ATTENUATION COEFFICIENT ESTIMATION USING
EQUIVALENT DIFFRACTION POINTS WITH MULTIPLE
INTERFACE REFLECTIONS
T.P. Lerch1 and S. P. Neal2
Industrial and Engineering Technology Department, Central Michigan University, Mt.
Pleasant, MI 48859
Mechanical and Aerospace Engineering Department, University of Missouri - Columbia
Columbia, MO 65201
2
ABSTRACT. The ultrasonic attenuation coefficient of a fluid or solid material is an acoustic parameter
routinely estimated in nondestructive evaluation (NDE) and biological tissue characterization. In this
paper, a new measurement and analysis technique for estimating the attenuation coefficient as a function
of frequency for a fluid or solid is described. This broadband technique combines two established
concepts in attenuation coefficient estimation: (1) frequency spectrum amplitude ratios of front surface,
first back surface, and second back surface reflections from interfaces of materials with plate-like
geometries, and (2) equivalent diffraction points within the transducer wave field. The new approach
yields estimates of the attenuation coefficient, reflection coefficient, and material density without the
need to make diffraction corrections. This simplifies the overall estimation process by eliminating the
transducer characterization step, that is, by eliminating experimental characterization of the effective
radius and focal length of the transducer which are required when careful calculated diffraction
corrections are applied. In this paper, attenuation coefficient and reflection coefficient estimates are
presented for water and three solids with estimates based on measurements made with two different
transducers.
INTRODUCTION
The ultrasonic attenuation coefficient of a medium is an acoustic parameter
routinely estimated in nondestructive evaluation (NDE) and biological tissue
characterization. Knowledge of the ultrasonic attenuation of a given material is useful to
the NDT field inspector searching for flaws in various structural materials, the material
scientist characterizing the mechanical properties of the material, and the biologist
investigating the acoustic properties of various types of biological tissue.
One of the challenges associated with making accurate attenuation coefficient
measurements is to separate the energy loss due to absorption and scattering within the
medium from other possible sources of energy loss including those due to reflection and
transmission at interfaces, diffraction of the transducer's wave field, measurement system
inefficiencies, and misalignment of the transducer and specimen. In this paper, we will
consider four attenuation coefficient estimation approaches (see Table 1): 1) a Classical
Approach driven by the ratio of magnitude spectra from two interface reflections; 2) the
Papadakis Approach which eliminates the need to make explicit corrections for reflection
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
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TABLE 1. Summary of attenuation coefficient estimation approaches.
Classical
Papadakis
Equal
Diffraction
New
Approach
input
output
input
Water
Attenuation
cancel
cancel
input
Diffraction
Corrections
input
input
output
input
cancel
System
Effects
cancel
cancel
cancel
Solid
Thickness
input
input
input
Wavespeed
in Solid
input
input
input
Solid
Density
input
output
input
R&T
Coefficient
cancel
input
input
output
cancel
and transmission losses by utilizing three interface reflections [1]; 3) an Equal Diffraction
Point Approach which adjusts the water path to eliminate the need for diffraction
corrections [2-4]; and 4) and a New Approach which combines the Papadakis and Equal
Diffraction Point approaches to simultaneously estimate reflection, transmission, and
attenuation coefficients without making diffraction corrections. Corrections are, however,
required for water attenuation due to variable water path lengths. The water attenuation
coefficient is easily calculated based on the widely accepted work of Pinkerton [5].
Conversely, correcting for transducer diffraction requires full characterization of the
transducer's parameters (radius and focal length) across the transducer's useful bandwidth.
Transducer characterization can be a very time- and labor-intensive process. Since each
transducer has its own unique set of parameter values, the characterization process must be
implemented for each transducer used to make a measurement.
This paper will proceed with a model-based review of three existing attenuation
coefficient estimation approaches introduced above. Models which describe the New
Approach for the estimation of solid and fluid attenuation coefficients will then be
presented. Results will be shown for attenuation and reflection coefficient estimation for
water and for three solids. The paper concludes with a brief discussion section.
REVIEW OF ATTENUATION COEFFICIENT ESTIMATION APPROACHES
Classical Approach
Consider a solid material sample of plate-like geometry interrogated at normal
incidence in an immersion mode in water. A Classical Approach for estimation of the
attenuation coefficient for the solid involves measurement of a first back surface reflection
along with a front surface reflection and/or a second back surface reflection. Using a linear
time-invariant system modeling approach, the Fourier transform of the measured front
surface reflection can be modeled as:
= p(f)Rwsc(2Zwf,f)exp(-2Zwfaw(f))
(1)
We adopt a simplified notation throughout the remainder of the paper with frequency
dependence implicit and with each symbol representing the absolute value of its associated
complex quantity. The Fourier transform of the front surface reflection, F ( f ) , becomes:
F = j3Rc(2zwfy
(2)
where f i , the system efficiency factor, accounts for all transducer and electronics related
effects, R — Rws is the water-to-solid reflection coefficient, zwf is the water path length
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for the front surface reflection experiment, C\2zwf J accounts for beam diffraction in the
water, and aw is the attenuation coefficient in the water. Noting that the product of
r\
transmission coefficients, T^STSW, can be written as l-R , the first and second back
surface reflections can be modeled as:
Bl = /?(l - R2 ]RC(2zwbl }e~2z^a™ C(2zs >T2z ^
(3)
B2 = j3l-R2R3C(2zwb2)e-2z^a-C(4zs)e^z^
(4)
where zs is the plate (solid) thickness, as is the attenuation coefficient in the solid, and
zwbl and zwb2 are the water path lengths for the first and second back surface reflections. In
Equations (1) - (4), we assume that (3 - fif - fi^i - fib2 •
The solid attenuation coefficient can be estimated using any two (or all three) of the
measured signals. The diffraction terms are often calculated for the water/solid case by
replacing the two diffraction terms in (3) or (4) by a single diffraction term, C(2zwe ) , with
the equivalent water path length, zwe , calculated as follows:
c
\ve ~ zw ~*~ uc zs
w
z
c
^z\vebl ~ ^zwbl " • uc*
"^zs
w
^Zweb2
=
c
^Zwb2 ~ ^ u~c ^ z s
w
v^/
where zw = z^ = zw^; - zw^2 f°r fixed water path, cs and c^ are the wave speeds in the
solid and water, respectively, and zwe - z w since z5 = 0. We can now solve for as using
F and BI or using BI and B2 as follows:
F
C(2zw2
)
- ____
——
D
.Oi
or
C(2zwM)(l-R2)
<* =
1l ,n C(2zwM]
—
C(2zweb2)R2
The front surface reflection is corrected for diffraction in the water, and the back surface
reflections are corrected for interface losses and for diffraction in the water and solid.
Papadakis Approach
The Papadakis Approach uses the front surface and the first two back surface
reflections to eliminate fi and simultaneously estimate R and as . The ratio of spectra
corrected for diffraction is used to yield two new quantities denoted Ml and M2 by
Papadakis.
l-R2
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F
ii transducer
it
T fr
1 1
Hi
(
1 2Azw
2V
Motor
Controller
water
T
?
m
ill
J|
t
?
WiiKi
Hill
i
FIGURE 1. Typical immersion system depicting the measurement approach for the New Technique. The
transducer is not translated laterally as the figure implies.
These two equations are then solved for R and as as follows:
M1-M2
1 + M1-M2
R=
1
Ml
a, =- In2z p 1 + M1-M2
(8)
Equal Diffraction Point Approach - Solid Attenuation Coefficient Estimation
The Equal Diffraction Point Approach involves adjusting the water path (see Fig.
1) so that the equivalent water path length is the same for each reflection. The penalty is
that the aw must be known, and a correction of form exp(2zwaw) must be applied to each
reflection. With the water path for the front surface reflection used to dictate the value for
zwe (that is, zwe = zwf), the equalities given in Equation (5) can be used to solve for the
required
water
path
for
B\
as
zwbi = zwf-(cs/cw)zs
and
for
B2
as
Z
wb2 ~ zwf ~(cs/cw)^zs • The * superscript is introduced to indicate that the water paths
are associated with an equal diffraction point approach. The change in water path (Fig. 1)
*
between
reflections,
Az,,
by
successive
is
given
*
/
/
\
*
z
*
z
*
Z
*
wf -- zwbl
wbl =
=- wbl ~ \vb2 w = (cs /cw)zs = zwf
Incorporating these ideas, the water
attenuation terms for the back surface reflections can be re-written as follows:
-2z*wb2aw
=
-2
(9)
The key to this approach is that the diffraction terms, C\2zwe], are equivalent for each of
the three measured reflections. Folding the equalities in Equation (9) into Equations (1) (3), canceling the common diffractions terms, and solving for as yields:
F
2z
-In-
or
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2z,
(10)
Note that the relatively difficult to implement diffraction corrections in Equations (6) and
(7) are replaced by water attenuation corrections in Equation (10).
Equal Diffraction Point Approach - Fluid Attenuation Coefficient Estimation
The same basic approach used for estimation of as can be used to estimate the attenuation
coefficient in a fluid using a quartz specimen as the solid with known, essentially zero,
attenuation. With as = aq « 0 and exp(2zqaq)-^ 0 , solution for aw yields:
I
o A
D
1
^log———LT-
2Azw
r-'/i
or
r»/\
7
?
aw =——^log—2y
^ A
F(l-R )
(ii)
2Azw
A NEW ATTENUATION COEFFICIENT ESTIMATION APPROACH
Application to Attenuation Coefficient Estimation for a Solid
By using the front surface reflection and the two back surface reflections, with
measurements made at equal diffraction points, we can eliminate J3 and simultaneously
estimate R and as without making diffraction corrections. We start with the three
reflections, each corrected for water attenuation. With slight notational changes to indicate
that equal diffraction point measurements are being used, we then follow the Papadakis
approach as given in Equations (7) and (8) to reach the new estimation form for as :
"*
V
7?Z
-2z^h'yava
BI
l-R*
2
1-R
B,
Ml -Ml
\l + Ml*-M2*
'
- = ^r-
1
2zs
02)
Ml*
1 + M1*-M2*
The New Approach yields estimates of R and as ; however, the diffraction corrections in
Equation (7) are replaced by water attenuation corrections in Equation (12).
Application to Attenuation Coefficient Estimation for a Fluid
The same basic approach can be used to estimate the attenuation coefficient in a fluid given
a solid sample with known attenuation. Again, for illustrative purposes, we use water and
quartz with the following equations yielding estimates of R and aw:
M1
»=^
*——^-
*=^ = ———,——
M2
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(14)
i,i »,<,
M7 -M2 =———-
D
M1*-M2*
JR = J————*———sr
*
*
I , 1 + M1*-M2*
a wW =———In————————
M l *
(15)
EXPERIMENTS AND DATA ANALYSIS
The New Approach was used to estimate the reflection and attenuation coefficients
for water and three solids of plate-like geometry: stainless steel (zs = 1.28 cm), fused
quartz (zs = 0.64 cm), and plastic (zs= 0.73 cm). The apparatus employed for these
measurements is typical of most ultrasonic immersion inspection systems (see Fig. 1). All
equipment is commercially available. The transducer is driven by a pulser/receiver unit
and positioned with the motor controller. The rf signals are captured by the data
acquisition card on the PC and ultimately transferred to a work station for data analysis.
Three wave trains, each containing the A-scan time pulses from the front, first back,
and second back surface reflections, are digitally captured. The measurement process
begins by setting the water path at the desired length for the front surface reflection. At this
water path, the wave train is digitized and stored on the data acquisition PC. The
transducer is then axially translated toward the specimen a distance equal to Azw to place
the first back surface reflection at an equivalent diffraction point to that of the front surface
reflection. The resulting wave train is digitally captured and stored. The transducer is
again axially translated a distance of Azw toward the specimen in order to place the second
back surface reflection at the equivalent diffraction point for the first two reflections. As
before, this wave train is digitized and stored.
Data analysis is performed with software written and stored on a separate
workstation. Inputs include the three, digitized wave trains measured at equivalent
diffraction points, the wave speeds of the water and the solid, the water attenuation (when
the attenuation of a solid is measured), and the thickness of the specimen. Individual
signals are extracted from the wave train with a rectangular window and then transformed
into the frequency domain with a standard FFT routine. Equations (12-13) or (14-15) are
used to determine the reflection and attenuation coefficients, each as a function of
frequency, based on the magnitude spectra of the three reflections.
DISSCUSION OF RESULTS
The results of the series of measurements implementing the New Approach are
shown in Figures 2 and 3, where Fig. 2 summarizes the results for water attenuation
measurements and Fig. 3 summarizes the results associated with attenuation coefficient
estimation for fused quartz, stainless steel, and plastic. Reflection and attenuation
coefficients for each material are measured with two unfocused-transducers: a 10 MHz, 1/4"
diameter transducer and a 15 MHz, V£" diameter transducer.
Figure 2 shows the experimental reflection coefficients for the water-fused quartz
interface and the attenuation coefficient for water, each as a function of frequency. The
experimental reflection coefficients found with both the 10 MHz %" and 15 MHz V2"
transducers are basically constant across the useful bandwidth of each transducer and
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Water/Quartz Reflection Coefficient
Water Attenuation Coefficient
0.15
0.35
0.1
0.9
0.05
0
0.75
-0.05
0.7
-0.1
Reflection Coefficient with Diffraction Error
0.15
Attenuation Coefficient with Diffraction Error
0.1
0.05
0
-0.05
-0.1
0.65
0.6
10
12
Frequency (MHz)
10
12
Frequency (MHz)
FIGURE 2. Experimental results for fluid attenuation coefficient estimation using the New Approach.
compare well to the theoretical value. The attenuation coefficient estimates shown in the
upper right graph compare well to one another and to Pinkerton's widely accepted result
[5]. The three water paths used to achieve equal diffraction measurements are 25.4, 22.8,
and 20.2 cm for the front, first back, and second back surface reflections, respectively.
These water paths were used for both transducers to further demonstrate the robustness of
the approach. For the 10MHz /4" transducer, these water paths place the measurement
point its far field, while for the 15MHz W transducer, the water paths correspond to the
near field.
The lower two graphs in Fig. 2 demonstrate what happens when incorrect
equivalent diffraction points are chosen. Notice the deviation from theory, especially the
frequency dependence, in the experimental reflection coefficient which has been caused by
the diffraction error. In this instance, the diffraction error creates an additional perceived
loss of energy which the data analysis assigns to the water attenuation coefficient, resulting
in an overestimation of the water attenuation coefficient as shown in the lower right graph.
As seen in Fig. 3, the experimental reflection coefficients for the water-stainless
steel and water-fused quartz interfaces are also relatively constant across the useful
frequency spectra of both transducers. Although slightly oscillatory in nature, the
experimental reflection coefficients for the plastic also tend to be constant. Attenuation
coefficients for the three different solids are also shown in Fig. 3. These solids were
chosen because of their relatively wide range in attenuations, from fused quartz with no
apparent attenuation to a plastic with a substantial attenuation coefficient. Because
attenuation is very sensitive to material properties such as grain size and alignment, it
becomes very difficult to compare these results to a generally accepted standard. Notice
however the robustness of the new technique in returning consistent attenuation
coefficients estimates for the two transducers, without transducer characterization or the
formal application of diffraction corrections.
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Reflection Coefficients vs. Frequency
Attenuation Coefficients vs. Frequency
0.9
fO.B
I 0.7
o
I 0.6
u
1 0.5
0.4
0.
Q.G
O.B
1
1.2
Frequency (Hz)
1.4
x
1.6
^
FIGURE 3. Experimental results for solid attenuation coefficient estimation using the New Approach. Data
acquired with the 10 MHz transducer is represented with 'o'; the 15 MHz transducer is represented with '•'.
CONCLUSIONS
A new measurement and analysis technique for estimating the attenuation
coefficient as a function of frequency for either a fluid or solid is described. By acquiring
and analyzing the front surface, first back surface, and second back surface reflections at
equivalent diffraction points, diffraction corrections due to the beam spread of the
transducer are no longer necessary. The new technique greatly simplifies the overall
estimation process by eliminating the need for transducer characterization.
Attenuation and reflection coefficients are experimentally determined with the new
technique for water and three solids. The measurements are made with two different
transducers at different regions in their wave fields (near field, far field). The attenuation
coefficients for water correspond very well to previously published values. The attenuation
coefficients for stainless steel, plastic, and fused quartz computed from the two transducers
show very good agreement.
ACKNOWLEDGEMENTS
This research was supported in part by the Cancer Research Center (CRC),
Columbia, MO, the Department of Radiology at the University of Missouri-Columbia
(MU), and the National Science Foundation. A portion of this research was carried out
while Terry Lerch was a Postdoctoral Fellow in Mechanical and Aerospace Engineering at
the University of Missouri-Columbia.
REFERENCES
1.
2.
3.
4.
Papadakis, E. P., J. Acoust. Soc. Am. 44 (3), 724 (1968).
Ophir, J., Maklad, N. F., and Bigelow, R. H., Ultrasonic Imaging 4 (3), 290 (1982).
Insana, M. F., Zagzebski, J. A., and Madsen, E. L., Ultrasonic Imaging 5, 331 (1983).
Margetan, F. M., Thompson, R. B., and Yalda-Mooshabad, L, in Review of Progress in
QNDE, Vol. 12, eds. D. O. Thompson and D. E. Chimenti, Plenum, New York,
1993, p. 1735.
5. Pinkerton, J. M. M., Proc. Phys. Soc. London B62, 129 (1949).
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