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NATO ASI Series Advanced Science Institutes Series
A series presenting the results of activities sponsored by the NA
TO Science Committee, which aims at the dissemination of advanced
scientific and technological knowledge, with a view to
strengthening links between scientific communities.
The Series is published by an international board of publishers in
conjunction with the NATO Scientific Affairs Division
A Life Sciences B Physics
C Mathematical and Physical Sciences
o Behavioural and Social Sciences
E Applied Sciences
G Ecological Sciences H Cell Biology
Plenum Publishing Corporation London and New York
Kluwer Academic Publishers Dordrecht, Boston and London
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
The ASI Series Books Published as a Result of Activities of the
Special Programme on SENSORY SYSTEMS FOR ROBOTIC CONTROL
This book contains the proceedings of a NATO Advanced Research
Workshop held within the activities of the NATO Special Programme
on Sensory Systems for Robotic Control, running from 1983 to 1988
under the auspices of the NATO Science Committee.
The books published so far as a result of the activities of the
Special Programme are:
Vol. F25: Pyramidal Systems for Computer Vision. Edited by V.
Cantoni and S. Levialdi. 1986.
Vol. F29: Languages for Sensor-Based Control in Robotics. Edited by
U. Rembold and K. Hormann. 1987.
Vol. F 33: Machine Intelligence and Knowledge Engineering for
Robotic Applications. Edited by A.K.C. Wong and A. Pugh.
1987.
Vol. F42: Real-Time Object Measurement and Classification. Edited
by A. K. Jain. 1988.
Vol. F43: Sensors and Sensory Systems for Advanced Robots. Edited
by P. Dario. 1988.
Vol. F44: Signal Processing and Pattern Recognition in
Nondestructive Evaluation of Materials. Edited by C. H. Chen.
1988.
Series F: Computer and Systems Sciences Vol. 44
Signal Processing and Pattern Recognition in Nondestructive
Evaluation of Materials
Edited by
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Published in cooperation with NATO Scientific Affairs
Division
Proceedings of the NATO Advanced Research Workshop on Signal
Processing and Pattern Recognition in Nondestructive Evaluation of
Materials, held at the Manoir St-Castin, Lac Beauport, Quebec,
Canada, August 19-22, 1987.
ISBN-13:978-3-642-83424-0 e-ISBN-13:978-3-642-83422-6 001:
10.1007/978-3-642-83422-6
Library of Congress Cataloging-in-Publication Data. NATO Advanced
Research Workshop on Signal Processing and Pattern Recognition in
Nondestructive Evaluation of Materials (1987: Saint-Dunstan-du
Lac-Beauport, Quebec) Signal processing and pattern recognition in
nondestructive evaluation of materials 1 edited by C. H. Chen. p.
cm.-(NATO ASI series. Series F., Computer and systems sciences;
vol. 44) "Proceedings of the NATO Advanced Research Workshop on
Signal Processing and Pattern Recognition in Nondestructive
Evaluation of Materials, held at the Manoir St-Castin, Lac
Beauport, Quebec, Canada, August 19-22, 1987"-"Published in
cooperation with NATO Scientific Affairs Division." ISBN-i3:
978-3-642-83424-0 (U.S.) 1. Non-destructive testing-Congresses. 2.
Signal processing-Congresses. 3. Pattern perception-Con gresses.
I. Chen, C. H. (Chi-hau), 1937- II. North Atlantic Treaty
Organization. Scientific Affairs Division. III. Title. IV. Series:
NATO ASI series. Series F, Computer and system sciences; vol. 44.
TA417.2.N371987 620.1·127-dc 19
This work is subject to copyright. All rights are reserved, whether
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1985, and a copyright fee must always be paid. Violations fall
under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1988 Soitcover reprint of the
hardcover 1st edition 1988
2145/3140-543210
Preface
The NATO Advanced Research Workshop on Signal Processing and
Pattern Recognition in Nondestructive Evaluation (NOE) of Materials
was held August 19-22, 1987 at the Manoir St-Castin, Lac Beauport,
Quebec, Canada.
Modern signal processing, pattern recognition and artificial
intelligence have been playing an increasingly important role in
improving nondestructive evaluation and testing techniques. The
cross fertilization of the two major areas can lead to major
advances in NOE as well as presenting a new research area in signal
processing. With this in mind, the Workshop provided a good review
of progress and comparison of potential techniques, as well as
constructive discussions and suggestions for effective use of
modern signal processing to improve flaw detection, classification
and prediction, as well as material characterization.
This Proceedings volume includes most presentations given at the
Workshop. This publication, like the meeting itself, is unique in
the sense that it provides extensive interactions among the
interrelated areas of NOE. The book starts with research advances
on inverse problems and then covers different aspects of digital
waveform processing in NOE and eddy current signal analysis. These
are followed by four papers of pattern recognition and AI in NOE,
and five papers of image processing and reconstruction in NOE. The
last two papers deal with parameter estimation problems. Though the
list of papers is not extensive, as the field of NOE signal
processing is very new, the book has an excellent collection of
both tutorial and research papers in this exciting new field. While
most signal processing work has not yet been integrated into
practical NOE systems, as pointed out by Dr. L. J. Bond at the
Workshop discussion session, the future direction clearly shows
greatly increased use of signal processing in NOE.
I am grateful to all participants for their active participation
that made the Workshop very productive, and to NATO Scientific
Affairs Division for support. The Workshop format is indeed ideal
for a research meeting like this that brings together an
interdisciplinary group of researchers. I am confident that this
publication can be equally successful in helping to foster
continued research interest in NOE signal processing.
C.H. Chen Workshop Director
Group photo of some Workshop participants at the front entrance of
Manoir St-Castin, Lac Beauport, Quebec, on August 22, 1987.
Table of Contents
Preface by C.H. Chen
RESEARCH ON INVERSE PROBLEMS
1. S.J. Norton, J.A. SiDlllOns, A.H. Kahn and H.N.G. Wadley,
"Research inverse problems in materials science and
engineering"---------------l
2. L.J. Bond, J.H. Rose, S.J. Wormley and S.P. Neal, "Advances in
Born
inversion"----------------------------------------------------------23
DIGITAL WAVEFORM PRDCESSING IN NDE
3. S. Haykin, "Modern signal
processing"-------------------------------39
4. V.L. Newhouse, G.Y. Yu and Y. Li, "A split spectrum processing
method of scatterer density
estimation"-----------------------------49
5. N.M. Bilgutay. J. Saniie and U. Bencharit, "Spectral and spatial
processing techniques for improved ultrasonic imaging of
materials"----------------------------------------------------------71
6. J. Saniie, N.M. Bilgutay and T. Wang, "Signal processing of
ultrasonic ba.ckscattered echoes for evaluating the microstructure
of materials - a
review"--------------------------------------------87
7. C.A. Zala, I. Barrodale and K.I. McRae, "High resolution decon
volution of ultrasonic
traces"-------------------------------------101
8. P. Flandrin, "Nondestructive evaluation in the time-frequency
domain by means of the Wigner-Ville
distribution"------------------109
9. D. Kishoni, "Pulse shaping and extraction of information from
ultrasonic reflections in composite
materials"---------------------117
EDDY CURRENT SIGNAL ANALYSIS
10. S.S. Udpa, "Signal processing for eddy current nondestructive
evaluation"--------------------------------------------------------129
11. L.D. Sabbagh and H.A. Sabbagh, "Eddy current modeling and
signal processing in
NDE"-------------------------------------------------145
VIII
PATTERN RECOGNITION AND AI IN NDE
12. C.H. Chen, "High resolution spectral analysis NDE techniques
for flaw characterization prediction and
discrimination"---------------155
13. R.W.Y. Chan, D.R. Hay, J .R. Matthews and H.A. MacDonald,
"Automated ultrasonic system for sulxnarine pressure hull
inspection"----------175
14. V. Lacasse, J.R. Hay and D.R. Hay, "Pattern recognition of
ultrasonic signals for detection of wall
thinning"----------------------------189
15. R.B. Melton, "Knowledge based systems in nondestructive
evaluation"--------------------------------------------------------199
3-D AND 2-D SIGNAL ANALYSIS IN NDE
16. K.C. Tam, "Limited-angle image reconstruction in nondestructive
evaluation"--------------------------------------------------------205
17. M. Sm.unekh, "The effects of limited data in multi-frequency
reflection diffraction
tomography"---------------------------------231
18. R.S. Acharya, "A 3-D image segmentation
algorithm"-----------------241
19. X. Maldague, J .C. Krapex and P. Cielo, "Processing of thermal
images for the detection and enhancement of subsurface flaws in
composite
materials"-----------------------------------------------257
20. C.H. Chen and R.C. Yen, "Laplacian pyramid image data
compression using vector
quantization"-----------------------------------------287
PARAME."I'ER ESTIMATION CONSIDERATION
21. J.F. BOhme, "Parameter estimation in array
processing"------------307
22. F. El-Hawary, "Role of peak detection and parameter estimation
in nondestructive testing of
materials"------------------------------327
LIST OF
PARTICIPANTS--------------------------------------------------343
Abstract
S. J. Norton, J. A. Simmons, A. H. Kahn and H. N. G. Wadley
Institute for Materials Science and Engineering, National Bureau of
Standards
Gaithersburg, Maryland 20899, USA
The role of inverse problems in the characterization of materials
is discussed. Four such problems
are described in detail: deconvolution for acoustic emission,
tomographic reconstruction of temperature
distribution, electrical-conductivity profiling and inverse
scattering. Each exploits a priori information in
a different way to mitigate the ill-conditioning inherent in most
inverse problems.
Introduction
The importance of inverse problems in the characterization and
processing of materials has increased
considerably with the recent growth of advanced sensor technology.
Frequently, the quantitative infor
mation of interest must be extracted from a physical measurement
(or more typically, a set of physical
measurements) that by itself may be only indirectly related to the
information desired and thus difficult
to interpret. For instance, sensor measurements often yield some
form of spatial and/or temporal average
of the desired information; such an average may, for example, be
expressed mathematically in the form of
an integral equation (e.g., a convolution) or a system of linear
equations (e.g., as in tomography), where
the mathematical relationship is derived from a knowledge of the
physics governing the measurement.
In an inverse problem one attempts to extract the desired
information from measurements containing
noise on the basis of an idealized model of the measurement
process. The problem is made more difficult
since inverse problems are characteristically ill-conditioned; that
is, small errors in the measurement
typically lead to large errors in the solution. However, nowadays
we know that the key to mitigating such
ill-conditioning is the judicious use of a priori information. The
incorporation of such a priori information
often takes the form of constraining the solution to a class of
physically reasonable possibilities, or it may
take the form of incorporating a priori probabilistic information
about the solution or the statistical
distribution of measurement errors. The use of a priori information
necessarily introduces an element
of subjectivity into the problem, since often the choices of a
priori constraints (or how they are best
incorporated) are not clear cut; such choices are usually decided
by experience derived from real problems.
This paper emphasizes the point that inverse problems in materials
science often offer an unusual
abundance of physically-motivated a priori constraints; certainly
the possibilities appear greater than in
many other fields where inverse problems have traditionally played
an important role, such as in medical
NATO AS! Series, Vol. F44 Signal Processing and Pattern Recognition
in Nondestructive Evaluation of Materials Edited by C. H. Chen ©
Springer-Verlag Berlin Heidelberg 1988
2
imaging and geophysical prospecting. As a result, the
nondestructive characterization of materials based
on ultrasonic and electromagnetic sensors offers an unusually
fertile area for innovation in inverse-problem
development and application. In this paper we will see several
examples of the use of a priori information in
problems that have arisen in our work on acoustic emission,
ultrasonic and electromagnetic nondestructive
evaluation.
In the analysis of acoustic emission signals, a problem of central
importance is the deconvolution of
the acoustic-emission source signal from the transducer response
(characterized by the transducer impulse
response) and propagation effects (characterized by the
temporally-dependent elastic Green's function of
the material). With this problem in mind, a new and robust approach
to deconvolution was developed
that is particularly well suited for deconvolving causal signals
[1]. This approach is described in the
next section. In the area of ultrasonics, we describe a technique
based on time-of-Hight tomography for
reconstructing two-dimensional temperature distributions in hot
metallic bodies [2]. In this problem,
a priori heat-How information is utilized to help mitigate the
effects of severe ill-conditioning in the
inversion. The third example is drawn from the area of
electromagnetic NDE, in which we describe
the problem of reconstructing one-dimensional conductivity profiles
from variable-frequency impedance
measurements [3]. We conclude with a description of a new iterative
approach to the exact, nonlinear
inverse-scattering problem [4]. A significant result reported here
is the use of an exact expression for the
gradient of the measurements with respect to the scattering model.
The exact gradient leads to a mean
square-error minimization algorithm with better stability and a
higher rate of convergence compared with
most other proposed iterative inverse-scattering schemes.
1. Deconvolution for Acoustic Emission
Acoustic emission may be regarded as naturally generated ultrasound
produced by sudden, localized
changes of stress in an elastic body. The analysis of acoustic
emission signals is complicated by the
fact that the observed signal is the two-fold convolution between
the source signal, the elastic Green's
function characterizing the propagating medium, and the detecting
transducer's impulse response. In
principle, the latter two response functions can be calculated or
measured. The problem then reduces
to deconvolving the source signal from the transducer and material
response functions in the presence of
noise.
A wide variety of numerical deconvolution schemes have been
proposed over the years by researchers
in disciplines ranging from seismology to astronomy. Most modern
deconvolution methods exploit some
form of regularization to reduce the sensitivity to measurement
errors of an inherently ill-posed inversion
problem. A widely-used regularization approach is to impose some
generalized form of smoothing con
straint, of which Tikhonov regularization is the prototype [5]. The
latter approach has the undesirable
side effect of destroying the causality of the deconvolved signal.
The algorithm described below, however,
not only preserves causality, but may be thought of as yielding the
"best" causal estimate of the original
(deconvolved) signal in a least-squares sense [1]. The method
exploits the fact that the roots of the Z
3
transform (or the related Y transform defined below) of a discrete
(sampled) signal are preserved under
convolution. Recent progress in the development of polynomial
root-finding algorithms has now made
this powerful approach practical for time series exceeding several
thousand samples.
Consider the deconvolution of two discrete-time (i.e., sampled)
waveforms represented by the fi
nite time series {a",k = O,I, ... ,N-l}, {b",k = O,I, ... ,N-l},
and their convolution {c",k =
0,1, ... , 2N - I}, where
" c" = L: a"_j bj • (1.1) j=O
Note in particular that the time series we are concerned with here
are causal, i.e., are zero for negative
k.
One way of representing convolution utilizes a simple modification
of Z transforms, which we shall
call the Y transform. We define the Y transform a(y) of an infinite
causal time series {a} by the formal
power series
a(y) = L:anyn. (1.2) n=O
For any finite segment of a causal time series, the Y transform is
a polynomial. Here, we want to
examine the convolution equation (1.1) in terms of Y transforms,
where it can be shown to take the form
c(y) = a(y) . b(y), (1.3)
that is, the convolution of two time series becomes multiplication
of their Y transforms. If we wish to
deconvolve {b} from {c}, when {a} is known, the formal solution
should, in principle, then be
b(y) = c(y)/a(y). (1.4)
Unfortunately, the division algorithm seldom works in practice
because of noise in the data. Due to
noise, a(y) does not exactly divide c(y), and the division process
magnifies the errors exponentially with
increasing terms in the time series.
A second approach is to divide the fast Fourier transforms (FFT's)
of the two functions. This idea
may also be explained in complex function language as follows. The
well-known Cauchy theorem, applied
on the unit circle, gives
1 f c(y) dy b" = 2m a(y)yk+l ' (1.5)
which is the Taylor's series (i.e., causal time series) for {b}. If
we evaluate this integral numerically by
sampling on the unit circle at the points
for l = 0, 1, ... , N - 1, (1.6)
equation (1.5) leads directly to the FFT division formula.
4
In the deconvolution problem, we shall assume that a(y) is given
and that c(y) is measured in the
presence of noise. We shall further assume that a(y) and c(y) are
both causal, and hence b(y) is causal,
where b(y) is to be determined.
The difficulty in using the FFT division method for determining the
series {b} stems from the fact
that a(y) often has roots inside the unit circle (typically about
N/2 such roots). In the integral (1.5)
the roots of a(y) become poles and consequently, from complex
function theory, equation (1.5) will not
in general give the wanted Taylor's series, but rather a Laurent
series (a non-causal series which is only
valid within an annular region of convergence bounded by the
nearest poles bracketing the unit circle).
Only if all the roots of a(y) happen to lie outside the unit circle
is the method exact.
We cannot do anything about the location of the roots of a(y),
since they are characteristic of the
series {a}. The transform of c(y), which is formally the product of
a(y) and (the unknown) b(y), should
have among its roots all those roots of a(y), including those lying
inside the unit circle. In the absence
of noise and if the calculation were perfect, these roots of c(y)
would exactly cancel the roots of a(y) [the
poles of l/a(y)] in the division (1.4). The result would be a
causal b(y) with a Taylor's series expansion.
The reason this approach generally fails is that noise in the
measurement of c(y) perturbs the location
of the roots so that they are not exactly divided by those of a(y).
This suggests that a new and robust
deconvolution method could be developed based on a procedure for
adjusting c(y) so that its roots include
all of those of a(y) inside the unit circle; in this case the FFT
division [equations (1.5) and (1.6)] should
give a stable, and causal, result.
To adjust c(y), let y be any complex number and consider the
N-dimensional vector
where T denotes transpose. Similarly, we can represent the series
{a}, {b} and {c} as the vectors ~, k.
and £ of appropriate dimension. Then the dot product of!! and 1l.
is a(y), and if Yl is a root of a(y) [Le.,
a(yIJ = 0], this means that l!.l is orthogonal to g. Therefore, if
we can find all the roots of a(y) inside the
unit circle, we can use powerful least-squares projection methods
to adjust the series £ to a new series f.
The new series f can be selected to be the closest one to £, in a
least-squares sense, which is orthogonal
to all the geometric root vectors 1l., where 1l. are the roots of
a(y) inside or on the unit circle. To put this
another way, we select the new series f which minimizes the
distance between f and £, i.e., (£ - fjT (£ - f),
subject to the constraints
k = 1,2, ... ,K,
where 14. are the roots of a(y) in and on the unit circle. The
latter problem can also be interpreted as
selecting f as the projection of £ onto a subspace orthogonal to
the space spanned by the geometric root
vectors l!.k. This approach can easily be generalized so that the
new series f can be selected with time
5
or frequency weighting to take advantage of a priori information
about the signal and noise statistics.
Now the common roots of ely) and a(y) will divide exactly, and the
resulting series obtained by FFT
division, i.e., by using e(y)/a(y) in equation (1.5), will be the
"best- causal estimate of {b}. We call {e}
the root-projected series and the resulting {b} the root-projection
deconvolution (RPD) estimate of {b}.
We have previously developed a singular-value matrix method (SVD)
as an alternative approach for
solving the deconvolution problem, and this method is quite
powerful [6]. However, it requires selecting
a best guess filtered answer and frequently that is difficult to
do. Also, the frequency transform of
the estimated answer often has unnecessary errors, even in those
frequency bands where there is much
information, because the eigenfunctions which are built by the
method to represent the answer do not quite
reflect the exponential functions used in a frequency
representation. Because the particular decomposition
of the answer differs for SVD (singular-vector representation) and
RPD (frequency representation), the
information (signal) and noise are distributed differently over the
orthogonal "channels- corresponding
to the particular basis functions utilized in that representation.
For example, typically the SVD estimate
will show some of the most prominent high frequency features, but
will have reduced low frequency
fidelity. The RPD estimate, on the other hand, will tend to have
good low frequency features, but will
have reduced high frequency features and greater end noise in the
time representation. This suggests
that the two ·complementary- approaches, SVD and RPD, can be
combined to exploit the best features
of each. One strategy that has been successfully demonstrated on
numerous simulation problems consists
of the following. SVD and RPD are each applied independently to
produce a first estimate to the inverse.
The data residuals generated by each algorithm, conservatively
filtered to avoid extraneous features, are
then fed into the other algorithm. What one of these algorithms may
discard as noise can contain useful
signal when decomposed using the other algorithm. Taking the
average of the final two estimates yields
an estimate that is not only more accurate, but more robust than
the result of using either separately.
The process of combining SVD and RPD in this manner we call the
cross-cut deconvolution algorithm,
which has been successfully applied to a variety of extremely
ill-conditioned deconvolution problems,
employing both simulated and experimental signals [1].
2. Ultrasonic Measurement of Internal Temperature
Distributions
The development of a sensor for measuring the internal temperature
distribution in hot metallic
bodies has long been identified by the American Iron and Steel
Institute (AISI) as a fundamental goal
in improving productivity and quality and optimizing energy
consumption in metals processing. As
a consequence, the AISI and the National Bureau of Standards
initiated a joint research program to
develop such a temperature sensor based on ultrasonic velocity
measurements. Potential applications
include measuring the internal temperature distribution in steel
ingots during reheating and monitoring
the temperature profiles of steel strands produced by continuous
casting. The temperature sensor is
based on the tomographic reconstruction of sound velocity from
ultrasonic time-of-flight measurements
- a particularly ill-conditioned inverse problem.
6
The operation of the sensor relies on measuring changes in the
velocity of sound through a hot metallic
object and exploiting the strong, almost linear, dependence of
ultrasonic velocity on temperature [21. For
example, 304 stainless steel exhibits a change of longitudinal
velocity of about -0.68 m/sec per degree
Celsius. IT the relationship is sufficiently linear, we may write
in general,
v(r, t) = vref + .B(T(r, t) - Tref ), (2.1)
where the space and time dependence, (r, t), of the velocity v has
been explicitly indicated since the
temperature T is in general a function of rand t. The constants
Vref, Tref and .B are presumed known
from prior measurement.
Ultrasonic velocity is measured by recording the time-of-flight
(TOF) of transmitted ultrasonic pulses
through the sample. This provides a measure of the average velocity
along the propagation path, which,
in turn, can be converted to a measure of the average temperature
along the path using a previously
calibrated, velocity-temperature relationship of the form (2.1).
Moreover, an actual image, or map, of
the temperature distribution can be derived using tomographic
reconstruction algorithms if a sufficient
number of TOF measurements are made along multiple overlapping
paths.
The TOF of an ultrasonic pulse along a ray path through an object
is the line integral of the reciprocal
sound velocity along that path, i.e.,
Tm =/ ~, v(r) (2.2)
where Tm is the TOF along the path Lm.
In principle, in tomographic image reconstruction, at least as many
TOF measurements are needed
as pixels in the image. In practice, errors in the TOF and
path-length measurement combine with
inherent ill-conditioning in the tomographic inversion to require
considerable measurement redundancy,
in which case least-squares techniques could be employed to best
estimate the temperature field. A priori
information can be used both to reduce this dependency on redundant
information and to mitigate the
sensitivity of the inversion (ill-conditioning) to measurement
errors.
The most important a priori constraint available to us is the
assumption of symmetrical heat flow,
which is often reasonable in bodies of simple geometric shape
(e.g., of circular or rectangular cross
section). Knowledge that the temperature field is symmetrical
drastically reduces the number of un
knowns characterizing the temperature field, and thus reduces the
number of required measurements by
a comparable amount. Furthermore, heat flow is well modeled by the
thermal conductivity equation (a
diffusion equation). Because temperature is a solution to this
equation, it is, in effect, being subjected
to a low-pass spatial filter whose spatial-frequency cutoff
decreases in proportion to the square root of
the cooling time. Stated another way, rapid spatial temperature
fluctuations disappear with time due
to thermal diffusion. This limit on the spatial frequency bandwidth
(smoothness) of the temperature
field implies the existence of a limit on the density of data
sampling (and hence on the number of TOF
7
x-ray tomography, effectively far coarser spatial resolution is
sufficient to reconstruct the temperature
field.
In practice, because of time constraints and experimental
complications involved in coupling ultra
sound in and out of a hot body, a relatively small number of TOF
measurements are feasible. As a
consequence, it is absolutely necessary to exploit object
symmetries as well as the property that the
temperature distribution rapidly assumes a smooth shape due to
thermal diffusion. The possibility of
reconstructing reasonably accurate temperature profiles with a
small number of measurements relies
crucially on the incorporation of such a priori information.
The constraint that the temperature field cannot be arbitrary, but
must obey the thermal conductiv
ity equation, suggests that we look for distributions in the
general form of the solution to this equation.
In an axially-symmetric object (in which the heat How is assumed
uniform or zero in the axial direction),
the general solution reads
T(r, t) = Tamb + Len Jo(anr)e-.. a!t, (2.3) fI.=1
where Jo(-) is the Bessel function of order zero. In this equation,
Tamb is the ambient temperature
(presumed known), en are unknown constants determined by some
initial (and unknown) temperature
state, an are unknown constants determined by the boundary
conditions, and If. is the thermal diffusivity
(presumed known). For a square geometry (again assuming constant
heat How in the z-direction), the
general solution is
00 00
T(x, y, t) = Tamb + L L enm cos (an x) cos(amy)e-.. (a!+a!'lt,
(2.4) n=lm=l
where, once again, the (unknown) constants enm and an are
determined by the initial and boundary con
ditions. A reasonable approach would be to use the ultrasonic TOF
measurements and the relationships
(2.1) and (2.2) to fit the unknown parameters in the above
temperature models, namely the en and an in
equation (2.3) or the enm and an in equation (2.4), for a finite
number of terms in the sum. Because we
know that the lower-order terms dominate in a short time due to the
exponential time dependence that
increases rapidly with order, a reasonable first approximation
would be to retain only the single lowest
order terms in the sums (2.3) and (2.4). This approach has formed
the basis of a practical inversion
scheme that has been successfully checked against experiment [2J.
When only the lowest-order terms are
kept, the above temperature models simplify as follows.
For axial symmetry,
(2.5)
where, for convenience, we have dropped the subscript one on a and
renamed the first coefficient, el,
by defining el = Tc - Tamb. In the above temperature model note
that Tc = T(O,O) corresponds to the
8
axial temperature at an initial time when t = o. In the above model
there are only two undetermined
parameters: T. and a.
For a square cross-section, keeping the lowest-order term in
equation (2.4) similarly yields
T(z, y, t) = Tamb + (T. - Tamb) cos(az) cos(ay)e-2ICa' t ,
(2.6)
where, once again, T. and a are the two undetermined parameters,
and T. = T(O,O,O).
For purposes of illustration, consider the model (2.6) for a square
cross-section. (The general pro
cedure extends, of course, to the axially-symmetric case.) Suppose
the TOF measurements are made
through a square block along M parallel paths at heights Ym and at
times tm • Suppose further that
the length of the side of the block is 2a. Inserting equation (2.1)
into (2.2) gives the "model-generated"
measurements, If,
m= 1,2, ... ,M, (2.7)
where T(z, y, t) is defined by equation (2.6) and M is the number
of measurements. The parameters T.
and a are then chosen to minimize the mean-square error
M
E = L [r(Ym, tm ) - If(Ym' tm )]2, (2.8) m=l
where r(Ym' tm ) is the measured TOF value at position Ym and time
tm , and If(Ym' tm ) is the computed
TOF value using equation (2.7). The numerical minimization of
equation (2.8) with respect to T. and a
may be performed using well-known nonlinear least-squares
algorithms.
Both the cylindrical and rectangular versions of the above
reconstruction scheme were applied to
TOF measurements made through, respectively, a 6 inch diameter
cylinder and a 6 X 6 inch square block,
both composed of 304 stainless steel. The TOF measurements were
performed at temperatures ranging
from 25° C to 750° C. Thermocouples embedded in the steel samples
were used as an independent check
of the temperature derived from the TOF measurements. Agreement
between the thermocouple readings
and the reconstructed temperature distribution was generally within
10° C, well within the experimental
error expected from the estimated uncertainty in the TOF and
path-length measurements. A detailed
description of the experimental apparatus and the resulting
temperature reconstructions are given in [2].
3. Determination of Electrical Conductivity Profiles from
Frequency-Sweep Eddy Current Measurement
The problem of measuring a spatially-varying electrical
conductivity profile in the interior of a
conducting body has only recently been addressed in electromagnetic
NDE, although this inverse problem
has received some attention in geophysics. Several approaches to
the conductivity inversion problem in
the geophysical context were reported by Weidelt [7], Parker [8],
and Parker and Whaler [9] in their work
on depth-profiling the earth's conductivity from measurements of
the time dependence of surface currents.
The work reported below is an adaptation of Parker's [8] inversion
scheme to problems in NDE [3].
9
The penetration of an ac magnetic field into a body of uniform
conductivity is exponentially atten
uated with a characteristic decay distance given by the well-known
formula for skin depth,
5 = .../2/O'wJl.O, (3.1)
where 0' is the conductivity, w the angular frequency, and Jl.o the
(free space) permeability. A measure
ment of impedance at the surface of the body will give a
determination of the electrical conductivity.
H the conductivity is allowed to vary with depth into the body, one
could attempt to reconstruct the
conductivity profile by performing surface impedance measurements
at many frequencies. High frequency
measurements would respond to the conductivity near the surface,
whereas low frequency measurements
would reflect conductivity values at greater depth. Thus, one would
feel intuitively that an inversion pro
cedure based on multi-frequency measurements could allow a
reconstruction of an arbitrary conductivity
profile without invoking specific a priori models (e.g., an assumed
single surface layer).
The complexity of the general problem requires, however, that we
limit the discussion to profiling
planar stratified material, i.e., material in which the
conductivity is a function only of the depth, z, below
a planar surface. For computational convenience, we also assume
that the material is terminated by a
perfect conductor at a depth h. This assumption does no harm if h
is much greater than the skin depth
corresponding to the lowest measuring frequency available. This
forces the electric field to vanish at the
terminating conductor, allowing a solution in terms of discrete
eigenfunctions. However, this requirement
of an E-field node is automatically satisfied for all frequencies
at the central plane of a symmetric plate,
provided equal H-fields are applied to both sides. Also, an
equivalent condition (E = 0 for r = 0) is
automatically satisfied at all frequencies for a cylinder in a
uniform H-field parallel to the cylindrical axis,
provided the conductivity is a function of the radius only. Thus,
the condition of a fixed E-field node can
be assumed in many common NDE configurations.
The differential equation for the time-dependent electric field,
E(z, t), with the depth-dependent
conductivity O'(z), is
a2 E aE az2 = Jl.oO'(z)i/t:.
H we assume single-frequency excitation of the form E(z,w)
exp(-iwt), we have
(3.2)
(3.3)
and an equation for the magnetic field, H, of similar form. An
implicit integral equation for E and its
derivative, E'(z), can be obtained by integrating equation (3.3)
once, giving
z
(3.4)
10
E(z) = Eo + Eo' z + iwj.lo i (z - z')E(z')u(z') d:l, o
where Eo and Eo' are constants of integration and are fixed by the
boundary condition at z = 0.
(3.5)
We now approximate the conductivity profile as a weighted set of N
infinitesimally-thin, parallel
conducting shells at depths zt and conductivities .,.., i.e., we
let
N
u(z) = L .,..o(z - zt). (3.6) i=1
Between shells the magnetic field is constant, and thus the
electric field varies uniformly with z. The
current in each shell is proportional to the electric field at the
shell and induces a jump in the magnetic
field of an amount .,..E(zt). This causes a corresponding jump in
the derivative of the electric field of an
amount iWj.lo.,..E(zt). That is, across the i-th shell, E(z) is
continuous, but
(3.7)
by
A principal quantity of interest in the inversion problem is the so
called admittance function, defined
E(z,w) c(z, w) = E'(z, w)· (3.8)
This function can be measured at the surface z = ° from
measurements of the electric and magnetic fields
E(O, w) and H(O, w) as a function of frequency. c(w) can
equivalently be derived from surface impedance
measurements and knowledge of H(O,w) [101. The surface admittance
is
_ E(O,w) _ E(O,w) c(w) = c(O,w) = -E'( ) -. H( ). O,W IWj.lo
O,W
(3.9)
From equation (3.4) and (3.5), and in view of the definition (3.8),
we see that in propagating from
zt-l to z. between conducting shells, where no conductors are
present, the admittance undergoes a change
On the other hand, in propagation across the shell at zt, it
follows from equations (3.7) and (3.8) that
the change is given by
1 c(zt+,w) = ------
. 1 IWj.lo"'. + ( )
C Z,-,W
Noting that c(h,w) = ° (since the electric field vanishes at h), we
can apply these rules successively to
obtain a continued-fraction representation of the surface
admittance:
11
iWPOTl + ----------- 1
-hl + ------- 1
iWPOT2 + --- I
···+-h - N
(3.1O)
where the", = Zi+l-Z. are the spatial separations between the
shells. When the above continued-fraction
representation is rationalized it reduces to the ratio of two
polynomials of degree N. This polynomial
ratio can then be expanded in a sum of partial fractions,
giving
( ) = f. An(w) c w L...., \ .,
n=1 An. - \W (3.U)
where the An are real and An(w) are polynomialfundions in w. Thus,
equation (3.10) has been cast in the
form of a spectral density function. As written, equation (3.U)
implies that c(w) has N poles lying on the
positive imaginary axis. This can be independently verified as
follows. A set of real normal mode solutions
to the eddy current equation (3.2) are the exponentially-damped
functions E(z, t) = un(z) exp(-Ant).
Inserting these modes into equation (3.2) results in
(3.12)
where the eigen-solutions, Un, are subject to the boundary
conditions un(h) = 0 and 8un/8z = 0 at
z = o. The boundary conditions generate a discrete set of normal
modes, corresponding to the real
eigenvalues An, n = 0,1, ... , which decay in time, each with its
own time constant An. Now, the Green's
function for the eddy current equation (3.3) obeys the
equation
(3.13)
subject to the boundary conditions G(h, z') = 0 and 8G{zlz')/8z = 0
at z = o. Performing the expansion
of the Green's function in terms of the eigenfunctions of equation
(3.12), un(z), we have
(3.14)
With the help of equations (3.12) and (3.13) and Green's theorem,
it is easy to verify that
G(OIO) = - ;~~,;) = -c(w). (3.15)
Comparing equations (3.14) and (3.15) to (3.11) shows that the
finite shell problem corresponds exactly
to the spectral expansion (3.14) truncated at N terms. This
confirms that the An in equation (3.11) are
real and that the poles of c(w) lie on the imaginary axis.
The proposed scheme of obtaining the conductivity profile is as
follows:
12
1. From impedance measurements, obtain c(w) at numerous values of
the (real) frequency w, in a range
such that the skin depths span the dimensions of interest.
2. From the measurements of c(w), obtain a best fit to a truncated
expansion of the form of equation
(3.11). The task of performing this fit with incomplete and
imprecise data has been treated by
Parker [8] and Parker and Whaler [9].
3. Transform the partial fraction form to the model of conductive
shells by performing the expansion
into the continued fraction form (3.10). The locations and
strengths of the shells can, in principle,
be picked off by inspection. Algorithms for this computation have
also been implemented by Parker
and Whaler [9]. This gives a profile in terms of o-function
shells.
4. Spread the conductances of the o-function shells into the space
between the shells. We arbitrarily
bisect the regions between shells and spread the strength of each
shell uniformly between the neigh
boring bisecting planes. This procedure gives the profile in the
form of a series of flat steps. This
last procedure is based on the concept that each o-function shell
obtained in the inversion process
represents continuously distributed conductance.
The simplest realizable arrangement in which a uniform field may be
applied to a sample is that of
a long solenoid with a cylindrical core. H the conductivity depends
only on the radial coordinate, and if
we may neglect end effects, the problem may be transformed into the
form of the previously treated case
of the planar stratified medium. In the cylindrical case the
admittance is defined by [10]
E(R) c(w) = iw~oH(R)
(3.16)
where R is the radius of the sample. In addition, the inversion
algorithm for a set of shells carries over
from the planar case by the transformation:
c(w) -+ Rc(w)
Experimental tests were performed on several cylindrical samples,
including a solid brass rod, a
brass tube with a copper center, and a brass tube with a tungsten
center. Impedance data were acquired
after inserting the metal cylinders into a cylindrical coil. A
detailed description of the experimental
arrangement and the resulting conductivity profiles may be found in
[3]. In the tests on metal cylinders,
good qualitative experimental agreement was achieved; in
particular, the locations of the discontinuities in
conductivity at the interface between the different metals were
accurately reproduced. Unfortunately, the
quantitative agreement between the true and reconstructed
conductivity values was quite inconsistent.
The latter result may reflect limitations of the shell model
[equation (3.6)] as well as the severe ill
conditioning inherent in the conductivity inversion problem. To
improve the method, other geometric
13
arrangements might be used so that low frequency interrogating
fields would penetrate the entire sample.
Appropriate methods of reconstruction would have to be
developed.
One motivation for the present approach is that the shell model
permits an exact solution to the
reconstruction problem by means of the continued-fraction
representation of the admittance. There are,
however, a variety of other approaches for solving the inverse
conductivity problem which, although
perhaps less elegant analytically, offer some advantages. One such
method is based on iterative nonlinear
least squares. An example of this approach applied to the
inverse-scattering problem is given in the next
section. Although in the latter case the Helmoltz equation replaces
the eddy-current equation (3.3), the
inverse-conductivity problem can be formulated in an essentially
identical fashion. One notable virtue of
the iterative least-squares method is its great flexibility, both
in the choice of permissible basis functions
used to represent the unknown profile and in the ease with which a
priori information can be incorporated.
These points are discussed at greater length in the next
section.
4. Iterative inverse scattering
The acoustic inverse-scattering problem has found applications in
many disciplines, including medical
ultrasonic imaging, seismic imaging, and ultrasonic NDE. For our
purposes the inverse-scattering problem
may be defined briefly as the problem of reconstructing the
interior of a scattering object (i.e., the
distribution of some material scattering parameter) from scattered
waves observed outside the object.
For convenience, the interior of the object may be defined as a
bounded inhomogeneous region embedded
in an infinite homogeneous medium. We do not consider here the
related problem of reconstructing
the shapes of "hard" objects that do not permit penetration of the
waves. As a matter of terminology,
the "forward-scattering problem" is defined as the problem of
computing the scattered wave gillen the
scattering object and the incident wave.
We outline in this section an iterative approach to the exact
inverse-scattering problem which requires
the repeated numerical solution of the forward problem. Most
current inversion schemes are derived from
linear approximations to an exact, nonlinear inverse-scattering
theory. That is, such schemes are derived
under the assumption that the scattering measurements and the model
(by which we mean the unknown
scattering distribution) are linearly related. Born inversion,
which is based on first-order perturbation
theory, is a well-known approach of the latter type, and succeeds
when the scattering is sufficiently
weak. However, such methods fail to account for the distortion of
the internal wave field interacting
with the scattering medium. As a result, in any linearized
inversion scheme, multiple reflections and
refraction effects are almost always ignored. The assumption that
the internal field distortion is negligible
is occasionally justified in medical ultrasound, is poor for many
composite materials encountered in
ultrasonic NDE, and is rarely justified in seismology. Such
considerations have motivated the development
of an iterative approach to inverse scattering designed to fit the
model to measurements while employing
a more exact description of wave propagation. Iterative approaches
also have the advantage of permitting
the incorporation of a variety of a priori information, e.g.,
preventing the solution from straying too far
14
from an a priori model and/or using covariance operators that take
into account the statistical structure of
measurement errors and their correlation. In addition, such a
priori constraints playa fundamental role in
regularizing the inversion, that is, in significantly decreasing
the sensitivity of an otherwise ill-conditioned
problem to noisy and sparse data.
A reasonable and popular approach is to cast the problem in the
form of a minimization of the
mean-square-error between the measured data and data generated by
the current estimate of the model,
subject also to a priori constraints. In the work reported here,
the conjugate-gradient algorithm was used
to minimize this mean-square error [4J. This algorithm, unlike
quasi-Newton methods, avoids the need
to invert a large matrix at each iteration containing
second-derivative information.
In any iterative scheme, one needs to know something about the rate
of change, or gradient, of
the data with respect to the model, so that one can tell in what
"direction" to iteratively adjust the
model such that the measured data and the model-generated data
eventually coincide (within the limits
imposed by possible a priori model constraints). In existing
iterative schemes (e.g., [11,12,13]), a linearized
approximation to the gradient is almost always used, in which case
the gradient is correct only to first
order in the model. Weston [14J, however, has obtained an exact
expression for the gradient correct to
all orders in the model. This result is important since the
neglected higher-order terms are responsible
for all multiple-scattering and refraction effects. Weston derives
his exact gradient for the special case of
monochromatic, plane-wave illumination and far-field (plane-wave)
detection. We have, however, been
able to generalize Weston's gradient to the case of time-dependent
fields of arbitrary form and to point
source illumination and near-field detection [4J.
The importance of Weston's result, and its generalization, stems
from the fact that the exact gradient
will always give a descent "direction" in the mean-square error at
the current model estimate, i.e., an
incremental change in the model along the gradient direction will
guarantee a decrease in the mean square
error even if the scattering is strong (e.g., when the Born
approximation fails). In other work on iterative
inverse-scattering schemes [11,12,13J, the usual procedure has been
to derive an approximation to the
gradient by first linearizing the nonlinear measurement-model
relationship [i.e., the Lippmann-Schwinger
equation; see equation (4.3)J and then "differentiating" the data
with respect to the model. However,
it is important to realize that the linearized gradient derived in
this way may not lead to convergence
if the scattering is strong. That is, the approximate gradient will
not in general quarantee a descent
direction in the mean-square error unless the scattering is known
in advance to be sufficiently weak.
Thus, Weston's result should improve the stability and rate of
convergence of any descent algorithm
(including, for example, Newton-like methods). This is illustrated
in [4J for the special case of a one
dimensional, nonlinear inversion problem using the steepest descent
and conjugate-gradient algorithms.
In that simulation, both the conventional linear approximation to
the gradient and Weston's gradient
are used and the latter is shown to improve noticeably both the
stability and rate of convergence of the
minimum mean-square error algorithm. In particular, the simulations
in [4J show that, in the example
15
considered, a 20 percent velocity excursion is sufficient to
prevent the convergence of an iterative scheme
employing the linearized gradient, while the exact gradient leads
to rapid convergence.
Generally speaking, any iterative scheme must solve the
forward-scattering problem many times each
iteration. The forward-scattering algorithm is needed to accurately
compute the field distribution on the
basis of the current model estimate. The forward algorithm also
needs to be as fast as possible since in
most schemes, as well as the one proposed here, the
forward-scattering problem must be solved at least
Ns + NR times per iteration, where Ns and NR are, respectively, the
number of sources and receivers.
The forward algorithm can be performed in either the time or
frequency domains, although for simplicity,
we confine the present discussion to the frequency domain. A
general formulation that encompasses both
the time- and frequency-domain cases is given in [4].
For simplicity, scalar-wave propagation is assumed here, although
the formulation can be readily
generalized to more complex and realistic models, including
multiple-parameter models characterized
by unknown variations in velocity and density (or, for example,
variations in density and two Lame
constants, for an isotropic elastic model). In this discussion, we
assume that the acoustic velocity c(d is
the unknown scattering parameter of interest, where c(d = Co =
constant outside of a bounded scattering
region D.
We now illuminate the region D with an incident monocromatic field
u,w (d. The total field U w Cd (incident plus scattered fields) is
assumed to obey the Helmholtz equation
which can be rearranged to read
(4.1)
where
{ II
(4.2)
defines the "model" to be estimated and Co is the (constant)
velocity outside of D. Employing standard
techniques, the solution to the wave equation (4.1) can be cast in
the form of the integral equation
(4.3)
(4.4)
I W "'1 exp(iwlr - tl/co). 4,.. r. -_ (4.5)
In equation (4.3) u;., is the incident field that obeys the
homogeneous form of equation (4.1) (i.e., with
tI = 0). Now let rs denote the location of a point source outside
the scattering region. Then equation
(4.3) may be written
u...(r.rsjv) = u;.,(r,rs) + L dt G .. (r!t)tI(r')u...(r',rsjtl),
(4.6)
For clarity, the dependence of the field u...(r.,rsjtl) on the
source location rs and on the model tI has
been explicitly indicated.
Let R denote the scattered wave measured at the point !R outside of
the scattering domain D. Then
(4.7)
Thus,
(4.8)
in view of equation (4.6).
For brevity, define the observation vector ~ == (rR'rs,w), so that
R(rR,rs,Wjtl) = R(~jtl). In
general, we will make measurements for many values of~. Now let V
(r) represent the true model, so that
the measurement consists of R(~j V) + t over some domain of ~,
where t denotes any measurement
error. For a given estimate v(r) of V(r), define the measurement
and model residuals
e(~; v) == R(~; v) - R(~; V)
e(e.; v) == tI(r.) - o(r),
(4.9a)
(4.9b)
where o(r) is an a priori estimate of VCr). In the following, VCr)
is assumed real; this assumption
simplifies the derivation but is not strictly necessary.
We now define the mean-square error E(v) to be minimized with
respect to tI:
(4.10)
where the functions W and IV are assumed real and non-negative. W
and IV incorporate probabilistic
information about the measurements and model and can be optimally
chosen to selectively emphasize
more reliable data or weight the importance of the a priori model
oCr).
To find tI that minimizes the mean-square error E(tI), the gradient
of E(tI) with respect to tI is
needed. If we call this gradient g(r), iterative algorithms for
minimizing E(v) can be defined according
to the general scheme
17
(4.11)
where the choice of Inb:) can be made to define the method of
steepest descent or conjugate-gradient
descent [15J, as follows
(4.126)
In equations (4.12), 9,,(r.) is the gradient evaluated at the n-th
model estimate v,,(r) and a" and p" are
constants that vary with the iteration number n. The optimal
choices of a" and p" are discussed in [4J.
To determine the gradient or "rate of change" of a model-dependent
field quantity u(r; v) with respect
to v, we use the (Gateaux) differential du(r; v, J) defined
as
d ( I) I· u(r; v + aJ) - u(r; v) u !:jV, = 1m ,
a-+O Q (4.13)
in which the function I(r.) may be thought of as an incremental
change in the model v(r.). Equation
(4.13) may be viewed as a kind of functional derivative of u(r; v)
in the "direction" I evaluated at v. To
derive the gradient of E(v), take the (Gateaux) differential of
equation (4.10) with respect to v, giving
dE(v,J) = lim E(v+aJ)-E(v) a-O Q
= 2Re f d~ W(~)e(~; v)dR(~; v, 1)* + 2 f dr W(r.)e(r; v)f(r) ,
(4.14)
where Re means real part and· denotes complex conjugate.
The next important step is to evaluate the quantity dR(~; v, J),
that is the "derivative" of the
data R with respect to the model v. As mentioned earlier, this is
usually achieved by linearizing the
functional R(v) with respect to v, which results in an
approximation to the gradient of R(v) correct
only to first order in v. For monochromatic, plane-wave
illumination and far-field detection, Weston has
obtained an exact expression for the gradient of R correct to all
orders in v. As noted earlier, we have
generalized his result to the time domain and to point sources and
receivers. We state below the result
for monochromatic, point-source illumination and point detection.
Complete details of the derivation are
contained in [4J.
Define the field 11", (r, rR; v) as the solution to
(4.15)
18
where G",(dt) is the adjoint of G.,(dt). The physical
interpretation of the solution u., is a wave
propagating back in time from the receiver coordinates rR to the
interior point r. In [4], we show that,
for any two models Vl(r) and 112(r), the following result
holds:
R(rR,rS' Wj vd - R(rR,rs,Wj 112) = ! dr' [Vl(r') - l12(r')ju..(t,
rRj vdu",(r', tgj 112), (4.16)
where U.,(r, rRj vd is the solution to equation (4.15) with model
Vl(r) and U.,(r,rsj 112) is the solution to
equation (4.6) with model V2(r). R is defined by equation (4.7).
The four equations (4.6), (4.7), (4.15)
and (4.16) represent the generalization of Weston's result
[14].
Equation (4.16) is now used to obtain the gradient of the data R
with respect to the model v as
follows. Setting Vl = v + al and 112 = v in equation (4.16),
substituting R for u in equation (4.13) and
taking the limit, results in
dR(rR,[g,WjV,f) = ! dr' I(r') u",(r', rRj v)U.,(r',[gj v).
(4.17)
From equation (4.17), [u", (r,rRj v)U., (r,rsj v)]· may be
interpreted as the gradient of R(rR' [g,Wj v) with
respect to v, and the integral in equation (4.17) represents the
change in R( v) in the direction I at v.
This important result, which will be used below, is correct to all
orders of v.
For brevity, define
where ~ = (rR,[g,W), so that equation (4.17) may be written
dR(~jv,f) = ! dr' l(r')U(r',~jv). (4.19)
Finally, substituting equation (4.19) into (4.14) and interchanging
orders of integration results in
dE(v, f) = 2! dr' J(r')g(r'j v),
where
g(rj v) == Ref d~ W(~)e(~j v)U(r, ~j v)* + W(r)e(rj v) (4.20)
is the desired gradient of E(v). No linear approximations have been
made in arriving at equation (4.20).
The gradient given by equation (4.20) may now be inserted into
equations (4.12). The only remaining
task is to determine the constant parameters an and f3n appearing
in equations (4.11) and (4.12). For
strongly nonlinear problems, the parameter an can be chosen at the
n-th iteration to minimize E(vn ) =
E(Vn-l + anln) for a given In. To achieve this, one approach is to
perform a numerical line search in the
direction In(x) for the an that minimizes the mean-square error
[15]. This can be numerically expensive
19
since a line search requires the evaluation of E( I)) at many
points along the line. IT the surface defined by
E(I)) is approximately quadratic (which it is exactly in linear
problems), an can be expressed explicitly in
terms of In and gn' This is also the case for f3n in the
conjugate-gradient algorithm. Explicit expressions
for an and f3n are derived in [4].
It is interesting to note that, in principle, as many as five
degrees of freedom may exist in the dataj
that is, ~ == (!:R, rs, w) can denote a five-dimensional vector if
rR and rs are each allowed to vary
independently in two dimensions. On the other hand, the modell)(~)
can have as many as four degrees of
freedom, three in space and one in frequency. The latter variable
can be used, for example, to incorporate
frequency-dependent attenuation effects into the model. IT one
disregards sampling considerations, the
inversion problem in the extreme case of a five-dimensional data
set and a four-dimensional model is over
determined. This presents no difficulty in the least-squares
formulation, where redundant data is often
helpful in reducing sensitivity to random errors.
We conclude by commenting on the meaning of the weighting functions
W(~) and W(d that appear
in the mean-square error integral (4.10). In fact, equation (4.10)
should be regarded as a special case of
the generalized mean-square error defined by
E(v) = 1 1 d~d~' e(~j 1))'W(~,~')e(~'j I)) + 11 drdr' e(rj
I))W(r,r')e(r'j I)), (4.21)
where W(~, ~') is a complex weighting function with conjugate
symmetry and W (r, r') is a real, symmet
ric weighting function. W (~, ~') may be interpreted as a
generalized measurement covariance function
(the continuous analogue of the inverse covariance matrix in the
discrete case) and W(r,r') as the model
covariance function. The functions Wand W may be optimally selected
to take into account probabilis
tic information about the reliability of the data and the
importance of the a priori model. Information
regarding the correlation of the data, as well as the correlation
between different points within the model,
is also taken into account. For example, a "nondiagonal" W(r, r')
[i.e., in which W of 0 if r of r'] may
imply that the model is smooth in some sense. Accordingly, W may
also be interpreted as playing the
role of a spatial filter operating on the model. In general, the
value of I) that minimizes E( I)) will rep
resent some compromise between a model consistent with the data and
one that is not too far from the
a priori estimate ii. The relative weighting between these two
extremes is of course controlled by the
choice of functions Wand W. The derivation of this section can be
readily generalized to the problem of
minimizing equation (4.21) rather than (4.10)' but at the expense
of a major increase in computational
cost.
The minimization of the generalized mean-square error defined by
equation (4.21) [or equation (4.10)]
can be given a more formal probabilistic justification by noting
that minimizing E( I)) is equivalent
to maximizing the a posteriori probability density function
p(l)]data) (i.e., the conditional probability
density of the model I) given the available data) when the data and
model obey multivariate Gaussian
statistics. This criterion, much used in modern estimation theory,
provides an intuitively satisfying
definition of "optimum" since it gives the most probable model on
the basis of the available data and a
20
priori model information. From probability theory, P( vldata) = c
P(datalv) P( v), where c is a normalizing
factor independent of v and P(v) is the a priori probability
density function for the model. Maximizing
P(vldata)/c with respect to v is equivalent to maximizing
In[P(vldata)/c] = In[P(datalv)] + In[P(v)],
which, in turn, is equivalent to minimizing equation (4.21) when
the data and model are Gaussian. For the
Gaussian case, In[P(datalv)] can be formally identified with the
first term on the right in equation (4.21)
and In[P(v)] with the last term. In this interpretation, Wand Ware
the continuous analogues of inverse
covariance matrices in the discrete case for the data and model,
respectively. When a priori information
is not used, that is, when the last term in equation (4.21) is set
to zero, the solution corresponds to the
maximum-likelihood estimate.
Conclusion
In this paper four inverse problems are discussed that have arisen
in materials NDE: deconvolution,
electrical-conductivity profiling, tomographic reconstruction of
temperature and inverse scattering. The
ill-conditioned nature of the inverse problem must be dealt with to
allow meaningful inversions from
noisy measurements. A priori constraints of one form or another can
be applied to reduce this inherent
sensitivity to measurement error. For example, in the deconvolution
problem, causality is a powerful
constraint that was imposed in the inversion algorithm. In the
conductivity profiling problem, the con
traint of an E-field node at the center of the sample (justified
from symmetry) reduced the basis set
representing the unknown profile from a continuous to a discrete
set of eigen-functions. Moreover, the N
conducting-shell model reduced the dimensionality of the problem to
the finite value N. The conductivity
reconstruction can then be carried out with a minimum of N surface
impedance measurements. In the
temperature tomography problem, the temperature distribution was
constrained to match the general
form of a solution to the thermal-conductivity equation defined by
the geometry of the sample. Unde
termined parameters in the thermal-conductivity solution could then
be estimated on the basis of the
ultrasonic velocity measurements. In the inverse-scattering
problem, a variety of a priori information can
be incorporated in the minimum mean-square error formulation. Such
a priori constraints can reflect
confidence in the measurements based on noise statistics, and can
impose a priori smoothness bounds on
the scattering distribution to be reconstructed.
Acknowledgment
The authors would like to acknowledge the partial financial support
of the NBS Office of Nonde
structive Evaluation.
21
References
1. Simmons, J. A.: New methods for deconvolution and signature
analysis of causal and transient time
series. To be published.
2. Wadley, H. N. G., Norton, S. J., Mauer, F., and Droney, B.:
Ultrasonic measurement of internal
temperature distribution, Phil. Trans. R. Soc. Lond. A 320, 341-361
(1986).
3. Kahn, A. H., Long, K. R., Ryckebusch, S., Hsieh, T., and
Testardi, L. R.: Determination of elec
trical conductivity profiles from frequency-sweep eddy current
measurement. Review of Progress in
Quantitative Nondestructive Evaluation, Vol. 5B, Thompson, D.O.,
and Chimenti, D. E., eds., pp.
1383-1391 (1986).
4. Norton, S. J.: Iterative seismic inversion, Geophys. J. Roy.
astr. Soc (submitted).
5. Tikhonov, A. N.: Solution of incorrectly formulated problems and
the regularization method, Sov.
Math. Dokl. 4, 1035-1038 (1963).
6. O'Leary, D. P., and Simmons, J. A.: A
bidiagonalization-regularization procedure for large scale
discretizations of ill-posed problems, Siam. J. Sci. Stat. Comput.
2, 474-489 (1981).
7. Weidelt, Z., Geophysik: The inverse problem of geomagnetic
induction 38, 257-289 (1972).
8. Parker, R. L.: The inverse problem of electromagnetic induction:
existence and construction of
solutions based on incomplete data, J. Geophys. Res. 85, 4421-4428
(1980).
9. Parker, R. L. and Whaler, K. A.: Numerical methods for
establishing solutions to the inverse problem
of electromagnetic induction, J. Geophys. Res. 86, 9574-9584
(1981).
10. Kahn, A. H., and Spal, R.: Eddy current characterization of
materials and structures, ASTM Special
Tech. Publ. 722, Birnbaum, G., and Free, G., eds., American Society
for Testing and Materials, pp.
298-307 (1981).
11. Tarantola, A.: Inversion of seismic reflection data in the
acoustic approximation, Geophysics 49,
1259-1266 (1984).
12. Fawcett, J.: Two dimensional velocity inversion of the acoustic
wave equation, Wave Motion 7,
503-513 (1985).
13. Nowack, R. L., and Aki, K.: Iterative inversion for velocity
using waveform data, Geophys. J. R.
astr. Soc. 87,701-730 (1986).
14. Weston, V. H.: Nonlinear approach to inverse scattering, J.
Math. Phys. 20, 53-59 (1979).
15. Luenberger, D. G.: Linear and Nonlinear Programming, 2nd Ed.,
New York: Addison-Wesley 1984.
ABSTRACT
ADVANCES IN BORN INVERSION
L J Bond, J H Rose*, S J Wormley* and S P Neal*
NDE Centre, Department of Mechanical Engineering
University College London, Torrington Place, London, WCIE 7JE
England
The I-D Born Inversion Technique is well established as a method
which
gives defect radii from pulse-echo ultrasonic measurements.
Recent
developments give the diameter of a flaw from measurements in the
Born
Radius/Zero-of-Time Shift Domain (BR/ZOTSD) without the explicit
need to
select a correct zero-of-time for the inversion. A signature for
the flaw
is obtained by plotting the estimated flaw radius as a function of
a
certain time shift, (shifting the zero-of-time). The signature
does
depend on transducer bandwidth, but the resulting diameter is, to a
larger
extent, insensitive to the bandwidth of the transducer employed.
A
corresponding BR/ZOTSD signature has been obtained for sizing
voids. This
work represents a unification of many of the features considered in
earlier
studies of Born Inversion with those found in time-domain sizing
techniques,
such as 'SPOT'. The accuracy to which a flaw size estimate can now
be
given is significantly improved using this extension to the process
of
I-D Born Inversion and this is demonstrated with analytical,
numerical
and experimental data.
*Ames Laboratory, USDOE, Iowa State University, U.S.A.
NATO AS! Series, Vol. F 44 Signal Processing and Pattern
Recognition in Nondestructive Evaluation of Materials Edited by C.
H. Chen © Springer-Verlag Berlin Heidelberg 1988
24
INTRODUCTION
Signal processing techniques in general and inversion techniques
in
particular have for many years been widely used for information
retrieval
in the fields of radar, sonar and seismology. The problem of
defect
detection, location and characterisation encountered in ultrasonic
non
destructive testing are in many ways similar to those in fields
where
signal processing is well established. Over the last decade there
have
been increasing demands for the development of quantitative NDT
techniques
for flaw detection and characterisation. Of primary interest is
the
positive identification of flaws, above some particular size
threshold,
with a minimum of false calls. The flaw characterisation is then
required
to be in a form where it can be combined with stress and material
data for
use in what is variously called 'damage tolerance', 'retirement for
cause'
and 'remaining life' analysis.
The key parameters sought for flaw characterisation are its type,
(is it
a crack, a void or an inclusion of some particular material), as
well as
its specific location, its shape, orientation and size. One scheme
for
flaw sizing which has been under investigation for almost a decade
is
known as "Born Inversion" (1,2,3). In its simplest one-dimensional
form
this method can be used to provide a flaw radii, measured along the
view
ing axis, from a single broadband pluse-echo ultrasonic
measurement.
Although the scheme was originally developed for weak spherical
scatterers,
such as inclusions with low acoustic impedance contrast to the
matrix
material (1), it has also been shown to apply to sizing of
strong
scatterers such as spherical voids. (2,3,4,5).
Born Inversion is however a quantitative sizing technique which as
been
the subject of some debate (2,6,7). When measurements have been
made on
known features, such as single volumetric voids and inclusions with
radii
of the same order as the wavelengths used, it has been shown to be
capable
of giving radii estimates to within ± 10%; however this is not
always the
case and data needs careful treatement if rogue results are to be
avoided,
Rogue results may be due to the effects of a poor signal to noise
ratio,
however even when an apparently adequate signal to noise ratio
exists
there are still cases when poor sizings have been found to
result.
25
The reasons for this 'poor' data have been considered by several
groups
and would appear to be due to (7,8);
i. A mismatch between the flaw radii and the transducer bandwidth
and
hence the wavelengths which carry the energy in the pulse used;
and/or,
ii. Errors introduced due to the incorrect selection of the
"Zero-of-Time",
(ZOT).
It was in an effort to understand, and it was hoped find methods
to
overcome or at least limit these effects that the current work
was
undertaken.
The practical problem which has attracted most attention has been
obtain
ing an adequate probe bandwidth/flaw match, if the flaw size is not
known
a priori. It has been proposed that a minimum 'ka' range of
0.5 < ka < 2.5 is required for good sizing (9); where a is
the defect
radius k is the wave number, (k=2~/A), and ~ is the wavelength for
a
particular frequency component in the pulse. It is found that
for
typical defects, which are less than a mm in diameter, the
transducer
bandwidths are on the limits of those which are commercially
available.
This limitation can in part be overcome by a strict methodology.
(7,10)
However such an approach is both time consuming and cumbersome.
Even
when given the improved transducer bandwidths which can be
expected
using such material as PVDF and other more piezolectrically
active
polymers, there remains a need to reduce the uncertainty attached
to the
selection of the correct ZOT and also provide the best radius
estimate
in the extraction of data from the resulting characteristic
function.
This paper considers the later problem, that is; how does one infer
the
true flaw radius or diameter when the correct ZOT is unknown? A new
and
more robust methodology for the implementation of this inversion
has
been developed which gives the flaw diameter (D) with a higher
degree of
confidence than the radius is given with the current practice. It
does
not require the explicit selection of a 'correct' ZOT and it can
be
implementetl using the range of commercial transducers that are
currently
available.
26
I-D BORN INVERSION
An inversion algorithm is a scheme which enables a prediction of a
flaw's
characterisitics features to be made from its application to data
collected
in the scattered wavefield.
Prior to the work reported in this paper an investigation was
performed
to review the capability of the I-D Born Inversion. This review of
the
theory and current practice has now been published (7), so only a
brief
outline is given here.
The Born Inversion algorithm is designed to determine the geometry
of a
flaw from the ultrasonic scattering data. The geometry of voids
and
inclusions can usefully be described by the characteristic
function,t(r'),
which is defined to be 1 for r' in the flaw and zero outside. For
spheri
cally symmetric flaws the Born Inversion algorithm estimates
the
characteristic function by;
rt r ) = const J:k sin 2kr Re lA(k~ o 2kr (1)
Here A(k) denotes a longitudinal, L.L, far-field scattering
amplitude for
a flaw in an otherwise isotropic, homogeneous and uniform space.
Experi
mental data are of course not available for all frequencies and it
is
always necessary to evaluate (Eq. 1) in a band-limited form. The
result
is a smoothed estimate for the characteristic function.
The two most commonly used techniques for radii estimates from
the
characteristic function are:
(i) the radii at 50% of the peak of the characteristic function;
and,
(ii) the radii corresponding to the total area under the
characteristic
function, divided by the peak height.
27
However these radii estimates are only possible if A(k) is
known
accurately. Typically ultrasonic measurements only allow one to
infer A
to within an unknown overall phase error, i.e. A(k)exp(ikc~. Here c
is
an unknown phase error where c denotes the longitudinal wave
velocity.
Finally ~can be understood as representing an unknown time shift in
the
time domain. The inference of the scattering amplitude A(k)given
A(k)exp
(ikc1j with ~ unknown is referred to as the ZOT problem. Clearly
a
definitive method which selects the right ZOT or preferably avoids
the
need to select this 'correct' ZOT for estimating the radii is
desirable.
In essence the practical implementation of the technique is based
on a
single wide-band pulse-echo ultrasonic measurement made with a
digital
ultrasonic system. If measurements are made in the frequency band 5
to
20 MHz, 8 bit digitisation is required at a sampling rate of 100
MHz. The
system impulse response is then required and this is obtained from
the
pulse-echo signal for the flaw which is gated and digitised. The
system
characteristics are removed by a deconvolution with a reference
signal
obtained from a reflection at a flat surface, set at the same range
as
the flaw and taken using the same system settings. The resulting
decon
volved time domain signature is then the flaw's bandlimited
impulse
response function. The scattering amplitude is obtained by
Fourier
Transform of the impulse response function. (7)
The Born algorithm is then applied to the impulse response function
and
this requires the selection of a ZOT. For a spherical weak
scatterer,
in a wide band system, the correct ZOT is then the mid-point
between the
two 'delta function' surface responses and a step shaped
characteristic
function is obtained at the right ZOT. For real flaws, measured
using
bandwidth limited systems, the time signatures are more complex, as
is the
resulting characteristic function which is more complicated than
the
simple step; hence the problems associated with ZOT selection.
The
'correct' zero-of-time is conventionally sought when a nominal ZOT
has
been selected, from the time domain signature, by a series of
iterations
which look at features in the characteristic function; and small
changes
in the value selected for the ZOT are made.
28
.06 o .06 pSec
Fig. 1. Born Radius/ZOTS Domain signatures a. 0.1 to 4.2 ka (-)
and
b. 0.1 to 10 ka.~.
A feature of these band limited responses is that the Born
Radius
estimate given at the correct ZOT, for many cases, is not the
correct
radius but can have an error of 10% or more; by good fortune for
at
least some of the transducer/flaw~radiuska ranges that have been
used in
previous practical measurements the ka match is such that good
estimates
of the true radius are given when the correct value of ZOT has
been
identified.
BORN RADIUS/ZERO-OF-TIME SHIFT DOMAIN, WEAK SCATTERERS
To investigate the system response and the functions shown in Fig 1
two
further sets of data were produced. The impulse response function
was
simulated for the case of a 0.1-127 ka system, (0.5-614 MHz) for a
200~m
radius flaw and compression wave velocity 5960 m sec-', the
Born
algorithm was then applied to this data and its characteristic
function
29
obtained for a range of time shifts. The area function, impulse
response
function and characteristic function at zero time shift are shown
as Fig 2.
The radius estimate given as the correct ZOT for this bandwidth
were 197
and 198~m by the area under the curve and the 50% contour
techniques
respectively.
a.~ .1\-
Fig.2. a) The Area Function b) impulse response and c)
characteristic
functions for scattering by a weak spherical inclusion with a
bandwidth
0.1 to 127 ka.
The same process as the application of the Born algorithm was
also
performed using analytical integrations for the case of infinite
band
width (8,11).
eick't' ] (2)
for each value of the time shift~ one obtains an estimate for
flaw
radius, a "t' .
30
(3)
Here (r max.1r) is the maximum value of the characteristic function
for
a given time shift.
An exact analytical result for these equations is then given as
(8);
1-(x/2 1n(1+x/1-x)) 0<x<3/4
a(t)/re (l-x)/(l-2x) [1-(x/2 1n(1+x/l-x))] 3/4<x<1 (4)
2 (l-x) [1-(x/2 1n(x+1/x-1))] x>l
where x=ct/(2re ) is a scaled time. As seen in the fig 3 the
resulting
function is discontinuous at x=3/4 and assumes a minimum value
there.
An essential result is then the plot a(t) which gives two
independent
estimates of the radius. The first is the value of a at T=O; the
second
estimate is determined from the measurement of the time separation
of the
two minima.~T. The readius estimate is then a = c~T/3. The
first
estimate is most closely associated with the low-frequency part of
the
signal; the second is most closely associated with the delta
functions
which correspond to the flaws front and back surface echos. (which
is also
the data used in time domain sizing techniques such as SPOT).
This system of equations (4) was evaluated at flaw radius of 200~m
and
the ZOT varied. The data obtained for the two large bandwidth
calculations
together with the 0-10 ka data are shown as Fig 3. The separation
of the
two null points are in all cases found to be close to 0.105~sec.
When the
analysis is considered this null separation corresponds to a
time
~T = 3(a/c) and for this case a value of T=0.105~sec. is in good
agreement
with that seen in Figs 1 and 3; the observed null is at 3/4 of the
flaws
Born Radius.
.06 o .06 p5ec
Fig 3. Born radius/ZOTSD signatures for 200~m weak scattering
spheres for
various bandwidth. a. 0.1 to 10 ka (---), b. 0.1 to 127 ka (x)
and
c. infinite (.).
If the analysis of the response of a weak scatterer is considered
in terms
of the derivation of the impulse response from the areas function
and the
impulse response as its second derivative the signatures shown in
Fig 4
are obtained (2)
In the application of Born Inversion, followed by the determination
of the
BR/ZOTSD signature from the impulse response function, as
shown
above, it is seen that two integrations are involved. When the
connection
between the impulse response function and the new BR/ZOTS domain
signature
is considered one finds that the area function (see ref. 2) can be
related
to ~(r,1r) by (11);
" d "t'
Area ('t') const r dk e ik1: A(k) (5)
_to kl.
r 3r
Fig. 4. Pulse-echo scattering from a weak scattering sphere: a.
Area
function. b. first derivative of (a), c, second derivative of (a),
which
is the impulse response function.
The area underr(r,~ which we use to obtain the estimate for the
radius,
can be written in a closely similar form to the area
function.
Re lACk) e ik~
(6)
Only the case of the weak scatterer has been considered
analytically,
experimentally and using simulated scattering data. Both simulated
and
experimental data has been considered for both voids and strong
scattering
inclusions; in both cases the resulting signatures in the BR/ZOTSD
are
characteristic of the type of scatterer involved and can be related
to
feature dimensions.
33
The signatures obtained for a 200~m void, using simulated
scattering data
for limited and large bandwidth are shown as Fig. 5. Two
significant
differences between the weak (Fig 1) and this strong scattering
case
should be noted; in the band limited data one minima remains of the
same
form as that for a weak scatterer however the second minima becomes
a
'transition singularity'. What is more important is that the
separation
between the minima increases in the case of voids and this
indicates
either a change in the wave velocity and/or inpath taken by the
second
impulse.
When the bandwidth for spherical void data is increased to cover 0
to
30 ka the resulting BR/ZOTSD signature is as~own in Fig 5b; it is
also
an inverted parabola which is similar to the weak scattering case.
However
the area divided by the peak estimate at the correct ZOT is now
signifi
cantly below the correct estimate. The 50% contour radius estimate
gives
a value very close to 200~m. For the case of voids the contribution
due
to creeping waves is being investigated. These and other cases will
be
reported more fully in due course.
Radius
o -.1 o .1 -.1 o .1
Fig 5. BR/ZOTSD signature for a 200~m void with various
bandwidths;
a. 0.5 to 15 MHz. and b. 0.0 to 30.0 ka (0 to 100 MHz)
U sec
34
It has been found that the quality of data which is examined can
be
measured by the inter-comparison between the various Born Radii
estimate
techniques. Such measures for data quality have been found to be
useful
in identifying poor data, but are not able to significantly
increase the
confidence levels given to a selected Born Radii estimate.
When the selected ZOT is varied away from the true ZOT by a small
shift,
the Born radius estimate given varies. Data in this "Born Radius
Zero-of
Time Shift Domain" is now considered further and it is used to give
flaw
size.
The data for a weak scatterer in the Born Radius/Zero-of-Time Shift
Domain
has been reported previously (7,9) and similar data for a 0.1 to
4.2 ka
bandwidth for a 200~m inclusion (0.5 to 20 MHz) is given as Fig 1.
At
the correct ZOT a radius estimate of 179~m is found. When the
bandwidth
is increased to cover zero to 20 MHz the two radius estimates are
then 207
and 190~m by the area under the curve and the 50% contour
techniques
respectively.
When Fig. 1 and the curves obtained for a range of ka values are
examined
various observations are made. First, the detailed shape of the
BR/ZOTS
function is found to vary in a regular way with the transducer
bandwidth
and second, it has the functional form of (sin x)/x, band-limited
and
inverse Fourier Transformed. This functional form, although not
imme
diately recognised, is not to be unexpected as the kernel of the
Born
Inversion (see eqn 1) includes the function (sin x)/x, and it is
band
limited by the system as well as there being a Fourier Transform in
the
process.
When the bandwidth for waves incident on a 200~m inclusion is
increased
to cover 0.1-10 ka (0.5 to 100 MHz) the second curve, the
inverted
parabola, shown on Fig 1 is obtained. The two null points for each
of
the two ka ra
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