COMPUTATIONAL STUDY OF GRAIN SCATTERING EFFECTS IN
ULTRASONIC MEASUREMENTS
Anxiang Li, Ron Roberts, Pranaam Haldipur, Frank J. Margetan, and R. B. Thompson
Center for Nondestructive Evaluation, Iowa State University, Ames, IA 50011, USA
ABSTRACT. A two-dimensional scalar model has been established to study the scattering caused
by an inhomogeneous medium with granular microstructure. Time domain signals are calculated
using a 64-node PC cluster. Special test problems are set up to compare the effects of single and
multiple scattering and to quantify errors introduced by the Born approximation. Computed results
for backscattered RMS grain noise results are compatible with recent experimental measurements.
INTRODUCTION
The elastic inhomogeneity of materials introduces scattering noise in ultrasonic
measurements. Backscattering at grain boundaries produces the familiar noise floor seen
almost universally in ultrasonic measurements. Scattering in the forward direction produces
noise usually seen in signals generated by other scattering events, such as scattering by
flaws, or back wall reflections. For the backscattered noise, approximate models are
available which make a single-scattering assumption and employ the Born approximation
[1,2]. For forward scattering, no similar analytical models exist, and numerical
computation models must be pursued. In previous work [3,4] we developed a twodimensional computational model to simulate the distortion of an ultrasonic beam
propagating through a random medium. The computational procedure was based upon a
Neumann iteration scheme to solve the volume integral equation in a 2-D geometry. Using
this technique, we previously studied forward scattering effects including the generation of
phase aberrations, beam amplitude fluctuations, and flaw signal variations [3].
In this study we extend our model calculations to broadband time domain signals
using a 64-node PC cluster. Both forward and the backward scattering are considered. The
main objective of this work is to perform computational studies of microstructure induced
ultrasonic noise, and to assess the errors that are introduced by making single-scattering or
Born approximations. Our results also offer insights into the origins of experimentally
observed forward and backward scattering phenomena.
PROBLEM FORMULATION
Consider the propagation of a time-harmonic sound beam through a medium in
which the speed of sound c varies with position. The formal solution of the scalar wave can
be written as [3,4]:
= <{>"'(x')+£02 [(/>(x)V(X)G(\X-x'\)dx
(1)
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/$20.00
117
y
W = -T-—W
c0
c(xy
(2)
(/>(x):
the total pressure field at vector position x'.
m
(/) ( x ) :
the incident pressure field.
V(x):
the scattering potential.
c(x):
the spatially random varying wave speed of the medium.
c0:
the mean value of c ( x ) .
G(|JC - jc'|): the Green's function for the homogeneous mean medium.
The Neumann series expression for (j)(x) can be written as:
0 B+1 (jt') = (j)in(x') + co2 ^ ( / ) n ( x ) V ( x ) G ( \ x - x'\)dx
(3)
ln
where 0° = (/) . For an adequately weak scattering potential, the Neumann series will
converge after multiple iterations.
In addition to calculating the total pressure field within the inhomogeneous
medium, we also present results for the backscattered grain noise (voltage response) that
would be observed in a pulse-echo measurement.
This is done using Auld's
electromechanical reciprocity relation [5], which requires both the undistorted incident
wave field and the calculated distorted wave field. We assume the incident beam to be a
planar or focused Gaussian beam, and we arbitrarily set the system efficiency factor for
conversion of electrical energy to sound to be unity.
Broadband cases are solved by a series of time harmonic computations. We
generally assume the incident pulse to be a raised cosine function with the form:
- cos
cos
-" \~'t7*/
vv
J
0
„ . .3.0
~ —" —
r
otherwise
where t is the time in microseconds and f is the center frequency of the transducer in MHz.
We first obtain the spectral components of the input pulse by taking a discrete Fourier
transform, and we then perform the time harmonic calculation outlined in Eqs. (l)-(3) for
each of the equally-spaced discrete frequencies. The time domain signal is then recovered
by taking the inverse Fourier transform.
To optimize computational efficiency, parallel computation is implemented on the
64-node PC cluster at the Center for NDE of the Iowa State University. Two parallelization
schemes have been used. For calculations that involve a fixed microstructure, we can
divide the time harmonic calculations at different frequencies evenly among the 64 nodes.
Alternatively, when considering many related but different microstructures ("ensembles"),
we can divide ensembles equally among the PC nodes.
SPECIFICATION OF THE RANDOM MEDIA
In previous studies on forward-scattered noise [3] we represented the material
microstructure as a spatial distribution of sound velocities, with individual grains (regions
of constant velocity) modeled as rectangles whose edges were perfectly aligned with one
another (Figure la). Within each rectangle there were a fixed number of sampling points
(pixels) that defined the calculational grid. The aligned rectangular grain array is not
proper for modeling backscattered noise. Unphysical coherent reflections occur at each
aligned "row" of grain boundaries; these produce a series of voltage spikes which are
equally-spaced in time and not seen in UT measurements of grain noise in metals. Two
118
Tim el .Tim e 2, T im e 3
Total wavefield
tor homogeneous
medium.
Total wavefield for
random mecfum.
(a) ±4% vel ocit y
variation with
regular array of
pboels
(b) Large scale
variation: + 2%, sm all
scale variation + 2%
Total variation: + 4%
(c) ±4% velocity
vari ation with
random variations
in pixel location.
Total wavefield for
random medium.
AnplitudexlOO.
FIGURE 2. Modification of a propagation sonic
pulse by a random medium
FIGURE 1. Examples of model velocity variation
methods were developed to deal with this problem: 1) superimposing additional small-scale
random variations on the basic aligned, rectangular grain structure (Figure Ib); and 2)
randomly shifting the locations of the leading edge of each of the rectangular grains (Figure
Ic). Both methods effectively eliminate the coherent reflections.
Figure 2 displays one example of a model computation showing how microstructure
affects the forward propagating beam and generates backscattered noise. The assumed
random medium is that of Figure Ib. The pulse, generated by a focused transducer,
propagates from top to bottom and is shown at three successive times. In the lower panel,
the pressure amplitude has been multiplied by 100 to enable the weak backscattered
components of the field to be seen. The overall appearance is similar to the wave pattern
generated when a boat, traveling at the speed of surface waves, motors through seaweedinfested waters.
COMPARISION OF RESULTS FOR SINGLE SCATTERING, MULTIPLE
SCATTERING, AND THE BORN APPROXIMATION
Established models for characterizing and predicting grain noise from metal
microstructures are usually based on the Born (weak) scattering assumption and, in
addition, ignore multiple scattering effects [1,2]. Two basic questions arise. How
significant are the errors associated with the Born approximation itself? How large are the
multiple scattering effects? We defined several test problems to compare our full numerical
result to Born scattering (which is equivalent to ending our computation after the first
iteration). In the following examples, the term "full calculation" will refer to the converged
result of Eqs. (l)-(3) after multiple iterations, while "Born approximation" will refer to the
result of the first iteration only.
The calculations were carried out over an 8mm x 8mm square domain. The
assumed wave velocity of the host medium was 6000m/s, and the incident sound beam was
an unfocused Gaussian beam with half width w=2mm at the initial edge of the domain
(x=0). The incident pulse was assumed to be a raised cosine function with a central
frequency of 10 MHz. Two scatterers, denoted A and B, having different wave speeds from
119
O.OOE+O 5.00E-07 1.00E-06 1.50E-06 2.00E-06 2.50E-06 3.00E-06
0
Two scatterers located along the
beam axis. One (B) has wave speed
5% lower than host. Other (A) has
wave speed 5% higher than host.
Tlme(s)
8.GO&17 - ———————————————————————————
6.00E-174.00&172.00&17-
IQOO&OO-200&17-
Difference between multiple and
single scattering: (FAB-(FA+FB))
———vfr——
2.00E-16
1.00E-16
$ O.OOE-100
-1.00E-16-
•4.00&17-
—I^Hlr——
-2.00E-16 -
-6.00&17-
-aoo&i7
Difference between full
calculation and Born approx.
(FAB-BAB)
i
-..,-.,
,
,
,
0.00&0 500E07 1.00E06 1.80E06 200&06 250E06 300E06
0
O.OOE+0 5.00&07 1.00&06 1.50E-06 2.00E-Q6 2.50&06 3.00E-06
0
TimB(s)
Time(s)
FIGURE 3. Setup and results for the first calculation
the host medium were placed either along the incident beam axis (x direction) or
symmetrically off the beam axis (i.e., parallel to the y direction). The scatterers themselves
were 0.2mm x 0.2mm squares. When the scatterers were aligned along the x direction, the
coordinates of their lower left corners were (2.6, 4.0) and (5.2, 4.0) respectively in mm
units. When the scatterers were aligned along the y direction, the coordinates of their lower
left corners were (4.0, 5.2) and (4.0, 2.6). In both cases, the distance between the two
scatterers was 2.4 mm. We first calculated the wave fields by performing the first iteration
only (the Born approximation) and then continued the iterations until convergence was
obtained. In the figures below we use the following notations: FA (Full calculation when
only scatterer A is present), FB (Full calculation for scatterer B alone), FAB (Full
calculation when both A and B are present), BAB (Born approximation for both A and B
present), BA (Born approximation for scatterer A alone).
In Figure 3 we show the results from one of the test calculations. The scatterer
closest to the transducer has sound velocity 5% lower than that of the host medium while
the other scatterer has sound velocity 5% higher than the host.. One sees that the full
calculation (FAB) is dominated by the direct echoes from the two scatterers. Note that the
direct reflections from A and B are out of phase with one another due to the sign difference
in the velocity perturbations. The principal effect of including multiple scattering (in the
full calculation) is a waveform modification (FAB - (FA + FB)) that is large at the same
arrival time as the direct pulse from the deeper reflector (A). The Born solution leads to
inaccuracies in the single scattered amplitudes, as is evident from the error seen for the
early - arriving pulse (from B). The Born solution also neglects multiple scattering in that
the resultfor two scatterers is just the sum of the Born results for each scatterer separately
(BAB=BA+BB). The overall error resulting from the Born approximation (FAB-BAB) is
shown in the lower right panel of Figure 3.
120
2.00E-15 •
1.50E-15 •
Multiple scattering (FAB)
1.00E-15 5.00E-16 •
1 O.OOE-fOO -5.00E-16 -1.00E-15 •
-1.50E-15 •
-2.00E-15 •
0.(XDE+ 5.00E00
07
Two scatterers with velocity 12% higher
than the host are aligned in the y
direction.
1.00E-16
1.50E06
2.00E06
2.50E06
3.00E06
Tlme(s)
2.00E-15 -
2.00E-16
1.50E-16
1.00E06
Difference between multiple and
single scattering: (FAB-(FA+FB))
5.00E-17
1.00E-15 •
5.00E-16 -
————|j|W-————
! O.OOE400
| O.OOE+00 -
-5.00E-17
-5.00E-16 -
-1.0GE-16
-1.00E-15 -
-1.50E-16
O.OOE+O 5.00E0
07
Difference between full
calculation and Born approx.
1.50E-15 •
-1.50E-15 -
1.00E06
1.50E- 2.00E- 2.50E- 3.00E06
06
06
06
-2.00E-15 0.0()E+ 5.00E- 1.00E- 1.50E- 2.00E- 2.50E- 3.00E00
07
06
06
06
06
06
Time(s)
Time (s)
FIGURE 4. Setup and results for the second test calculation
Figure 4 shows results for a second test case in which both scatterers are at nearly
the same distance from the transducer, and both have a velocity 12% higher than the host
medium. The directly-scattered pulses from A and B arrive at the transducer at nearly the
same time and are in phase with each other. The principal effect of including multiple
scattering in the full calculation is a waveform modification (FAB - (FA + FB)) which is
large at the arrival time required for sound to travel the path from probe -> A -> B ->
probe. The Born solution leads to inaccuracies in both the single scattered amplitudes (as is
evident from the error seen for the early-arriving pulse), and completely misses the later
arriving (multiply scattered) pulse.
COMPUTATION OF RMS NOISE LEVELS SEEN IN FOCUSED-TRANSDUCER
INSPECTIONS.
In Figure 5 we present simulation results of the backscattered RMS grain noise
level for model microstructures with different levels of velocity variations. We assumed a
focused Gaussian beam with half width w = 4mm (6dB width). The broadband incident
pulse was again chosen to be a raised cosine function with a central frequency of 10 MHz.
The random medium occupied a 16mmx8mm rectangular region, and individual grains
measured 0.2mmx0.2mm. We assumed the sound velocity distribution to be uniform
within a fixed range, and performed calculations for three cases having velocity variations
of ±1%, ±2% and ±5%, respectively. For each of up to 500 microstructure "ensembles",
backscattered RF signal voltages (A-scans) were calculated using Auld's reciprocity
relation. From the A-scans, the rms noise level (at a fixed time t) was then obtained in the
usual way [2] by squaring the voltage values at time t for each ensemble, averaging the
squares, and extracting the square root. The resulting rms-noise-vs-time curves for the
121
Beam focus
(RMSFull-RMSBorn)/(RMSFull)
O.OOE+00
O.OOE+00
2.00E-06
4.00E-06
Hme(s)
(a)
FIGURE 5 (a) Computed RMS noise levels for 500 ensembles having ±5% random velocity variation,
(b) Fractional error in computed RMS backscattered noise when using the Born approximation.
"full" and "Born" calculations are shown in Figure 5a for the case with ±5% velocity
variations. The model curves peak at the round-trip travel time to the focal zone, as is
typically seen experimentally [2]. In Figure 6 we show the fractional difference of RMS
noise levels between the Born and the full results. Note that the Born model error increases
systematically with distance away from the focal depth.
QUALITATIVE COMPARISONS BETWEEN THE MODEL SIMULATION AND
RECENT EXPERIMENTAL MEASUREMENTS
The results of the RMS noise level computations are consistent with recent
backscattered noise measurements performed as part of a survey of the UT properties of
Nickel alloy billets [6]. A specimen from the center of an IN718 billet was found to have
auniform, eqiaxed, untextured microstructure. Backscattered noise A-scans were analyzed
to determine the so-called Figure-of-Merit (FOM), a frequency-dependent measure of the
noise generation capacity of the microstructure equal to the square root of the backscatter
power coefficient. The analysis procedure used to extract the FOM from noise data
assumed a single-scattering model for noise generation [2]. It was found that the deduced
FOM values, which should be independent of the details of the measurement, were in fact
sensitive to the time gate chosen for the analysis. The experimental setups and results are
summarized in Figure 6. In the first experiment the inspection water path was fixed and
grain noise data was gathered using a 15-MHz focused transducer. The resulting RMSnoise-versus-time curve is shown in Figure 6b, together with four time gates that were
subsequently used for FOM determination. The FOM-versus-frequency curves, deduced
by analyzing the spectral components of the backscattered noise within each time gate, are
122
(d)
RMS Noise Levels and Analysis Gates
RMS Noise Level and Analysis Gates
0.014 -i
0.012 i
Shallow Focus ___Deep Focus
(b)
6
8
10
12
14
Time (usec)
0.10 -i
Specimen GFM-A (Axial) Site 0"
7
8
9
10
Frequency (MHz)
11
12
4
(e)
(f)
5
6
7
Time (usec)
Specimen GFM-A (Radial) Site 0"
6
8
10
12
14
16
Frequency (MHz)
FIGURE 6. FOM Measurements for a Nickel alloy specimen, (a) (d): Measurement setups, (b) (e): RMS
grain noise profiles, with analysis gates indicated, (c) (f): Resulting FOM values.
shown in Figure 6c. One sees that backscattered noise data beyond the focal zone generally
led to a larger deduced FOM value than noise data from the focal zone. This is consistent
with the noise computations of the previous section, where the difference in noise levels
predicted by the full model (including multiple scattering effects) and the single-scattering
model was found to increase as one moved away from the focal zone. In the experimental
analysis, any increase in noise level beyond that expected from single-scattering alone
results in a larger deduced FOM value. To further investigate the dependence of deduced
FOM on gate choice, a second round of measurements was made using the same
transducer. As depicted in Figure 6d, the water path was varied to alter the depth of the
beam focus in metal. Shifting the beam focus downward caused the peak in the rms grainnoise level to shift to later arrival times, as expected. For each of the three water paths, the
123
FOM-vs-frequency curve was deduced using a time gate enclosing the focal maximum, as
shown in Figure 6e. The resulting FOM curves (Figure 6f) were very similar to one
another, and in good agreement with the earlier result that used noise data from the focal
zone. In addition to ignoring multiple-scattering events, the model used to extract FOM
values from measured noise data assumes that the transducer behaves like an ideal,
focused, piston probe. The observed dependence of FOM on the time gate shown in
Figure 6 may be due to multiple scattering events and the manner in which they contribute
to the observed noise in and away from the focal zone. Alternatively, the gate dependence
may be due to the accuracy of the "ideal piston probe" assumption; i.e., the transducer's
actual radiation pattern may be similar to that of a piston probe near the focal zone, but
depart from it systematically as one moves away from the focal zone.
SUMMARY
Time domain backscattered signals from simulated 2D metal microstructures were
calculated using a Neumann iteration scheme and Auld's reciprocity relation. Simple test
problems were considered to illuminate the errors that are introduced when multiple
scattering effects are ignored or the Born approximation is used. Model calculations of rmsgrain-noise-versus-time curves showed that Born results are systematically in error by an
amount that (1) increases with the level of velocity variability; and (2) increases as one
moves away from the focal zone (i.e., as the beam diverges). The model calculations are in
qualitative agreement with recent experimental RMS grain noise data.
ACKNOWLEDGEMENTS
This material is based upon work supported by the Federal Aviation Administration
under Contract #DTFA03-98-D-00008, Delivery Order #029 and performed at Iowa State
University's Center for NDE as part of the Engine Titanium Consortium program through
the Airworthiness Assurance Center of Excellence.
REFERENCES
1. J. H. Rose, "Ultrasonic Backscatter from Microstructure", Review of Progress in
Quantitative Nondestructive Evaluation, Vol. 11B, edited by D. O. Thompson and D.
E. Chimenti, (Plenum Press, New York, NY, 1993), p 1677.
2. F. J. Margetan, R. B. Thompson, and I. Yalda-Mooshabad, "Modeling ultrasonic
microstructural noise in titanium alloys," Review of Progress in Quantitative
Nondestructive Evaluation, op.cit. Vol. 12B, p. 1735. (1993).
3. A. Li, R. Roberts, F. J. Margetan, and R. B. Thompson, "Influence of Forward
Scattering on Ultrasonic attenuation Measurements", Review of Progress in QNDE,
op.cit. Vol. 21 A, p. 51. (2002)..
4. S. Ahmed R. Roberts and F. Margetan, "Ultrasonic Beam Fluctuation and Flaw Signal
Variance in Inhomogeneous Media", Review of Progress in QNDE, Vol. 17B, op.cit.
(2000) p. 985.
5. B. A. Auld, "General Electromechanical Reciprocity Relations Applied to the
Calculation of Elastic Wave Scattering Coefficients", Wave Motion 1, 1979, p3.
6. P. Haldipur, F. J. Margetan and R. B. Thompson, "Correlation between Local
Ultrasonic Properties and Grain Size within Jet-Engine Nickel Alloy Billets", these
proceedings.
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