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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
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© Springer-Verlag Berlin Heidelberg 1988 Soitcover reprint of the hardcover 1st edition 1988
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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
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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