Complex Spectral Decomposition via Inversion Strategies

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Complex spectral decomposition via inversion strategiesDavid C. Bonar and Mauricio D. Sacchi, University of Alberta

SUMMARY

Spectral decomposition is a time-frequency analysis tool

widely used in seismic data interpretation. Unlike conven-

tional frequency analysis, such as the Fourier transform, spec-

tral decomposition estimates the frequency content of a sig-

nal at any particular time. Thus, the frequency content is de-

fined on a local, not global, scale. An alternative method for

spectral decomposition is proposed in which the seismic signal

is deconvolved with a dictionary of different frequency com-

plex Ricker wavelets. By posing the underdetermined problem

through a mixed 2 - 1 norm cost function, greater resolution

is obtained in the time-frequency map as the frequency distri-

bution is constrained to be sparse. Examples on synthetic data

are presented to illustrate the proposed method for two differ-

ent mixed 2 - 1 norm solving algorithms.

INTRODUCTION

The frequency content of a time series, such as a seismic sig-

nal, is commonly found using the Fourier transform. However,

the frequency information obtained from this method relates to

the entire time series and contains no information about how it

varies locally. Knowledge of how the frequency content of a

signal varies in time can be significant, as illustrated by a mu-

sical score which denotes particular frequencies (i.e. notes) to

be played at specific times. The local variations of frequency

in a signal can be acquired through time-frequency analysis,

also known as spectral decomposition, which decomposes a

one dimensional signal into a two dimensional space (i.e. timeand frequency). Resolving how the frequency content of a sig-

nal changes with time has been studied in a large range of

fields such as geophysics, quantum mechanics, engineering,

sound analysis and speech recognition, radar, and musicogra-

phy (Gardner and Magnasco, 2006).

In seismic interpretation, spectral decomposition has been em-

ployed to study aspects such as the reservoir characterization

of thin beds (Partyka et al., 1999) and the low frequency shad-

ows associated with hydrocarbons (Castagna et al., 2003). This

type of analysis has commonly been performed using short

windowdiscrete Fourier transforms(Partyka et al., 1999) where

the reflectivity sequence, which represents the geological se-

quence, can no longer be considered white, or completely ran-dom. Therefore, the spectral attributes seen in the amplitude

spectrum of the short-window Fourier transform are a com-

bination of the amplitude spectrums of the seismic wavelet

and reflectivity sequence. It is this interference attribute of

the wavelet and reflectivity amplitude spectra that has been ex-

ploited in earlier time-frequency analysis exercises. Various

other approaches have also been developed for time-frequency

analysis such as the continuous wavelet transform (Sinha et al.,

2005), S-transform (Stockwell et al., 1996), Wigner-Ville dis-

tribution (Wu and Liu, 2009), and matching pursuit algorithms

(Mallat and Zhang, 1993), (Chakraborty and Okaya, 1995).

The alternative proposed method takes a different approach to

the spectral decomposition problem by stating it as an inverseproblem, similar to Portniaguine and Castagna (2004). A dic-

tionary of different frequency complex Ricker wavelets is de-

convolved from the seismic trace, producing a time-frequency

map. By using an 1 norm as the regularization parameter in

this underdetermined problem, the time-frequency map is con-

strained to provide a sparser solution that has higher resolution

as seen in Figure 1. Additional information about the seismic

signal may also be acquired through the phase of the complex

time-frequency map.

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Time(s)

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Figure 1: Synthetic trace spectrally decomposed using2 - 2

norms and 2 - 1 norms (left to right: synthetic trace,2 least

squares solution, IRLS solution, FISTA solution)

THEORY

Complex spectral decompositionThe convolutional model states that a seismic trace,s, is com-

posed of the convolution of a wavelet, w, with the reflectivity

sequence,r, of the Earth as seen in Equation 1.

s = wr (1)

Equation 1 can also be represented as the linear system of

equations, s = Wr, whereW represents the convolutional ma-

trix of the wavelet. As developed in Bonar et al. (2010), this

14EG Denver 2 1 Annual Meeting 2 1 SEG


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Complex spectral decomposition

methodology can be expanded to study how multiple wavelets

can be represented in the seismic trace. Using a dictionary of

different Ricker wavelets that are uniquely defined by a cen-

tral frequency, the seismic trace can be decomposed into its

different frequency components at a specific time through the

employment of deconvolution. IfNdifferent frequency Ricker

wavelets comprise the Ricker wavelet dictionary, the seismic

trace can be constructed from its frequency components by

s =`

W1 W2 . . . WN0BBB@

r0r1...

rN

1CCCA= Dm (2)

where Wi and ri refer to the wavelet convolution matrix for

the Ricker wavelet with central frequency fi, and its associ-

ated reflectivity sequence. The matrixD represents the dictio-

nary of Ricker wavelets and, finally,mis used to represent the

vector containing all the pseudo-reflectivity sequences. This

procedure can be accomplished in the Fourier domain to in-

crease computational efficiency. Essentially, the seismic sig-nal is decomposed into several traces that are uniquely deter-

mined from a singular Ricker wavelet and its related reflec-

tivity structure. The time-frequency analysis can then be ob-

tained with the prior knowledge of the frequency content of

the individual Ricker wavelets. Thus, the seismic signal can

be transformed into a time-frequency map by deconvolving a

Ricker wavelet dictionary from the seismic signal to obtain a

pseudo-frequency dependent reflectivity structure that can be

portrayed as a time-frequency map.

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Time(s)

Frequency (Hz)20 40 60 80 100

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Figure 2: Comparison of using a real Ricker wavelet dictionary

(center) to a complex Ricker wavelet dictionary (amplitude of

frequency displayed) (right)

Real

Imaginary

Envelope

Real

Imaginary

Envelope

Figure 3: Complex Ricker Wavelets (Top: positive frequencies

retained; Bottom: negative frequencies retained)

The Ricker wavelets of the aforementioned dictionary can be

replaced by complex Ricker wavelets to force the time-frequency

map to be complex as well. The benefits of a complex time-

frequency map are that it can display both amplitude and phase

information about the frequency content, whereas a real time-

frequency map suffers from the influence of the polarity ofthe original seismic trace. This effect is portrayed in Figure 2

where the example trace is decomposed using a real and com-

plex Ricker wavelet dictionary. However, care must be taken

when constructing the complex Ricker wavelet dictionary to

ensure full frequency coverage.

A complex Ricker wavelet is constructed by taking the Hilbert

transform of the real Ricker wavelet, which essentially takes

the Fourier transform of a signal, removes the negative fre-

quency portion, and then converts the signal back into its orig-

inal domain. If the entire complex Ricker wavelet dictionary

was constructed in this manner, negative frequency compo-

nents of the signal could not be accounted for as the dictio-

nary contains no negative frequency components. One couldalso create a complex Ricker wavelet using this concept with

maintaining only negative frequencies, not positive ones. The

complex Ricker wavelet obtained from these two methods are

displayed in Figure 3 and only differ in the polarity of the

imaginary part. When summed together, the imaginary parts of

these wavelets will cancel each other. To ensure that the com-

plex Ricker wavelet dictionary contains both positive and neg-

ative frequencies, it must be constructed using pairs of com-

plex Ricker wavelets that maintain the positive or negative fre-

quencies.



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Applying sparsity constraints for higher resolution

Resolution in the time-frequency map can be increased by ap-

plying an 1 norm regularization term to the inverse problem.

The 1 norm promotes a sparser solution when compared to the

2 norm. By imposing that the solution, or pseudo-frequency

reflectivity structure, be sparse, a minimal amount of Ricker

wavelets will be required to represent the seismic signal. This

in turn will make the pseudo-frequency reflectivity structureand thus, the time-frequency map, have a greater resolution.

The inverse problem can then be posed by the cost function

J, as seen in Equation 3, where the 2 norm is applied to fit

the data and the 1 norm is applied to promote sparsity in the

solution and is controlled by the trade-off parameter .

J=||sRm||2 +||m||1 (3)

The non-quadratic nature of the 1 norm causes the minimiza-

tion of the cost function J to become computationally more

expensive than if an 2 norm was used. The mixed 2 - 1

problem was solved using two different techniques, the iter-

atively reweighted least squares (IRLS) method (Scales andGersztenkorn, 1988) and the fast iterative soft thresholding al-

gorithm (FISTA) (Beck and Teboulle, 2009). The IRLS algo-

rithm solves the cost function Jin a similar way that the regu-

larized least squares method solves the 2 - 2 norm problem

and can be described by the iterative nature of Equation 4,

mk+1=

DTD+Q1

RDTs (4)

where the matrix Q implicitly depends upon the original model

mk. A conjugate gradient algorithm was employed to solve the

inversions required in this method to improve computational

efficiency. The FISTA algorithm is based upon the idea of it-

eratively solving simpler cost functions that are always greaterthan or equal to the original complicated cost function using

a soft-thresholding operator. This algorithm is described by

Equation 5, where is a constant that must be greater than or

equal to the maximum eigenvalue ofDTD, and hkis a clever

update of the model estimate mkreducing computational time.

mk+1= sof t

hk+

1

DT (sDhk) ,

2

(5)

The soft-thresholding operator,sof t , is defined by Equation 6

for complex numbers.

s oft

Aei,=((A)ei ifA >

0 ifA .(6)

EXAMPLES

The example trace used in Figures 1 and 2 was contaminated

with 15% noise and subsequently spectrally decomposed. As

seen in Figure 4, the recovered time-frequency map of the

FISTA solution is less prone to recover artifacts in frequency

content caused by the addition of noise for a solution recovered

with a comparable computational time from IRLS. However,

both solutions recover the frequency content of the example

trace quite favourably while exhibiting the ability to attenuate

noise.

By using a complex Ricker wavelet dictionary, the proposed

spectral decomposition method can also be applied to complex

signals as illustrated in Figure 5. For a real signal, the imag-inary components of all of the complex Ricker wavelets used

to represent the signal must sum together to zero. This means

that the time-frequency map is symmetric about the zero fre-

quency line. When the signal contains both real and imaginary

components, this symmetry is broken as an imaginary compo-

nent needs to be modelled. Similar to the Fourier transform,

only half of the frequencies need to be displayed for real sig-

nals while all of the frequencies are required to be displayed

for complex signals. A possible application for the inclusion

of a complex signal is spectrally decomposing the vertical and

horizontal components of a seismogram together.

CONCLUSION

An inversion method incorporating complex Ricker wavelets

was proposed for the spectral decomposition problem. The

inclusion of complex wavelets removed seismic amplitude po-

larity effects on the time-frequency map witnessed when the

method was applied with real wavelets. Sparsity constraints

were imposed to increase the resolution of the time-frequency

map and applied using the IRLS and FISTA algorithms, pro-

ducing similar results. The sparsity promoting algorithms were

also able to adequately acquire the frequency content of signals

contaminated with random noise and provide a possible tool

for noise attenuation. By generalizing the proposed spectral

decomposition method to include complex wavelets, signals

of both a real and complex nature are able to be modelled.

ACKNOWLEDGEMENTS

We would like to thank NSERC for funding this research as

well as the members of SAIG (Signal Analysis and Imaging

Group) at the University of Alberta for their continued support.



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Figure 4: Results of complex spectral decomposition on a

noisy signal (top row left to right: noisy signal, IRLS solu-

tion, FISTA solution; bottom row left to right: noiseless signal,

IRLS reconstructed signal, FISTA reconstructed signal)

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Figure 5: Comparison of complex spectral decomposition for a

real and complex trace (from left to right: real trace, IRLS so-

lution, FISTA solution, complex trace, IRLS solution, FISTA

solution).



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EDITED REFERENCES

Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2010

SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata foreach paper will achieve a high degree of linking to cited sources that appear on the Web.

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Bonar, D., M. Sacchi, H. Cao, and X. Li, 2010, Time-frequency analysis via deconvolution with sparsityconstraints: Geo-Canada2010.

Castagna, J., S. Sun, and R. Siegfried, 2003, Instantaneous spectral analysis: Detection of low-frequencyshadows associated with hydrocarbons: The Leading Edge, 22, no. 2, 120, doi:10.1190/1.1559038.

Chakraborty, A., and D. Okaya, 1995, Frequency-time decomposition of seismic data using wavelet-based methods: Geophysics, 60, 19061916, doi:10.1190/1.1443922.

Gardner, T. J., and M. O. Magnasco, 2006, Sparse time-frequency representations: Proceedings of theNational Academy of Sciences of the United States of America, 103, no. 16, 60946099,doi:10.1073/pnas.0601707103.PubMed

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Portniaguine, O., and J. Castagna, 2004, Inverse spectral decomposition: SEG Expanded Abstracts.

Scales, J., and A. Gersztenkorn, 1988, Robust methods in inverse theory: Inverse Problems, 4, no. 4,10711091, doi:10.1088/0266-5611/4/4/010.

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Complex Spectral Decomposition via Inversion Strategies