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ATTENUATIONCOEFFICIENT ESTIMATION USING
EQUIVALENTDIFFRACTION POINTS WITH MULTIPLE
INTERFACE
REFLECTIONS
T.P. Lerch
1
and S. P.Neal
2
Industrialand Engineering Technology Department, Central Michigan University, Mt.
Pleasant, MI 48859
2
Mechanicaland
Aerospace Engineering Department, University
of
Missouri
-
Columbia
Columbia,
MO
65201
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 afunction
of
frequency
for a fluid or solid is described.
This
broadband technique combines two established
conceptsinattenuation
coefficient
estimation: (1)
frequency
spectrum amplitude ratiosof
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
Th e
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 searchingfo r flaws in
various
structural materials,the material
scientist
characterizing the mechanical
properties
of the material, and the
biologist
investigating
th eacoustic propertiesof
various
typesof
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,
andmisalignmentof thetransducerandspecimen. Inthis paper,w ewill
consider four attenuationcoefficient estimation approaches (see Table 1): 1) a
Classical
Approach
driven by theratioofmagnitude spectrafrom two
interface
reflections; 2) the
Papadakis
Approach
which eliminatestheneedto m akeexplicit correctionsfor reflection
CP657,
Review
o f
Quantitative Nondestructive Evaluation
Vol.
22 ,ed. by D. O. Thompson and D. E. Chimenti
2003Am erican Instituteof Physics 0-7354-0117-9/03/$20.00
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TABLE
1 Summary
of attenuationcoefficient
estimation
approaches.
Classical
Papadakis
Equal
Diffraction
Ne w
Approach
System
Effects
cancel
cancel
cancel
cancel
Solid
Thickness
input
input
input
input
Wavespeed
in Solid
input
input
input
input
Solid
Density
input
output
input
output
R&T
Coefficient
input
output
input
output
Water
Attenuation
cancel
cancel
input
input
Diffraction
Corrections
input
input
cancel
cancel
andtransmissionlossesbyutilizing 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 aN ewApproach which combines the Papadakis and
Equal
Diffraction
Point approaches to simultaneously estimate reflection, transmission, and
attenuation
coefficients without
making
diffraction
corrections. Correctionsare,however,
required
for water attenuation due to variable
water
path
lengths.
Th e
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 thetransducer's
useful
bandwidth.
Transducer characterization can be a very
time-
and labor-intensiveprocess. Since
each
transducer has its own un ique set of parameter values, thecharacterization processmu st be
implementedforeachtransducer usedtomakeameasurement.
This paper will proceed with a model-based review of three existing attenuation
coefficient estimation
approaches introduced above. Mo dels 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 forthree solids. The paper concludes
with
abriefdiscussion 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 measurementof a
first
back surface reflection
along withafront surface reflection and/or asecond backsurface reflection. Usingalinear
time-invariant system modeling approach, the Fourier transform of the measured
front
surface reflection
can be
modeled
a s:
=
p f)R
ws
c 2
Zwf
,f)exp -2
Zwf
a
w
f))
(1)
We adopt a simplified notation throughout the remainder of the paper with
frequency
dependence implicit andwith
each
symbolrepresenting the absolutevalueof its associated
complex
quantity. The
Fourier
transform of the front
surface
reflection, F f ) , becomes:
F=
j3Rc 2z
wf
y
(2)
where
f i
,
the
system
efficiency factor,
accounts
for all
transducer
an d
electronics related
effects, R
R
ws
is the
water-to-solid reflection
coefficient,
z
w
f
is the
water path length
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for the front surface
reflection
experiment, C\2z
w
f J accounts forbeam
diffraction
in the
water, and a
w
is the
attenuation coefficient
in the
water. Notin g that
the
product
of
r\
transmission
coefficients, T^
S
T
SW
, can be written as
l-R
, the first and second back
surface reflections
can be
modeled
as :
B
l
=/?(l
-
R
2
]RC 2z
wbl
}
e
~
2z
^
a
C 2z
s
>T
2z
(3)
B
2
=j3l-R
2
R
3
C(2z
wb2
)e-
2z
^
a
-C(4z
s
)e
z
^
(4)
where
z
s
is the plate (solid) thickness,
a
s
is theattenuation coefficient in the solid,and
z
wb l
and z
wb 2
are thewater path lengthsfor the first andsecond
back
surface
reflections. In
Equations
(1)
- (4), weassume
that
3- fif - fi^i - fib2
Th e solid attenuationcoefficient can beestimatedusingany two (or allthree)of the
measured signals. Th e
diffraction
terms are
often
calculated for the
water/solid
case by
replacing
the twod iffraction
terms
in (3) or (4) by a
single
diffraction
term,
C 2z
we
),
with
the equivalent water path length,z
we
,
calculatedas
follows:
c c c
z
\ve
~
z
w ~ ~
z
s ^
z
\vebl
~
^
z
wbl
^
z
s
^
Z
web2
=
Z
wb2 ~ ^ ~ ^
z
s v^ /
c c* c
w
u
w
u
w
where
z
w
=
z
=
z
w
^;-
z
w
^2
f
r
fixedwaterpath,
c
s
andc^ are the
wavespeeds
in the
solid
and
water, respectively,
and
z
we
-
z
w
since
z
5
=
0.
We can now
solve
for
a
s
using
F and
B I
orusing
B I
and
B
2
as
follows:
F
.Oi
C 2z
w
)
1
,
C 2z
wM
]
-
2
or
< =
l n
____
C 2z
wM
) l-R
2
)
C(2z
web2
)R
2
The
front
surface reflection is corrected for
diffraction
in the water, and the back surface
reflections arecorrectedfor interfacelossesand fordiffraction in the
water
and
solid.
Papadakis Approach
Th e Papadakis Approach uses the front
surface
and the
first
tw o back surface
reflections
toeliminate fi and simu ltaneously estimate
R
and
a
s
. The ratio ofspectra
corrected for
diffraction
is
used
to yield two new
quantities
denoted Ml an d M2 by
Papadakis.
l-R
2
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otor
ontroller
T
V
ii
it
t r ansducer
f
r
ii i
T
z
w
t
ll
J
water
ll
FIGURE
1
Typical immersion systemdepictingth emeasurement approachfor theN ew
Technique.
Th e
transduceris not
translated
laterallyas the
figureimplies.
These
two equations are then solved for
R
and
a
s
as
follows:
R =
M1-M2
1+
M1-M2
a,=-
1
In -
Ml
2z
p
1 +M1-M2
(8)
Equal
D iffraction
PointApproach
Solid Attenuation
Coefficient
Estimation
Th e
Equal
D iffraction PointApproach involves adjusting thewaterpath(see Fig.
1) so
that
the
equivalent water
path
length
is the same for
each reflection.
Th e
penalty
is
that
the
a
w
mustbe
known,
and acorrectionof
form
exp(2z
w
a
w
)
must
be
applied
toeach
reflection. With thewaterpath for the
front surface
reflection used todictatethevaluefor
z
we
(that
is,
z
we
=z
w
f),
the
equalities given
in
E quation
(5) can beusedto
solve
for the
asequired water path for
B \
as z
wb
i =
z
w
f- c
s
/c
w
)z
s
and for B
2
Z
wb2
~
z
wf
~
c
s/
c
w)^
z
s Th e
*
superscript
is
introduced
to
indicate
that the
water paths
are associated
with
an
equaldiffraction
point approach. Th e change inwater path (Fig.
1)
*
between
successive
reflections, Az,,
is given
by
/ /
\
w
=
c
s
/
c
w)
z
s
=
z
wf -
z
wbl =
f
-
z
wbl
=
z
wb l
~
Z
\vb2
-
Incorporating these ideas,
the
water
attenuation termsfor theback
surface
reflections can bere-writtena s
follows:
-2z*
wb2
a
w=
-2
(9)
The key to
this approach
is
that
thed iffraction
terms, C\2z
we
],
are
equivalent
for
each
of
the three measured
reflections. Folding
theequalitiesin
Equation
(9)
into Equations
(1) -
(3),
canceling the common
d iffractions
terms,
and solving fora
s
yields:
2z
- In-
F
or
2z,
(10)
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Notethat therelativelydifficult toimplementdiffraction corrections in Equations (6) and
(7) arereplacedbywater
attenuation
correctionsin Equation (10).
Equal DiffractionPoint Approach Fluid AttenuationCoefficient Estimation
The same basic approach
used
for estimation ofa
s
can be used to estim ate the attenuation
coefficient in a
fluid using
a quartz
specimen
as the solid
with
known,
essentially zero,
attenuation. With
a
s
=a
q
0
and
exp 2z
q
a
q
)-^ 0
,
solution
for
a
w
yields:
7 ?
^ l og LT- or a
w
=
^log2y
(ii)
o
r- /i
r/\ ^
2A z
w
F(l-R
) 2Az
w
ANEW ATTENUATION COEFFICIENTESTIMATION APPROACH
Applicationto Attenuation
Co efficient
Estimationfor aSolid
By using the
front surface
reflection and the two back
surface
reflections, with
measurementsmade
at
equal
diffraction
points,
we can
eliminate J 3
and
simultaneously
estimate R and
a
s
without
making
diffraction
corrections. We
start
with the three
reflections,
each corrected
for water
attenuation. Withslightnotational changes
to
indicate
that equal diffraction point measurements arebeing used, wethen follow thePapadakis
approach
as
given
inEquations(7) and (8) to
reach
the new
estimation
form fora
s
:
B I l-R*
B,
*
-2z^h ya
va
7?
Z
=
r-
02)
Ml
-Ml
1 Ml*
1-R
2
\l
+Ml*-M2* 2z
s
1 +M1*-M2*
TheNew
Approach
yields estimates of R anda
s
;
however,
the
d iffraction
corrections in
Equation
(7) are
replaced
by water
attenuationcorrections
in
Equation
(12).
Applicationto AttenuationCoefficient
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
quartzwith thefollowingequations yielding estimates ofR anda
w
:
M1=
* ^ -
M2
*=
=
,
(14)
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i,i
, < ,
D
M1*-M2* I ,
1 +
M1*-M2*
M7 -M2 = - J R = J * s r a
w
= I n ( 1 5 )
* *
W
M l *
EXPERIMENTS
AND DATA ANALYSIS
Th eNew Approach wasused to estimate the reflection and
attenuation
coefficients
for
water
and
three solids
of
plate-like
geometry:
stainless steel
z
s
=
1 . 2 8 c m ) , fused
quartz z
s
= 0 . 6 4 c m ) ,
and plastic
z
s
= 0 . 7 3 c m ) .
Th e
apparatus employed
for
these
measurements
is
typical
of
mostultrasonicimmersioninspectionsystems ( s e e F i g . 1).
All
equipment iscommercially available. The transducer is driven by apulser/receiver unit
and positioned with the m otor controller. The rf signals are captured by the data
acquisition
card on the PC and
ultimately
transferred to a work
station
for
dataanalysis.
Threewavetrains,each containingthe
A-scan
time pulses
from
the
front, first
back,
and
second back surface
reflections,
are digitally captured. The measurement process
beginsbysettingthewater
path
at thedesired lengthfor thefront
surface
reflection. Atthis
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
A z
w
to
place
the firstback surface reflection at an
equivalent
diffraction
point
tothatof the front surface
reflection.
The
resulting wave train
is
digitally captured
and
stored.
The
transducer
is
again axially
translated
a distance of
A z
w
toward
the specimen in order toplacethe second
back surface reflection
at the
equivalent diffraction point
for the
first
two
reflections.
As
before,thiswavetrainis digitized and stored.
Data analysis
is
performed
with software
written
an d
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 m easured), 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 FF Troutine. 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 F i g .
2
summarizes
the results for
water attenuation
measurements
and
F i g .
3
summarizes
the
results associated with attenuation coefficient
estimation for fused
quartz, stainless
steel, and plastic. Reflection and
attenuation
coefficients foreach materialaremeasured withtwo unfocused-transducers: a 10 M H z ,
1
/4
diametertransducer
and a 15 M H z , 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 V
2
transducersa rebasicallyconstant across the
useful bandwidth
of each
transducer
and
1 7 6 4
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Water/Quartz Reflection
Coefficient
0.35
0.9
0.75
0.7
0.15
0.1
0.05
0
-0.05
-0.1
WaterAttenuationCoefficient
ReflectionCoefficient
with
Diffraction Error
0.65
0.6
10 12
Frequency MHz)
0.15
0.1
0.05
0
-0.05
-0.1
Attenuation Coefficientwith Diffraction Error
10 12
Frequency
MHz)
FIGURE
2.
Experimental results
fo r
fluid
attenuation
coefficient
estimationusing
th e
New Approach.
compare well
to the
theoretical value.
Th e
attenuation coefficient estimates
shown in the
upper right
graph
compare well to one another and toPinkerton's
widely
accepted result
[5].
Th e
threewater paths used
to
achieveequal
diffraction
measurements
are 25.4,
22.8,
and
20.2
cm for the front, first
back,
and
secondback
surface reflections,
respectively.
Thesewater
paths
wereused for both transducers to
further
demonstrate therobustness of
the approach. For the
10MHz
/4
transducer, these
water
paths place the measurement
point its far
field,
while for the 15MHzW transducer, the water paths correspond to the
near
field.
The lower two
graphs
in
Fig.
2
demonstrate what happens when incorrect
equivalent diffraction points arechosen. Notice the deviation
from
theory,especially the
frequency
dependence, in the experimentalreflection coefficientwhich hasbeencaused by
the
diffraction
error. In
thisinstance,
the
diffraction
error
creates
an additional
perceived
losso fenergy whichth edataanalysisassignsto thewaterattenuationcoefficient,resulting
in anoverestimationof the water attenuation
coefficient
as shown in the
lower
right graph.
A s seen in Fig. 3, the experimental reflection coefficients for thewater-stainless
steel
and water-fused quartz interfaces are
also relatively constant
across the useful
frequency spectra of both transducers. Altho ugh slightly oscillatory in nature, the
experimental
reflection coefficients for theplastic
also
tend to be
constant. Attenu ation
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 theseresults to a generally accepted standard. Notice
however
the robustness of the new technique in returning
consistent
attenuation
coefficients estimates for the two
transducers
withouttransducer characterization or the
formal
application
of
diffraction
corrections.
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Reflection C o e f f ic i e n t s v s F r e q u e n c y Attenuation
C o e f f ic i e n t s v s F r e q u e n c y
0.9
f O . B
0.7
o
0.6
u
0.5
0.4
0.
Q G
O B 1 1 2 1 4 1 6
F r e q u e n c y Hz )
x
FIGURE3 .
Experimental results
for solid
attenuation
coefficient
estimation using
th e
N ew
Approach.
Data
acquired
withthe 10 MHz
transducer
is
represented
with
'o';
the
15
MHztransduceris
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
processby
eliminating
the need fortransducer characterization.
Attenuation and reflection coefficients are experimentally determined
with
the new
technique for
water
and
three
solids. Th e measurements are made with two
different
transducers
at
different regions
intheir wave
fields (near field,
far
field).
Th e
attenuation
coefficients forwater correspond verywell topreviouslypublishedvalues. Th eattenuation
coefficients forstainless
steel,
plastic,and fused quartz
computed
from the two transducers
show verygoodagreement.
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
ou t
whileTerryLerchwas a Postdoctoral Fellow in Mechanical andAerospace Engineeringat
the
University
of Missouri-Columbia.
R F R N S
1.
Papadakis,E. P.,J.
Acoust.
Soc. Am. 44(3),7 24(1968).
2. Ophir,
J .,
Maklad,
N. F., and
Bigelow,
R. H.,
Ultrasonic Imaging
4
(3),
290
(1982).
3. Insana,
M.
F.,
Zagzebski,J.
A.,
and
Madsen,
E. L.,
UltrasonicIm aging
5,
33 1(1983).
4. Margetan,F .M., Thompson,R. B., andYalda-Mooshabad,L , inReviewo fProgress in
QNDE,Vol. 12 ,eds.D. O.Thompson and D. E.Chim enti,
Plenum,
NewYork,
1993,
p. 1735.
5.
Pinkerton,
J. M.
M.,Proc.Phys. Soc.London
B62, 129
(1949).
1766