To be published in: Review of Progress in Quantitative
Nondestructive Evaluation, Vol. 13, eds. D.O. Thompson
and D.E. Chimenti, Plenum Press, NY, 1994
SPECTRAL REDUNDANCY IN CHARACTERIZING SCATTERER STRUCTURES
FROM ULTRASONIC ECHOES
Kevin
D.
Donohue,
Department
of
Electrical
University
of
Lexington,
KY
Nihat
M.
Varghese
Engineering
Kentucky
40506
Bilgutay
Department
Drexel
Tomy
of
Electrical
and
Computer
Engineering
University
Philadelphia,
PA
19104
INTRODUCTION
Broadband ultrasonic back-scattered energy contains information regarding material structures from which it was scattered.
Signal processing techniques use charac-
teristic changes in the back-scattered signals to detect scatterers or reflectors that represent flaws and impurities within a manufactured part.
A-scans from materials such as
stainless steel, titanium, and composites comprised of dense layers or fibers, exhibit significant scattered energy at positions throughout the material.
Consequently, signal proc-
essing methods are needed to filter out signals characteristic of normal material echoes
and enhance echoes from defective structures.
Several signal processing techniques have been applied to the problem of material noise suppression and flaw detection.
Among them are the Wiener and matched fil-
ters, frequency compounding algorithms associated with split-spectrum processing, and
maximum-likelihood estimation.
In cases where scatterer size differences exist between
defective structures and material scatterers, the spectral content of their echoes differs
due to the frequency dependence between scattered energy and scatterer size [1].
A opti-
mal approach in this case is the Wiener filter which operates on spectral differences between the ultrasonic flaw echo and the material noise.
The Wiener filter application re-
quires a priori knowledge of the power spectral density (PSD) characterizations of both
noise and defect echoes, along with a sensitivity factor related to the expected signal-tonoise ratio (SNR).
The filter achieves optimal performance when an accurate SNR fac-
tor and PSDs are used.
The matched filter also uses the PSD characterization for the noise, however, the
flaw signal is modeled deterministically (using both magnitude and phase spectrum).
The requirement for a deterministic signature practically limits the matched filter to
flaws modeled by a coherent scatterer, where a linear pattern in the phase spectrum can
1
be used.
It is expected that the matched filter will perform better than the Wiener filter,
since the matched filter uses additional phase information.
The performance of the
matched filter, however, is very sensitive to the accuracy of the phase spectrum [2].
In
cases where the flaw comprises several scattering centers, a consistent model for the
phase spectrum is practically impossible to obtain.
Split-spectrum techniques also exploit differences in the spectral energy from
flaw and material scatterers in addition to using simple phase patterns [3].
Several non-
linear processing algorithms exploit linear phase patterns resulting from coherent flaw
echoes.
The many parameters for split-spectrum processing methods allow for greater
flexibility, however, significant performance degradation can occur for small parameter
deviations outside an acceptable parameter range.
This is particularly true for the non-
linear algorithms. Parameters are usually set by empirical optimization on a training set
of data, however, adaptive and robust methods have been proposed to limit the problem
of parameter sensitivity [3].
The maximum-likelihood estimator (MLE) and the matched filter both model the
flaw signal parameters deterministically.
The MLE approach describes the material and
material-plus-flaw echoes with a Gaussian distribution.
The parametric signal model
was used to develop an adaptive MLE implementation that estimates the PSD components over local A-scan regions to adjust for nonstationary noise characteristics due to
the attenuating pulse [4].
trix used in the MLE.
The PSD values comprise the diagonal of the covariance ma-
Performance results comparable to that of the matched filter are
obtained for enhancing echoes from coherent flaw scatterers.
A significant part of the SNR improvements for the above techniques result from
distinctions between the PSD’s of the material and flaw echoes.
The frequency empha-
sis in the PSD for the most part characterizes the scatterer size, while phase spectra relate to structure in the scatterer configurations.
The direct use of phase information,
however, is limited by the sensitivity of phase spectra to small changes in relative scatterer positions.
Thus, reliable phase characterizations are desirable that can discriminate
between scatterer structures related to the flaw and material echoes.
This paper considers a particular phase characterization introduced in an extension of the MLE that included spectral correlation.
A comparative study on A-scans
from flat-bottom holes embedded in stainless steel demonstrates 3 to 6 dB flaw signal
improvement for the MLE that used a phase characterization as opposed to the MLE that
simply used the PSD of the noise [5].
The phase characterization involves correlating
discrete Fourier transform (DFT) components from different frequency bins.
For a wide-
sense stationary (WSS) noise process, the correlation between different frequency bins is
zero [6].
The WSS process implies the signal has no structure (i.e. a uniform distribu-
tion for the phase spectrum).
Thus, non-zero DFT correlation terms from different fre-
quency bins (referred to as spectral redundancy [7]) indicate the presence of structure or
consistency in the echo signature.
While the performance improvements reported in [5] were attributed to the presence of structure, the paper provided no general descriptions of changes in the unresolvable scatterer configurations and their effect on spectral correlation.
This paper demon-
strates a relationship between the average number of uniformly distributed scatterers per
resolution cell and spectral correlation.
In addition, cases are considered where structure
is embedded into the unresolvable scatterers in the form of quasi-periodic spacings.
2
The next section describes a scattering model and presents an analytical relationship between scatterers spaced with a Gamma distribution and the resulting spectral correlation in the received signal.
The following section presents simulation results to dem-
onstrate a relationship between the number of scatterers per cell and spectral correlation.
Finally, conclusions are presented regarding the use of spectral redundancy in ultrasonic
NDE signal processing.
SCATTERER SPACING AND SPECTRAL CORRELATION
In order to understand the relationship between scatterer structures and spectral
correlation, this section presents an ideal scatterer model where only variations in scatterer strength and relative position are considered.
Contour plots of the resulting spec-
tral autocorrelation (SAC) function will demonstrate the relationship between spectral
correlation signatures and the scatterer spacings. Results in this section do not include effects from the system or frequency dependent attenuation.
Simulation results in the next
section, however, will include such effects.
Denote the strength and position of scatterers over a finite interval time T
as a
train of scaled and delayed Dirac Delta functions:
() =
g t
M
∑ m δ(
m=
a
t
− τm)
(1)
1
m
where a
and
τm
represent the scattering strength and delay associated with each of the
M scattering centers.
T(
G
f
The Fourier transform of Eq. (1) with respect to t becomes:
) =
M
∑m
m=
a
(−j2πf τm)
exp
(2)
1
T(
The SAC function is obtained by correlating G
f
) with
its complex conjugate:
M M



ST(f1, f2) = E [ GT(f1) GT(f2)] = E ∑ ∑ an am exp(−j2π (f1τn − f2τm))


n=1m=1



∗
(3)
⋅
where f1 and f2 are coordinates in the bifrequency plane, and E[ ] is the expected value
operator.
Note that SAC function values along the diagonal of the bifrequency plane
(where f1 = f2) are equivalent to the PSD values.
Off-diagonal SAC function values
include a non-zero phase component, which is the phase difference between Fourier components at different frequencies.
To consider the effects of regular scatter spacing within an A-scan segment, let:
∆m =
where
τm
τm
(4)
m
is the time corresponding to the m
of the finite segment.
While
∆m
th
scatterer position relative to the beginning
th
is not the actual scatterer spacing at the m
3
position , it
is a good approximation for small
scatterer spacing variance and leads to a tractable
solution for the expected value.
Substitute
τm out of
Eq. (3) to obtain:
M M

ST(f1, f2) = E ∑ ∑ an am

n=1 m=1



exp(−j2π (f1n∆n − f2m∆m))



In the deterministic case where all scatterer spacings are equal,
∆ can
.
(5)
be factored out to
obtain:
M M

ST(f1, f2) = E ∑ ∑ an am

n=1 m=1

Note that when
(f1n − f2m)
phase, and a local


exp(−j2π ∆(f1n − f2m))



is an integer multiple of
maximum occurs.
bifrequency plane when f1 and f1
For Gamma distributed
− f2
∆is
1⁄∆,
.
(6)
all terms in the summation are in
In general, local maxima occur throughout the
are integer multiples of
1⁄∆.
the expected value has been computed in [5].
Figure 1 presents a contour plot for the magnitude of the SAC function with a standard
deviation for scatterer spacing equal to 10% of the mean scatterer spacing.
In this case
local maxima are expected both along the PSD and throughout the bifrequency plane at
−
positions f1 and f2 where f2 f1 are integer multiples of
bring out the detail in the off-diagonal area.
shown.
1⁄∆.
This plot was thresholded to
Thus, variations along the PSD are not
In this case, however, the PSD was almost flat except for an observable maxi-
mum at 10 MHz. The expected local maximum at 20 MHz (f
to the variance in the scatterer spacing.
= 2⁄∆ )
is not observed due
For decreasing scatterer spacing variance,
local maxima in the SAC function become more distinct [5].
the
Note in Figure 1 that local
maxima occur at the correlation points for 10 and 20 MHz, which correspond to the
proper scatterer spacing of .1
µs
-1
= (20 MHz - 10 MHz)
spreading around the PSD region at 10 MHz.
Finally, take note of the
For more regular scatterer spacing the
magnitudes in this area will increase, which are seen in the simulation results in the next
section.
When system effects are considered, the finite bandwidth limits the window in
the frequency domain.
by the pulse width.
As a result the minimum detectable scatterer spacing is limited
In general, the resultant SAC of the received signal is the product of
the SAC functions for the system response and the scatterers.
If the segment length T is
several times greater than the pulse width and includes resolvable scatterer spacings, the
computed SAC functions will resemble the one in Figure (1) with noticeable peaks in
the off-diagonal areas.
If T is chosen on the order of the pulse width (as in [5]), then the
only observable effect will be the correlation properties close to the PSD.
The next sec-
tion considers the effect of regular scatterer spacings when the scatterers are unresolvable.
While distinct peaks cannot be observed in this case, the over all correlation
around the PSD region can be quantified.
4
25
20
z 15
H
M
10
5
5
15
10
20
25
MHz
Figure
1.
SAC function magnitude for Gamma distributed scatterer spacing with
µs, and standard deviation equal to 10% of the mean.
mean 1
UNRESOLVABLE SCATTERER STRUCTURES
This section describes a simulation that demonstrates a relationship between the
average number of scatterers per resolution cell and a spectral correlation measure.
Since the effect of unresolvable scatterers is considered, SAC functions are computed
over segments lengths, T, equal to the pulse width of the interrogating energy.
The SAC
function corresponding to an A-scan region centered at t is computed from overlapping
A-scan segments given by:
M⁄2
T(
S
k1, k2 ; t
T( i
where G
tered at
) =
k ; t
T
t+m ⁄J,
∑ T(
G
m=−M⁄2
+ mT⁄J)
k1 ; t
is the i
th
 j2π(k1 − k2)
T
T
T 
+ m ) G∗T(k2 ; t + m ) exp
(τN − )
J
J
N
2


DFT component from a time segment of duration T cen-
and J denotes the degree of overlap between adjacent segments.
SAC functions computed in this section, J was set to 4 (75 % overlap).
factor with parameter
(7)
τN represents
For the
The exponential
a phase correction.
If a relatively strong maximum occurs within the window, a linear phase pattern
is generated whose slope is proportional to the maximum’s position relative to the center
of the windowed segment.
In cases where only a few scatterers per segment exist, this
effect slows down the convergence of the SAC function over the averaging process.
Thus, for each segment the position of the maximum value,
τN,
is determined for the
phase correction factor, which cancels out the linear phase effect by shifting the maximum to the center of the window.
The SAC function was computed over each simulated A-scan by applying Eq.
(7) over the entire scan.
An average spectral correlation measure was then computed ac-
cording to:
5
N2
N2
ρµ(t) =
∑
1
N
µ
S^T(k1, k2;t)
∑
(8)
)
k1 = N1 k2 = N1 √
) ST(
T(
k1≠k2
^
S
k1, k1;t
^
k2, k2;t
where N1 and N 2 correspond to the 6 dB cut-off points of the A-scan spectrum, and N
the total number of off-diagonal components used in the summations.
µ
is
The normalization
based on the PSD values eliminates variations due to power changes from A-scan to
A-scan.
The A-scans were simulated for a 10 MHz transducer with 6 MHz bandwidth, and a
transducer pulse with a Rayleigh shaped envelope.
The material comprised scatterers of
5.8 km/s.
A 100 MHz sampling rate was applied.
0.32 mm in diameter with the velocity of sound at
The resolution cells consisted of 44 time domain samples (4 DFT components
in 6 dB bandwidth) and each A-scan contained 30 resolution cells.
Frequency dependent
attenuation was also included in the simulation.
Figure 2 presents the average spectral correlation measures for uniformly distributed scatterers.
Each point on the graph corresponds to the average number of scatterers
per cell, which were Poisson distributed with means 1, 2, 3, 4, 5, 6, 7, 8, 10, 15, and 20.
Each point represents the results from 35 simulated A-scans.
viation of
ρµ were computed and plotted on
the errorbars,
Based on the widths of
it appears that cells with scatterer numbers less than 4 can be consistently
discriminated from those cells with 10 or more scatterers.
scatterers
The mean and standard de-
the errorbar graph.
ρµ converges near 0.2.
For an increasing number of
Ideally this should go to zero, however, spectral en-
ergy exists in the off-diagonal regions due to spectral leakage from the finite windows.
If this leakage can be reduced through more efficient windows, better discrimination
may be possible.
Figure 3 presents the average spectral correlation when the resolution cell scatterers have Gamma distributed scatterer spacings.
the spacing regularity.
Note in these cases that
Figures 3a and 3b represent increasing
ρµ
increases more abruptly in the transi-
tion from 2 to 3 scatterers than for the uniformly distributed case.
Thus, only cells with
1 and 2 scatterers can be consistently discriminated from cells with 3 or more scatterers.
Also note in Figures 3a and 3b that as the regularity increases, another local
maximum occurs.
The reason for this is seen in Figure 1.
tral correlation around the PSD region near the
1⁄∆
point.
Recall the increase in specIn the simulation the mean,
was adjusted to allow for proper number of scatterers per cell.
∆,
As a result, this high cor-
relation area moved through the frequency window around 10 MHz, which corresponded
to the transducer center frequency.
Cells with average scatterer numbers 5 through 8
have high correlation regions corresponding to 7.2 MHz through 11.6 MHz, respectively.
While this result is interesting, in that inferences can be made concerning order
in the unresolvable structures, it presents problems in trying to determine scatterer concentrations based on
ρµ.
For example, in Figure 3b cells with an average of five scatter-
ers may have high enough correlation energy to be taken for 2 scatterers per cell.
ther work is needed to develop methods for eliminating ambiguity in these cases.
6
Fur-
0.45
Average Spectral Correlation
0.4
0.35
0.3
0.25
0.2
0.15
0
Figure
2.
5
10
15
20
Average Number of Scatterers Per Resolution Cell
25
Average spectral correlation vs. average number of scatterers per resolution
cell for uniformly distributed scatterer positions and Poisson distributed scatterer numbers.
CONCLUSIONS
This paper demonstrated a relationship between two special cases for scatterer
structure and spectral correlation.
For decreasing numbers of uniformly distributed scat-
terers per resolution cell, more structure is apparent in the echo signatures, which results
in increased spectral correlation.
This result can be useful for detecting flaws with multi-
ple scattering centers (less than 3 or 4) embedded in grain structures that are comprised
of significantly more scattering centers.
When the unresolvable scattering centers have
regular spacing (as may be the case for some composites), ambiguity can occur in discriminating based on scatterers per cell.
The MLE described in [5] is an example of a signal processing algorithm that incorporates spectral correlation.
The MLE approach is limited, however, by requiring a
deterministic model for the flaw signal.
Other algorithms need to be developed that
model complex flaws as random variables and incorporate spectral correlation.
ACKNOWLEDGMENTS
This material is based on work supported in part by the National Science Foundation under Grant No. MIP-8920602, and the National Cancer Institute and National Institutes of Health Grant No. CA52823.
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0.45
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Average Spectral Correlation
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Average Number of Scatterers Per Resolution Cell
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