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university 6659

 

 

 

 

 

This is to certify that the

dissertation entitled

The Determination of Attenuation-Velocity
Products in a Layered Homogeneous Medium

presented by

Joseph Nodar

has been accepted towards fulﬁllment
of the requirements for

Doctoral Electrical Engineering

degree in

 

 

(339%?

Major/professor

Date October 27, 1989

MS U i: an Afﬁrmative Action/Equal Opportunity Institution 0- 12771

 

 

 

PLACE IN RETURN BOX to move this checkout from your record.
TO AVOID FINES return on or before date duo.

DATE DUE DATE DUE DATE DUE ll
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MSU Is An Affirmative Action/Equal Opportunity Institution

THE DETERMINATION
OF
ATTENUATION-VELOCITY PRODUCTS
IN A

LAYERED HOMOGENEOUS MEDIUM

By

Joseph Nodar

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1989

:4 WM

(pm
,4 K)

.

ABSTRACT

THE DETERMINATION OF ATTENUATION-VELOCITY PRODUCTS
IN A LAYERED HOMOGENEOUS MEDIUM

By

Joseph Nodar

In the ﬁeld of acoustic imaging, many techniques have been forwarded to permit the
creation of images by means of the attenuation properties of the object under study.
Each of these has its own advantages and drawbacks. In this thesis, a new method
of accomplishing this type of amplitude-based imaging is proposed, using attenuation-
velocity products as an imaging index. It is shown that this choice has the advantage
of providing high contrast between various media, which can allow the identiﬁcation of

the inner materials comprising an object to be performed remotely.

Using a bidirectional interrogation of a one-dimensional object model, it is shown
that the solution of the N -layer object problem can be found uniquely, and that separ-
ation of the effects of reﬂection coefficient and attenuation can always be made. No
assumptions are necessary concerning the values of the various parameters, with the
exception of the need to know the loss factors of the two outermost layers. An
algorithm is presented that has successfully eliminated the deleterious effects of

multiple reﬂections on the solution, and it also solves for the number of layers within

Joseph Nodar

the object, often an unknown quantity, when given just the experimental data
acquired during the bidirectional interrogation of the object, i.e. the left echoes, the
left-right transmission, the right echoes, and the right-left transmission. It is demon-
strated that only this set of signals provides for sucha solution. Rationale is given
for the apparent uniqueness of this solution even in the presence of the extraneous
multiple reﬂection signals. Also, a detailed consideration of the components required
to implement such a measurement system is made, with appropriate error analyses
performed. Preliminary experimental results are presented, and the reasonable errors
that have been achieved are compared with the standard published values for the
materials utilized. Finally, suggestions for certain future work to extend the results

are outlined.

Copyright by
JOSEPH NODAR

1989

This thesis is dedicated to my family.

ACKNOWLEDGMENTS

The author would like to thank Dr. Bong Ho, Department of Electrical Engineering,
for his support and discussions during the work on this thesis, Dr. Pavol Meravy,
Department of Mathematics, for his considerable help and sincere interest, and Drs.
Mani Azimi, Donnie K. Reinhard, and H. Roland Zapp, Department of Electrical
Engineering, for their valuable constructive criticism of this work. Special thanks are

also in order to Gayle J. Sears, for his assistance above and beyond the call of duty.

TABLE OF CONTENTS

LIST OF TABLES ..................................................................................................... page ix
LIST OF FIGURES ............................................................................................................ x
I. INTRODUCTION ...................................................................................................... 1
II. BACKGROUND ...................................................................................................... 10
2.1 Basic Acoustic and Mechanical Properties and Wave Behavior ................... 10

2.2 General Acoustic Imaging ............................................................................. 16

2.3 The Target ...................................................................................................... 17

III. ATTENUATION IMAGING .................................................................................. 24
3.1 Attenuation ..................................................................................................... 24

3.2 Attenuation Imaging ...................................................................................... 27

3.3 Attenuation-Velocity Products ....................................................................... 32

IV. DETERMINATION OF ATTENUATION-VELOCITY PRODUCTS ................. 34
4.1 Basic Goals of This Section ........................................................................... 35

4.2 Problem Statement and Definition of Terminology ....................................... 36

4.3 Assumptions to Solve the One-Dimensional Model ...................................... 38

4.4 The Unknown Quantities ............................................................................... 39

4.5 Solution of The Problem ................................................................................ 40

4.5.1 Case 1: Single-Sided Purely Reﬂective Interrogation ...................... 40

4.5.2 Case 2: Pure Transmission Interrogation .......................................... 43

4.5.3 Case 3: Bidirectional Interrogation ................................................... 44

4.6 Finding the Attenuation-Velocity Products of the N-Layer Object ............... 52

4.7 Multiple Reﬂection Elimination and Finding the Number of Layers ............ 54
4.7.1 Finding N, the Number of Layers ..................................................... 54

vii

viii

4.7.2 Finding the Primary Data Set in the Presence of

Multiple Reﬂections ......................................................................... 55
V. SYSTEM ANALYSIS AND ERRORS ................................................................... 61
5.1 Introduction .................................................................................................... 61
5.2 The Test Object .............................................................................................. 61
5.3 The Transducer .............................................................................................. 64
5.4 The Transmitter and Receiver ........................................................................ 72
5.5 Signal Acquisition .......................................................................................... 75
5.6 Signal Processing ........................................................................................... 82
5.7 Sensitivity of Equations (4.20) and (4.21)

to Errors in the Amplitudes of the Primary Data ........................................... 90
5.8 Additional System Requirements .................................................................. 95
5.9 Conclusions .................................................................................................... 97
VI. SIMULATIONS AND EXPERIMENTAL RESULTS ........................................... 98
6.1 Simulations and Results ................................................................................. 98
6.2 Experimental Results ................................................................................... 103
VII. RECOMMENDATIONS AND CONCLUSIONS ................................................. 120
7.1 Recommendations for Future Investigation ................................................. 121
7.1.1 Extension to Higher Dimensional Objects ...................................... 121

7.1.2 Extension of the Alpha-V Method to Transmission
Tomography .................................................................................... 123

7.1.3 Extension to Electromagnetic (and General Wave)
Situations ........................................................................................ 126
7.2 Conclusions .................................................................................................. 127
APPENDIX A Program: TWOSIDE ......................................................................... 131
APPENDIX B Program: ALPHA-V .......................................................................... 136
APPENDIX C Program: TWOCHANNEL ............................................................... 140

BIBLIOGRAPHY ........................................................................................................... 145

Table 3.1

Table 6.1

Table 6.2

Table 6.3

Table 6.4

Table 6.5

Table 6.6

LIST OF TABLES

Attenuation-velocity products of various. media

at a 1 MHz transducer frequency ......................................................... page 33
The results of program TWOSIDE on the example of Figure 6.1 .............. 101
Physical constants for some materials, at 2.25 MHz .................................. 108

Summary of the experimental results for the single layer

plexiglass sample ........................................................................................ 112
The experimental results for the aluminum sample .................................... 114
Summary of the results for the ﬁve layer object ......................................... 117

(a) The experimental data, and (b) the calculated
results for the liver sample ......................................................................... 119

ix

Figure 1.1

Figure 1.2

Figure 2.1
Figure 2.2
Figure 2.3

Figure 2.4

Figure 2.5

Figure 4.1

Figure 4.2

Figure 4.3
Figure 4.4

Figure 4.5

Figure 4.6

LIST OF FIGURES

An example layout of an ultrasound imaging system ........................... page 4
The driving characteristics of a piezoelectric transducer .............................. 6
Wave behavior at an impedance discontinuity ............................................ 14
The pulse-echo imaging technique .............................................................. 17
An example test object, and its one—dimensional model ............................. 18

An example steplike impedance proﬁle for an isotropic
one-dimensional object bounded by discontinuities ................................... 21

Example of multiple imaging artifacts in an object ..................................... 23

The object model to be studied and the notation
conventions used in the derivation of the equations in

this chapter ................................................................................................. 37
Example of left side echo data. .................................................................... 40
The deﬁnition of Ri with respect to Li .. ...................................................... 45
Example of right side input echo returns ..................................................... 46

The algorithm described in this section that ﬁnds the set
(N, k, r, t} and eliminates multiple reﬂections .......................................... 56

Illustration of the difference between the boundary
properties when viewed from opposite sides .............................................. 58

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 5.7

Figure 5.8
Figure 5.9
Figure 5.10
Figure 5.11
Figure 5.12

Figure 5.13

Figure 5.14

Figure 5.15

Figure 6.1

xi

Detailed block diagram of the components of the attenuation-
velocity product measurement system to be analyzed in this

chapter ......................................................................................................... 62
The geometry for ﬁnding the effective receiver area when the

beam illuminates an angular planar reﬂector .............................................. 65
Plot of Equation (5.2), the effective receiver area ....................................... 67

The geometry used to determine the transducer beam

pattern and beam width ............................................................................... 68
Plot of Equation (5.3), the transducer receiving beam pattern .................... 70
An electrical equivalent circuit for a piezoelectric transducer .................... 71

The use of a time/gain ampliﬁer to boost the lower level

echo amplitudes for increased accuracy and dynamic range ...................... 74
The spectrum of the function of Equation (5.10) ........................................ 76
Plot of Equation (5.16), the quantization error ............................................ 81
The software preprocessing for the algorithm of Figure 4.5 ...................... 83
The window detector in action ................................................................... 84
The windowing algorithm used to produce Figure 5.11 ............................. 84

The effect of sampling nearly at the Nyquist limit on
the accuracy of peak amplitude determination .......................................... 86

Plot of the effects of cubic splines on the peak

amplitude approximation ........................................................................... 89
The various sensitivity relations derived in Section 5.7 ............................. 92
The example object to be considered, with N=4 ....................................... 100

Figure 6.2 The results of program Alpha-V using the

data shown in Table 6.1 ............................................................................ 102

Figure 6.3
Figure 6.4

Figure 6.5

Figure 6.6

Figure 6.7
Figure 6.8

Figure 6.9

Figure 6.10

Figure 7.1

xii

The layout of the experimental system ...................................................... 104
The example setup to illustrate the experimental procedure ..................... 109
(a) The experimental time signals, and (b) the

set {L, TLR’ R, TRL} experimental for Figure 6.4 ................................. 110
The experimental setup for the aluminum sample .................................... 113
The experimental data for the aluminum sample ...................................... 114
The conﬁguration of the ﬁve layer object ................................................. 115

(a) The experimental time signals, and (b) the
echo data for the ﬁve layer object of Figure 6.8 ....................................... 116

The experimental setup for the liver sample ............................................ 118

The type of object proposed in Section 7 .1.2 ............................................ 124

CHAPTER I

INTRODUCTION

In many circumstances, there exists a requirement for non—invasive and non-
destructive probing of the internal characteristics of an object under study. Such ob-
jects range from the living tissue structures that are encountered in a medical setting,
to manufactured industrial products that need veriﬁcation of their internal or composi-
tional soundness prior to shipping from the factory. In each of these instances, it is
undesirable or impossible to disassemble or damage the object in question in any
manner, yet some means must be found to acquire information regarding the internal
structures, and in some cases to even identify the actual material from which a partic-
ular part is constructed. In the clinical setting, this problem is further exacerbated
since no prior knowledge of the exact makeup or dimensions of any internal form is
possible, in contrast with the industrial situation wherein any manufactured compo-
nent is usually thoroughly designed beforehand and is therefore precisely deﬁned in
comparison.

Several techniques have come to the forefront of this very basic problem, namely
X-ray tomography, nuclear magnetic resonance imaging (MRI), and ultrasound.
Each has certain advantages, and each presents its own problems, both technical and

practical. For example, both X-ray tomography and MRI imaging systems

are capable of producing excellent pictures of a widely diverse class of test objects,
and in some cases can even be used to give a three-dimensional reconstruction of
these objects. Also, an excellent clarity of the boundaries separating the internal
parts is the norm, which is invaluable in medical diagnostic tests, particularly in
delicate areas where exploratory surgery is not feasible or possible without patient
risk. Commercial systems are rather well developed, with many features available,
and are quite widespread, currently in operation at many sites worldwide. There are,
however, certain problems with these systems, for example the extremely high cost,
which can usually exceed several million dollars, and the admittedly unknown health
risks posed by exposure to the high intensity energy emissions used to interrogate

the object being probed.

In contrast with the above technologies, ultrasound is relatively underdeveloped.
The acoustic measurement and imaging systems that are presently available commer-
cially are rather expensive when one considers their somewhat more limited capabili-
ties in comparison with X-ray tomographic and MRI imagers; the typical acoustic
system can cost over two hundred thousand dollars (1989) and cannot as yet deliver
images that stand up to even an ofﬂrand judgement against the other methods. This
situation has resulted in ultrasound being relegated to a somewhat less signiﬁcant
position among the various imaging technologies; yet ultrasound possesses features
that the other methods will always lack, namely comparatively lower cost and a
greater potential for safety in operation. The latter is evident because while both

X-ray tomography and MRI imaging rely On narrow duration high intensity

electromagnetic energy emissions to penetrate the object--resulting in the
aforementioned short- and long-term safety risks-ultrasound uses only high
frequency mechanical wave propagation and interaction with the object to form con-
clusions about the internal structure. This type of energy intromission has the possi-
bility of being safer to apply since peak and average power levels can be kept signiﬁ-

cantly lower than the amounts employed in the other imaging schemes.

The standard ultrasound system is comprised of several key components and units:
(1) some form of energy emitting transmitter, which originates the high frequency
mechanical waves used to interrogate the object under study, (2) some form of
receiving device that is used to recapture the emitted wave energy after it has
passed into and/or through the object, (3) some means of saving the information
contained in this received signal, and (4) some scheme of processing and presenting
the signal/data for display and information dissemination to the system operators.
Depending upon the ﬁnal intent of the application, it is usually advantageous for the
basic components to take various forms, particularly with respect to how the trans-
mitter and receiver are deﬁned and oriented in relation to the object. Figure 1.1

shows a common way to set up such a system.

In many ultrasound systems, it is common for the transmitter and receiver functions
to be combined into one device. Typically, a piezoelectric effect device is employed in
this position, since this physical phenomenon directly relates electrical and mechanical

stimuli and responses in a bilateral manner-~i.e. a single unit can be electrically

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excited in an appropriate manner and thereby emit mechanical vibrations, and likewise
the same device can perform the conjugate function of transforrrring the subsequently
received mechanical waves into corresponding electrical signals. This conciseness
usually allows the electromechanical portion of the system to be compact in form,
possibly handheld, and precludes the potentially nasty problems of alignment and the
matching/calibration that is necessary in the case of a separate transmitter and
receiver. The fact that these piezoelectric devices are electrically and mechanically
coupled gives rise to another prominent characteristic, that of narrow frequency
bandwidth operation. This is due to the resonant nature of the physical structure and
materials from which the unit is constructed, commonly a thin crystalline material such
as quartz. The implications of this are several: (1) the unit has a limited frequency
response, (2) the unit will tend to resonate at its own natural frequency when stimu-
lated, from both electrical and mechanical sources, (3) the fact that the bandwidth is
narrow tends to lower the received noise, (4) it is difficult to produce very narrow
transmitted pulses, which are usually necessary for high resolution imaging, and (5)
the narrowly tuned nature of these devices makes matching among several of them
difﬁcult in systems that use multiple units, and causes arrays of transducers to

exhibit beam patterns that are somewhat unpredictable [8]. It should be noted that
although these consequences of using piezoelectric devices will sometimes have a
negative effect on system performance, they also may be used to advantage in other

instances.

In order to create the requisite mechanical waves that are the foundation of any

acoustic measurement, some form of electrical stimulation is commonly used. Most
often, a high amplitude voltage pulse of extremely narrow time duration is

applied to the transmitting transducer, the result being that such a driving signal
"shock excites" the transducer unit into natural resonance, thus beginning the gen-
eration of mechanical output waves as a consequence of the piezoelectric effect.
Figure 1.2 shows a typical transducer natural response to such a driving signal.

After these vibrations are created, they begin to propagate away from the trans-
ducer, and usually ﬁrst encounter some type of coupling medium that ( 1) serves as an
impedance match, (2) has well known acoustic properties, and (3) ﬁlls all the "bumps
and lumps" (even microscopic) that comprise the outer surface of the the object.
Water is popular--being inexpensive, readily available and relatively safe--and it
surrounds the test object under its own volition, always a convenience. This ensures

that the mechanical waves will reach the object intact (air usually presents problems

Input signal output signal

Figure 1.2 The driving characteristics of a piezoelectric transducer.

for ultrasound propagation since it is highly attenuative at the frequencies commonly
needed for clear imaging). After initial wave impingement upon the surface of the
object, interaction in a mechanical sense then occurs internally, i.e. reﬂection, trans-
mission, and absorption of the acoustic energy, with the eventual return of some por-
tion of the initially transmitted pulse energy to the receiver of the system. This
signal is then converted back into electrical signals and is processed by the remaining
system components, eventually destined for presentation to the system user. It is
important to realize that this sequence of events will ultimately comprise only a single
position or point of view with regard to the object being imaged; in order to obtain a
two- or three- dimensional image, the transducer must be moved or scanned to
various positions about the object, and the entire sequence repeated each time.
Performing this operation repeated will enable a complete multi-dimensional recon-

struction to be created.

With only minor variations, the above description of the operation of an ultrasound
imaging system would sufﬁce for most examples in use today. Typically, thicknesses
on the order of a millimeter or less can be distinguished, but lateral resolution can vary
greatly, being dependent on transducer size, beamwidth, and the interval over which
the scanning motion is conducted; also, the object geometry has an inﬂuence on the
lower bound of this parameter. Limitations on the performance of acoustic imaging
systems arise from many sources, and will be discussed in some detail in the next
section, but many systems self-impose restrictions on their own abilities by virtue

of simply not using most of the information contained in the received signals.

The data that can be extracted is rich with information concerning the acoustic
properties of the object interior, since the sound has passed through this region and
has therefore been affected by its mechanical and physical qualities. In particular,

the amplitude information of the received signals is commonly ignored by many
systems, even though [48] this is a source of much more quantitative information than
is available from only a consideration of the time data, and has hidden within a record
of the acoustic impedances, reﬂection coefﬁcients, and attenuations encountered
during its time of ﬂight. In the past, electronic technology was not sophisticated
enough to permit the older ultrasound systems to capture for processing the returning
signals in a form ﬁom which this data could be gleaned; today, however, it is possible
to record these signals very accurately using high—speed analog-digital conversion
and fast, large semiconductor memories, placing the entire waveform at the disposal
of a digital computer for processing. This allows possibilities of data processing,
interpretation, and display that simply were not feasible a short while ago, all at very
affordable costs and with reasonable operational speeds. The question reduces to

one of how to process these captured signals in a useful manner, and it is to this end

that this thesis work has been addressed.

As previously mentioned, it is extremely desirable to not only identify whether a
boundary is internally present or not in an object, but also to be able to determine
what sort of material is between those boundaries, and to eventually be able to identi-
fy the material that comprise the regions within the object. Examples of applications

that might use such a capability are numerous and obvious. For many years, a safe

and reliable method of tumor detection/identiﬁcation has been sought by the medical
community, and a capable ultrasound system that is equipped with the sophistication
necessary to allow material identiﬁcation would certainly ﬁll this need [43]. Other
medical uses include instrument positioning during delicate surgical procedures, and
location of foreign objects in a wound site. In the industrial sector, ultrasound has
been extensively explored for use in testing manufactured products, such as composite
materials, which are commonly formed of layers of woven cloths such as carbon ﬁber
and embedded in some type of resin for binding and rigidity. These materials are
hoped to eventually be used in place of metals in many applications since they can be
lighter and stronger, but in order for this to be true, the layered structure of the com-
posite must be intact. Delamination of the layers of cloth is at present difﬁcult to
detect by any means, particularly in shaped composites installed permanently. A
simple method of commercially useful testing is needed before these materials can be
placed into practical service [12]. An ultrasound imaging system with the capability
to detect and identify nonuniformities or the existence of foreign substances inside an
object would deﬁnitely be of immense value in these and other such applications that

require the safer operation and lower cost possible by acoustic methods.

In Chapter 2, we examine some preliminary acoustic phenomena and background
information, then discuss acoustic imaging in general, reserving attenuation to be

the focus of Chapter 3.

CHAPTER II

BACKGROUND

In order to discuss the research undertaken in this thesis, it will be necessary to
momentarily digress to review certain preliminary concepts of acoustic imaging. We
ﬁrst describe some basic properties of acoustic interaction in a material, including
boundary behavior, ﬁnally covering in some detail the more complicated properties

that frequently are considered non-idealities, such as scattering, in the context of the

broader considerations of acoustic imaging in general.

 

When mechanical waves propagate through a material medium, certain properties

are always of interest; we now describe these as they relate to the imaging problem.

Normally for the purposes of imaging, we desire the acoustic energy to propagate
in a single direction exclusively, usually in the manner of longitudinal vibration, i.e. all
the mechanical motion is conﬁned to the path of propagation. Due to the net motion
of energy, the particles from which the medium is composed are displaced from their
rest locations, and this disturbance moves through the medium. Such motion is natu-

rally resisted by the elastic binding forces that hold the medium together as an entity,

10

11

and this results in the exchange of potential and kinetic energy along the direction of
energy propagation. Since we expect a controlled wave motion in our medium, we
normally consider the wavefront to consist of a series of plane pressure waves of in-
finite extent, for the purposes of a mathematical description. If we deﬁne a particle of
medium as simply a unit of matter that is small enough so that any physical and mech-
anical quantities of interest are constant along its extent, and allow for monochromatic
(single frequency) periodic medium excitation, then the particle displacement about its

mean location in the lattice is given by

x(t) = x0 + A1 sin(2 11: (2.1)

fwave 9

where x(t) is the particle displacement, x0 is the particle rest position, A1 is the

amplitude of the oscillation, and fwave is the wave oscillation frequency. In order to

ﬁnd the particle velocity and acceleration about its rest position, we simply differenti-
ate Equation (2.1). Of more interest to us is the velocity of the wave, which may be
considered to be a constitutive parameter of the medium in which propagation is

occurring; this quantity is contained in the function

u(x,t) = “0 + A2 sin[2 1t fwave (t - x / vwaven (2.2)
which describes the plane wave both in vibration and translation, where u(x,t), "0’
A2, and fwave are deﬁned in a manner similar to the particle, x is the wave positional

location at a particular time t, and v is the velocity of wave translation.

an6

12

It is pertinent to realize that since we have assumed unidimensional wave propa-
gation, any distance we may compute from the experimental data resulting from any
measurement will be assumed to lie along this same line of motion. Usually, we use

the common simple relation

(2.3)

x = vwave tmeasured

«where x is a distance, v is again the wave velocity, and (measured is any time

wave
that we may have measured in a given experimental situation--to calculate a dis-
tance, at least when we know the wave velocity beforehand. Alternatively, we may

ﬁnd a value for Vwave when we have no such prior knowledge.

The velocity of wave propagation represented by our quantity v is governed

wave
by such physical quantities as temperature, medium density and stiffness, and the
frequency of wave oscillation. For normal use, we consider a region made from a

single type of material to be isothermal, so in this case Vwave

vwave = V 1" p (2.4)

where k is the material stiffness (elastic constant) and p is the material density in

is given by

units of inverse cubic distance, for the one dimensional model that we have deﬁned.
This relation could be made much more complex by including other factors such as

temperature, large signal constitutive parameters (which can be required under the

13

high amplitude drive conditions present in e. g. ultrasonic welding), and coupling of

wave energy into other vibration modes.

Another quantity of importance is 2., the wavelength of the wave in the medium,

which of course depends on the wave frequency and velocity as

A = vwave/ fwave (2'5)

This quantity is of great importance when choosing a frequency for which to use in
imaging, since (1) the transducer, being piezoelectric in action, must be resonant at
this frequency, and (2) the resolution of the image will depend inversely on A [49].

We will consider this later when we discuss acoustic imaging.

The intensity of an acoustic wave may be attenuated by several loss mechanisms
as the wave propagates through a medium, such as heating loss, lateral beam
spreading, and partial scattering, but these are not of sufﬁcient interest to discuss in
detail here. In the next chapter, we will describe the present state of the attenuation
literature, and some mention will be made about these types of problems. For our

purposes though, the total effects of these loss actions may be combined into a single

\

parameter a with the units of inverse distance [13].

Another quantity that is commonly deﬁned in conjunction with a particular material
is the characteristic acoustic impedance, which is analogous to the same quantity that

is deﬁned in the case of distributed electrical networks, and is given by

14

Z

characteristic = p vwave (2-6)

where the quantities p and v are as deﬁned before. The characteristic impedance

wave
of a medium arises from the dual-quantity nature of the pressure and velocity in

mechanical systems, again analogous to electrical voltage and current.

By means of the concept embodied in Equation (2.6), we can now investigate the
action of a wave at a boundary of two media, particularly ones with differing character-
istic impedances. Using Snell’s Law and the continuity of both pressure and velocity

at a boundary [2], we can write for Figure 2.1 that

vi sin 9i - Vr sin 01. = vt sin 9t
. (2.7)
pi + pr = pt
medium 1 medium 2

incident wave
Pi . vi

transmitted wave
pt ' vt

 

reﬂected wave
pr ’ Vr

 

Figure 2.1 Wave behavior at an impedance discontinuity.

15

where the subscripts i, r, and t stand for "incident", "reﬂected", and "transmitt "
respectively, p is the wave pressure, v is the wave velocity, and Z is the characteristic

impedance of the medium. From the dualistic nature of pressure p and velocity v, we

know that the relationship
V = zcharacteristic p (2'8)

must hold, again in a manner similar to electrical networks. By substitution of
Equation (2.8) into the lower Equation (2.7), and then solving the system in Equation

(2.7) simultaneously, we arrive at the following ratios as the pertinent solution:

 

 

 

ﬂ
pr ZzSinei'Zl Sinet
pi ' ZQSin0i+lein6t
‘ (2.9)
pt 2 a sin 9i
pi - ZzsinOi-t-Zl sinOt
where we have used the fact that 9i = 61. to simplify the equations. Usually, the
equations given in (2.9) are called the reﬂection and transmission coefﬁcients,
respectively--that is, we deﬁne the reﬂection coefﬁcient to be
r = Pr/Pi (2.10)

and the transmission coefﬁcient to be

t = pt/pi (2.11)

16

We note in passing, by inspection of Equation (2.9), the interesting fact that
t = 1 + r (2.12)

a fact that we will use often later. Also, in the case of normal incidence, where
0i = 01. = 0t = 90°, the deﬁnitions of the reﬂection and transmission coefﬁcients

reduce to the more commonly known

22'21

72+zr

222
Z2+Zl

(2.13)

 

where Equation (2. 12) naturally holds as well.
2 2 Q l i . I .

In its most basic form, the method of acoustic imaging is rather similar to the oper-
ating principle of a radar system--signals are transmitted from the source into a medi-
um which might have objects embedded within that must be detected, and the
presence or absence of such an object is indicated by the existence or lack of an echo
return, or reﬂection, from the target. This idea for medium imaging is usually called
pulse-echo imaging, and is illustrated in Figure 2.2. The distance of the object from

the transmitter is computed in a simple manner, by

object distance = (propagation velocity)(time of ﬂight) / 2 (2.14)

17

V

Transmit

 

i _

Receive * '
t=0 t=TimeofFlight

Figure 2.2 The pulse-echo imaging technique.

and naturally this deﬁnition of displacement requires the prior knowledge of the

velocity of propagation in the medium being interrogawd.

Another signiﬁcant fact about the pulse-echo imaging technique is that in order to
image a volume, as opposed to the one-dimensional interrogation performed in Figure
2.2, the transmitter must be scanned or positioned so as to allow data to be taken in
different measurement directions, as mentioned in Chapter 1, and then this informa-
tion must subsequently combined into an appropriate picture after suitable pro-
cessing. We will discuss this processing in more detail in later chapters, but it now
sufﬁces to say that the time and amplitude information present in the echo are usually
handled independently from each other, and both are strongly dependent upon the

wave characteristics of the objects from which the echoes were reﬂected.
2.3 The Target

As we mentioned earlier, some assumptions are usually necessary to be made

concerning the physical structure of an object under test, in order to simplify the
analysis of the internal acoustic wave behavior and potential sources of problems with

the accuracy of the resulting image. We will now discuss these in limited detail.

The test object is usually considered to be composed of regions of different media

as shown in Figure 2.3. In a particular measurement direction, the object is thought

multi-region
object

   

g
\

  
 

transducer

acoustic
beam

 

 

Figure 2.3 An example test object, and its one-dimensional model.

19

of as a series of layers arranged in one dimension, as shown at the bottom of Figure
2.3, where the different layers have been shaded in accordance with the upper part
of the ﬁgure. Usually, we desire that the boundaries of such a test object be rather
smooth, and also be perpendicular to the acoustic beam. There are several reasons

for this, one being obvious from Figure 2.1, we showed that any such incident angle

other than 90° will cause both the reﬂected and transmitted waves to leave the path of
the transducer line of sight; in the expanded case of multiple layers and angular inci-
dence, we can be almost completely sure that the beam will be lost to the receiver,
unless it is positioned at a different angle than the transmitter with respect to the
transmitter line of sight. In order to reconstruct an image in this case, we must scan
both the transmitter and the receiver individually and over all angles so as to capture
both the entire object and all of the returning energy reﬂected at every angle. This is

discussed in greater detail in Chapter 7.

The planar layer assumption discussed above is usually considered valid when
lateral spatial changes in object features are gradual, this being judged with respect to
the transducer wavelength in the incident medium. Naturally, this situation does not
always happen, e.g. at sharp corners in a boundary. Additionally, we cannot expect
that all the inner boundary surfaces are ﬂat and normal; in fact, by examining Figure
2.3 one can see that as the scanning angle of the transducer is changed around the
example object in the ﬁgure, the apparent number of layers will change in each of the
one-dimensional models resulting from each position, and we can expect that at

some angles the acoustic beam will encounter a steeply curved boundary surface.

20

At such locations, diffraction and angular scattering must occur, and unless a separate
receiver is employed at a different angle to intercept this energy, the imaging system
will usually present erroneous data at this location. There has been some research to
done with regard to this problem, but not much success has been achieved, primarily
because of uncertainties in the proﬁle of the acoustic beam emitted from the transmit-
ter. This is discussed further in Chapter 5, but we can mention here that this

proﬁle is not a column of uniform intensity and phase, as usually assumed, but rather
a complex variation of width (due to focusing and radial pressure divergence), and
amplitude and phase (which both depend on the coupling medium and the transducer
design). This proﬁle can vary widely from transducers from even the same production
lot, making ﬁxed speciﬁcation of the beam shape difﬁcult to perform The end result of
this problem is that a great uncertainty exists concerning exactly what occurs acousti-
cally inside a sample when it is being interrogated. For the purposes of this thesis,
however, we will not address this problem further, other than to discuss the implica-

tions of its effects on theattenuation-velocity product technique.

The example object of Figure 2.3 is composed of layers of differing media, but we
usually cannot expect these layers to be isotropic in their acoustic properties. For
example, the attenuation of in-vitro tissue can vary signiﬁcantly from the center of an
organ toward the outer boundary [35], and we would like in principle to be able to
observe this information non-invasively as well as the discontinuities of the boundary
surfaces. Some research has been done to investigate the problem of reconstructing

such an arbitrary spatial variation of acoustic properties [e.g. 23, 25, 29, 30, and 40],

21

but only in the case of internally boundaryless objects. In Chapter 7, we discuss the
possibility of extending the work in Chapters 4, 5, and 6 to handle the cases of objects
with spatial property variations, even with multiple regions bounded by discontinuities
of, for example, acoustic impedance, but for the main purposes of the work done here,
we will consider only the. case of isotropic object regions bounded by discontinuities.
In such objects, the one-dimensional model we construct in each measurement
direction is then presumed to be made of layers of uniform media bounded by discon-
tinuous impedance transitions, implying that the acoustic constitutive parameters of
the object, when plotted versus distance within the object, form a steplike proﬁle

like that shown in Figure 2.4. This assumption can also be made in cases of spatial
parameter variation, in which case it will result in a calculation of a sort of average

value for the parameter between its bounding discontinuities.

Another problem that arises in a target object is that the acoustic parameters

deﬁned for the boundaries and layers, such as reﬂection coefficients and attenuations,

Z
1 .
Ii
20
. Z3
Figure 2.4 . An example steplike impedance proﬁle for an isotropic
‘ one-dimensional object bounded by discontinuities.

Z4

 

 

 

22

are functions of the frequbncy of the acoustic signal propagating in the medium, and
therefore these parameters depend on the transducer frequency as well. For example,
acoustic attenuation occurs in direct proportion to frequency (to a power greater than
unity) in many materials, thus limiting the upper range of useful operating frequencies
[17]. As we mentioned earlier, the resolution of an imaging system is a direct func-
tion of this frequency, so consequently the smallest detectable vertical distance is
limited to some fraction of the wavelength given by Equation (2.5). Some work has
been done to improve this resolution limit, including methods such as the deconvolu-

tion of the echo return signals [19], but the problem still stands as a fundamental limit.

Possibly the most perplexing difﬁculty of acoustic imaging is the problem of
multiple reﬂections and transmissions. This phenomena is a consequence of the
boundary behavior of an incident wave that we investigated in the derivation of
Equation (2.9) in relation to Figure 2.1. Put simply, we must expect that any wave
travelling inside an object will see an impedance discontinuity as a boundary whether
it is incident from the left or the right. This is of immense importance to pulse-echo
imaging in cases where aitest object is constructed of several layers, like in our
Figure 2.3, since echoes that return to the receiver from the deeper layers will encoun-
ter the outer boundaries on their right on their return trip, and will be re-reﬂected back
into the object, as illustrated by Figure 2.5. This situation will occur repeatedly, and
in theory continues indeﬁnitely, which makes the phenomenon difficult to analyze in a
formal manner. The end result is that instead of a single echo returning from each

boundary, there will be a cavalcade of superﬂuous signals that will only cause great

incident wave >

primary reﬂection

primary reﬂection

multiple reﬂection
(externalized)

 

/

\ primary transmission

 

/
V \ multiple transmission
g (externalized)

\ internal multiple reﬂections

Figure 2.5 ,. Example of multiple imaging artifacts in an object.

confusion about which echoes are real and which are artifacts, at best, or at worst

will obscure the real echoes so that the ultrasound system cannot detect them at all.

In the latter case, there is little that can be done at present but to accept the data as

received and present it aS‘best as possible, but in the former case, presumably, we

can perform some sort of ﬁltering operation to reduce or eliminate these obfuscating

effects on the imaging quality [16]. We consider this further in Chapter 4, where an

algorithm is presented that has been rather successful at performing this ﬁltering

operation.

One final property of interest in target objects is attenuation, but we reserve this

discussion for Chapter 3.;

CHAPTER III

A'I'I'ENUATION IMAGING

In the previous chapter, we considered the basic physical phenomena that inﬂuence
acoustic energy, particularly for the conditions surrounding acoustic imaging, with the
exception of loss mechanisms. We now brieﬂy discuss the considerable subject of
mechanical wave loss mechanisms in solids and liquids, and take a look at previous

efforts to apply these phenomena to the imaging problem.

3.1—Anew

As a mechanical wave propagates through a transmission medium, the intensity of
the wavefront maxima will reduce as a function of distance travelled; this result can be
brought about from various sources, but the total effect is usually spoken of generally
as "attenuation". We must distinguish this particular type of diminution of wave
amplitude from other sources of such, for example, transmission through an impedance
discontinuity, as mentioned in Section 2.1 and Figure 2.1, where the amplitude
of the incident wave is modiﬁed by the boundary behavior that was therein discussed.

Loss of amplitude of this type is not attenuation in the sense we have deﬁned above,

24

25

and in practice, the inﬂuence of this behavior complicates accurate identiﬁcation of the
portion of the amplitude loss due to actual attenuation. We will discuss the

separation of the effects of wave reﬂection at discontinuities and actual attenuation

in the following chapter, where we will show that these two phenomena can in fact

be decoupled even in a multi-layered measurement situation (i.e. one with many such
boundaries). The discussion that follows is based on those given in Wells [2], and

Herzfeld and Litovitz [14].

One mechanism of attenuation that results from beam nonuniformities is the
change in apparent energy per unit beam area that is a consequence of the deviation
of the beam from parallel [49]. Recall we mentioned earlier that typically the beam
is assumed to consist of plane wavefronts (or at least constant intensity over each
beam cross section); in effect, we thus also have presumed that the beam does not
converge on (focusing) or diverge from (spreading, or defocusing) the line of sight.
Attenuation from this mechanism may be computed by simple calculation [36], under

the assumption that the phase fronts were initially constant.

A second class of acoustic absorption can be delineated from energy conversion
phenomena, such as viscosity and hysteresis losses, and heat conduction. These
mechanisms have been thoroughly considered in the literature, and so will not be
covered here (see, e.g., Stumpf [13]). In addition to these phenomena can be
added losses due to wave oscillation mode conversion [52], lattice/particle vibration

resonances [51 and 53], and relaxation absorption [50 and 54]. With the exception of

26

mode conversion, which can be considered a macroscopic phenomenon, these are
microscopic properties of solids and liquids, and all may share the energy present in a
vibrating mechanical system. Loss occurs when the stored energy is returned out of

phase with the desired dominant propagation mode, in our case longitudinal.

We note that the total loss observed in a particular sample medium may in fact be
a combination of some of these individual effects. For the purposes of our work here,

we can lump the separate mechanisms together and deﬁne as in [13]

a = loss per unit distance (in units of inverse distance) (3.1)

to be a quantity that relates the "lossiness" of a particular material to the general
imaging problem. Using the above deﬁnition of attenuation, we can also deﬁne a

factor, as in [14],

k=exp[-ax] (3.2)

to be a "loss factor (or parameter)" that is of interest for a speciﬁc object, where or
is the attenuation per unit distance as deﬁned in Equation 3.1, and x is the distance

travelled during the propagation of the wave. Note that k e (O, 1], a fact that we will

use later in Chapter 4.

One last concept that is of experimental importance is that attenuation is a
frequency-dependent quantity; by this, we mean that the measurable loss of a partic-

ular material is different for different acoustic wave oscillation frequencies. For

27

example, biological materials, such as liver and muscle, exhibit an inverse
dependence of loss with increasing frequency (see e.g. Bamber and Hill [39]). Other
inorganic media exhibit a proportional dependence of attenuation with frequency for a
wide range, then behave differently outside of that band [14]. These sorts of depen-
dencies make measurement of attenuation difﬁcult, since many of the techniques
available to accomplish this employ pulsed ultrasound to interrogate a sample; in
fact, a standard set of data for common materials has yet to be decided upon. In this
thesis, we will not tangle directly with this problem, but consideration of the worries

of gathering accurate experimental data in Chapter 5 will touch upon it again.

11 ' m 'n
The literature involving imaging and related problems by means of loss parameters
is varied, both in application and extent. Here we endeavor to give the reader a
glimpse of the more salient aspects of this past work. In the process, a view of the
current state of acoustic imaging will be forthcoming, and the chapter will conclude
with some comments on the motivation of the work undertaken in this thesis in this

regard.

Acoustic attenuation has been a topic of interest from the beginnings of research in
the ﬁeld of sound. For example, Fry (1952) [46] discussed in analytical detail the
mechanisms that cause attenuation to occur in tissue structures; his early work was
the basis for many of the later papers on such phenomena. The industrial uses of

ultrasound also attracted researchers to investigate how lossy materials might be

28

processed by acoustic means; such applications are amply discussed in older
textbooks such as Blitz (1967) [1], and new uses are constantly being forwarded,

many related to materials processing and manufacture.

The imaging problem has long been acknowledged in medical and industrial
research, but using attenuation to accomplish this is a relatively recent endeavor,
primarily because of the reducing cost of computer equipment. Among the more prom-
inent of the earliest work is ~that ofgreenleaf ct aLin [24, 25, 29, and 30}, where7
various techniques are described to accomplish acoustical computed tomography
in a manner similar to that of X-ray diagnostic equipment. The work centered on the
soft tissues of the human breast, with an intention to ﬁnd a means of early tumor
detection. Other researchers, for example Clement at al. in [56], were involved
in applying arrays of transducers to similar uses. The promise of arrays is still unful-
ﬁlled, but as techniques of integrating more sophisticated acoustoelectric materials
on a common substrate become more ﬁnancially possible, perhaps these methods will
be more widely investigated. Addison et al. in [55] describe an interesting
alternative to electronically scanned arrays of physical transducers, using an optical
laser to generate thermal stress on the target surface and induce mechanical waves.

In the above work, examples of a noncontact system are given, and mention is made
of the possibilities of simulating arrays of point sources in complex conﬁgurations.
This work has yet to see a more widespread acceptance however, and it certainly will
be some time before the complex and delicate optics required to perform the requisite

signal reception are of sufﬁcient durability to satisfy non-research applications.

29

Typical of the attenuation measurement systems reported is that of Klepper et al.
in [28]. Their intention is to ﬁnd points of in-vitro tissue pathology, speciﬁcally for
cardiac infarction. Their experimental set up is muchulike we have already described,
utilizing a pair of diametrically opposed transducers that are computer controlled in
rotational position about the object. The transducers they employ are of acousto-
electric nature, which gives the receiver the property of being somewhat phase insen-
sitive with respect to the incoming signal [23]. Reconstruction of the object is done
by means of one of several imaging parameters, including attenuation and time-of-
ﬂight. In the above paper, a mathematical model is formed along straight lines that
pierce the object in the various measurement directions, thus ignoring possible
angular scattering. The results indicate a reasonably good quality of image for the
object model used, and the authors report that they were able to detect the tissue
abnormality in most of the cases they investigated. Additional experiments performed
by imaging with the attenuation slope between two neighboring frequencies seems to
have given even more accurate images. The authors concluded that such effort had
potential for in-vivo mammography, although no experiments have appeared subse-

quently to substantiate this opinion.

Glover and Sharp [7] reported an imaging system based on time-of-ﬂight pro-
jections along straight lines through the object, again using a pair of rotatable trans-
ducers and the transmission tomography technique. The interest in this work is
centered about the clarity of the images obtained and their marked similarity to the

actual physical object. The conclusions the authors arrive at is that transmission

30

signals are far less affected by the amplitude distorting effects we will consider in
Chapter 5, and that these signals are of prime importance to an imaging system that

will investigate varied objects.

Other methods intended to pursue this goal abound. Farrell [48] proposes an
iterative ﬁltering technique to combat the image distortions associated with speckle,
a by-product of the transducer position scanning that is necessary in the pulse-echo
technique. Kuc et a1. [32] exploited the approximate linear relation between the
transducer frequency and the acoustic absorption to form imaging parameters based
on the frequency slope of the attenuation. Data in this work is taken at several
distinct operating frequencies, as opposed to using Fourier analysis; they propose
that certain statistical measures be used for best advantage of the interpretation of
any results that such a technique might yield. Parker and Waag [42] forwarded
results of experiments they performed to process backseattered ultrasound signals
from tissue boundaries to determine the medium loss as a function of frequency. They
describe in detail the steps necessary to compensate for the distortion introduced by
employing a time- gain compensator (see Chapter 5), and proceed to compute the
attenuation within a selected portion of the structure, which they suggest would be
useful for diagnostic clinical situations. Dameron [40] discusses the case of continu-
ous media, and discusses the enhancement of attenuation images by means of
correction of the distortion introduced from a nonuniform propagation velocity within
the sample. Greenleaf et al. in [43] consider the effects of digital signal processing

kernels on the results obtained from electronically scanned arrays, in order to obtain

31

synthetic focusing of the transducer beam. Studies were presented that indicated a
greatly increased lateral resolvability for their system, and mention is made of the
compensation for beam diffraction in limited instances, primarily in simulated tests.
Further work in this vein was reported in Greenleaf et al. in [26] and [30], where
speciﬁc attempts to apply attenuation imaging to production of human mammograms
was undertaken in-vivo. Methods were described for the solution of various acoustic
parameters of interest, with some results forwarded indicating that automated identi-

ﬁcation of tumor structures is conceivable.

Much of the reported research has emphasized the engineering aspects of
building such systems, so it is only natural that some assumptions have been made in
pursuit of the goal of a practical and affordable system. Often, the authors are not
explicit when discussing the application of their theories, and the means by which
they acquired their experimental results are often unclear. For example, as we
demonstrate in the following chapters, simply using the amplitudes of the echo
returns or transmission signals is not theoretically adequate to determine the attenu-
ation of a medium under a wide range of circumstances, since other mechanisms can
inﬂuence the amplitude data. Most papers neglect to mention how such inﬂuences
are removed from necessity of consideration, which limits the usefulness of these
works to others. The work in the following chapters attempts to clearly list the
assumptions and limitations, as well as the implications, of each item, tedious as this

might be for the reader and writer.

32

3.3 Attenuation-Velocity Mums

In an effort to reevaluate the existing methods of attenuation imaging, the author
arrived at the conclusion that in general it is impossible to determine the value of a
in an experimental situation where nothing is known about the object internal makeup
beforehand. We will see this clearly in Section 4.6, where we show that the velocity
and attenuation of a material are inseparably bound together when using the pulse-
echo imaging technique. It was therefore convenient to speak in terms of the product
of these quantities as being a desirable unknown. Further investigation led to the
discovery that the range of values for av is very large, as shown in Table 3.1, and
this wide spread permits a high degree of contrast between different media. In fact,
such a wide difference would permit material identiﬁcation to be made, given a
predeﬁned table of values for various media, perhaps in conjunction with the
reﬂection coefﬁcient information calculated by the solution method of Chapter 4. This
would be extremely useful for many of the imaging situations that we have previously

discussed.

In Chapter 4, we begin discussion of the theoretical speciﬁcs to determine the

quantity av in a general one-dimensional object. In following chapters, a more

pragmatic viewpoint is taken, and application methodology and error sources are

more closely evaluated.

33

 

 

 

 

 

 

 

 

 

Material a (dB/cm) v (m/sec) av (dB/sec)
water 0.0022 1480 325.60
aluminum 0.018 6400 . __1gl_,__,_52,0.0
plexiglass 2.00 2680 536,000.0
air 12.00 331 397,200.0
castor oil 0.950 1500 l42,500.0
mercury 0.00048 1450 69.60
polyethylene 4.70 2000 9.4 x 106
fat 0.630 1450 91,3500
brain 0.850 1541 130,985.0
liver 0.940 1549 145,606.0
blood 0.180 1570 28,2600
skull bone 20.00 4080 8.19 x 106
muscle 1.30 1585 206,050.0
kidney 1.00 1561 156,100.0
lens of eye 2.00 1620 324,000.0
Table 3.1 Attenuation-Velocity Products of various media at a

1 MHz transducer frequency (based on Wells [2] ).

 

CHAPTER IV

DETERMINATION OF ATTENUATION-VELOCITY PRODUCTS

Up to this point in this thesis, we have considered both the principles and
problems that are involved in acoustic measurements, and have also discussed the
available literature in the ﬁeld of attenuation-type acoustic imaging. As we have
seen, there have been many attempts, both theoretical and practical, to address this
difﬁcult application of ultrasound, and there have been varying degrees of success in
achieving this goal. After reviewing this literature, the author is of the Opinion that
different concepts about what constitutes attenuation imaging seem to exist, and no
real formalization of this technique has yet become accepted by the research commun-
ity at large. This is unfortunate, since applications and non-specialist users will only
be suspicious and remain unconvinced that what they themselves expect in their

diagnostic tool is what is really being done inside the "black box".

One of the basic motivations of undertaking the research delineated in this
thesis simply was to investigate this very same inconsistency in attenuation imaging
and perhaps help to create a more formal approach to visualizing the conditions of the
problem. In particular, it was intended to examine the importance of each aspect of

an ultrasound system as related to computing the attenuation of the regions in a

34

35

test object, with an awareness of the need for identiﬁcation of solvable and insoluble
difﬁculties, and to reﬂect on possible compromises to these problems. One item

that becomes apparent upon perusal of the literature is a tendency to neglect rigorous
and thorough listing of the assumptions made and the exact techniques employed to
create the results and images that are presented, and the end result of this oversight
can only be that the usefulness of this work is limited to other researchers. No signif-
icant progress can be made in any ﬁeld without the cooperation and precise interaction
of many people, and in a subject such as acoustics, with such a broad base of pheno-
mena and applications, each demanding individual expertise, this need for clear
interfacing between researchers and others is even further crucial. The intention here

is to clearly identify the assumptions and their effect on the generality of the results.

The discussion that follows in this and subsequent chapters will take the a pro-
gressive outline, in the interest of presenting the new material earliest so that later
examination of the impact of other difﬁculties on the concepts will be feasible; with this

in mind, we start at the beginning...

4 l B si 1 f Thi S ti n

Our primary intention is to investigate the experimental data that is necessary to
allow computation of a quantity related to acoustic attenuation in a layered test object,
while keeping in mind the assumptions and efﬁcacy of any method/computations we
propose to achieve this goal. For example, in practice we are not likely to know

beforehand the number of layers that comprise the test object in a particular direction,

36

so any technique we devise should solve for this number, as well as any other such
unknown quantities. Additionally, we ask what importance the reﬂection coefficients

of each boundary may have on the computation of the attenuation in the object layers.

4 l n f' ' ' n f T ' 1

We set up the situation shown in Figure 4.1. This is a simple one-dimensional
model that is intended to represent the internal layout of our test object in a particular
direction. Note that there are N layers of up to N distinct materials comprising the
object, and that we have allowed for the possibility of two transducers (the maximum
number possible in a one—dimensional situation) being used in our measurement.
Since our goal is to accomplish attenuation-like imaging, by inspecting Figure 4.1 we
see that we are then required to determine the N values of k that represent the

loss present in each layer. We should point out that each ki may in fact be a contin-
uous function of position, i.e. ki(x) where x is a position within the layer, but any

discontinuities are not permissible within the layer boundaries since, as we have

noted in the derivation of Equation (2.9) and in Figure 2.1, any discontinuity of imped-
ance (or any other constitutive parameter, for that matter) will cause an impinging
wavefront to exhibit the type of boundary behavior we described at that time in
Chapter 2. Also, any precipitous values of the derivative(s) in the constitutive
parameters, when compared with a wavelength of the wavefront at the transducer
operating frequency, are disallowed, since such a occurrence can cause behavior

not unlike that of a discontinuity (in fact, this latter can be considered to be a deﬁnition

37

 

 

 

   

left .1, g . right
layer ;
transducer I;::gf:E_ transducer
ri+r rN—r
where

N = the number of object layers.

the loss parameter of the i-th layer.

= exp[—ai Vi til

the attenuation in layer i in units of (distance)'1.

0‘1
Vi = the propagation velocity in layer i in units of (distance/time).
ti = the propagation time delay in layer i in units of (time).

ri = the reﬂection coefﬁcient between boundaries i-l and i.

Figure 4.1 The object model to be studied and the
notation conventions used in the derivation of the
equations in this chapter.

38

of a discontinuity, for somewhat inexact but practical purposes) [51].

4. Assum tions to Solve th -Dimensional N-la er b'ect
In order to successfully attack the N-layer problem, we will make the following

assumptions, and then later examine their implications:

(1) We will use the pulse-echo type of measurement to probe the
object; this will allow us to rather easily measure the amplitude of
each echo in the signal, by e. g. a local peak detector.

(2) Unidimensional wave propagation is the only mode allowed, in
order to remain consistent with our one dimensional model. This
precludes angular scattering.

(3) Homogeneous layer composition, or simply that ki(x) = ki, in order

simplify the variables to constants. We must ﬁrst ask whether this
case can be solved, before tackling the more difficult spatial
variation problem.

(4) Coplanar layers and boundaries are necessary in order to remain
consistent with Assumption (2) above; implicit in this is the
added condition that the boundaries be perpendicular to the line
of sight along the beam path of the transducers.

(5) No multiple reﬂections are present in the data we will manipulate

to solve for the loss parameters It. This is not unreasonable since

we will later discuss an algorithm to accomplish this very feat.

39

(6) The number of layers, N, is a known quantity; although this too
appears to be unreasonable, it will be shown later, as in Assump-

tion (5), that this can in fact be done before processing the data.

The assumptions above are not inconsistent with our one-dimensional object model,
and yet do not strongly limit the generality of possible values for the constitutive
parameters, within reasonable limits. For example, extreme attenuation in a particu-
lar layer, even under the best circumstances, will most likely cause difﬁculty in obser-
ving underlying layers, perhaps to the point of total obfuscation. With such cases
suitably discounted beforehand, we can now begin to investigate the solution of the

N-layer problem.

4 1 II ll 1 Q . .
By examining Figure 4.1 and considering the assumptions given above, we anive

at the following list of unknown quantities pertaining to the N-layer object:

(1) The N values of ki’ the loss parameters of the layers.
(2) The N-l values of ri, the reﬂection coefﬁcients of the boundaries.

(3) The N values of ti, the propagation time delays in the layers.

Taken as a whole, we can represent the object as a set (N, k, r, t}, where N is the
number of object layers in this direction, k is an N-vector that is comprised of the loss

parameters of the layers, r is an (N-1)-vector with as its elements the N-l reﬂection

40

coefﬁcients of the boundaries, and t is an N-vector that is comprised of the time

delays ti of the layers. We note that by specifying the set {N, k, r, t} we completely

and uniquely describe the object in this direction, and thus to ﬁnd a unique solution

to the posed N-layer problem we must ﬁnd all the elements in this set.

4,5 Sglugg’ n of the mblem
We now undertake to consider several arrangements of experimental data that the
conﬁguration given in Figure 4.1 can provide to us, subject to our stated assumptions.

The possibilities and problems inherent in each will be explored.

4.5.1 Case 1: Sin gle-Sided giggly Reﬂective Intgrggatign

In most ultrasound imaging systems, only a single transducer is used to probe the
test object, and this is done only from a single side during the entire measurement
procedure, even when the transducer is used in B and C scans. For example, consider
the "left" transducer of Figure 4.1 (noting that we may use a like discussion for the
"right" side, if we so desired). In this situation, with the assumptions above in mind,

we derive a series of echoes, one ﬁom each boundary, as shown in Figure 4.2.

L1 L LN-2

ﬁ“.

02 O'N_1 tika

t=0 01 j

 

0' GIN-2 I
L
L 2 N-l

Figure 4.2. Example of left side echo data.

41

In Figure 4.2, we see that what we measure is a series of N-l echoes Li which each

reﬂect from a boundary, speciﬁcally the i-th, and return to the receiving transducer at

times Oi, where we allow i to range from 1 to N-l. As we have assumed, we know

the value of N, the number of layers in the object, beforehand, ﬁom prior consider-
ation by means of our yet to be discussed algorithm.

In order to compute ti, the propagation time delay in the i-th layer, and thus ﬁnd

the N-vector t, we simply can do the following:

ti = (o- - on) / 2 (4.1)

if we will deﬁne 0'0 = 0 and allow i = 1 to N-l. We note in passing that if we had

known what the acoustic velocity in each layer was, we could then compute the

physical thicknesses of the layers, or vice versa, using also these values of ti, but

since these quantities are unavailable, we cannot do this.
It can’be rather easily veriﬁed, by ray tracing and use of Equation (2.13) (recall that
we have assumed normal incidence here), that the equation describing the amplitudes

of the NI left echoes is
_ 2 2 2 2 2
Lian r) - [input/tea "0 ri(1-r1 > In ---<1 “Ii-1 Hem (42>
where i=1 to N-l and Ainput/left is the initial left-input pulse amplitude sent out. Also

time(Li)=2(tO+t1+...+ti) ' (4.3)

42

where i=1 to N-l, is the time delay of the left echo, where we have used the deﬁnition

of ti given in Equation (4.1). We note, by inspection of Equation (4.2), that
sign (ri ) = sign ( Li) (4,4)

where i=1 to N-l, which is an important fact that we will use later.
We are thus presented with a problem that we cannot solve, since we only have

the N-l equations given in Equation (4.2) and the N—l experimental values of Li that
are illustrated in Figure 4.2, but have many more unknowns to ﬁnd, speciﬁcally the

values of Ainput/left’ k0, . . . , N-l’ r1, . . . ’ N-l , which total to 2N unknown quanti-

ties. We should note that we actually need only ﬁnd the magnitudes of ri since the

signs of the reﬂection coefﬁcients are already known, by Equation (4.4). In some

cases, the value of Ainputﬂeft

might be known, but this is generally not possible in
practice since the initial pulse amplitude is a function of temperature, damping, etc.,
all of which are quantities that are difficult to hold ﬁxed even under laboratory con-
ditions. We naturally would prefer to not deal with this value at all, and later we
show how it is possible to eliminate it from consideration completely.

From the above discussion, we can only decide that attenuation imaging is im-
possible to achieve from a single direction and merely single sided data echoes. The
best that we can do is to compromise on strictness and speak only in terms of

quantities that are vaguely proportional to attenuation, such as rk products. Even this

is not acceptable, since the value of the rk product for a speciﬁc material will change

43

in different measurement conﬁgurations, e.g. different adjacent materials. This is
due to the fact that r is dependent on two adjacent layers and not only on a single

layer’s characteristics; therefore this is not a good choice for an imaging index.

4.5.2 Case 2: Pure Transmission Interrogation

In the manner of Greenleaf et al. [43], we now investigate the situation of trans-
mission imaging. The discussion here is limited to consideration of the effects of
r and k on an single measurement line (direction), one of the many used to construct
an image in the paper above. It can be shown, in like manner as in Equation (4.2),

that the left-to-right transmission pulse amplitude is given by

TLR(k, r) = Ainput/left k0 (1 '1’ 1'1) k1 . . . (l + rN-1)kN-1 (4.5)

and that the time delay of this single pulse is given by
time(TLR)=to+tl+...+tN_l (4.6)

where the quantities are as deﬁned for Case 1 above. It is signiﬁcant to realize that
in this instance, multiple reﬂections are never a problem since we are only interested
in this single pulse, and it is the first to arrive at the receiver. Note that this experi-
mental situation requires that we use both the left and right transducers in Figure 4.1,
oppositely aligned and either matched or calibrated to be so. Also, from an argument
similar to that taken in Case 1, we know immediately that this single Equation (4.5) is

woefully inadequate to solve for the same 2N unknowns we listed in Case 1. In the

work of Greenleaf et al., this difﬁculty is circumvented by two means: (1) the reﬂec-
tion coefficients are assumed to be negligibly small (when compared to unity), and (2)
deﬁning the object in a different manner than we have, as a continuous variation of im-
pedance and attenuation. The latter is advantageous in their case since they intend to
make a series of measurements from many vantage points around the object, and

then synthesize an attenuation map of the inside by means of a CT scanner-like
solution of a large linear system. We can, by inspection of Equation (4.5), see the
negative effect of the reﬂection coefﬁcients on the results of their calculations, which
in the present author’s experience can be considerable at times. Assuming that the
reﬂection coefﬁcients are negligibly small severely limits the scope of their ranges,

as well as the classes of permissible objects. In the next chapter, we brieﬂy discuss
how to extend the method presented in this thesis to include the work of Greenleaf

et al. and allow loss-like imaging in objects with both spatial variation of attenuation

and non-negligible reﬂection coefﬁcients.

153 C 3.5.1. i 11 .
With the set up of Figure 4.1, having two oppositely placed transducers, we can
generate additional data by probing the object from both the left and the right sides
(sequentially, not simultaneously), and since the data from these measurements
arises from like physical situations, we can use all of it simultaneously to solve for r
and It. By an argument similar for Equations (4.2) and (4.3), we ﬁnd that the right
side echo return amplitudes due to an independent right side input pulse are given

by the equation

4S

2 2 2 2 2
Ri(k’r)=Ainput/rightki ('ri)(1"i+l ”8+1 '“(l'rN-l )kN-l

(4.7)
and that the time delays of these echoes are given by
time(Ri)=2(ti+ti+1+...+tN_1) (4.8)

where we have deﬁned Ri to be the right-input echo that bounces off of boundary 1,
using the numbering given in Figure 4.1, in correspondence with Li as shown in
Figure 4.3, and where Ainput/right is the amplitude of the initial right input pulse and

i = 1 to N-l . In general, we cannot expect that Ainput/left and Ainput/r‘ight are

identical quantities, for the same reasons given in Case 1 regarding the irnpracticality

of knowing A One consequence of the numbering scheme we chose in

input/left
Figure 4.1 and Equation (4.7) is that the right-input echoes come out in the reverse

indexed order than do the left-input echoes given in Figure 4.2; this is illustrated by

 

[{i

Figure 4.3 The deﬁnition of Ri with respect to Li'

46

 

RN-r
R2
t=0 :
5N—1 I i 62

Figure 4.4 Example of right side input echo returns.

Figure 4.4, which shows an example compatible with Figure 4.2, the left side input
echo example.

Note that we can use Equation (4.1) to derive the time delays of the layers t, but
for completeness we note that by using the right side echo arrival times shown in

Figure (4.4), that

ti= (SN—i-l " 5N—i) / 2 (4.9)

where 5i is as indicated in Figure 4.4, i = l to N-l, and we must deﬁne 5N = 0.

In a manner similar to Equation (4.4), we see that by inspection of Equation (4.7)

that
sign (ri ) = - sign ( R1 ) (4.10)

and using Equation (4.4) we arrive at a simple but important fact we will make use

use of later also, that

47

sign(Li)=-sign(Ri) (4.11)

The negative sign in Equations (4.10) and (4.11) arises from the fact that a reﬂection
coefﬁcient viewed from the right has the opposite sign of the same viewed from the
left side; one may see this readily from either Equations (2.9) or (2. 13).

As we stated before, the 2N -2 left and right echoes that give us the 2N-2 values

of experimental data Li and Ri are derived from similar physical situations in the object

and thus we may use these data simultaneously in our solution.. There now exist

2N+1 unknowns (now including Ainput/right)’ so we“ see that the problem is still

underspeciﬁed to allow a solution. In order to accomplish this, we must either
(1) reduce the number of unknowns, or (2) increase the number of equations/experi-

mental data pairs. Certainly from a practical standpoint the values of k0 and kN—l

are usually known to us (or are measurable independently) beforehand, since these
layers are almost always water or some other type of acoustic couplant, and therefore
are identiﬁed materials with accessible properties (e.g. by tables). In order to solve
this dilemma of not enough equations/data, we will use the latter fact, along with the
added information available from the transducer opposite to the transmitter, i.e. the

transmission data as well, namely Equations (4.5) and (4.6) for TLR and, for the

right-to—left transmission pulse, that

TRL(k,r) = Ainput/right k0 (1 ' 1'1) k1 . . . (I ' rN_1) kN-l (4.12)

and

48

timC(TRL)=t0+tl+...+tN_l (4.13)

=timC(TLR)

It is interesting to note that Equations (4.5) and (4.12) are not identical; once again,
these differing signs are the results of the left and right viewpoints of the reﬂection
coefﬁcients. The implication of this distinction in the two equations is that the ampli-
tude of the left-right and the right-left transmission pulses are different in value, a
fact which seems non-intuitive. To understand this, however, one need simply to
look at a single boundary, as in Figure 2.1, and examine Equations (2.10) and (2.11);
it becomes readily apparent that even in the single boundary case of Figure 2.1, the
transmission coefﬁcient is different when looking from either the left or right side.
We expect then that the N-layer case will behave similarly, and our conclusions
about Equations (4.5) and (4.12) are valid. Additionally, this left and right difference
has been substantiated in laboratory experiments.

With the addition of the two extra transmission equations, we now have 2N
pairs of equations/data to use in solving the N -layer problem. We will denote this set

by {L, TLR’ R, TRL}primary' where we intend the subscript "primary" to distinguish

this set of data without multiple reﬂections from {L, TLR’ R, TRL’experimental’

which is the set of all the data that we derive from the object, including any multiple
reﬂections that occur. The primary data is then a "ﬁltered" version of the experimental
data, and the algorithm we present later is expected to accomplish this ﬁltering before

we perform any solution. The number of unknowns has been reduced to 2N-1, since

49

we have assumed that we know k0 and kN-l’ but if we inspect Equations (4.2), (4.5)
(4.7), and (4.12), we will see that itis not possible to, solve for either Ainput/left or

Ainput/right’ and we must deal with this, or else we cannot solve the problem at all.

We can approach this as follows:

 

 

 

 

 

Form the ratios
' Li+1(k’r) '-
Hi(k,r) =
Li(k,r)
(4.14)
Rider)
H ' (k,r) =
N+r-2
Ri+l(k")
where i = 1 to N-2. Irnportantly, we note that the above equations do not involve
either Ainput/left or Ainput/right' If we deﬁne new equations to solve, i.e.
G (k ) Hl(k ) Li+1(experimental) T
. r = . r -
r ’ r .
Li(experrmental)
(4.15)
G Ri(experimental)
-_ (k,r) = H -_ (k,r) -
N+l 2 N+l 2 Li+l(CXPcrimental)

 

where again i = 1 to N -2, thus forming a system of 2N-4 equations. We notice that

the deﬁnition given in Equation (4. 15) is made so as to force the solution of the

50

problem when we set G(k,r) = 0. Substitution of Equations (4.2) and (4.7) into
Equation (4.15) yields
r- l - Li+1(experimental)

2 2 +
Gi(k,r) = 1ti (1 -ri ) T‘i— Limpcﬁmmm)

 

(4.16)

ri Ri(experimental)
i+1 R1+1(experimental)

 

 

2 2
Ohm-20‘") = "i (1 “i+1 )

 

_J

where i = l to N2. This is a fascinating result, since it clearly shows that the
uniqueness of the solution depends only on the values of r and not at all on those of

k; in effect, we have successfully decoupled the single problem into two smaller ones,
and the problem of solving for k is simply a matter of ”setting G(k,r) = 0, once we have
the values of r. We should also note that our solution has the implicit assumption that

the values of kc and kN-l are known, as we discussed before. Additionally, it seems

that the uniqueness of the solution depends only on the magnitudes of the values of r,
since we have the signs of r by either of Equations (4.4) or (4.10). In order to ﬁnd the
values of r, we perform the following steps:

Using Equations (4.7) and (4.12), form the product
TLR(k,r) TRL(k,r) = (4.17)
2 2 2 2 2
Ainput/left Ainput/right 1‘0 (1 ‘ r1 “‘1 - - - (1 ' rN-l )kN-l

We note the similarity between this product and the echo equations. Also form

51

Li(k,r) Ri(k,r) = (4.18)
2 2 2 2 2
Ainput/left Ainput/rightk0 ki (' ri ) (1 " r1 ”‘1 ° ' °

2 2 2 2 2 2
--(1"i-1 )ki-l (1414.1 ”i+1 °°°(l'rN-l )kN-l

where i = 1 to N-l. Comparison of Equations (4.17) and (4.18) allows us to write

that

 

Li(krr) R108" ) = TLRO‘ar) TRI‘(kJ')

where i = 1 to N-l. Using the experimental values for L and R, we can solve the

above equations for the values of ri

 

' Ri Li I . (4.20)

TLRTRL'RiLi

 

ri = sign ( ri)

where i =1 to N-l. This is the result that we desired, and it is unexpectedly simple,
involving only the echoes from the i-th boundary and the transmission amplitudes.
Recall that we can ﬁnd the signs of r by use of either Equations (4.4) or (4. 10). It is
worth noting that the roots in Equation (4.20) always exist, since the radicand is

always positive, which is easy to see by Equation (4.11), and the fact that both TLR

and TRL are always positive as well, by inspection of Equations (4.5) and (4.12). We

52

can now solve for the vales of k1 . . . N-2 (recall that we assumed that we know the

values of kc and kN- 1) by using either of Equation (4.16); i.e.

 

Li+1 1.i

2
Li 1ri+1 (1 ' ri )

 

J.“
ll

(4.21a)

or the similar result involving the right echo amplitudes

 

_ . r.
ki = R1 ”1 (4.21b)

2
Ri+1 Ii (1 “i+1 )

 

where for both Equations (4.21a) and (4.21b) we allow i = 1 to NZ. The combination

of Equations (4.20) and (4.21) is the solution to the N—layer problem we desired.

 

The solution that we derived in the previous section for the one-dimensional
N-layer problem is completely general in use, since we have assumed no restrictions
on the range of values that may be taken by any r or k in the object. After conducting
a pair of left and right input measurements, yielding the experimental data set that we

have called {L, TLR’ R, TRL} tal which may include multiple reﬂections,

experimen
we then apply the algorithm which is described in the following section which provides

the ﬁltering of this experimental set into the set {L, TLR’ R, TRL} which does

primary

53

not contain multiple reﬂections, and has implicitly contained in it the value of N, the
number of layers in the object in this particular direction (this value is easily found
from the primary data set by simply counting the number of echoes in either L or R,
and adding one). Application of Equations (4.1), (4.4), (4.10), (4.20), and (4.21) will
then provide the solution to the posed N—layer object, in the form of set {N, k, r, t} ,
which as we have stated earlier is sufﬁcient to uniquely deﬁne the object model in the
measurement direction we have chosen.

Once having acquired the proper set {N , k, r, t] , we then would proceed to compute
the imaging parameters that we would use for output and/or graphic display. For ex-
ample, it is possible to use the time delays t in the layers as a parameter for this
purpose. Perhaps of more interest would be a display, of the reﬂection coefﬁcients, or
even a display of the characteristic impedances of the layers. We can accomplish this
easily because of our knowledge of the impedances of layers 0 and N-l (or we might
simply normalize these values to unity for convenience), and by making use of a

rearranged version of Equation (2.13a), i.e.

1+5 _ «23

 

where i = 1 to N-2 and we know 20 and ZN-l already. This can be used in impedi-

ography, as ﬁrst suggested by Jones [41] , only the calculations here will be more
accurate than in the above paper since we decoupled the effects of k and r on the echo

amplitudes.

54

As the title of this thesis suggests, we are primarily concerned with ﬁnding the
attenuation-velocity products of the layers for use as imaging parameters, so we now

describe this computation. As shown in Figure 4.1, the deﬁnition of kl contains the

product of ai and vi, which are thus inseparable since, even though we know ti, we are
not aware of the layer thicknesses and thus cannot compute Vi separately. However,

if we would rather use the attenuation-velocity products, we can solve for them simply
by rearrangement of the deﬁnition of ki’ i.e.

aivi = -ti'11nki (4.23)

where i =0 to N1. We then possess the quantities we originally sought, and can pro-
ceed to image, perhaps with a gray-scale or color graphics display to enhance the

different regions of the object.

4.] Multiple Reﬂection Elirninetien and Finm' g the Number of Layers
The ﬁnal issues left to consider in this chapter are: (1) the determination of N, the
number of object layers in the measurement direction, and (2) the elimination of the

multiple reﬂection artifacts in (L, TLR’ R, TRL} tal to get the set of main

experimen

echoes {L, TLR’ R, TRL} which is lacking in these artifacts.

primary’

4.7.1 Finding N, the Number of Object Layegg

In the case where there are no multiple reﬂections in the experimental data, this

task is just limited to counting the number of echoes in either L or R and adding one.

55

However, when these artifacts are present, and experience seems to dictate that they
are present in experimental data derived from even the simplest target objects, ﬁnding
the value of N becomes more complex; in fact, it is inseparably tied together with the

problem of ﬁltering out the multiple reﬂections from the experimental data set.

42 FininthPrim Dt in fMl'lRﬂ tion
Initially, this problems appears to be insoluble, but an algorithm was developed
in conjunction with the solution given by Equations (4.20) and (4.21) that has been
successful in choosing the primary data set from the experimental data, even in the
presence of severe multiple reﬂections (e.g. if the number of multiple reﬂections is
comparable to the number of primary echoes present in the data). This algorithm is
diagrammed in Figure 4.5. Inspecting this, we see that the upper bound on the value

of N to which the trial value Ntrial is set is taken from the larger of the number of

echoes in either L tal or R tal' This will result in one of the follow-

experimen experimen

ing three conditions being true: (1) Ntrial is identical with N a1 (i.e. is the right

actu
value, since it is the same as the actual number of layers in the object), (2) the value

of Nuial is larger than Nactual (which will occur in the presence of multiple reﬂec-
tions in either or both Lexperimental and Rexperimental)’ or (3) the value of Ntrial is
less than the value of N actual (which can occur when deep-lying echoes from layers

remote in the object are so diminished in amplitude that they are overlooked by the
echo detection apparatus that precedes the algorithm). The latter case has a solution,

which is discussed in more detail in Chapter 5, where we consider the effects of the

56

 

Given: {L9 TLR’ Ra TRLlcxpcﬁmcntal

 

 

 

 

Set Ntrial = 1 + max [ # in Lexperimental’ # in Rexperimentaﬂ

V__, # in L
for each exp left echo combination

 

 

 

 

ll

 

 

for each # in Rexp right echo combination
0 V Ntﬁﬁl ‘1

and for each Nuial decremented down to 1

 

 

 

 

 

 

If for i=1 to Ntrial - 1 both
sign( Li ) = - sign( Rl )

v

 

and

 

timC( Li ) '1' timC( RI ) = timC( TLR ) "I" timC( TRL )

 

 

 

 

 

 

v > Then ﬁnd {N, k, 1', ”trial by Equations (4.20)
A and (4.21), and form {L, TLR’ R, TRUsimulated

[.__ Test if simulation and experimental sets agree.
"OK" :

DONE: either have {N, k, r, t]

 

 

 

 

 

H 0 n .
no solutron 4, or have "no solution".

 

 

 

 

Figure 4.5 The algorithm described in this section that ﬁnds the set
{N, k, r, t] and eliminates multiple reﬂections.

57

system components on the performance of the attenuation-velocity method.

In each of the three cases listed above, the algorithm of Figure 4.5 has been
successful in either ﬁnding the correct primary echoes and set of {N, k ,r, t}, or has
correctly reported that no solution exists for the given experimental data set, e. g. in
case 3, where some of the primary echoes are missing from the experimental data.
What has proven to be particularly fascinating is that for even very complicated tar-
gets, with large N, only a single (correct) solution appears to exist for {N, k, r, t},
even in the presence of many multiple reﬂections! This surprising result leads to
the optimistic speculation that perhaps the corruptingeffects of these artifacts can be
eliminated, or at least reduced in number, in the more useful (and difﬁcult) three-
dimensional object measurement situation, at some future date. The possibility of
extending the attenuation-velocity product method to higher dimensions will be con-
sidered brieﬂy in Chapter 7.

As a ﬁnal note on the algorithm in Figure 4.5, we should comment on the method
of simulating the trial model of the object. This is not difﬁcult to perform, and the
use of Equation (2.13) is necessary, along with the fact that the reﬂection coefﬁcient
of a boundary is opposite in sign, and a transmission coefficient is (1 - r) instead
of (1 + r), if viewed from the opposite direction than they were originally deﬁned. This
concept is illustrated in Figure 4.6, and may be veriﬁed by consideration of Figure 2.1
and the derivation following that ﬁgure. In order to produce the simulated data set

{L, TLR’ R, TRL}simulated’ which is a set that is intended to correspond with the

set of actual experimental data {L, TLR’ R, TRL} the following steps

experimental’

58

left incident

wave
r t=l+r
r I O O
nght rncrdent
wave
t = 1 " r - 1-

Figure 4.6 Illustration of the difference between the boundary
properties when viewed from opposite sides.

are performed:

(0) Given: {N, k, I', ”trial

(1) Select the minimum amplitude % tolerance to allow in the simula-

 

tion by taking the minimum of
smallest echo in Lexperimental
largest echo in Lexperimental
or
smallest echo m R “glimm—

 

largest echo in Rexperimental

(2) Select the maximum simulated echo time by ﬁnding

maximum I: time( TLR(CXPCﬁmnta-l» t

time( TRL(experimental)) ]

(3)

(4)

(5)

(6)

59

Input a left transmit pulse on the left boundary of the object

defined by {N, k, 1', ”trial' Follow this pulse from boundary

to boundary, creating a reﬂected and a transmitted wave at each
boundary encountered, using Figure 4.6 as a guide. Follow all these
pulses likewise, modifying their amplitudes and times according to

the values in {N, k, r, ”trial When either the minimum amplitude

or maximum time limits are exceeded by a pulse, then eliminate it
from further consideration. Eventually, pulses will reach the left
side (boundary 0) and are considered to be left echoes and are put

into Lsimulated and eliminated, or reach the right side (boundary N)

and be considered left-to-right transmissions and are put into

TLR(simulated) and eliminated. After a time depending on

the tolerances and N all the pulses will have been eliminated.

trial’
Repeat (3) for a right input pulse at boundary N and for

R(simulated), TRL(simulated), etc.
Sort the pulses in {L, TLR’ R, TRLlsimulated into time order.
Scale each pulse in Lsimulated and TLR(simulated) by the factor

amplitude ( L] (experimental))
amplitude ( L1(simulated))

 

and scale each echo in Rsimulated and TRL(simulated) by the like

factor involving R1(experimental) and R1(simulated).

After performing these steps, we Will have the 86! {L, TLRr R. TRUsimulated’

which is just a simulated version of {L, TLR’ R, TRL) tal that has used

experimen

{N, k, r, ”trial instead of {N, k, r, ”actual

Finally, the trickiest part of applying the algorithm in Figure 4.5 is in making the
comparison between the simulated and experimental data sets to determine if a good

set {N, k, r, ”trial has been determined. This is not as easy as it ﬁrst appears, since

what constitutes a good ﬁt between the two sets is difficult to deﬁne, particularly

since the dimensions of the vectors L, TLR' R, and TRL in either set may be of a

different order (different dimensions, or number of echoes). In the trials done for this
thesis, a simple requirement was employed that expected a ﬁxed percentage of the
non-primary echoes to be within a certain tolerance in amplitude and time. Naturally,

more sophisticated methods of comparison could be used, such as pattern recognition.

In the next chapter, we investigate in more detail the requirements that implement-
ing the attenuation-velocity product method would impose on a ultrasound system,
and explore in some detail the sources of error that we can expect from each of the
components in such a system. Also, some of the techniques that could be used to
preprocess the experimental data before using the algorithm of Figure 4.5 will be

discussed.

CHAPTER V

SYSTEM ANALYSIS AND ERRORS

.1 In ' n

In the previous chapters, we have investigated the concepts and possibilities that
the method of attenuation-velocity product imaging possesses; now we will turn to
the more practical problems presented to one who would implement such a system.
In particular, we are interested in what types of hardware and software components
might be required, what features these must possess, and how the errors that each
section and functional block introduce affect the accuracy of an acoustical measure-
ment performed with such a system. For the purposes of our analysis, we will use
the conﬁguration diagrammed in Figure 5.1; the components indicated will be common
in many ultrasound systems, and our queries will be enlightening even for the cases
where a particular system might not follow this diagram literally. We will center our

discussion on the issues that would confront an attenuation measurement system.

5.2 The Te§t Object

In Chapter 2, we approached the nature of acoustic wave behavior in a propagation

medium from two points of view. First, we began with the mechanical wave and

61

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63

discussed from its point of view how certain object conditions affect its dynamics, e.g..
the behavior at an impedance discontinuity. This allowed us to devise basic equations
to use in further developments, which we approached in the second half of the chapter.
From the transducer point of view, we considered the implications of certain object
features and forwarded some assumptions that are usually introduced in order to
simplify the analysis of the performance of a system. In this section, we will recon-
sider these and other items as they speciﬁcally relate to an attenuation measure-

ment system.

We have already discussed the particular effects of the basic mechanical properties
of acoustic waves on the amplitude of the received signals in Chapter 2, but in the
discussion on scattering, we intentionally omitted a complication that can cause
difﬁculties in imaging objects that are on the order of the beam width of the transducer.
Normally, for objects that are large in this respect, the scattering takes the form of
specular, or mirrorlike, reﬂection; i.e. the shape and spatial coherence of the incoming
wavefronts are maintained by the scattering body (surface). However, for small ob-
jects, the acoustic beam is split into many smaller beams that travel in many separate
directions, i.e. diffuse scattering [20]. Obviously, this situation is a problem for an
imaging system, since the separate angular directions of scattering can be (mis)inter-

preted as separate scattering sources, and at present the resolution of this is unclear.

We also previously considered the effects of angular planar reﬂectors on the

one-dimensional acoustic beam, and here we mention the interaction of this with

the echoes that return to the receiver from deeper lying layers. It is apparent that the
reﬂection coefﬁcients will be in error, as given by Equations (2.9) and (2.10), but less
obvious is that the values of the loss factors k are adversely inﬂuenced as well, simply
because the beam path length is increased due to the angular propagation. This will
add to the error in the calculation of the attenuation-velocity products in Equation
(4.23), of course. We mention once more that this angular propagation of the beam is
more likely to send the deeper echoes in angled directions that are outside the beam

pattern of the transducer, which we consider in the next section.

W

As we have mentioned in our previous discussion on the typical assumptions that
are made in acoustical imaging, we usually consider the beam of mechanical vibrations
that are emitted from the transmitting transducer to be a column of uniform sound in
phase and amplitude, both axially and longitudinally. This of comse is not the case,
and most manufacturers of transducers provide some form of information with regard
to this deviation. Analysis of this problem is pursued in the literature, and will not be
considered further here, except to note that this nonuniformity of the beam proﬁle will
cause the efﬁciency of the mechanical to electrical signal transforming action of the
receiving transducer to vary with different target distances, even for inﬁnite planar
planar targets. This problem is difficult to compensate, since the beam pattern can
vary signiﬁcantly between electrically-similar transducers and therefore the measure-

ments necessary to learn the shape of the beam proﬁle are not simple [6].

65

Another factor governing the efﬁciency and accuracy of the transforming action of
a receiving transducer is the amount of received beam energy that actually intercepts
the full receiver cross sectional area, e.g. in cases of reﬂectors that are not exactly
perpendicular to the direction of propagation. To investigate this phenomenon, we will

use the geometry of Figure 5.2, where we allow a uniform column beam to bounce off

transducer transmitter/receiver
face, with Area = 1: r2

2 returning
d tan 0 ill . ating
beam

20

 

 

 

Q‘QQQQ‘QQQ‘Q“

dtan20

-““““‘..“ C.

perfect plane

reﬂector
= the effective receiver area

 

Figure 5.2 The geometry for ﬁnding the effective receiver area
when the beam illuminates an angled planar reﬂector.

a perfectly reﬂective plane surface, which is positioned at an angle 9 to the path of
beam propagation. Note also that the transducer is located a distance d from the

plane reﬂector, and has a radius r (usually transducers are round in shape), which

gives us a maximum receiver area of 1: r2. By simple trigonometric relations, we can

ﬁnd that the lateral displacement of the reﬂected beam which arrives at the transducer

is d tan 29 , where we have used 0i = 9r to determine the angle of reﬂection.

By integrating the effective receiver area, we ﬁnd that

Effective receiver area = (5.1)

2

rtr2 - 4dtan20[r2- %-tan220]1/2

- 8123in’1[ _____]d;a:26

we can deﬁne a percentage of area lost by the angled return beam, i.e.

Effective receiver area
% area loss - 1 - Actual receiver area (5'2)

and this is plotted in Figure 5.3. We see that the inﬂuence of the angled reﬂector has
a very signiﬁcant effect on the accuracy of the amplitude information in a received

signal, and thus is a major source of error when imaging nonideal objects.

Another important effect that the transducer exhibits is caused by the interaction of
the materials from which it is constructed with the coupling medium. Typically the

piezoelectric crystal that is the heart of the transducer is embedded in an epoxy

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that is necessary for protection of the fragile crystal, but has a characteristic imped-
ance that is several times larger than the couplant medium (e.g. water), which results
in the appearance of a rather large reﬂection coefﬁcient at the interface between the
transducer active surface and the couplant [6]. By means of Equation (2.9), we can
determine the effect of this on the receiving gain of such a transducer, as shown in
Figure 5.4, where we allow a uniform beam to illuminate the transducer surface at an
incident angle 0, with both being identical in cross sectional area. The coupling medi-

um has a characteristic impedance 21’ and the transducer has an impedance of 22,
which as we have said is usually several times greater than that of the couplant. If
we neglect the inﬂuence of the transmitted angle (i.e. set at = 900) in Equation (2.9),

we can arrive at an expression for the transmission coefficient into the transducer that

  
   

incomin column beam
If. \ g

identical in area to transducer face

coupling medium impedance = 21

transducer impedance = 2’2

Figure 5.4 The geometry used to determine the transducer
beam pattern and beam width.

69

involves just the incident angle, i.e.

2 sin 0
t = . (5.3)
srn 0 + 21 /Z2

 

Note that this has been slightly rearranged to show the ratio of 21 and 22, which is

usually much less than one, and that this function is plotted in Figure 5.5. We

see that the largest value of transducer receiving gain is

2

maximum receiver garn = TTz—l—I-ZZ- (5.4)
which occurs at normal incidence, and that the pattern half-beamwidth, found by
setting Equation (5.3) to -3 dB or .707 of Equation (5.4), is given by

e 5 90 - sin'l ”07 z1 / Z2 (5.5)

 

which for the case of Figure 5.5 is approximately 79°, which is quite a broad receiving

angle, but we have assumed no inherent beam pattern, which is only idealized.

As we mentioned in the introduction in Chapter 1, the transducer of many acoustic
systems is piezoelectric in nature, which results in a rather speciﬁc natural resonant
frequency being present in its characteristics. The exact value of this resonant
frequency depends mainly on the physical geometry of the crystalline material used as

the active element in the transducer, although the crystals lattice properties are also

70

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71

Q = (iii—EFL

Figure 5.6 An electrical equivalent circuit for a piezoelectric
transducer [6].

 

important. Electrically, the input port of this type of device appears to be a resonant
tank, which can be modeled as shown in Figure 5.6. From electric circuit theory [4],

we know that the resonant frequency of this network is given by

(0 = — (5.6)
O L C

which is identical with the transducer frequency, and the bandwidth is

 

1
= _ - 5.7
B R C ( )
and the Q is
2 maximum energy stored
Q - 1: energy lost per period of oscillation (5 8)

72

The signiﬁcance of Q is that a larger value of this parameter implies that the circuit
will ring longer for the same input energy; by looking at Equation (5.8) above we see
then that the value of R must be fairly small to limit the amount of ringing, and thus R
can act as a damping control. Normally, this is made a variable resistor and provided
as a front panel control for just this purpose, and usually the damping is made large to
reduce the pulse width (by limiting the number of ringing cycles). If necessary, we can

modify the transducer input impedance at resonance by adjusting R [4].

5.4 The Tgnsmitter and Reeeiver

In commercial systems, the transmitter and receiver are usually combined into a
single unit which has controls available to the user for making various adjustments.
As was mentioned earlier, the transmitter is required to output a narrow hi gh-voltage
pulse to the transducer, in order to initiate the output mechanical wave. From the
derivations done in Chapter 4, we know that some means of controlling the peak
amplitude of this output pulse is necessary in order to avoid having the situation
where any of the echo amplitudes are either too large for the receiving circuitry or too
small for the level of noise present in the system. Again, we emphasize that the exact

value of Ainput/left and Ainput/right are not important, as long as the receiver will

not be overtaxed, so the controls that determine these’values need not have ﬁne

resolution.

Another control on the transmitter/receiver that is important to system operation is

the damping adjustment that is needed since the transducers are "tuned" devices; i.e.

73

they will ring after the transmitter applies the output pulse to the electrical input port.
The length of time that the ringing lasts will affect resolution of the echo detector,
discussed later, and may be long enough to cause overlapping of subsequent echoes.
Also, if the transducer damping is insufﬁcient, the amplitude of the transducer ringing
may be very large, since the Q of the resonant nature of the transducer is conse-
quently large, and this may affect the receiver adversely. We see then that the
amplitude and damping controls interact with each other, and therefore it is re-
commended that the damping be set ﬁrst in such a manner to allow only one peak

half-cycle of ringing to occur, at an amplitude that does not overdrive the receiver.

The receiver is usually a simple linear ampliﬁer with some form of calibrated
gain adjustment provided to allow setting the echo amplitudes to a reasonable value.
Typically gains from 0-1000 are necessary, at a bandwidth exceeding the transducer
resonant frequency, and with reasonably low noise to allow detection of low level
echoes. One useful feature of a receiver ampliﬁer is the ability to vary the receiving
gain during the measurement cycle, particularly in the case of deep-lying echoes that
are usually small in amplitude, perhaps sufﬁciently so to evade detection or cause
introduction of error due to quantization, which we discuss in the next section. The
effect of a Time-Gain ampliﬁer is illustrated in Figure 5.7. The difficulty of using
such an ampliﬁer in attenuation computation systems is the need to know the exact
gain variation with time during the measurement, which if not undone with accuracy
can be a source of signiﬁcant error in the values of echo amplitudes.

In the interest of completeness we should mention the impact of dynamic range

74

signal before:

 

/ \
Gminimum Gmaximum
gain variation:

: t i i t I

Figure 5.7 The use of a time/gain ampliﬁer to boost the lower level
echo amplitudes for increased accuracy and dynamic range.

on the system performance. We deﬁne the dynamic range as [3]

largest allowed signal (before distortionL

dynarmc range mm = smallest detectable signal (above noise)

(5.9)

Usually in linear systems the noise ﬂoor determines the minimum detectable signal
level, but our system is not completely linear, since as we discuss in the next section

the effect of quantization is both nonlinear and extremely disruptive to the dynamic
range, thereby reducing the sensitivity of our receiving system signiﬁcantly. However,
the inﬂuence of the receiving ampliﬁer on this performance ﬁgure should be considered
as being of similar signiﬁcance. In the next section, we will consider this matter in
more detail, particularly as to how it relates to the type of signal acquisition unit that

is based upon an analog to digital conversion scheme.

75

5,5 Sigegl Amm’sitien

In a modern ultrasound system, it is most common to ﬁnd some combination of
high speed analog-to—digital (AID) converter(s) and fast semiconductor memory
arrays that are used to form a time-domain waveform capture unit. The features and
costs of such units can vary quite extensively (take, for example, the many expensive
multiple channel digital storage oscilloscopes with built in computational features that
are now on the market from many manufacturers), but for the purpose of conducting
acoustic measurements, our requirements are less demanding, particularly since we
intend to accompany the signal acquisition unit by a microcomputer; in fact, we can
reduce these needs to four: (1) the A/D sampling rate must be greater than the
Nyquist rate for the transducer signal and bandwidth, (2) the size of the storage
memory should be sufﬁcient to ensure that the length of consecutive time that can be
acquired is long enough to save the particular signals of interest, at the A/D clock rate,
(3) the number of bits (quantization levels) in the output word of the A/D converter
is sufﬁcient to guarantee a useful value of system dynamic range, and (4) the
triggering capability of the signal acquisition unit is accurate. One additional feature
that might be useful is multiple channel (simultaneous) acquisition; sometimes this
can be replaced by an analog multiplexing scheme that precedes the receiver of the
system, c. g. in the case of large transducer arrays, where the sheer number of
channels required prohibits the resulting cost and system size from implementing

the receivers and signal acquisition in a separate manner.

In order to select the sampling rate for the signal acquisition system, we must

76

naturally ﬁrst consider the nature of the signals that we are likely to be capturing.

As we already know from our previous discussions, the echo signals of the pulse-
echo method are comprised mostly of emptiness, and only a relatively small fraction
of the total time is ﬁlled with echo signals. As we showed in Figure 1.2, these echoes
are similar in form to a sinusoid at the transducer frequency amplitude modulated by a
gaussian pulse envelope; for simplicity, we can model this type of signal by [31]

A sin ( 21: f t)
80) = transducer (510)

2“ ftransducer t

 

which is shown in the frequency domain in Figure 5.8 [3].

8(0

_A_.
2ftrulsdncer

 

 

 

' ftransducer O ftransducer

Figure 5.8 The spectrum of the function of Equation (5.10).

For this function, it is obvious by looking at the spectral characteristics that it is

bandlimited to a frequency of ftransducer’ and in order to satisfy the Nyquist criterion

we would need to sample this signal at least 2 ftransduc In practice however, the

Cl"

spectral characteristics of the transducer signal are not ideally bandlimited, and it is

77

common practice to oversample by several times the Nyquist limit in order to
guarantee an accurate representation of the time signal in the sample memory. In the
next section, we will consider this issue further in regard to the measurement of the

peak value of the echo.

The memory capacity in words of the storage unit will directly inﬂuence the length

of time that we can capture at once; for a single channel unit, this length of time is

 

given by
time captured = # “13mm” ““13 (5.11)
sample rate
where fsample rate is the sampling frequency. For example, a single channel system

with a memory size of 16k (i.e.. 16384) words, used at a sampling rate of 20MHz (or
20 million samples taken per second, i.e. "20MSPS"), will hold a time waveform of
819.2 microseconds in duration. As a ﬁnal comment on this issue, we should note
that some AID units only are able to capture repetitive (periodic) signals at a high
sampling rate, and this type of acquisition method is not very convenient for the more
unpredictable type of signals usually encountered in acoustic measurements. Gener-

ally, this sort of unit is not used for this purpose.

The number of bits in the output word of the A/D converter used in the signal
storage unit of a system will inﬂuence both the required memory word width (in
number of bits needed per word), and also the accuracy of the recorded waveform

amplitude information. In a binary output AID converter with 11 bits in the output word,

78

the number of quantization levels is given by

# quantization levels - 2n (5.12)

for the full positive to negative input range allowed. Normally, for the signals we
encounter in acoustics, we are concerned with only the time varying (AC) component
of a signal, and the average (DC) value is not important. In this case, we require that
the ND converter be able to quantize both the positive and negative going portions of
an echo pulse, and this will force us to offset the zero level to the midrange of the
allowable input voltage, which causes us to have only one half of the number of levels

given by Equation (5.12) to be available for both the positive or negative peaks, i.e.

# bipolar quantization levels = 2n-1 (5.13)

For example, an 8 bit output word gives 127 quantization levels (from 0 to the
maximum positive or negative value allowed. Note also that we must expect that the
input signal to the A/D converter does not exceed these maximum values, causing an
overrange condition that will lessen the accuracy of the captured data. If this occurs,
we must reduce the amplitude of the input signal by reducing the receiver gain or
by lowering the output level of the transmitter unit).

Another signal acquisition issue of importance to attenuation measurements is the
system dynamic range, as we discussed in an earlier section in this chapter. In the
A/D unit, we deﬁne the dynamic range similarly as in Equation (5.9), and we ﬁnd, for

the linear scale it bit binary output converter, that

79

AD dynamic range in (dB) = 201og10 2“'1 (5.14)

E 6(n-1)

For the 8 bit example, this gives a dynamic range of about +42 dB, which is not very
large (only about 120 to 1), and by itself is usually insufﬁcient for attenuation
measurements. As we discussed in the receiver section, a time/gain compensator
(TGC) may be used to increase the dynamic range, provided that the amplitude dis-
torting effects of this unit are undone prior to subsequent processing. Another meth-
od of increasing the dynamic range, which is more limited in usefulness than a TGC,
is to average repetitive measurements performed in the same position on a sample-
by-sample basis [10], using ﬂoating point arithmetic (note this action will also tend

to reduce the uncorrelated noise that arises from measurement to measurement in

the system, thus making echo identiﬁcation easier). If we average 2a times, then

effective dynamic range (dB) = 2010g10 2““"1 (5.15)

E 6(n+a-l)

causing the system to appear to have an AID converter with 2n+a total quantization
levels. One problem with this technique is that it is necessary to align the individual
time waveforms with each other before averaging them together, otherwise distortion
will invariably occur. By this we mean that the exact position of the echoes in each
signal will vary somewhat due to uncertainties in the time of transmit pulse output,

and unless steps are taken to correct this, we will cause more trouble than

80

we are remedying. Even with such alignment being done, the fact that we are
sampling at almost the Nyquist in many situations can cause the additional problem
of introducing phase distortion into the averaged version of the time signal, since the
sample data is sparsely distributed over each cycle (at the transducer frequency) of
the signal (this effect is considered in more detail in the next section). We can
conclude therefore that this type averaging is rather difﬁcult to perform in such cases,
unless special care is taken, possibly interpolating values between the actual

samples for use in computing the average time signal.

One problem of quantizing the input signal in an attenuation (amplitude) measure—
ment system is that the error in the echo amplitudes measured will depend upon the
the magnitude of those echoes. This of course is a consequence of the limited resolu-
tion that is available for small input signals. The percentage error introduced by quan-
tizing an input signal with amplitude A with an n-bit linear span A/D converter that

has a maximum voltage limit V is

 

 

 

 

max

Vm . n

2n integer [ 2 V ] - A
% quantization error = ‘ L—
‘ A
Vm . . n
= n rnteger[ ] - 1 (5.16)
2 A max

which is plotted for our 8 bit example with Vmax = 1 volt in Figure 5.9.

81

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82

We can clearly see that the error introduced is much more pronounced for smaller
values of A, which means that more error is introduced for these smaller values of

echo amplitudes when they are quantized by the AID converter.

The fourth and ﬁnal consideration for the signal capture unit is the need for stable
and reliable triggering of the storage action, particularly if signal averaging is being
contemplated. If this requirement is not met, then it will be difﬁcult to measure
absolute time delays of the echoes, although relative time will be unaffected, and it
will be impossible to accurately ﬁnd the value of transmission pulse time delay, e. g..

time (TLR) in Equation (4.6). In most commercial AID systems, this triggering is

a parameter well speciﬁed by the manufacturer, and usually then the acoustic system
designer need only be aware of the problem, so that the system user can be assured

of the time measurement accuracy.

5,5 Siggﬂ Emeessing

After the transient waveform has been digitized and captured, it is usually made
available to a digital computer for processing prior to storage and display. Normally,
some form of computations are performed before storage of the information in the
signal, since saving the actual samples can require excessive memory or disk space,
especially for the many distinct measurements that must be done in order to create
a multidimensional image. Figure 5.10 indicates a possible block diagram of the
necessary software to process the signals prior to application of the attenuation-

velocity product algorithm of Figure 4.5; i.e., the routine that creates the set

 

83

 

 

 

    
 

time waveforms

 

 

 

;-~ . ; echo identiﬁcation
'- avera ‘
3mg » and windowing

l

L, TLR, R, TRL

   

 

 

 

ll

 

 

 

 

detection of cubic spline

peak amplitude " curve ﬁt
Within windows

 

and time

 

 

 

 

 

 

 

V

{L’ TLR’ R’ TRL}experirrrental

 

Figure 5.10 The software preprocessing for the algorithm of
Figure 4.5.

{L, TLR' R, TRL} experimental from the various time waveforms.

We have already considered the problems involved in signal averaging in the
previous section, so here we will not discuss it directly; however, as we mentioned
before, in order to do this operation properly without introducing distortion, it is
necessary to interpolate between the sampling points given and align the peak values
of the echoes. The theory and use of cubic splines will be discussed shortly, for the
purpose of improving the accuracy of the reported peak echo amplitudes, and the
discussion given there can naturally be adapted for the signal averaging technique.

The ﬁrst task of the preprocessing software is to identify the location of pulses in

84

 

__.l_|___l'—L 1—1

Figure 5.11 The window detector in action.

within a particular time waveform, which all will be of the form depicted in Figure 1.2.
Such a waveform is shown in Figure 5.11, along with detected echoes, indicated by the
presence of a logic "1" in the lower waveform. This waveform is intended to indicate
the windows in time within which the echo signals are supposed to exist, and are used
to partition the given total signal into regions of interest within which we will process
the signal further. Regions outside of the windows are ignored from further con-

sideration. The algorithm used to produce this result is depicted in Figure 5.12.

 

 

/\,in——Hls(t)l

V

window

> cam

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(set peak detector ‘
threshold) (time co)nstant retriggerable
= 't at
‘ 1 (set hysteresis) one ShOt

 

(pulse width = 1:2)

Figure 5.12 The windowing algorithm used to produce Figure 5.11.

85

This algorithm has been reasonably successful in detecting echo locations within the
the time waveforms encountered during the laboratory experiments, but it has been
seen to be susceptible to impulse noise. An obvious improvement would be the
inclusion of a frequency monitor that only allows windows to occur when the signal
is approximately equal to the transducer frequency; this can be done with zero
crossing measurements. The latter change would help to eliminate false window
generation. A disadvantage of the detector in Figure 5.12 is that pulses that are rel-
atively close together cannot be distinguished. Methods such as deconvolution [10]
of the time signal are sometimes effective in this situation, but are rather time
consuming and have the tendency to introduce many false echoes into the detected

pulse train.

To ﬁnd the peak value of an echo pulse, we need to simply determine the largest
value reached by the signal, within each time window produced by the window
detector. This operation is intuitively simple, but is prone to error in cases where the
A/D sampling rate is close to the Nyquist limit. Due to this situation, it is rather
likely that the samples procured by the AID unit were not acquired at the time of the
exact peak of the echo pulse, instead being of lesser amplitude than the actual signal.
This problem naturally is related to the ratio of transducer frequency and sampling

rate, i.e.

fsample rate (5 17)

% oversampling = 2 f
transducer

86

sample point

\

s(t) = cos (2 1: ft].

/ ansducer t )

 

 

0 \
l Tsample/2

:l‘h—T

“I

 

 

sample

Figure 5.13 The effect of sampling nearly at the Nyquist limit on
the accuracy of peak amplitude determination.

and is illustrated in Figure 5.13. The worst error that will occur is also shown in this
ﬁgure, where two subsequent sample points have straddled the peak of the input
sinusoid; the maximum error (neglecting quantization error) is then given by

% maximum amplitude error = cos (1: M) - 1 (5.18)

sample rate

which for the case of a 2.25 MHz transducer frequency and a 20 MSPS (20 MHz)
sampling rate (approximately 8 samples taken per cycle, which is only about four
times the N yquist rate) amounts to about -6.2% error, which is rather large. In
addition to this potentially severe inaccuracy in the amplitude is the equally trouble-
some fact that the time of occurrence of the peak amplitude is uncertain to within

one sampling period width, T

sample' This fact can be seen by contemplation of Figure

5.13 above, if one allows the location of the depicted sampling points to vary.

87

A remedy for both of these problems can be found in the use of splines to curve
ﬁt a continuous function between subsequent extremum samples, such as those
indicated in Figure 5.13, and approximate the actual peak echo amplitude/time by the
peak value/time of the spline curve within that particular sampling period. For such

an application for a function s(t), we can use cubic spline functions of the form

Sl(t)=at3+bt2+ct+d for t8[t0,t1) (5.19)

where Sl(t0)=s(t0)= at03+ bto2 +ct0+d

3+ bt12 +ctl+d

8101) = $01) = at]
S1’(t0)= 0 =3at02+2bt0 +c
__ 51.80) = 0

is used to deﬁne SI on the indicated interval, and elsewhere

 

6ato +2b

Si(t) = Si_1(t) + ci (t — ti)3 for te [ ti, ti+1 ) and i = 2...n
where n = the # of sampling points in the window of interest
to = the starting time of the pulse window

ti = the i-th sample point in the pulse window
ti - ti-l = sampling period = 1 / ( fsampling rate )

tn = the ending time of the pulse window
50m) ' Si-1(‘i+1)

( ti+1 ‘ t i )3

and c- =

 

The intricacies of applying these spline functions are best left to the references; see
for example [57] for a good practical discussion of this type of spline interpolation, and

a derivation of an expression for the maximum error bound for certain types of sampled

88

functions. For our purposes, we would use this formulation to generate a set of
piecewise continuous functions within each window provided by the echo detector, and
investigate these for a maximum, which we would take to be an estimate of the
maximum amplitude of the original signal in the window; the time that this estimated
amplitude maximum occurs at is also given as well by the spline functions, and the
uncertainty error in this time is much less than one sampling period, which is an
improvement over the case of such without splines. Splines will give us this
amplitude and time in ﬂoating point number form which will increase the accuracy of
the calculations in Chapter 4 over the use of integer numbers. Note that this will

help to reduce somewhat the pessimistic error estimate we found in Equation (5.16)
and shown in Figure 5.9, although it will still not completely alleviate the problem of
estimating the amplitude of low level echoes. In Figure 5.14 is a plot of the errors in
estimation of a sinusoid before and after spline interpolation, for the previous example
we used, with ftr

= 2.25 MHz and fs = 20 MSPS; clearly the error

ansducer ample rate
is reduced signiﬁcantly. As a ﬁnal note on splines, we mention that the piecewise

continuous functions Si(t) can be used to simulate an increased sampling rate on the

original function s(t), with very high rates of multiplication possible by simply
choosing values of the spline curves between actual samples as virtual sample
points. By selecting the virtual samples uniforrrrly in time, higher simulated ﬁxed
sampling rates are possible, but only if the Nyquist rate for the original s(t) was
exceeded by the actual sample data. Then these estimated/simulated samples will be

within a reasonable error band, as can be seen in Figure 5 .14. This use of splines is

89

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o

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90

handy for increasing the resolution of digital signal processing algorithms.

After applying the preprocessing software of Figure 5.10, to each of the time
waveforms L, TLR' R, TRL we produce the data set {L, TLR’ R, TRL} experimental:
which is the set of (amplitude/time) pairs for all the pulses in the time waveforms.

At this point, the data is ready for processing by the algorithm of Figure 4.5, and the

discussion in Chapter 4 now applies.

ni'vi f uati n 4.2 4.21 4

L9 Em in the Amplimdee ef the Primary Date

In order to investigate the effect of inaccurate values of the measured amplitudes

on the calculations of Chapter 4, we will use the deﬁnition of sensitivity [58], i.e.

s)’

X

sensitivity of y(x) with respect to x

fractional change in y(x) for a fractional change in x

Ay(x) / y(X)
Ax / x

x Ay(X)

3(7) Ax (5.20)

This quantity is a measure of how dependent y(x) is to changes in x. Note that we
can interpret the differentials as a (partial) derivative of y with respect to x. If the

sensitivity is less than unity (+ or -), then the function y is rather indifferent to
changes (or uncertainties) in x; the opposite is true when Sxy is greater than unity.

We will now investigate Equations (4.20), (4.21), and (4.23) in this manner.

91

For Equation (4.20), we see that the deﬁnition of r is symmetrical with regard to
both Li and Ri’ so we need only evaluate either sensitivity since SLr = SRr. Using

the definition we gave in Equation (5.20), we ﬁnd that

sLr = (1 -r2)/2 (5.21)

_ 1'
_SR

and in a similar manner for TLR and TRL’ by symmetry again we ﬁnd that

ST1r = (r2 -1)/2 (5.22)

for both TLR and TRL' By inspection of Equation (2.13a), we see that r e [-1, 1], so

it is apparent that Equations (5.21) and (5.22) are always less than unity in
magnitude, which is desirable since this implies that the computation of r by Equation

(4.20) is rather insensitive to errors in any of L, R, TLR’ or TRL‘ To study the inter-

action of errors in these variables requires the use of higher order sensitivities, which
we will not pursue here. In Figure 5.15 is plotted the sensitivity relations found in

Equations (5.21) and (5.22).

For Equations (4.21a) and (4.21b), we need only investigate the sensitivities of
either, since they are symmetrical and the answers will be identical again. Applying

Equation (5.20) to Equation (4.21a), we ﬁnd that the sensitivity of k to ri+1 is given

by

92

 

 

 

 

 

 
    

   
   

l
s r 0 i
L r
-.5 A
r r
4 ST
0 l
1
k I >
Sr . 0 i k
5 k av
: Sk

 

 

 

 

 

 

V

Figure 5.15. Plots of the sensitivities found in Section 5.7.

93

= -1 /2 (5.23)
i+1

_ k
'SL.
1

(by symmetry with ri +1) (5.24)
So we have in these cases low error sensitivity as well. Also

st = 1/2 (5.25)
i+1

which is also pleasant. Finally, we have that

r-2

1
Srk = — + -—1—2- (5.26)
i 2 (l - ri )

which is not good, since ki is very sensitive to ri if Iril is close to unity. Note that

Equation (5.26) is also plotted in Figure 5.15.

For Equation (4.23), which is the computation to ﬁnd av given the values of k and

the layer propagation times T, we determine that

81“" = [ ln k 1'1 (5.27)
and that

s,“v = -1 (5.28)

For Equation (5.27), the plot is also shown in Figure 5.15, and we see that the

94

calculation for av is very sensitive to k when k is close to unity (by inspection of
Equation(3.2), we see that k e [0, 1] since a, v, and t are always positive).

The computation for ki is not made very inaccurate by errors in ti because this time is

usually very certain since the AID converter is almost always driven by a crystal-
controlled, high-stability oscillator for a sample frequency clock, and the frequency
error of this type of oscillator is quite small, on the order of 100 parts-per-million or

less.

To conclude this section, we restate that the values of r are relatively insensitive
to amplitude errors, but when the magnitude of r is close to unity, the computation of
k can suffer if r is not accurate. This situation is not often likely to occur, since deeper
layers will be obscured by such a large value of reﬂection coefﬁcient. The calculation

of k is rather insensitive to errors in the amplitudes, and we can say that

k _ r k
SL1 - SLi Sr
= (1 + r2) / 2 (similarly for R1) (5.29)
(by use of the chaining property of sensitivities [57]) is always less than unity, so
Equation (5.29) implies that k is insensitive to errors in the echo amplitudes, even

though it is not to errors in r. Note that Equation (5.29) is plotted in Figure 5.15 also.

Finally, we can ﬁnd that

av_ av k
SL ‘Sk SL

95

= (1 + r2) / (2 1n k) (5.30)

which still looks like Equation (5.27) in general, so we cannot say that av is insen-
sitive to errors in the amplitude when k is close to unity. Fortunately, when k is
near unity, the attenuation is small, so we tend to get a larger signal from that layer,
which can help to alleviate this problem. For k small (close to 0), the attenuation is

large, and the calculation of k is rather insensitive to errors in the echo amplitudes.

5.8 Addig'onﬂ 5ystem Rguﬂ' ments

As a ﬁnal note to this chapter on system considerations, we must mention the
additional hardware requirements that can be expected for an acoustic measurement
system of the type discussed in this thesis. We have mentioned in the previous
sections that a microcomputer (or better) is required to perform the complicated
calculations necessary to implement the algorithm of Figure 4.5 and to incorporate
the software outlined in this chapter. The operating speed of the computer selected
will directly inﬂuence the time required for an image to be reconstructed, and this can
be considerable for even a one-dimensional object problem if the number of layers is
large (and there are many multiple reﬂections to contend with). This will tend to tip
the trade-off of speed versus cost in the favor of higher hardware expense in many
cases. The storage requirements, as mentioned in the previous section can be
excessive, running into the tens of megabytes, for a multi-dimensional image if the
entire time waveforms are saved prior to processing; this however might be the

course of choice for a fast imaging system for use in e. g. clinical diagnostics. In slower

96

systems where processing time is not as critical, preprocessing and preconditioning
of the data could be done, e. g. saving only the windowed signal fragments on disk,
but naturally this will slow the system operation down by increasing the time spent
between measurement cycles. This type of trade-off must be decided in light of the
particular application for which the system is ﬁnally destined, weighing the target
cost appropriately.

In order to effectively display attenuation-type images, some form of high
resolution graphics display is desirable, either in gray scale or color. For a gray scale
type of display, usually 8 bits of data per pixel is adequate, and provides near
photographic quality to the average human eye, if the number of pixels per line and the
number of lines per frame are sufficiently high. A color display must have a wide
palette of colors available, and each color should have a large luminance range (e. g.
8 bits per color in an RGB system, meaning 24 bits per pixel). Because of the much
larger memory requirements for a color system, along with the higher quality monitor
(CRT) needed, the cost is much higher than the gray scale display, presently at least
three times the expenditure for gray scale, and this sort of cost differential is likely to
remain extant because of the added complexity of a color system. Again, the
beneﬁts and disadvantages of each display type must be considered against cost and
the intended system application.

From this discussion, however, we can see that in the future the cost and ability of
the hardware used in ultrasound systems is very likely to improve, and will be

commensurate with like improvements in semiconductor and computer technology and

97

manufacturing. Perhaps in the near future we will see such systems in greater
proliferation than at present; the low cost and inherent safety of acoustic systems can
only become more attractive to a wider variety of users, but only if the imaging quality

can be equally improved.

mm

In this chapter, we have considered the more esoteric needs of attenuation-type
acoustic imaging systems, and have elaborated on some of the problems that face
an amplitude-based measurement scheme. In the author’s opinion, all of the
difﬁculties we have covered-multiple reﬂections, hardware problems, operation com-
plexity, speed, imaging quality, sophistication of processing, etc.-—can all be overcome
eventually if not already so, but the most severe problem is that of angular/diffuse
scattering into angles outside of the receiving domain. This is really the only
difference that ultimately separates ultrasound techniques from X-ray methods, for
example. In Chapter 7, brief mention is made of future possibilities research in this
problem; in the next chapter, we present some experimental ﬁndings of the method

of attenuation-velocity product imaging.

CHAPTER VI

SIMULATIONS AND EXPERIMENTAL RESULTS

In the preceding chapters, we have developed and discussed a means for the
determination of the quantity av for a medium, particularly under the conditions of
a test object composed of many layers of differing media. Generally, we have
constrained our considerations to a strictly one-dimensional situation; for consistency,
we continue this restriction into this chapter; in Chapter 7, we discuss possible exten-
sions to these results into higher dimensions, but here we attempt to investigate our

work in Chapters 4 and 5 in a more pragmatic light.

6.1 Simuletione and Results

In order to develop and investigate the results of Chapter 4, a variety of computer

simulations were performed using programs developed by the author. These and the

results will now be discussed.

The ﬁrst requirement of the work was to develop a means for understanding the

98

99

action of waves within a layered sample, with the added need of creating simulated
experimental data with which to test any algorithms developed for stability and
accuracy. To meet this, a stand alone program, TWOSIDE, was written in the BASIC
language and compiled for speedier use with a commercial BASIC compiler. The
function of this program, which is listed in full in Appendix A, is to simulate the
complete behavior of a pair of left and right input impulses to a one-dimensional user-
speciﬁed object. All reﬂections and transmissions at the internal boundaries are
followed, and any output signals are recorded and sorted in order of time of occurrence.

These lists of output impulses then form a set {L, TLR’ R, TRL} of echoes that can

include multiple output reﬂections and transmissions, and this set of signals can be

used for testing the algorithm of Figure 4.5; in fact, the use of TWOSIDE was instru-
mental in the development of this method. The detailed operation of TWOSIDE was
outlined in the discussion given in Section 4.7.2, where we considered the simulation

of a set {N, k, 1', ”trial produced during the implementation of Figure 4.5; the problem

is identical, and will not be repeated here.

In order to demonstrate the overall av method, we [will ﬁrst create a simple ex-
ample using TWOSIDE. The test object we will consider is shown in Figure 6.1.
Note that it is composed of four layers, with three boundaries separating the different
materials, each having the values of r, v, t, and a indicated. When TWOSIDE is run,
the program ﬁrst requests these values, inputted from left to right which is our

convention. The program then requests a value for the smallest amplitude desired in

100

the simulation; as we mentioned earlier, this is necessary to terminate the simulation.
The program then proceeds to simulate the left and right inputs, and eventually pro-
duces the data shown in Table 6.1; the primary signals have been indicated for the
convenience of the reader. We see that is a multiple reﬂection in each of L and R.
This set of data was then used as input information to the program ALPHA-V, which
is listed in partial form in Appendix B. The results of this are depicted in Figure 6.2;
we see that the algorithm has correctly selected the proper primary signals, and has
found the original values of {N, k, r, t} of Figure 6.1, but not before it has tried N=5
and N=4 once, both of which involve impossible conditions for Equation (4.11),
meaning that those trial choices of the primary data are self-inconsistent and therefore

incorrect.

r1 — r2 = r3 =
0.3 -0.4 0.6
layer 0 layer 1 layer 2 layer 3

 

 

 

 

 

Figure 6.1 The example object to be considered, with N=4.

101

 

 

# emplitude time path through boundaries
Lsimulated:
L1 .300 2 O, 1, 0
L2 -.29801 4 0, 1, 2, 1, 0

-.02928 6 0, 1, 2, 1, 2, 1, 0
L3 .20608 8 O, 1, 2, 3, 2, 1, 0
TLRsimulated:
TLR .83656 4 0, 1, 2, 3, 4

.08219 6 0, 1, 2, 1, 2, 3, 4

.11018 6 0, 1,2, 3, 2, 3,4
Rsimulated:
R1 -.6 2 4, 3, 4
R2 .1404958 4 4, 3, 2, 3, 4

.0185953 6 4, 3, 2, 3, 2, 3, 4
R3 -.0724677 6 4, 3, 2, 1, 2, 3, 4
TRLsimulated:
TRL .2627654 4 4, 3, 2, 1, 0

.0346101 6 4, 3, 2, 3, 2, 1, 0

Table 6.1 The results of program TWOSIDE on the example

of Figure 6.1.

102

Checking: N=5

No solution possible (signs not opposite in L(3) and R(2) ).

Checking: N=4
No solution possible (signs not opposite in L(3) and R(l) ).

Solutionll for:

N=4

L(1)= .3 R(1)= -.0724677

L(2) = -.2980181 R(2) = .1404958

L(3) = .2060803 R(3) = -.6000

TLR = .8365595

TRL = .2627654

K(O) = 1 T(0) = l
K(l) = .904 R(1)= .29995 T(1)= 1
K(2) = .7408 R(2) = -.410002 T(2) = 1
K(3) = 1 R(3) = .6 T(3) = 1
Simulating...

Checking:

Simulation = experimentalll

***** DONE *Illalﬂlnk

Figure 6.2 The results of program ALPHA-V using the
data shown in Table 6.1.

103

6.2 Expgimental Results

The attenuation-velocity method has also been tried in the laboratory, with the
setup as diagrammed in Figure 6.3. To conduct the measurements, a PC-based
AID converter board (the Markenrich Corp. WAAGII) was used, providing a pair of
simultaneous channels sampling at 20 MSPS with 8 bit resolution; up to 16k (16384)
points may captured in each channel, providing about 820 microseconds of captured
time in each single shot measurement. This unit is more than adequate for the sys-
tem proposed in Chapter 5. For the receiver/transmitter function, a pair of Paname-
trics, Inc. 5050PR model pulser units were utilized, each providing for separate trans-
ducer matching via the front-panel controls as previously discussed in Sections 5.3
and 5.4, by means of the damping and receiver gain adjustments. A pair of Paname-
trics 2.25 MHz resonant frequency piezoelectric transducers, 0.5" diameter, were
used to provide both the acoustic output waves and act as mechanical signal receiv-
ers; the pair selected was not initially well matched, but chosen as the best of the
several combinations available in the lab, providing an opportunity to investigate the
effect of transducer pair matching on the performance of the amplitude measurement

system.

To facilitate the experiment, a test ﬁxture was constructed from 1 cm plexiglass
sheet, which held the transducers ﬁrmly in position to ensure stability of their align-
ment. The center of the test ﬁxture was outﬁtted with a manually rotatable platform
to allow for ﬁne angular adjustment of the samples to be investigated. The entire test

ﬁxture, rotating platform, test sample, and pair of transducers were immersed in a

104

 

 

H Microcomputer

' (di la disks,
trrgger generator an 31,858

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

: wan)
”13.9w ............
two channel 20 MSPS
A/D board, 8 bits, 16k samples
per channel
Channel Channel

A B
v
pulser unit A pulser unit B

 

 

 

 

 

 

 

 

 

 

 

‘— 20 cm spacing _’

 

 

Figure 6.3 The layout of the experimental system.

 

105

plexiglass tank ﬁlled with water, which acted as a coupling medium for the acoustic

waves.

The software described in Chapter 5 (i.e. the preprocessing of Figure 5.10) was
implemented, and is included in Appendix C as program TWOCHANNEL. This was
also written in BASIC and compiled for speed, and the execution performance is
adequate for the types of measurements used in the experiments described here; a
more streamlined version would be necessary for use in rotational-scanning imaging,

for example.

The experimental procedure involves two main portions: (1) alignment and calibra-
tion of the transducers and test ﬁxture, and (2) alignment and measurement of the
sample object. The derivation leading up to Equations (4.20) and (4.21) show the

immunity of these formulations to the difference between Ainput/left and Ainput/right;

however, these equations expect that the amplitudes of the signals have been mea-
sured with accuracy. This implies that while we need not match the output pulse of
the two pulser units/transducers, we must expect that the transducers are calibrated
to produce the same peak electrical amplitude for identical mechanical signals-i.e.,
the pressure-to-voltage conversion constants for the pair must be identical. This is
another shortcoming of using more than a single transducer to probe a sample, for if
this calibration is not performed correctly, the numerical values of r and k will be in
error, as we learned in Section 5.7. In order to perform this calibration step, a bubble

of air trapped behind a stretched sheet of thin plastic ﬁlm was used to approximate a

106

perfect reﬂector (at 2.25 MHz, the value of Zc for air is .0003 times that

haracteristic
of water, which means that the interface between the two has a reﬂection coefﬁcient

close to -1, about -0.9995), and the following steps were taken to ensure calibration:

(1) One transducer (e. g. left) was selected, and the output was adjust-
ed to provide a moderate amplitude, low ringing signal, when the
output of the pulser unit/transducer was viewed on an oscilloscope.
To see the output signal, the planar air bubble reﬂector was
aligned in the outgoing wave path so as to return this signal
entirely to the transmitter. The amplitude of this echo was noted.

(2) The air bubble reﬂector was removed, allowing the beam to pass
uninhibited to encounter the far transducer (e. g. right). Using the
oscilloscope to monitor the output of that transducer/pulser, the
damping and receiver gain were adjusted to provide an identical
signal to the one observed in (1).

(3) The steps of (1) and (2) were repeated in the reverse order (e. g.

ﬁrst right, then left) to ensure accurate calibration.

It should be noted that the above procedure cannot guarantee that the transducers
will remain calibrated over the long term, and therefore this procedure should be
repeated often. We see then that this is a fundamental problem with all methods that
employ multiple transducers, including arrays; unfortunately, it appears that unless

the numerical values of the layer material constants are not of interest, the use of

107

an array of transducers will only aggravate this accuracy problem. The author has
seen no mention of this in the literature involving array measurements, and thus the

question is as yet left unanswered.

Once the setup of the test ﬁxture has been performed, we need to concern our-
selves with the test object. As we have amply mentioned, the work performed here
is limited to one-dimensional structures, and the objects investigated conform to
that requirement as well as the others listed in Section 4.3. For the test objects,
slabs of various materials and thicknesses were used, allowing easy construction and
alignment. The materials used, along with their relevant physical properties, are
listed in Table 6.2. In order to align the objects so that their surfaces were normal to
the acoustic beam, use of an oscilloscope was made to monitor the echoes returned
from the front faces, and the orientation of the object was adjusted to maximize the
amplitude of the echoes from these faces. This of course does not ensure alignment,
since the beam may not always constrain itself to the line of sight between the trans-
ducers; in fact, deviation in the transmission signal intensity was noted if the sending
transducer was axially rotated in place, which means that the beam is not symmetric
about the line of sight. No resolution of this problem will be made, since there is little
one can do to cause the acoustic beam to behave in a more controlled manner; how-
ever, the problem cannot be ignored and is certainly a major source of experimental

CITOI'.

To illustrate the measurement of a simple object, we will consider the setup of

Figure 6.4, where we attempt to interrogate a single layer of plexiglass. The relevant

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108

 

 

 

 

 

 

 

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layer 1
plexiglass
layer 0 . layer 2
C] § D
x0 = 5.79 cm \ x2 = 12.6 cm
v0 = 1350 m/sec ‘ ' v2 = 1350 m/sec
t0 = 39.15 usec 0 : t2 = 85.14 usec
X1 = 1.189 cm
V1 = 2680 m/SCC

t1 = 4.437 usee

Figure 6.4 The example setup to illustrate the experimental
procedure.

dimensions and ideal physical quantities are indicated. This sample was interrogated
in the manner we have outlined, producing both the graphic results and the numerical

values of the set {L, TLR, R, TRL} shown in Figure 6.5, inspection of

experimental’
which shows the presence of multiple reﬂections in all these time signals. The prima-
ry echoes are also identiﬁed in Figure 6.5b, and we can use these data to compute the

quantities of interest:

By Equation (4.20), we can solve for the values of r1 and r2, i.e.

h I = / (75x17)
1 (75)(17)+(90)(94)

 

 

llO

 

 

 

 

 

 

exp

V"
time = O 267 usec
Lexp Rexp:
L1 75 78.9E-O6 sec R2 56 1.691513-04 sec
L2 -21 87.3E-06 R1' -17 1.77558-04

15 1.5788-04 14 .0002562

-24 2.566E-04
TLRexp: TRLexp:
TLR 9O 128.3E-06 sec TRL 94 128358-04 see

Figure 6.5

(a) The experimental time signals, and (b) the

SCI {L, TLR, R, TRLlexperimcntal for Figure 6.4

111

lrl| = .3620

and

 

 

I I / (20(56)

1'

2 (21)(56) + (90)(94)
= .3490

Also, using Equation (4.21a), we ﬁnd that

 

 

 

75 .3490 (1 - (.3620)2) 3. £31.... I",( 1» I; )

= .5780

.. t" “M-.n.~ uguu-wgv—uﬂl-Ipwgp-M“. '

7" 3 .1 '7 a *3 ti"...
By means of Figure 5.6b and Equations (4. 1) and (4.23), we get that, S , I ‘ , ._

 

_. -1
. . "l
alvl = _ time(Ll) - time(LZ) ln(kl) , I“ .I.’ t; g
2 3. L C i ,i t‘

— ‘

 

..., (a

— —_1

78.9 sec - 87.3 ec
= - ll 2 us ln(.5780)

— ‘

 

 

 

123,421 nepers/sec

520 9' ’xf-
.lquDt/t')

Additionally, from the experimental data, and knowing x1 in Figure 6.4, we can

determine that

fin/15,).— @160

v1

(.01189 meter) / (4.437 usec)

2679.74 meters/sec , ID‘ A.

112

Finally, by using the above two results above, we can ﬁnd

(11 = 46.058 nepers/meter

The above information is summarized in Table 6.3 below, along with error comparison
to the reference information contained in Table 6.2. The errors indicated are reason-
able, in light of the difﬁculty of accurately calibrating the ﬁxture and transducers and
ensuring that the sample was normal to the acoustic beam. The results of this
measurement serve to point out the inherent tendency of such amplitude-based
techniques to be numerically imprecise, due of course to all of the problems we con—

sidered in Chapter 5, although for some applications the results would be adequate.

 

 

 

 

Quantity experimental actual % error
value value (in % )
Irll .3620 .3680 1.63 %
lr2l .3490 .3680 5.16 %
It .5780 .5400 7.02 %
av (nepers/sec) 123,421 138,850 11.1 %
v (m/sec) 2619.74 2680 0.037 %
0t (nepers/m) 46.058 51.81 11.07 %

Table 6.3 Summary of the experimental results for the
single layer plexi glass sample.

113

To further illustrate the method, we will consider the similar problem of determining
the same quantities for a sample of aluminum. This is actually a more difficult material
to probe acoustically since it has an extremely large reﬂection coefﬁcient with respect
to water, about 0.848; this causes the amplitude of deeper boundaries to be very
diminished, adding to the quantization inaccuracy in the manner of Equation (5.16).
The experimental conﬁguration and dimensions are shown in Figure 6.6, and the data
acquired in a manner parallel to the previous discussion is shown in Figure 6.7 and
Table 6.4. It is apparent that the errors are much greater in this example, and this
can be attributed to the difﬁculty of accurately measuring very small amplitudes, such
as L2 and R1; also applicable is the error given by Equation (5.26). Again, it is inter-
esting to note that while the absolute accuracy appears to suffer, the general trend of
the quantities is still to be roughly in the range of the correct answer, which may be

wholly acceptable for pictorial imaging.

 

 

 

 

 

 

 

layer 0 layer 2
water water
x0=8.3cm x2=7.99 cm
to = 56,08 “sec t2 = 53.99 [.1860
X1 = 3.71 cm
t1 = 5.79 “SOC

Figure 6.6 The experimental setup for‘the aluminum sarnple.

114

Lexp Rexp:

L1 114 117.16E-06 sec R2 85

L2 -16 133.74E-06 R1 - - 20
7 140.71E-06 4

TLRexp: TRLexp:

TLR 24 121.97E-06 sec TRL 28

110.98E-06 sec
120.56E-06
135. 14E-06

122.15E-06 see

Figure 6.7 The experimental data for the aluminum sample.

p.-

 

 

 

 

(X fij”" ( 1 \1! 158;”ij
Quantity Experimental Actual % error
value value

lrll .878 .848 3.538 %
lr2| .817 .848 3.656 % ‘
k .972 .983 1.12 %
av (nepers/sec) 4880.67 0 LL” C 2984.32 63.54 %
v (meters/sec) 6238.4 6400 2.52 %
0t (nepers/m) .7820 .4663 67.8 %

Table 6.4 The experimental results for the aluminum sample.

_ 0’1
Qﬂﬂd

(I.

_ u

(‘1

‘1

115

In order to illustrate the additional complexity of imaging an object with a larger
number of layers, we will use the case of Figure 6.8, which is comprised of two
identical layers of plexiglass separated by a layer of water. In this case, all the values
of the reﬂection coefﬁcients should be the same, and the middle layer of water pre-
sents the opportunity to investigate the end result of interrogating low loss regions.
Figure 6.9a and b depict the time signals and data, respectively, acquired for this
setting. The calculations proceed in the manner of the ﬁrst example, using Equations
(4.20), (4.21), and (4.23). The results of this are summarized in Table 6.5; interesting
to observe is that the error for most of the quantities is comparable to that of the
prior examples, with the exception of the attenuation value for the middle water layer,
which is rather erroneous. However, we again note that the trend of the magnitude

of all these quantities is close to that of the actual values.

layer 0 layer 1 layer 2 layer 3 layer 4
water plexiglass water plexi glass water

 

 

 

C; x0 = 4.25 cm x2 = 5.50 cm
to = 33 usec t2 =
X1=1.0 Cm
t1: 5.88 [1860

 

Figure 6.8 The conﬁguration for the ﬁve layer object.

 

116

 

 

 

 

 

 

 

v f
I LRexp
K r- 4
Rexp
IRLexp Jl
VQ— l IL .1
time = O 267 usec
Lexp Rexp:
L1 119 6.455E-05 sec R4 106 6.375E-05 sec
L2 -65 .0000732 R3 -41 7.245E-05
48 .0001311 -14 .0001269
32 1.3955E-04 9 1.3615E-O4
L3 40 .0001484 R2 18 .0001476
L4 -9 1.5715E-04 R1 -4 .0001539
11 .0001972 —16 1.9535E-O4
9 .0002061 10 .0002224
14 .000215
TLRexp: TRLexp:
TLR 46 .0001095 sec TRL 109 .0001113 sec
12 1.7585E-O6 12 1. 199513-04

Figure 6.9 (a) The experimental time signals and (b) the
echo data for the ﬁve layer object of Figure 6.8.

117

 

Table 6.5

 

 

 

Quantity Experimental Actual % error
value value

Irll .2945 .368 18.66 %
Irzl .4350 .368 20.17 %
Ir3| .4965 .368 37.15 %
lr4l .3998 .368 10.4 %
k1 .6363 .596 6.762 %
k2 .8155 .997 18.21 %
k3 .6090 .596 2.181 %
alvl (nepers/sec) 104,517.8 138,850.8 24.73 %
“2V2 (nepers/sec) 5424 87.32 6,116 %
'a3v3 (nepers/sec) 114,008.5 138,850.8 17.89 %
v1 (tn/sec) 2312 2680 13.75 %
v2 (In/sec) 1396.3 “[1480 5.68 %
v3 (m/sec) 2298.9 2680 14.22 %
a1 (nepers/m) 45.207 ‘___51.81 12.70 %
02 (nepers/m) 3.885 ;_:§_569 . 26,911 %
a3 (nepers/m) 49.594 , 51.81 4.26 %

Summary of the results for the ﬁve layer object.

118

As a ﬁnal example of the possibilities in choosing materials with which to experi-
ment, a section of common beef liver was suspended in the ﬁxture/tank, thus forming
a three layer object, as shown in Figure 6.10. The sample was interrogated as before,
with the experimental data resulting shown in Table 6.63; in Table 6.6b the computa-
tions and corresponding errors are shown as before. The reﬂection coefﬁcient of the
liver-water interface is extremely small, about 0.051 in magnitude, and this accounts
for the difﬁculty in measuring the echo amplitudes from these boundaries. An addi-
tional difﬁculty is manifested in the surface of the liver which is somewhat rough, as is
characteristic of biological samples; this surface acts as a diffuse scattering source and
this action too adds to the error present in the data. Nevertheless, the percentages of

error in this measurement are not much different than for the previous experiments.

layer 0 layer 1 layer 2
water liver water

 

 

 

x0=8cm x2=11.5cm
t0 = 54,05 usec t2 = 77.7 usec
x1 = 0.5 cm
t1 = 3.23 usec

Figure 6.10 The experimental setup for the liver sample.

119

 

Lexp: Rexp:

L1 20 115.35E-06 sec R1 22 165.3E-06 sec

L2 -9 121.4E-06 R2 -8 171.8E-06

TLRexp: TRLexp:

TLR 74 138.75 E-06 sec TRL 126 130.55E-06 sec
Quantity Experimental Actual Percentage

value value error

lrll .130 .0540 140.7 %

lr2| .143 .0540 164.8 %

k1 .6520 .7839 16.83 %

alvl (nepers/sec) 66,829.8 37,718.2 77.18 %

v1 (m/sec) 1607.3 1549 3.764 %

a1 (nepers/m) 41.58 24.35 70.76 %

Table 6.6

 

 

 

(a) The experimental data, and (b) the computed
results for the liver sample.

CHAPTER VII

RECOMMENDATIONS AND CONCLUSIONS

In the preceding chapters, we have, from basic principles, built up to a detailed
understanding of the possibilities and problems that confront attenuation imaging,
both at present and in the future. Such an acoustic measurement technique must rely
on amplitude processing of the experimental data acquired during a measurement
situation, and this aggravates certain deﬁciencies in ultrasonic techniques that are
currently unsolved, such as angular (non-collinear with the direction of propagation)
scattering. In fact, based on the reported work in the literature and the results
reported here in this thesis, the author is of the opinion that almost all of the
discussed difﬁculties may in fact be surmounted eventually, at least with sufﬁciency
to permit useful imaging in many circumstances, with the possible exception of the
aforementioned angular scattering problem, which appears to be the most severe
limitation. It appears that this difﬁculty may also yield to a scanning/array technique,
such as that pursued by Clement et al. in [56], which showsvery promising
results. It is in this spirit that the recommendations of Section 7.1 are made, that
future work should center on investigation of higher dimensional models, interrogated

by means of a scanned set of transducers.

120

121

7.1 Remmgndatigns For Fugue Investigation

The work reported in this thesis has centered on a one-dimensional model for the
test object, which is not usually realistic except in certain cases, such as when
imaging composite materials. We now brieﬂy consider the possibility of extension to

other cases of interest.

In order to reconstruct the internal makeup of a test object with spatial dimensions
higher than one, the space curves (or surfaces) that deﬁne the boundaries of the
different regions must be speciﬁed. In practice, due to the discrete number of scan-
ning angles/positions available when performing the requisite measurements, it will
not be feasible to exactly measure the shape of these boundaries, so an approxima-
tion to these shapes must be made, perhaps by means of spline ﬁtting of either
curves or surfaces to the data. This method will be satisfactory only if sufﬁcient data
points are available to accurately determine the ﬁrst two spatial derivatives of the
boundary, in the case of cubic splines, implying that a rather large number of separate
measurements are necessary, as mentioned before in Chapter 5. However, the
main difﬁculty with this concept is that it is difﬁcult to decide which data points belong
to which boundary, since the number of regions can change with measurement position
and boundaries may even intersect. Once the boundaries are known, then steps can
be taken to correct for the effect of angular scattering on the calculation of the reﬂec-
tion coefficients, as mentioned in Chapters 4 and 5, and calculations similar to that of

Chapter 4 may be possible. In two dimensional slice measurements of a three

122

dimensional test object, there will be perhaps signiﬁcant error introduced from
assuming that all the acoustic energy is conﬁned to a region coplanar with the
measurement plane, in a similar vein to that discussed in Chapter 5 for the one-
dimensional test object model. Investigation of these higher-order objects must
answer the questions of uniqueness of the solution, whether multiple reﬂections can
be removed, and what form of experimental data is needed, all of which will directly
inﬂuence the system hardware complexity, computation time of the measurement, and
accuracy. A good starting place for such research would be to conﬁne the study to a
two-dimensional object, i.e. one that does not scatter acoustic energy into the third
dimension and out of the measurement plane, and reconsider the information neces-
sary to uniquely describe the object internal structure, forming an object set much like
the set {N , k, r, t} used in this thesis. For example, internal ray tracing may be help-
ful in accomplishing this goal, and a good assumption to make is that all of the scat-
tered energy is received and acknowledged by measurement system. Intuitively, it
appears to the author that this two—dimensional situation will respond in kind as has
the one-dimensional case, albeit in a more restricted or complex manner; normally,
we expect to see elements of simpler system behavior within the tOtal response of a
more complex object, which can be useful in attacking the more complex situation. It
is evident that the measurement needs of this two-dimensional object would entail
some sort of array/scanning procedure to ensure recapture of all the outputted acous-
tic energy, and much investigation into various schemes to do this is possible, with

many performance/complexity trade-offs permissible.

123

7.1.2_ Extension of the Alpha-V Method to Transmission Tomography

 

 

More speciﬁcally, we shall brieﬂy consider the potential of upgrading the method
of transmission tomography forwarded by Greenleaf et al. in [43]. As we have
discussed previously, this latter work is intuitively appealing in its similarity to X-ray
tomographic methodology, but suffers in accuracy from ignoring angular scattering
effects, which includes reﬂective backscattering if one assumes one-dimensional
energy propagation. We have shown that this mechanism is indistinguishable from
loss if only transmission data is used (see Section 4.5.2), and this will result in poor

accuracy in cases of large impedance discontinuities.

The alpha-v method can be used to help rectify this problem, and in order to show
how this might be accomplished, we will assume that the measurement situation is
likewise the same in Greenleaf et al.; here we have a two dimensional psuedocircular
object to image, composed of a single region with a continuous spatial variation of
acoustic attenuation. For our purposes, we can extend this model to the more general
case of such a test object, but one composed of various regions with continuous
spatial variation of attenuation, bounded by impedance discontinuities, such as that
shown in Figure 7.1. It is readily apparent that this type of object is a superset of the
type discussed in Greenleaf et al. and is the most general two-dimensional test
object that is worthy of imaging consideration. In the cited work, the measurement is
performed by modeling the object as a set of one-dimensional diametric slices, with

the experimental data acquired by rotating a pair of transducers about a common

  
 
 

receiver

D/

transducer
rotation

transmitter

E1

Figure 7.1 The type of object proposed in Section 7.1.2.

center, about which the test object is located. Each one-dimensional slice is treated
as a transmission imaging problem, like that described in Section 4.5.2 of Chapter 4.
As we mentioned, the main assumption in [43] is that the reﬂection coefﬁcients of

any boundaries present are negligibly small, so Equation (4.5) reduces to

TLR (k,r) = Ainput k1 k2 . . . kn (7.1)
= Ainput exp[-(0t1x1+ 0.2x2 + . . . + anxn)]
where n is taken to be a large integer (note that n is not the number of layers, since

it has been assumed that there are no discontinuities in the acoustic properties of the

object internal structure), and xi are a ﬁxed small distance which is given by

overall object length (7.2)

1 n

125

for i = 1 to n. The length represented by xi is considered to be the same as the length
of one pixel of the output screen display; therefore, all the xi are considered identical.

Thus, given a particular measurement direction, we proceed to measure the transmit-

ted amplitude TLR (experimental) and then can write that

“1+“2+“-+°‘n=‘1"[TLR/Ainput]/x (7.3)

If this is repeated from a sufﬁciently large number of angles, say n such distinct

positions, then we will have a set of n linear equations to solve for the n values of (xi,

giving us a discrete pixel map approximation of the variation of the attenuation within
the object. Implicit in this is that n is very large so as to allow adequate resolution,
and to furnish sufﬁcient data pertaining to each pixel to permit complete speciﬁcation

of the linear system.

As we stated, the premise of this method is to ignore the reﬂection information as
irrelevant; we have seen that in fact the reﬂected energy contains much more informa-
tion than the transmitted. Our proposal is to incorporate our knowledge of the results
of Chapter 4 into the above discourse. This may be done simply by following the
bidirectional interrogation scheme described therein, and performing the calculations of
Equations (4.20) to ﬁnd the reﬂection coefﬁcients of each boundary. Then we can use

the full form of Equation (4.5) and rewrite Equation (7 .3) more formally

"lnITLR/Ainput]
(1 +0 +...+a = .
1 2 n X(l+r1)(l+r2)...(l+rN) (74)

 

126

where N represents the number of apparent layers in the particular measurement
direction. Following the lead from this point, it should be possible to image internal
regions of an object bounded by impedance discontinuities even if these areas have a
continuous variation in acoustic properties. This would be a most valuable I
accomplishment, but it should be noted that the angular scattering problem must be
addressed as well before congratulations are in order; in effect, we must also

solve the proposed problem given in Section 7.1.1 as well. This topic, taken in its
entirety, appears to be most lucrative for further research, with the rewards of success

being inestimable.

7 xni th' _‘ 1101-31' 1!. ‘niﬂi. =.' 111011

The results of Chapter 4 are by no means limited to acoustic probing; in fact, many
systems can be described by the formulations forwarded therein. For example,
uniform transmission lines with characteristic impedance discontinuities ﬁt this type
of behavior, and the results of Equations (4.20) and (4.21) apply directly. Additional-
ly, the case of a single such transmission line is the epitome of the one-dimen-
sional situation, since the electrical energy has only an axial component (assuming no
radiation, unless one wishes to model this as a loss per unit length)! These equations
could be used to probe long electrical lines to ﬁnd the position and perhaps type of
fault that exist at some remote but unknown location; perhaps the results of Chapter 4
could be expanded to include lines with distribution branches as well. Unfortunately,

the practical application of these equations to radar and underground remote sensing

127

seems limited, since only one side of the test "object" is available to the investigator.

7.2 Conclusions
In this thesis, we have investigated a one-dimensional c0planar N-layered homo-
geneous test object being probed by acoustic means. On the way, we discovered the

following:

(1) The only experimental data that permits unique solution of the problem is
the bidirectional (two-sided) four-signal interrogation employed in
Chapter 4. This suggests that in order to uniquely solve an n-th order
spatial dimension object problem of this type, it may in fact be necessary
to investigate it remotely from a space with at least 2n degrees of
freedom, and perhaps viewing this object from a space of higher order

than 2n would not permit a unique solution to be found either, due to over-
speciﬁcation. For a three—dimensional object, this would mean that the
reﬂected and transmitted data at/from each spherical angle must be
recorded, possibly a vary large quantity of information to process in

practice.

(2) The one-dimensional N-layer problem can be decoupled into ﬁrst a
solution for r, then secondly a solution for k, implying that the uniqueness

of then problem is not governed by the losses of the layers.

(3) The solution we found in Chapter 4 always exists, for any combination of

128

data even with experimental errors.

(4) An algorithm was developed that demonstrates the possibility that the
solution of the N -layer one-dimensional object is unique even in the
presence of multiple reﬂections in the experimental data. Unfortunately,
this immensely important result remains to be rigorously proved at

present.

(5) The algorithm also ﬁnds N, the number of ' object layers, uniquely, given
experimental data that may or may not be corrupted with multiple

reﬂections, as long as the primary signals are present in this data.

(6) The above strongly suggests that the solution is unique even with multiple
reﬂections and prior unknown N. It appears possible that this condition
may be true for the similar classes of two- and three-dimensional objects
if the necessary and sufﬁcient experimental data can be acquired. The
solution of such problems will by no means be computationally simpler
than the already involved one-dimensional situation. Further research

should be directed at this topic.

(7) The system analysis of Chapter 5 has indicated the beneﬁts and difﬁculties
of an amplitude measurement system. This study has shown that the
solution equations derived in Chapter 4 for the N-layer case are rather

insensitive to experimental error, if certain system features and

(8)

(9)

129

corrections are incorporated prior to commencing the calculations indicated.
The major source of error was clearly shown to be scattering of energy into
non-measurement directions, which can only be corrected by some form of
array/scannin g method to allow complete recapture of the output energy,

and treating the test object as a two- or three-dimensional entity.

Experimental work has been performed to verify the correctness of the
derived solution, and to investigate the practical feasibility of amplitude-

based measurements. The results show reasonable agreement with

published values. Interestingly, the ow product appears to be extremely
varied for different materials, which supports the notion of using this
product as an index for imaging and/or material identiﬁcation. Unfortun-
ately, it has been found that the use of multiple transducers is very

difﬁcult, since pressure-to-voltage calibration (matching) is necessary.

This fact tends to reduce the attractiveness of transducer arrays, which is
decidedly unfortunate since mechanical scanning of a pair of transducers
will always remain the slower alternative. Additionally, target angle has
been found to be a strong inﬂuence on the accuracy of reﬂected amplitudes,
not having as large an inﬂuence on the transmitted pulses, even for slight

misalignment. This will deﬁnitely be a tOpic of ardorous contention, even

with array/scanning advancements.

Suggestions have been forwarded for further research areas related to the

130

work pursued here, centering primarily on expanding the results to objects
of higher spatial dimension. It is not crucial that all work performed in this
be of practical value, since much can be learned about what is or is not
possible by formulating a somewhat abstract situation an investigating it

in a reasonable manner. This is perhaps the author’s greatest criticism of
the literature in this ﬁeld, but this lack has left a wide range of theoretical

investigation open to nascent researchers.

APPENDICES

APPENDIX A

Program: TWOSIDE

1 Rem **** TWOSIDE: a bidirectional simulator for N-layer test objects
2 Rem

 

3 Rem BY: J.Nodar 1989 MSU
4 Rem

8 Rem --- initialize

9 Rem

10 Dim S(500,4),S$(500),Lout(30,2),Lout$(30),Rout(30,2),Rout$(30)
20 Dim K(lO),R(10),T(10)

30 Dim Lsim(30,2),Rsim(30,2),TLRsim(30,2),TRLsim(30,2)

40 Dim Lsim$(30),Rsim$(30),'ILRsim$(30),TRLsim$(30)

45 Ain=1 :Tol=.02*Ain

48 Rem

49 Rem ---- read input data ﬁle to get {N, k, r, t)
50 Rem

51 Input "Input ﬁle";A$ : Open A3 for input as #1
54 Input #1,N

55 For 1:0 to N-l : Input #1 ,K(I) : Next I

56 For I=1 to N-l : Input#l,R(I) : Nextl

57 For 1:0 to N-l : Input #1,T(I) : Next I

70 Close 1

90 Rem

92 Rem ---- simulate {N, k, r, t) using TWOSIDE algorithm
93 Rem .

95 Gosub 1000 : Rem -- now have [LTRTlsimulated

99 Rem

100 Rem ---- print results of simulation
101 Rem

200 Print : Print"Lsim:" : Print

210 For 1:1 to Lsim(0,0) : Print Lsim(I,1),Lsim(I,2),Lsirn$(I) : Next I

211 Print : Print"TLRsim:" : Print

212 For I=1 to TLRsim(0,0) : Print 'ILRsim(I,l),TLRsim(I,2),TLRsim$(I) : Next I
220 Print : Print"Rsim:" : Print

221 For 1:] to Rsim(0,0) : Print Rsim(I,1),Rsim(I,2),Rsim$(I) : Next I

 

 

 

131

132

230 Print : Print"TRLsimz" : Print

235 For I=1 to TRLsim(0,0) : Print TRLsim(I,1),TRLsim(I,2),TRLsim$(I) : Next I
240 Rem ’

250 Rem ---- print to line printer

 

 

 

255 Rem
260 Rem ---- save to disk ﬁle
800 Rem
850 Rem ---- done
900 Rem
999 End
1000 Rem
1010 Rem --- TWOSIDE simulation subroutine ------------
1020 Rem
1030 Rem
1040 Rem ---- input variables:
N .= number of layers in model
K() = loss of each layer
R() = reﬂection coefficient of each boundary
T0 = time delay in each layer
Ain = input pulse amplitude
tol -- minimum amplitude to use in simulation

1100 Rem ---- output variables:

Lsim(),Lsim$() = simulated left echoes, paths
TLRsim0,TLRsim$() = simulated left to right transmissions,paths

Rsim(),Rsim$() = simulated right echoes,paths
TRLsim(),TRLsim$() = simulated right to left transmissions,paths

NOTE: Lsim(0,0) = number of echoes in Lsim(), etc.

1200 Rem ----- internal variables:
80 = stack of internal waves
S$() = stack of internal wave paths
P = stack pointer
1,] = loop counters
Side$ = indicator for input pulse side

Lout(),Lout$() = left output wave list, paths

133

Rout(),Rout$0 = right output wave list,paths
Temp,Temp$ = scratch variables for sorting

NOTE: S(P,1) = wave amplitude
S(P, 2) = wave time delay
S(P, 3) = wave direction (0=left,1=1ight)
S(P, 4) = current boundary location (i. e. 0.. .N)

1250 Rem ---- initialize the simulation

 

Ain = Ain :Tol=Tol

1300 Rem ---- Input on the left side

 

Side$="left" : Gosub 1500

For I=l to Lout(0,0) : For J=1 to 2 : Lsim(I,J)=Lout(I,J) : Next J
Lsim$(I)=Lout$(I) : Next I : Lsim(0,0)=Lout(0,0)

For 1:1 to Rout(0,0) : For J =1 to 2 : TLRsim(I,J)=Rout(I,J) : Next J
TLRsim$(I)=Rout$(I) : Next I : TLRsim(0,0)=Rout(0,0)

 

1400 Rem ---- Then input on the right side
Side$="right" :Gosub 1500

For 1:1 to Lout(0,0) : For J=1 to 2 : TRLsim(I,J)=Lout(I,J) : Next I
TRLsim$(I)=Lout$(I) : Nextl : TRLsim(0,0)=Lout(0,0)

For 1:1 to Rout(0,0) : For J=l to 2 : Rsim(I,J)=Rout(I,J) : Next I
Rsim$(I)=Rout$(I) : Nextl : Rsim(0,0)=Rout(0,0)

1450 Rem ---- Done with both sides, so leave

 

Return : Rem -- goes back to calling routine

1500 Rem ---- Simulate the model from the speciﬁed side

 

Rem -- init this part
P=0 : Lout(0,0)=0 : Rout(0,0)=0 : S(P,1)=Ain : S(P,2)=0

if side$="left" then S(P,3)=l : S(P,4)=0 : S$(P)=STR$(0)
else S(P,3)=0 : S(P,4)=N : S$(P)=STR$(N)

134

1580 Rem ---- start the simulation

 

If P=-1 then 2000 : Rem -- stack is empty, so done!
If abs(S(P,l))<tol then P=P-1 : Goto 1580

1630 Rem -- left output wave?
if S(P,4)<>0 or S(P,2)=0 then 1700
Lout(0,0)=Lout(0,0)+1
Lout(Lout(0,0),l)=S(P,l) : Lout(Lout(0,0),2)=S(P,2) : Lout$(Lout(0,0))=S$(P)
P=P-l : Goto 1580

1700 Rem -- right output wave?
If S(P,4)<>N or S(P,2)=0 then 1800
Rout(0,0)=Rout(0,0)+1
Rout(Rout(0,0),l)=S(P,l) : Rout(Rout(0,0),2)=S(P,2)
Rout$(Rout(0,0))=S$(P)
P=P-1 :Goto 1580

1800 Rem -- left going wave?
If S(P,3)<>0 then 1900
S (P,4)=S (P,4)- 1
S(P,1)=S(P,1)*K(S(P,4))
S(P.2)=S(P.2)+T(S(P.4))
S$(P)=S$(P)+","+STR$(S(P.4))

If S(P,4)=0 then 1580 else S(P+1,1)=S(P,1)*(1-R(S(P,4))
S(P+1,2)=S(P,2)
S(P+1,3)=0
S(P+1,4)=S(P,4)
S$(P+1)=S$(P)
S(P,1)=S(P.l)*(-R(S(P.4))
S(P,3)=1
P=P+1 :Goto 1580

1900 Rem -- right going wave?
S(P,1)=S(P.l)*K(S(P.4))
S(P,2)=S(P.2)+T(S(P.4))
S(P,4)=S(P,4)+l
S$(P)=S$(P)+","+STR$(S(P.4))

If S(P,4)=N then 1580 else S(P+l,1)=S(P,1)*(1+R(S(P,4)))
S(P+1,2)=S(P,2)
S(P+1,3)=1
S(P+1,4)=S(P,4)

135

S$(P+1)=S$(P)
S(P,1)=S(P,1)*(+R(S(P,4)))
S(P,3)=0

P=P+1 :Goto 1580

2000 Rem ---- sort the output wave lists in increasing time order ..............

For I=l to Lout(0,0) : For J=l to Lout(0,0)-I
If Lout(J,2)>Lout(J+l,2) then temp=Lout(J,l) : Lout(J ,1)=Lout(J +1.1) :
Lout(J-l- 1 , 1 )=temp
temp=Lout(J,2) : Lout(J,2)=Lout(J+1,2) :
Lout(J +1 ,2)=temp
temp$=Lout$(J)C : Lout$U)=Lout$(J+1) :
Lout$(J+l)=temp$
Next 1,1

For 1:1 to Rout(0,0) : For J=l to Rout(0,0)-I

If Rout(J,2)>Rout(J+l,2) then temp=Rout(J ,1) : Rout(J ,1)=Rout(J +l,l) :
Rout(J+1,1)=temp
temp=Rout(J,2) : Rout(J,2)=Rout(J+1,2) :
Rout(J+1,2)=temp
temp$=Rout$(J) : Rout$(J)=Rout$(J+1) :
Rout$(J+1)=temp$

Next J ,l

2200 Rem ---- all done with this side

 

return : Rem -- goes back to either 1300+ or 1400+

APPENDIX B

Program: ALPHA-V

100 Rem **** Elimination of multiple reﬂections in {LTRT}experimental
105 Rem

 

106 Rem BY: J. Nodar 1989 MSU
110 Rem

112 Rem ---- initialize

113 Rem .

120 Rem Lexp(30,2),Rexp(30,2),TLRexp(30,2),TRLexp(30,2)
130 Dim Lsim(30,2),Rsim(30,2),TLRsim(30,2),TRLsim(30,2)
135 Dim Lsim$(30),Rsim$(30),TLRsim$(30),TRLsim$(30)
140 Dim Lpri(30,2),Rpri(30,2),TLRpri(2),TRLpri(2)

150 Dim K(30),R(30),T(30)

160 Dim S(500,4),S$(500),Lout(30,2),Lout$(30),Rout(30,2),Rout$(30)
180 Rem

185 amptol=.1 : timetol=.1

199 Rem

200 Rem ---- read data ﬁle into {LTRT}experimental arrays

 

205 Rem

206 Input "input ﬁle: "; A$ : Open A$ For Input As #1

208 Input #1, Lexp(0,0)

210 For I=1 to Lexp(0,0) : For J=1 to 2 : Input #1, Lexp(I,J) : Next J,I

215 Input #1, TLRexp(0,0)

218 For I=l to TLRexp(0,0) : For J=1 to 2 : Input #1, TLRexp(I,J) : Next 1,]
220 Input #1, Rexp(0,0)

225 For I=1 to Rexp(0,0) : For J=1 to 2 : Input #1, Rexp(I,J) : Next l,J

230 Input #1, TRLexp(0,0

235 For 1:1 to 'IRLexp(0,0) : For J=1 to 2 : Input #1, TRLexp(I,J) : Next 1,]
250 Close 1

299 Rem

300 Rem ---- begin processing
301 Rem "

302 Rem -- see if transmission data is present and get it

303 If TLRexp(0,0)=0 or TRLexp(0,0)=0 then Print"error!!" : Stop

 

136

137

304 Tsample==(TLRexp(l,2) + TRLexp(1,2))l2

305 TLRpri(l)=TLRexp(l,l) : TLRpri(2)=TLRexp(l,2)

306 TRLpri(1)=TRLexp(1,1) : TRLpri(2)=TRLexp(1,2)

307 Rem

308 Rem -- ﬁnd upper bound On the number of layers in object

309 Rem

310 If Lexp(0,0)<Rexp(0,0) Then N=Lexp(0,0)+1 Else N=Rexp(0,0)+1
311 Rem

 

312 Rem ---- main loop onN

313 Rem

320 While N>0 : Print "N=";N : Print

325 Rem

326 Rem ---- choose a combination of N-l left echoes as Lpri() --------------
327 Rem

330 For I=1 to Lexp(0,0) : If I>N-1 Then Lexp(I,0)=0 Else Lexp(i,0)=1 : Nextl
331 Rem

335 Rem -- test to see if it has N-l echoes in it

340 Count=0 : For I=1 to Lexp(0,0) : Count=Count+Lexp(I,0) : Next I

342 If Count=N-1 then Goto 400 : Rem -- has sufficient # of echoes in it

345 Rem

350 Rem -- get next left echo combination and test if N<1 (DONE?)

352 Carry=1 : For I=2 to Lexp(0,0) : Lexp(I,O)=Lexp(I,0)+Carry : Carry=0

354 If Lexp(I,0)=2 then Lexp(I,0)=0 : Carry=l

356 Nextl

357 If Carry=1 then N =N-1 : WEND : Print"no combinations workl?" : Stop

358 Goto 335

399 Rem

400 Rem -- put the left echo combination into Lprimary

410 J=1 : For I=1 to Lexp(0,0)

415 If Lexp(I,0)=0 then 420

417 Lpri(J,I)=Lexp(I,1) : Lpri(J,2)=Lexp(I,2) :J=J+1
420 Nextl : IfJ>N-l then Print"error!!!" : Stop
499 Rem

500 Rem ---- choose a combination of right echoes
510 Rem

530 For I=1 to Rexp(0,0) : If I>N-1 then Rexp(I,0)=0 else Rexp(I,0)=l : Next I
532 Rem

535 Rem -- test to see if right combination has N-l echoes in it

540 Count=0 : For I=1 to Rexp(0,0) : Count=Count+Rexp(I,0) : Next I

542 If Count=N-1 then 600 : Rem -- i.e. has sufﬁcient # of echoes

549 Rem

550 Rem -- get next right echo combination

552 Carry=1 : For I=2 to Rexp(0,0) : Rexp(I,0)=Rexp(l,0)+Carry : Carry=0
554 If Rexp(I,0)=2 then Rexp(I,0)=0 : Carry=1

 

138

556 Next I : If Carry=1 then 350 : Rem -- no right combinations left to try

558 Goto 535

599 Rem

600 Rem -- put the right combinations into Rprimary

610 J=l : For I=1 to Rexp(0,0)

615 If Rexp(I,0)=0 then 620

617 Rpri(J,l)=Rexp(I,1) : Rpri(J,2)=Rexp(I,2) :J=J+1

620 Next I : If J>N-1 then Print"Errorl" : Stop

649 Rem

650 Rem ---- test if left and right combinations are compatible

655 Rem

660 Rem -- check for opposite signs on echo amplitudes

665 For I=1 to N-l : If Lpri(I,1)*Rpri(N-I,l)>0 then 550 : Rem -- i.e. not OK

667 Nextl

668 Rem

669 Rem -- check the time delays

670 For I=1 to N-l : If Abs((Lpri(l,2)+Rpri(N-I.2))/2/r sample -1)>Timetol then 550

671 Next I

679 Rem

680 Rem ---- print echo combination found

681 Rem

682 For I=1 to N—l : Print "L(";I;")=";Lpri(l,1),"R(";I;")=";Rpri(l,1) : Next I

683 Print : Print

685 Print"TLR=";'ILRpri(1)

686 Print"TRL=";TRLpri(1)

689 Rem

690 Rem ---- Do alpha-v computations to ﬁnd {N, k, r, tltrial

695 Rem

699 Rem -- ﬁnd the reﬂection coefﬁcients

700 For I=1 to N-l

710 R(I>=Sgnaaﬁ<1.1>*Sqr(Lpﬁa.1)*Rpri(N-I.1)/(Lpri(1.1)*Rpri(N-I.1)-
TLRpri(1)*TRLpri(l)))

712 Nextl

715 Rem -- ﬁnd the loss parameters

720 K(0)=1 :K(N—1)=l : For I=1 to N-2

725 K(I)=Sqr(R(I)/R(I+1)/(1-R(I)"2)*Lpl’i(1+l.1)/Lp1'i(1.1))

727 Nextl

730 Rem -- ﬁnd the layer time delays

735 If N =1 then T(0)=Tsample else T(0)=Lpri(1,2)/2

737 For I=1 to N-2 : T(I)=(Lpri(I+1,2)-Lpri(I,2))/2 : Next I : T(N-1)=Rpri(1,2)/2

750 Rem

760 Rem ---- print the set {N, k, r, t}t1ial ---

770 For i=0 to N—l : Print "N=";N : Print

772 Print"k(";I;")=";K(I); : IfI=O then Print ,; : goto 774

 

 

 

 

139

773 Pﬁnt."r(";1;")=";R(I);

774 Print ,"t(";I;")=";T(I) : Next I

799 Rem

800 Rem ---- simulate the set {N, k, r, t}trial using TWOSIDE algorithm -------

805 Rem

810 Gosub 1000 : Rem -- now have the set {LTRT}simulated

880 Rem .

881 Rem ---- rescale the set {LTRT}simulated so that Ain(sim)=Ain(exp) -----

882 Rem

883 Print"Rescaling the simulated data set..."

884 If N>1 then Scale=Lexp(1,1)/Lsim(1,1) Else Scale=l

886 For I=1 to Lsim(0,0) : Lsim(I,1)=Lsim(I,1)*scale : Next I

887 For I=1 to TLRsim(0,0) : TLRsim(I,1)=TLRsim(I,1)*scale : Next I

888 If N>1 then Scale=Rexp(1,l)/Rsim(1,1)

890 For I=1 to Rsim(0,0) : Rsim(I,1)=Rsim(I,1)*scale : Next I

895 For I=1 to TRLsim(0,0) : TRLsim(I,1)=TRLsim(I,1)*scale : Next I

899 Rem .

900 Rem --- check if simulation and experimental data sets agree ------------

910 Rem

911 Print"Checking if exp is in sim..."

915 Rem

920 For I=1 to Lexp(0,0) : For J =1 to Lsim(0,0)

922 If abs(Lexp(i,1)/Lsim(J,1)—1))<amptol and abs(Lexp(l,2)/Lsim(J,2)-1)<timetol
then 924

923 Next I : goto 950 : Rem -- not OK

924 Next I : goto 963 : Rem -- is OK!

926‘ For I=1 to Rexp(0,0) : For J=1 to Rsim(0,0)

928 If abs(Rexp(I,1)/Rsim(J,1)-1)<amptol and abs(RexP02)/Rsim(l,2)-1)<timetol
then 930

929 Next J : Goto 950 : Rem -- not OK

930 Next I : Goto 963 : Rem -- is OK!

940 Rem

950 Print"Checking if sim is in exp..."

952 For I=1 to Lsim(0,0) : For J =1 to Lexp(0,0)

954 If abs(Lsim(I,1)/Lexp(J ,1)-1)<amptol and Abs(Lsim(I,2)lLexp(J,2)-1)<timetol
then 956

955 Next I : Print"Sim<>exp???" : Goto 550

956 Nextl

958 For I=1 to Rsim(0,0) : For J=l to Rexp(0,0)

960 if abs(Rsim(I,1)/Rexp(l,1)-1)<amptol and abs(Rsim(l,2)lRexp(I,2)-1)<timetol
then 962

961 Next J : Print"Sim<>exp???" : goto 550

962 Next I : Print"SIM=EXP!!!!!" : END

1000 Rem "" 1118611 TWOSIDE (lines 1000-) of Appendix A here --------------

APPENDIX C

Program: TWOCHANNEL

100 Rem “** TWOCHANNEL: Controls WAAGII board and gets [LTRL}exp.
110 Rem
120 Rem By: J. Nodar 1989 MSU
130 Rem
140 Rem ---- initialize
150 Rem
170 ’
175 Def FNLOBT(x) = x and &HFF
176 Def FNI-lIBT(x) = &HFF and ((x and &HFFOO) \256)
177 ’
180 RAM = &HO : REG = &H178 : REGO = REG+0 : REGI=REG+1
REGZ=REG+2 : REG3=REG+3 : SEGMENT=&HDOOO
185 ’
190 N=5333 : ’ ---- number of sample points to take
based on a sample rate of 20 MHz
and a 20 cm span test ﬁxture.

 

195 ’

199 DEFINT C,X

200 DIM oldflag(2),max(2),tmax(2),ﬂag(2),peak(2),shot(2),
ch(1,5333),x(1,5333),
impulse( 1 ,20,2),
Lexp(20,2),TLRexp(20,2),Rexp(20,2),TRLexp(20,2)

215 ’

220 A=0 : B=1 : ’---- channel marker ﬂags

222 ’

1000 ’ ---- main routine

1005 ’

1010 ’--- init

1011 ’

1020 Lexp(0,0)=0 : TLRexp(0,0)=0 : Rexp(0,0)=0 : TRLexp(0,0)=0

1030 ’

1100 ’ ---- loop for input on left, then right sides

1110 ’

 

140

141

1115 C13 : Locate 5,10 : Print "Two Channel for {L,TLR,R,TRL}exp." : Print : Print

1116 Input "Number of times to average: "; avnum : Print : Print

1120 Input "Theta: "; theta : Print : Print

1125 ’

1126 ’ ---- left input

1127 ’

1128 Print "Left input <CR>?" : input A$ : Gosub 13000 : Gosub 11000 : Gosub 12000

1129 ’

1130 For I=1 to impulse(A,0,0) : For J =1 to 2
Lexp(I,J)=impulse(A,I,J) : Next J : Next I
Lexp(0,0)=impulse(A,0,0)

1135 ’

1140 For I=1 to impulse(B,0,0) : For J=1 to 2
TLRexp(I,J)=impulse(B,I,J) : Next J : Next I
TLRexp(0,0)=impulse(B,0,0)

1145 ’

1200 ’ ---- right input

1210 ’

1220 Print "Right input <CR>?" : Input A$ : Gosub13000 : Gosub 11000 :
Gosub 12000

1225 ’

1230 For I=1 to impulse(A,0,0) : For J =1 to 2
TRLexp(1,J)=impulse(B,I,J) : Next J : Nextl
TRLexp(0,0)=impulse(A,0,0)

1235 ’

1240 For I=1 to impulse(B,0,0) : For J=l to 2
Rexp(IJ)=impulse(B,I,J) : Next J : Next I
Rexp(0,0)=impulse(B,0,0)

1245 ’

1300 ’ ---- save data to a ﬁle

1310 ’

1320 Input "ﬁle to save "; A$ : if A$="" then 9999

1330 open A3 for output as #1

1335 ’

1340 Print #1,theta

1350 Print #1,Lexp(0,0)

For I=1 to Lexp(0,0) : Print #1,Lexp(I,1),Lexp(I,2) : Next I

1360 Print #1,TLRexp(0,0)

For I=1 to TLRexp(0,0) : Print #1,TLRexp(I,1),TLRexp(I,2) : Next I

1370 Print #1,Rexp(0,0)

For I=1 to Rexp(0,0) : Print #1,Rexp(I,1),Rexp(I,2) : Next I

1380 Print #1,TRLexp(0,0)

For I=1 to TRLexp(0,0) : Print #1,TRLexp(I,1),TRLexp(I,2) : Next I

1390 Close 1

9999 CLS : END

142

 

10000 ---- Subroutine to get two channels of WAAGII data
10010 ’

10020 ’ ---- init control registers

10030 ’

10040 Def SEG=SEGMENT

10045 OUT REG3, &HOO : OUT REGZ, &H6F : ’ -- load trigger offset of 140.
10050 Poke 0,140

10055 OUT REG3, &H80 : OUT REG2, &H6F : ’ -- reset the control values
10060 DEF SEG

10070 ’

10100 ’ --- enable sampling

10150 ’

10170 Print : Print "Sampling....";

10180 OUT REG3, &H80 : OUT REGZ, &H6F : ’ -- init the control register
10190 OUT REGl, &HFF : OUT REGO, &HFF: ’ -- clear the byte counter
10200 OUT REGl, &HFF : OUT REGO, &HFF

10210 NUM=-(N+&HFF) : ’ -- compute the correct byte count

10215 OUT REGl, FNHIBT(NUM) : OUT REGO, FNLOBT(NUM)

10220 OUT REG], FNHIBT(NUM) : OUT REGO, FNLOBT(NUM)

10230 CONTROL=&l-IBFFC and &H800C : ’ -- engage the trigger mechanism
10250 OUT REG3, FNHIBT(CONTROL) : OUT REGZ, FNLOBT(CONTROL)
10260 ’

10270 OUT &H3E4, 0 : ’ ---- trigger the one shot on the motor control board
10275 ’

10290 WAIT REGZ,1 : ’ ---- wait until all data is read into sample memory
10300 ’

10310 ’ ---- read back the data into the array called ch()

10320 ’

10330 OUT REG3, &H80 : OUT REGZ, &H6F : ’ -- enable the ram for read
10340 VTA=2*(INP(REGO)+(INP(REGl)and &H3F) * 256)
10350 ’
10360 ’ ---- read the data into ch(A,,) and ch(B,,)
10365 ’
10370 Print "Loading data..."
10390 DEF SEG=SEGMENT
10400 I=0
10410 ch(A,I)=PEEK(VTA) : ch(B,I)=PEEK(VTA+1)
if VTA=&H7FFF then VTA=0 else VTA=VTA+2
I=I+l : If I<=N then goto 10410 else DEF SEG

10450 ’

10460 ’ ---- all done with this side
10470 ’

10480 return

10490 ’

143

 

11000 ’ ---- plot the results on the PC graphics screen
11010 ’
11015 CLS : Screen 2 : Key off
11020 Locate 1,1 : Print CHA:" : Locate 15,1 :Print "Cl-13:"
1 1040 Pset(0,35-35*(ch(A,O)/128-1)) : Pset(0,l35-35*ch(B,0)/128-1))
11050 ’
11060 For I=1 to N
Line( (I—1)/N*640 , 35-35*(ch(A,I-l)/128-1)) -
(I/N*640 , 35-35*(ch(A,I)/128-1) )
Line( (I-1)/N*640,35-35*(ch(B,I-1)/128-1)) -
(I/N*640 , 35-35*(ch(B,I)/128-1))
Next I : Beep : Locate 24,1
11090 Return

12000 ’ ---- use the delta detector to ﬁnd the amplitude and time of each echo -----
12010 ’
12020 average=128 : threshold=3 : samperiod=1l2e+7 : cutoff=7 : decay =8/9
12023 ’
12025 For I=A to B
ﬂag(I)=0: shot(I)=0 : peak(I)=0 : impulse(l,0,0)=0 :
Next I
12030 ’
12040 For T=1 to N
For Chan: A to B
oldﬂag(chan)=ﬂag(chan)
if abs(ch(chan,t)~average)-peak(chan) < threshold then 12080
else peak(chan)=abs(ch(chan,T)-average) :
ﬂag(chan)=l ;
12080 peak(chan)=peak(chan)*decay
if peak(chan) < cutoff then peak=cutoff :
ﬂag(chan)=0;

if ﬂag(chan)=1 then shot(chan)=8 : ’ -- trigger the one shot
if shot(chan) > 0 then ﬂag(chan)=1
shot(chan)=shot(chan)-1 : if shot(chan)<0 then shot(chan)=0

12100 if oldﬂag(chan)=ﬂag(chan) then 12120
if ﬂag(chan)=0 then 12110
max(chan)=0 : goto 12120

12110 impulse(chan,0,0)=impulse(chan,0,0)+1
impulse(chan,impulse(chan,0,0),1)=max(chan)*sgn(max(chan)-average)
impulse(chan,impulse(chan,0,0),2)=tmax(chan)*samperiod
goto 12200

144

12120 if ﬂag(chan)=0 then 12200
if abs(max(chan)) < abs(ch(chan,T)-average) then max(chan)=ch(chan,t)-

average) :
tmax(chan)=T
12200 Next chan
12205 ’ -- plot the square wave output of the detector to shown echoes
pset (T/N*640, 80-5*ﬂag(A))
pset (T/n*640,180-5*ﬂag(B))
12210 Next T
Beep : Input AS : return : ’ ---- done with this side’s plot

_’

 

13000 ’ ---- averaging subroutine
13025 Print "Averaging ";avnum; "time(s)..." : Print

if avnum=1 then gosub 10000 :
return

Gosub 10000

For chan=A to B : For T=1 to N :
x(chan,T)=ch(chan,t)

Next T : Next chan

for 2:2 to avnum

Gosub 10000: Print"Averaging..."

For chan=A to B : For T=1 to N
x(chan,’1')=(x(chan,T)*(Z-1)+ch(chan,T))/Z
ch(chan,T)=x(chan,T)

Next T,chan,Z

return

BIBLIOGRAPHY

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Haykin, Simon. W. Second Edition, John Wiley &
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Nilsson, James W. W. Second Edition, Addison Wessely

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Mason, Warren P. Pigzgglgtric Qgsgls and Their Application to
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Glover, G. H., and J. C. Sharp. "Reconstruction of Ultrasound Propagation
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Liang, Kenneth K. "Accuracy of Acoustic Impedance Determination

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Stumpf, F. B. Anslyg’cal Acoustiss. Ann Arbor Science Publishers Inc.,
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