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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

1759


2/8

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

1760


3/8

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

1761


4/8

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)

1762


5/8

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)

1763


6/8

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


7/8

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.

1765


8/8

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

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Cdr Pdfs Indexed 1759 1