underwater Electromagnetic Data. In thisVolume.

Li, Y. F., Chang, S. Y., Tzeng, W. C., and Huang, K. (2003). The Pseudo Dynamic Test ofRC Bridge Columns Analyzed Through the Hilbert-Huang Transform. Chin. J. Mech.A 19, 3: 373–387.

Loh, C.-H., Wu, T. C., and Huang, N. E. (2001). Application of the EMD-Hilbert Spectrumto Identify Near-Fault Ground Motion Characteristics and Structural Responses. Bull.Seismological Soc. Am. 191, 5: 1339–1353.

Long, S. R., Huang, N. E., Tung, C. C., Wu, M. L., Lin, R. Q., et al. (1995). The HilbertTechniques: An Alternative Approach for Non-Steady Time Series Analysis. IEEEGeoscience Remote Sensing Soc. Lett. 3, 3–11.

Lundquist, J. K. (2003). Intermittent and Elliptical Inertia Oscillation in Atmospheric Bound-ary Layer. J. Am. Meteorological Soc. 60, 2261–2273.

Magrin-Chagnolleau, I., and Baraniuk, R. G. (undated). Empirical Mode DecompositionBased Time Frequency Attribute. pp 4.

Montesinos, M. E., Munoz-Cobo, J. L., and Perez, C. (2002). Hilbert-Huang Analysis ofBWR Neutron Detector Signals: Application to DR Calculation and to CorruptedSignal Analysis. Ann. Nuclear Energy 30, 715–727.

Neithammer, M., et al. (2001). Time–Frequency Representations of Lamb Waves. J. AcousticalSoc. Am. 109, 5: Pt. 1, 1841–1847.

Olhede, S., and Walden, A. T. (2004). The Hilbert Spectrum via Wavelet Projections. Proc.R. Society A 460, 2044: 955–975.

Osegueda, R., Kreinovich, V., Nazarian, S., and Roldan, E. (2003). Detection of Cracks atRivet Holes in Thin Plates Using Lamb-Wave Scanning. Proc. SPIE 5047, 55–66.

Pan, J., Yan, X. H., Zheng, Q., Liu, W. T., and Klemas, V. V. (2002). Interpretation ofScatterometer Ocean Surface Wind Vector EOFs over the Northwestern Pacific.Remote Sensing Environ. 84, 53–68.

Phillips, S. C., Gledhill, R. J., and Essex, J. W. (2003). Applications of the Hilbert-HuangTransform to the Analysis of Molecular Dynamics Simulations. J. Phys. Chem. A107, 4869–4876.

Pines, D., and Salvino, L. (2002). Health Monitoring of One-Dimensional Structures UsingEmpirical Mode Decomposition and the Hilbert-Huang Transform. Proc. SPIE 4701,127–143.

Qu, J., and Jarzynski, J. (2001). Time Frequency Representation of Lamb Waves. J. Acoust.Soc. Am., 109(5) Part 1, 1841–1847.

Quek, S. T., Tua, P. S., and Wang, Q. (2003). Detecting Anomalies in Beams and Plate Basedon the Hilbert- Huang Transform of Real Signals. Smart Mater. Structures 12,447–460.

Quek, S. T., Tua, P. S., and Wang, Q. (2004). Comparison of Hilbert-Huang, Wavelet, andFourier Transforms for Selected Applications. In this Volume.

Salisbury, J. I., and Wimbush, M. (2002). Using Modern Time Series Analysis Techniques toPredict ENSO Events from SOI Time Series. Nonlinear Processes in Geophysics 9,341–345.

Schlurmann, T. (2002). Spectral Analysis of Nonlinear Water Waves Based on the Hil-bert-Huang Transformation. Trans. ASME 124, 22–27.

Schlurmann, T., and Datig, M. (2005). Carrier and Riding Wave Structure of Rogue Waves.In this Volume.




Shen, J. J., Yen, W. P., and Fallon, J. O. (2003). Interpretation and Application of Hilbert-HuangTransformation for Seismic Performance Analyses. Advanced Mitigation Technolo-gies 657–666.

Veltcheva, A. (2005). An Application of HHT Method to the Nearshore Sea Waves. In thisVolume.

Veltcheva, A. D. (2002). Wave and Group Transformation by a Hilbert Spectrum. CoastalEng. J. 44, 4: 283–300.

Wang, W. (2005). Decomposition of Wave Groups with EMD Method. In this Volume.Wen, Y. K., and Gu, P. (2004). Description and Simulation of Nonstationary Processes Based

in Hilbert Spectra. J. of Eng. Mech. 130, 8: 942–951.Wiley, A. P., Gledhill, R. J., Phillips, S. C., Swain, M. T., Edge, C. M., and Essex, J. W.

(2005). The Analysis of Molecular Dynamics Simulations by the Hilbert-HuangTransform. In this Volume.

Xiang, S. C., and Feng, L. Q. (2003). Hilbert-Huang Transform and Wavelet Analysis of TimeHistory Signal. Acta Seismological Sinica 16, 4: 422–429.

Xu, Y. L., Chen, S. W., and Zhang, R. C. (2003). Modal Identification of Di Wang BuildingUnder Typhoon York Using the Hilbert-Huang Transform Method. The StructuralDesign of Tall and Special Buildings 12, 21–47.

Yang, J. N., Lei, Y., Pan, S., and Huang, N. (2003a). System Identification of Linear StructuresBased on Hilbert-Huang Spectral Analysis. Part 1: Normal Modes. Earthquake Eng.Structural Dynamics 32: 1443–1467.

Yang, J. N., Lei, Y., Pan, S., and Huang, N. (2003b). System Identification of Linear StructuresBased on Hilbert-Huang Spectral Analysis. Part 1: Complex Mode. Earthquake Eng.Structural Dynamics 32: 1533–1554.

Yang, J. N., Lei, Y., Lin, S., and Huang, N. (2004). Hilbert-Huang Based Approach forStructural Damage Detection. J. Eng. Mech. 130, 1: 85–95.

Yang, J. N., Lin, S., and Pan, S. (2002). Damage Identification of Structures Using Hil-bert-Huang Spectral Analysis. 15th ASCE Eng. Mech. Conf., New York.

Yu, D., Cheng, J., and Yang, Y. (2003). Application of EMD Method and Hilbert Spectrumto the Fault Diagnosis of Roller Bearings. Mechanical Syst. Signal Process. 19, 2:259–270.

Zeris, A., and Prinos, P. (2005). Coherent Structures Analysis in Turbulent Open-ChannelFlow Using Huang-Hilbert and Wavelets Transforms. In this Volume.

Zhang., R. R. (2004). An HHT Based Approach to Quantify Nonlinear Soil Amplification.In this Volume.

Zhang, R. R., Demark, L. V., Liang, J., and Hu, Y. (2004). On Estimating Site Damping withSoil Non-Linearity from Earthquake Recordings. Int. J. Non-Linear Mech. 39,1501–1517.

Zhang, R. R., Ma, S., Safak, E., and Hartzell, S. (2003). Hilbert-Huang Transform Analysisof Dynamic and Earthquake Motion Recordings. J. Eng. Mech. 129, 8: 861–875.




ADDENDUM: PERSPECTIVES ON THE THEORY AND PRACTICES OF THE HILBERT-HUANG TRANSFORM

Recently there have been innovative applications and major improvement to EMD-HHT. These improvements have spread to general analytical approaches, hybridapplications, and bidimensional HHT.

13A.1 ANALYTICAL

Coughlin and Tung [2004] used EMD to extract the solar cycle signal from strato-spheric data. The authors highlighted some difficulties in using EMD — especiallythe influence of the end point in the sifting process. They addressed a method ofreducing the influence of the end effects on the internal solution. Coughlin and Tung[2004] extended both the beginning and end of the spline by introducing a waveequation of the form:

(13A.1)

The typical amplitude, A, and period p, are determined by the nearest local extrema.

(13A.2)

where max(1) and min(1) are the first two local extrema in the time series andmax(N) and min(N) are the last two local extrema. This approach reduces the largeswings in the spline calculation. This approach is quite different from the oneproposed by Dätig and Schlurmman [2004].

13A.2 HYBRID METHOD

Iyengar and Kanth [2004] used HHT to decompose India monsoon data. The authorsargued that the first IMF is nonlinear and the remaining IMFs are linear. This is alittle questionable, since it is not clear how the authors determined these properties.The authors used a neural network to forecast the nonlinear part of the data. Lee etal. [2003] explored the vibration mechanism of a bridge at both global and localbehavior. They proposed a combination of EMD, HHT, and a nonlinear parametricmodel as an identification methodology. The approach was successful in identifyingabnormal signals indicating the structural condition of the bridge decks based onthe time–frequency spectrum.

Wave Extension = + phase + local mAt

psin

2πeean

A

A N N

p

beginning

end

= ( ) − ( )= ( ) − ( )

max min

max min

1 1

bbeginning

end

time time

ti

= ( ) − ( )=

2 1 1

2

max min

p mme timemax minN N( ) − ( )




Loutridis [2004] used EMD to decompose vibration signals from gear systems.The author developed an empirical law that relates the energy of the intrinsic modesto the crack magnitude of the gear. Using this technique, the degree of crack can bepredicted based on energy obtained from the IMFs.

13A.3 BIDIMENSIONAL EMD

Nunes and coworkers developed a method for analyzing two-dimensional (2D) timeseries which was used for textural analysis [2003a]. The bidimensional siftingprocess they presented in an earlier paper [Nunes et al., 2003] is as follows:

• Identify the extrema of the image I by morphological reconstruction basedon geodesic operators.

• Generate the 2D “envelope” by connecting the maxima points with aradial basis function (RBF).

• Determine the local mean m_i, by averaging the two envelopes.• Subtract I – m_i = h_i• Repeat the process.

The authors used a radial basis function in the form:

(13A.3)

radial basis function (RBF);pm is mth degree polynomial in d variables;λ are RBF coefficients;xi are the RBF centers.

The stopping criteria for Nunes et al. [2003] approach is based on similar to 1Dcase used, using the standard deviation.

Linderhed [2002] developed a 2D EMD and presented a coding scheme forimage compression purposes. In the coding scheme, only the maximum and mini-mum values of the IMFs are used. Linderhed [2004] also developed the concept ofempiquency, short for empirical mode frequency as a frequency measure.Empiquency is defined as one half the reciprocal distance between two consecutiveextrema points. Linderhed [2004] developed a sifting process for 2D time series.The symmetry criterion on the IMF envelope is relaxed, and the stop criterion isbased on the condition that the IMF envelope is close to zero. Hence, there is noneed to check for symmetry. The new sifting process is as follows:

1. Find the positions and amplitudes of all local maxima and minima in theinput signal in_lk(m,n), where m and n denote spatial dimensions of theimage.

2. Create upper and lower envelopes by spline interpolation, denoted bye_max(m,n) and e_min(m,n).

s x p x x xm i i

i

N

( ) = ( ) + −( )=∑λ Φ

1




3. For each position, calculate the mean of the upper and lower envelopes:

(13A.4)

4. Subtract the envelope mean signal from the input signal:

(13A.5)

The process is terminated if the envelope mean signal is close to zero:

(13A.6)

for all (m,n). If it is not, the process repeated until:

(13A.7)

When the termination criterion is achieved, k = K, and the IMF is definedas:

(13A.8)

5. The residue is defined as r_l(m,n):

(13A.9)

6. The next IMF is found as follows:

(13A.10)

7. Just as the 1D series the signal can be represented as:

(13A.11)

Liu and Peng [2005] proposed a boundary-processing technique for bidimen-sional EMD. The authors proposed two techniques, but unfortunately the two algo-rithms developed by these authors are similar to algorithms proposed by Nunes etal. [2003] and Linderhead [2004].

em m ne m n e m n

lk ,, ,max min( ) =

( ) + ( )2

h m n in m n em m nlk lk lk, , ,( ) = ( ) − ( )

em m nlk ,( ) ≺ ε

h m n h m nl k lk+( ) ( ) = ( )1 , ,

c m n h m nl lK, ,( ) = ( )

r m n in m n c m nl l l, , ,( ) = ( ) = ( )1

in m n r m nl l+( ) ( ) = ( )1 1 , ,

x m n r m n c m nL j

i

L

, , ,( ) = ( ) + ( )=∑

1




13A.4 SOME OBSERVATIONS

The use of bidimensional EMD-HHT is beginning to emerge. Few results indicatethat the 2D HHT outperforms traditional wavelet and Fourier analysis. As with the1D HHT, there are difficulties regarding the type of spline and the effect of the endpoints on the overall sifting process. The use of different RBFs, such as thin platesplines, cubic spline, and multiquadrics, will be worth exploring in the bidimensionalHHT.

REFERENCES

Coughlin, K. T. and Tung, K. K. (2004). 11-Year Solar Cycle in the Stratosphere Extractedby the Empirical Mode Decomposition Method. Adv. Space Res. 34, 323–329.

Dätig, M. and Schlurmann, T. (2004). Performance and Limitations of the Hilbert-HuangTransformation with an Application to Irregular Water Waves. Ocean Eng. 31, 14/15:1783–1834.

Iyengar, R. N. and Kanth R. S. T. G (2004). Intrinsic Mode Functions and a Strategy forForecasting Indian Monsoon Rainfall. Meteorol. Atmospheric Phys. (online publica-tion).

Lee, Z. K., Wu, T. H. and Loh, C. H. (2003). System Identification on Seismic Behavior ofan Isolated Bridge. Earthquake Eng. Structural Dynamics 32, 1797–1812.

Linderhed, A. (2003). 2-D Empirical Mode Decomposition: In Spirit of Image Compression.SPIE Proc. 4738, 1–8.

Linderhed, A. (2003). Image Compression Based on Empirical Mode Decomposition. Proc.SSBA Symp. Image Anal. 110–113.

Linderhed, A. (2004). Adaptive Image Compression with Wavelet Packets and EmpiricalMode Decomposition. Dissertation No. 909 submitted to Linkoping Studies in Scienceand Technology.

Liu, Z. and Peng, S. (2005). Boundary Processing of Bidimensional EMD Using TextureSynthesis. IEEE Signal Process. Lett. 12, 1: 33–36.

Loutridis, S. J. (2004). Damage Detection in Gear Systems Using Empirical Mode Decom-position. Eng. Structures 26, 1833–1841.

Nunes, J. C., Bouaoune, Y., Delechelle, E., Niang, O. and Bunel P. (2003). Image Analysisby Bidimensional Empirical Mode Decomposition. Image Vision Comput. 21,1019–1026.

Nunes, J. C., Niang, O., Bouaoune, Y., Delechelle, E., and Bunel P. (2003a). BidimensionalEmpirical Mode Decomposition Modified for Texture Analysis. SCIA 2003, LectureNotes on Computer Science 2749, 171–177.


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The Hilbert-Huang Transform in Engineering