Development of optimized solution methods for inverse radiation transport problems involving the characterization of dis...

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Title:
Development of optimized solution methods for inverse radiation transport problems involving the characterization of distributed radionuclides in the large waste container
Physical Description:
xiii, 118 leaves : ill. ; 29 cm.
Language:
English
Creator:
Chang, Chin-Jen, 1959-
Publication Date:

Subjects

Subjects / Keywords:
Nuclear and Radiological Engineering thesis, Ph. D   ( lcsh )
Dissertations, Academic -- Nuclear and Radiological Engineering -- UF   ( lcsh )
Genre:
bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1997.
Bibliography:
Includes bibliographical references (leaves 116-117).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Chin-Jen Chang.

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University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 027588806
oclc - 37153717
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AA00022837:00001

Table of Contents
    Title Page
        Page i
        Page ii
    Dedication
        Page iii
    Acknowledgement
        Page iv
        Page v
    Table of Contents
        Page vi
        Page vii
    List of Tables
        Page viii
        Page ix
    List of Figures
        Page x
        Page xi
    Abstract
        Page xii
        Page xiii
    Chapter 1. Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
    Chapter 2. The nondestructive assay methods for characterizing radionuclide inventory in a nuclear waste container
        Page 7
        Page 8
        Page 9
        Page 10
    Chapter 3. The multiple energy peaks gamma detection system
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
    Chapter 4. Conjugate gradient with non-negative constraint method
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
    Chapter 5. Maximum likelihood expectation maximum method
        Page 24
        Page 25
        Page 26
        Page 27
    Chapter 6. Experiments and model verifications
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
    Chapter 7. Numerical simulations
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
    Chapter 8. Conclusions
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
    Appendix A. Derivation of conjugate gradient solution as a least squared solution of a quadratic function
        Page 108
    Appendix B. Derivation of scalar search factor for conjugate gradient algorithm
        Page 109
    Appendix C. Derivation of orthogonality factor for conjugate gradient algorithm
        Page 110
        Page 111
    Appendix D. Printout of the kernel response table of two energies and four detector positions for homogeneous base density case
        Page 112
        Page 113
        Page 114
        Page 115
    List of references
        Page 116
        Page 117
    Biographical sketch
        Page 118
        Page 119
        Page 120
Full Text










DEVELOPMENT OF OPTIMIZED SOLUTION METHODS FOR INVERSE
RADIATION TRANSPORT PROBLEMS INVOLVING THE CHARACTERIZATION
OF DISTRIBUTED RADIONUCLIDES IN THE LARGE WASTE CONTAINER













By

CHIN-JEN CHANG


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1997






























Copyright 1997

by

Chin-Jen Chang






























This research is dedicated to my wife and son.













ACKNOWLEDGMENTS


The author wishes to identify the contributions of the many people and the

organizations that have made this research successful.

The initial concept of this research, to develop an optimization solution scheme

for the characterization of radionuclide inventory in a nuclear waste drum is attributed to

Dr. Samim Anghaie, the research committee chairman. Dr. Anghaie's consistent advice

and accurate guidance enabled the author to accomplish the research goal in a very short

time period. Mr. David Henderson helped in the initial setting of the experiment,

including the testing of the electronic instruments for the gamma ray spectroscopy and

building the rotation device for the waste barrel. Dr. William G. Vernetson helped with

use of the equipment in the radiological laboratory in the nuclear science building.

Special thanks are given to Mr. Donald Munroe who assisted in cutting the Ir"' ribbon

source into small pieces and received more than 25 mrem radiation dose on his hand

during very short preparation time. The author sincerely appreciates the time spent by his

committee members including Dr. David. E. Hintenlang, Dr. Chihray Liu, Dr. W.

Emmett Bolch and Dr. Robert J. Hanrahan, in discussing and in making comments on

this research.

Two organizations also are worthy of grateful acknowledgement: the National

Science Council of Taiwan (The Republic of China) for awarding the author a research

fellowship from 1994 to 1997; and the Institute of Nuclear Energy Research of Taiwan








(the author's employer) for granting the author a leave of absence for three years.

Support from these two organizations allows the author to concentrate on this research

and enables its completion in a timely manner.














TABLE OF CONTENTS


p~gge_


A CK N O W LED G M EN TS ................................................................................................. iv

L IST O F T A B L E S ........................................................................................................... viii

LIST O F FIG U R E S ....................................................................................................... x

A B ST R A C T ...................................................................................................................... xii

CHAPTERS

1 IN TR O D U C TIO N ..................................................................................................... 1

2 THE NONDESTRUCTIVE ASSAY METHODS FOR CHARACTERIZING
RADIONUCLIDE INVENTORY IN A NUCLEAR WASTE CONTAINER ...... 7

Segmented Gamma Scanning Method ...................................................................... 7
Tomographic Gamma Scanning Method ................................................................. 8
Passive and Active Neutron Detection Method ........................................................ 9

3 THE MULTIPLE ENERGY PEAKS GAMMA DETECTION SYSTEM ............. 11

The Model of a Gamma Scanning System Using Four-Detector Positions ............ 11
The Concept of Using Multiple Gamma Energy Peaks .......................................... 11
Computer Modeling and Verification Procedure ................................................... 14

4 CONJUGATE GRADIENT WITH NON-NEGATIVE CONSTRAINT
M E T H O D .................................................................................................................... 18

5 MAXIMUM LIKELIHOOD EXPECTATION MAXIMUM METHOD ............... 24

6 EXPERIMENTS AND MODEL VERIFICATIONS ............................................ 28

Experim ental Setup ................................................................................................. 28
Detector Efficiency Calibration ............................................................................... 32
Region of Interest of Ir192 Gamma Spectrum .......................................................... 35
D ata Collection Process .......................................................................................... 37








Experiments of Distributed Ir'92 Sources under the Homogeneous Density
C ondition ....................................................................................................... . 39
Experiments of Distributed Ir'92 Sources under Heterogeneous Density
C on dition s .............................................................................................................. 4 5
Discussion of the Results of the Experiments ........................................................ 49

7 NUMERICAL SIMULATIONS .............................................................................. 59

The Density Perturbation Model ............................................................................ 60
The Model of Local Nonuniformity Noise ............................................................ 60
The Model of Counting Statistics ......................................................................... 61
Error Bound and Confidence Level ....................................................................... 62
The Naming System for the Numerical Experiment ............................................... 63
Homogeneous and Noise-Free Condition .............................................................. 64
Comparisons with the Segmented Gamma Scanning Method ................................ 66
Heterogeneous and Noise-Free Conditions ............................................................ 68
Heterogeneous and Noise-Perturbed Conditions ................................................... 72
Simulations of Biased Source Distribution Cases .................................................. 80
Simulations Using the Base Density Kernel Table ................................................. 83
Simulations Using Eight Detector Positions .......................................................... 88
Simulations Using Different Base Densities .......................................................... 96
Total Activity Estimation for the Full Length of the Waste Barrel ....................... 96

8 C O N C L U SIO N S ......................................................................................................... 102

APPENDICES

A DERIVATION OF CONJUGATE GRADIENT SOLUTION AS A LEAST
SQUARED SOLUTION OF A QUADRATIC FUNCTION .................................... 108

B DERIVATION OF SCALAR SEARCH FACTOR FOR CONJUGATE
GRAD IEN T A LG ORITH M ...................................................................................... 109

C DERIVATION OF ORTHOGONALITY FACTOR FOR CONJUGATE
GRA DIEN T A LG O RITH M ...................................................................................... 110

D PRINTOUT OF THE KERNEL RESPONSE TABLE OF TWO ENERGIES
AND FOUR DETECTOR POSITIONS FOR HOMOGENEOUS BASE
D EN SIT Y C A SE ....................................................................................................... 112

LIST O F R EFERE N C E S ................................................................................................. 116

BIO G RA PH ICA L SK ETCH ........................................................................................... 118













LIST OF TABLES


Table page

4-1. Results of the convergence test for different zero criteria of CGNN by using
10,000 random source distribution cases of homogeneous density .............. 23

5-1. Results of the convergence test for different iteration number of MLEM by
using 10,000 random source distribution cases of homogeneous density ......... 27

6-1. Characteristics of check sources and related efficiency calibrations ................ 33

6-2. Activity estimation errors for 18 homogeneous density experiments by using
multiple energy peaks in the CGNN and MLEM reconstruction algorithms .... 43

6-3. Summary of activity estimation errors for 18 homogeneous density experiments
by using multiple energy peaks in the CGNN and MLEM algorithms ...... 44

6-4. Location of different material channels inside the barrel for the study of Type A
and Type B density heterogeneity ................................................................. 46

6-5. Activity estimation errors for the heterogeneous density experiments by using
multiple energy peaks in the CGNN and MLEM algorithms ....................... 50

6-6. Summary of activity estimation errors for the heterogeneous density experiments
by using multiple energy peaks in the CGNN and MLEM algorithms ...... 51

6-7. Summary of activity prediction for 36 experiments .......................................... 53

6-8. Comparisons of ideal and observed detector responses for experiment case H1
and case H 3 .................................................................................................. 57

7-1. Confidence levels for total activity estimation by different optimization
algorithms under the homogeneous and noise-free condition ...................... 65

7-2. Comparisons of confidence levels between the SGS and the 2-energy CGNN
and MLEM reconstruction algorithms under the homogeneous and
noise-free condition ..................................................................................... 69









7-3. Confidence levels for total activity estimation by different optimization
algorithms under the heterogeneous and noise-free condition ...................... 71

7-4. Confidence levels for total activity estimation by different optimization
algorithms under the 20% heterogeneity and noise-perturbed condition .......... 74

7-5. Confidence levels for total activity estimation by different optimization
algorithms under the 50% heterogeneity and noise-perturbed condition ......... 78

7-6. Comparison of the confidence levels of 10,000 random source distribution cases
and 100 biased source distribution cases ..................................................... 82

7-7. Comparisons of confidence levels for activity reconstruction using perturbed
kernel table and base density kernel table under 20% heterogeneity
condition ..................................................................................................... . 86

7-8. Comparisons of confidence levels for activity reconstruction using perturbed
kernel table and base density kernel table under 50% heterogeneity
condition ..................................................................................................... . 87

7-9. Confidence levels for activity estimation using different reconstruction
algorithms and eight detector positions ....................................................... 89

7-10. Confidence levels for activity estimation using the base density kernel table
and eight detector positions ......................................................................... 94

7-11. Confidence levels of activity estimation for the full length of a 208 e waste
barrel under 50% heterogeneity and different noise-perturbed conditions ...... 101














LIST OF FIGURES


Figure page

1-1. A generic configuration of a distributed source S in a self-absorbing medium of
volume V measured by several external detectors .......................................... 3

3-1. Configuration of a four-detector positions measuring system in a vertical
segment of a 208 e nuclear waste barrel ....................................................... 12

3-2. ir'92 gam m a spectrum .......................................................................................... 15

6-1. Experimental setup for the 208 e waste barrel characterization ......................... 30

6-2. The acrylic-glass matrix container inside the waste barrel ............................... 31

6-3. The efficiency calibration curve for HPGe detector .......................................... 34

6-4. The ROI for a 316 keV energy peak of the Ir'92 gamma spectrum .................... 36

6-5. The ROI for the 468 keV and 588 keV energy peaks of the Ir192 spectrum ..... 38

6-6. Six materials of different density in the insertion channels and an experimental
heterogeneous configuration inside the waste barrel ................................... 40

6-7. Configurations of Ir'92 source distribution for 18 experiments under the
hom ogeneous density condition ................................................................... 42

6-8. Density distribution and configurations of Ir192 location for 9 experiments
under the Type A heterogeneity condition ................................................... 47

6-9. Density distribution and configurations of Ir'92 location for 9 experiments
under the Type B heterogeneity condition ................................................... 48

7-1. Distributions of relative prediction error for the 2-energy CGNN and MLEM
algorithms under the homogeneous and noise-free condition ..................... 67

7-2. Distributions of relative prediction error for the SGS method using 4 and 64
detector positions under the homogeneous and noise-free condition ........... 70









7-3. Distributions of relative prediction error for 2-energy CGNN and MLEM
methods under the heterogeneous and noise-free condition ......................... 73

7-4. Distributions of relative prediction error for 2-energy CGNN and MLEM
methods under the 20% heterogeneity and different noise-perturbed
condition ..................................................................................................... . 76

7-5. Distributions of relative prediction error for 2-energy CGNN and MLEM
methods under the 50% heterogeneity and different noise-perturbed
condition ..................................................................................................... . 79

7-6. Representative configurations for six types of biased source distribution ...... 81

7-7. Distributions of relative prediction error for 2-energy CGNN method using 100
biased source distribution cases under heterogeneity and noise-perturbed
conditions ................................................................................................... . 84

7-8. Distributions of relative prediction error for 2-energy CGNN and MLEM
methods (8 detector positions) under the 20% heterogeneity and different
noise-perturbed conditions ............................................................................ 91

7-9. Distributions of relative prediction error for 2-energy CGNN and MLEM
methods (8 detector positions) under the 50% heterogeneity and different
noise-perturbed conditions ............................................................................ 92

7-10. Confidence levels of activity estimations by using perturbed kernel table and
base density kernel table in the 2-energy CGNN method under different base
density conditions ......................................................................................... 97

7-11. Six relative error distribution functions used in the total activity estimation for
the full length of the w aste barrel ..................................................................... 100













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

DEVELOPMENT OF OPTIMIZED SOLUTION METHODS FOR INVERSE
RADIATION TRANSPORT PROBLEMS INVOLVING THE CHARACTERIZATION
OF DISTRIBUTED RADIONUCLIDES IN THE LARGE WASTE CONTAINER

By

Chin-Jen Chang

May, 1997

Chairman: Samim Anghaie
Major Department: Nuclear and Radiological Engineering

This research presents a new mathematical approach for finding an optimum

solution to a singular inverse radiation transport problem involving the nondestructive

assay of radionuclide inventory in a nuclear waste drum. The method introduced is an

optimization scheme based on performing a large number of numerical simulations which

account for the counting statistics, the nonuniformity of source distribution, and the

heterogeneous density of the self-absorbing medium inside the waste drum. The

simulation model uses forward projection and backward reconstruction algorithms. The

forward projection algorithm uses randomly selected source distribution and a first flight

kernel method to calculate external detector responses. The backward reconstruction

algorithm uses the conjugate gradient with nonnegative constraint or maximum

likelihood expectation maximum method to reconstruct the source distribution based on








calculated detector responses. Total source activity is determined by summing the

reconstructed activity of each computational grid. By conducting 10,000 numerical

simulations, the error bound and the associated confidence level for the prediction of total

source activity is determined. The superiority of using multiple energy peaks in

backward reconstruction algorithms is analyzed.

The accuracy and reliability of the simulation model are verified by performing a

series of experiments in a 208 t waste barrel. A number of Ir'92 seeds are used to simulate

unknown distributions of the source in the barrel. Density heterogeneity is simulated by

using different materials distributed in 37 egg-crate type compartments simulating a

vertical segment of the barrel. Four orthogonal detector positions are used to measure the

emerging radiation field from the distributed source. Results of the experiments are in

full agreement with the estimated error and the confidence level which are predicted by

the simulation model.

This research presents a way of defining the error bound and its associated

confidence level, which is inadvertently ignored by the traditional waste drum

characterization method. The concept of using multiple energy peaks with advanced

backward reconstruction algorithms also provides a more accurate prediction ability than

does the traditional method.














CHAPTER 1
INTRODUCTION

The inverse radiation transport problem discussed in this dissertation is to

characterize radiation sources inside a black box by using remote sensing methods. This

problem exists in such fields as in vivo radionuclide assays, remote sensing for geology

or astronomy, and chaos studies for heat conduction. In nuclear engineering, the most

prominent task related to the inverse radiation transport problem is the characterization of

radioactive sources inside a nuclear waste barrel. Before shipping the nuclear waste

barrel to disposal sites, barrel contents need to be determined as high or low level waste

based on the types and the amount of radionuclides inside. The cost of processing high

level waste is usually much greater than that of low level waste. Therefore, accurate

characterization of the waste barrels is not only essential environmentally but also

important economically for the nuclear industry.

Most nuclear waste, including commercial power plant waste and defense waste,

is stored in cylindrical steel drums (208 e waste barrel) about 57 cm in diameter and 88

cm in height.' Millions of waste barrels are waiting to be characterized in the United

States. Therefore, gamma ray detection time, speed of analysis, and accuracy of

evaluation are very important factors in satisfying the regulatory requirements in a timely

manner. For most applications, using far field measurement and assuming the drum as a

line or a point source are not practical because of the long measurement time associated








with the long measurement distance. For a near field measurement, the estimation error

for the total source activity is usually very large because the detection rate is very

sensitive to the source distribution and the density of the self-absorbing medium. Figure

1-1 shows a generic configuration of a distributed source S in a self-absorbing medium of

volume V and density p. A series of measurements at different detector positions provide

the needed information on the source distribution and its total activity. Using passive

gamma scanning method,2 the total response of a external detector at position i Di, to

the uncollided radiation from an isotropic source is

e-(P1p)'p'ti
D, = Gi .SV. dV (1-1)
V 47 r r i

where Gi is a local constant which includes all geometrical factors and the detector

efficiency with respect to detector position i; Sv is the volumetric source strength

(y's / cm3 s) ; 1u is the linear average attenuation coefficient; ti is the thickness measured

from Sv to the medium boundary; ri is the distance of Sv from the detector at position i;

and dV is the volume element. Total source strength, S,,, (y's / s), is calculated by

integrating Sv over the container volume.

S,, = fSv dV (1-2)
V

The inverse radiation transport problem which is considered in this research deals with

the solution of a series of integral equations similar to Equation 1-1 substituted into

Equation 1-2 for the calculation of total source strength Stot. In matrix form, the problem

can be represented as follows:


D=K.S


(1-3)






























l-J D.





x

Figure 1-1. A generic configuration of a distributed source S in a self-absorbing medium
of volume V measured by several external detectors


D I


D 2








where D is the vector representing the external detector responses (counting rate), K is the

matrix representing the point response kernel and all the geometrical and physical factors

involved in the detection process, and S is the vector representing the source distribution

in the drum. In most cases, the distribution of the radioactive source inside a nuclear

waste container is nonuniform. For a near field measurement, estimation of the total

activity is sensitive to (a) spatial resolution (i.e., locations) of the source, (b)

heterogeneity of the absorbing medium, (c) uncertainties of the detecting system, and (d)

the solution scheme used to determine S. Theoretically, if the number of detector

responses is equal to the total number of points over which the source is distributed,

Equation 1-3 is well posed and an exact solution can be obtained. However, due to the

variability of the source distribution and the heterogeneity of the self -absorbing medium,

Equation 1-3 is either highly singular or the response kernel K is so ill-conditioned that a

unique solution can not be found. Uncertainties involved in the detection process and

choice of solution method can also add to the measurement error. Previous attempts to

solve this problem have been limited to cases of considering a single point source, fixed

source distribution patterns, or a uniformly distributed source.2'-1 For other cases

involving a wider variety of source distributions and density heterogeneity, a satisfactory

solution to this problem has not yet been found.

This research develops a new mathematical approach for finding an optimum

solution to a singular inverse radiation transport problem involving the nondestructive

assay of radionuclide inventory in a large nuclear waste container. The method introduced

is an optimization scheme based on performing a large number of numerical simulations

which account for the counting statistics, the nonuniformity of source distribution and the








heterogeneous density of the self-absorbing medium inside the nuclear waste container.

The simulation model uses forward projection and backward reconstruction algorithms.

The container volume is first modeled to have several computational grids for a vertical

segment. Then, the forward projection algorithm uses randomly selected source

distribution for these computational grids and a first flight kernel method to calculate the

external detector responses. The backward reconstruction algorithm uses numerical

optimization methods to reconstruct source distribution based on calculated detector

responses.213 Total source activity is determined by summing the reconstructed activity

of each computational grid. By conducting 10,000 numerical simulations, the error

bound and the associated confidence level for the prediction of total source activity is

obtained. A relative error distribution function based on these 10,000 simulation results

is also determined. The accuracy and reliability of the simulation model are verified by

performing a series of experiments in a 208 t waste barrel. A number of Ir"' seeds are

used to simulate unknown distributions of the source in the barrel. Density heterogeneity

is simulated by using different materials distributed in 37 egg-crate type compartments

simulating a vertical segment of the barrel. Four orthogonal detector positions are used to

measure the emerging radiation field from the distributed source.

Chapter 2 describes published literature related to the nondestructive assay of

radionuclide inventory in a nuclear waste drum. Chapter 3 explains the modeling of the

208 e waste barrel and the idea of using multiple gamma energy peaks for the

optimization study in this research. Chapter 4 describes the conjugate gradient with







nonnegative constraint (CGNN) reconstruction method.2 Chapter 5 describes the

maximum likelihood expectation maximum (MLEM) reconstruction method.3

Chapter 6 details experimental setup, procedures of data collection, and results of

the experiment. Chapter 7 describes concepts and procedures for performing the

numerical simulations. Results of the activity estimation based on a wide spectrum of

simulated experimental conditions are presented. Effects of the density heterogeneity and

locally nonuniform source distribution on the simulation results are discussed. A

statistical sampling procedure to estimate total activity for the full length of the waste

barrel is demonstrated. Chapter 8 presents the conclusions of this research. The benefits

and the potential utilization of the results of this research are also presented.













CHAPTER 2
THE NONDESTRUCTIVE ASSAY METHODS FOR CHARACTERIZING
RADIONUCLIDE INVENTORY IN A NUCLEAR WASTE CONTAINER

Segmented Gamma Scanning Method


The segmented gamma scanning (SGS) method was developed in the early

1970s.- It was designed to assay the special nuclear waste material contained in low

density scrap and waste by measuring gamma emission rates. This method is widely used

by the United States Department of Energy complex and industry. The basic assumption

underlying this technique is that materials and radionuclides are distributed

homogeneously and uniformly within the waste barrel. Barrels are usually divided into

several vertical segments. Gamma measurement is done one at a time for each vertical

segment. During measurement, the waste barrel is rotated stepwisely or continuously and

the uncollided gamma emission rate is measured by a collimated germanium detector

located outside the barrel. In a continuous scan, the rotating sample moves past the

collimator at a constant speed. In a stepwise scan, the sample is positioned vertically,

counted, repositioned, counted again, and so on. The purpose of drum rotation is to

compensate for activity distribution asymmetries such that the uniform and homogeneous

assumption is maintained. Some installations use an external gamma source

simultaneously to measure the average density and correct for the nonhomogeneous

effect. In some applications, the measurement is performed at a very large distance








(mostly 150 cm to 300 cm away from the drum) to reduce the sensitivity to the source

distributions.4- Such applications are usually limited to nuclear waste drums with much

higher activity sources. In some cases involving small size waste barrels or low activity

levels, detectors are positioned at close distance (within 30 cm) and homogeneous source

distribution is assumed. For the estimation of total activity in these measurements, the

SGS is not capable of making an accurate assay and no confidence level is reported for a

specific high accuracy requirement.

Tomographic Gamma Scanning Method


Since 1990, many efforts have been devoted to the tomographic gamma scanning

(TGS) method, and at least, two groups of researchers have published preliminary results.

One was a Japanese group of researchers from the Hitachi Ltd. and the Toshiba

Corporation.67 The other one was a North American group of researchers from the

Lawrence Livermore National Laboratory and the Los Alamos National Laboratory.8-9

The TGS is essentially an extension of the SGS method. It emphasizes the fact that the

matrix inside the drum is heterogeneous in nature, and the density distribution of the

matrix should be determined by a transmission tomographic method in addition to the

emission spectrum. A representative TGS design is a prototype TGS system built at Los

Alamos.9 It uses a high energy gamma transmission source moved simultaneously with a

collimated HPGe detector. For the horizontal movement, the waste drum is rotated

stepwisely or continuously. A total of 9 or 10 vertical segment scannings are made.

Detailed information on density distribution can be obtained and is subsequently fed into

the traditional SGS method to correct for heterogeneous matrix effects.







Compared to the SGS method, TGS takes more scanning and analysis time in

order to get density information. For nearly homogeneous waste, sparsely distributed

solid waste, or low density waste, TGS has no significant advantage over SGS. However,

it does have a superior advantage if matrix density is heterogeneous and is as high as

2.0 g/cm3. High density waste usually consists of glassified or cemented waste forms that

are nearly homogeneous in density. Therefore, TGS does not offer much improvement

over SGS, unless the scanning time can be shortened or the backward reconstruction

algorithm can be improved.

Passive and Active Neutron Detection Method


The neutron detection method is specific to assays of waste drums containing

actinides, especially for the measurement of plutonium content. Three neutron assay

methods are reported: (a) the passive neutron assay, (b) the active neutron assay using an

external neutron source, and (c) the active neutron assay using an external gamma

source..0-11 The passive neutron assay method utilizes the spontaneous fission behavior of

some Pu isotopes (Pu238, Pu240, and Pu242) and detects these spontaneous fission neutrons

by He3 neutron detectors. The active neutron assay method needs an external source to

trigger the induced fission of the actinides. External sources can be an electron linear

accelerator that generates high energy neutron pulses or high energy gamma rays, or it

can be a Cf52 spontaneous fission neutron source. When using external neutron pulses,

these fast neutrons are thermalized as they pass through a moderator matrix surrounding

the waste barrel and cause fissile actinides to produce fast fission neutrons. When using

external gamma rays, high energy photons (usually as high as 7 MeV or above) cause








production of fast neutrons from all actinides by photo-fission and photo-neutron

reactions.

Neutron detection methods usually need a coincidence counting system and time

of flight detection method to distinguish between time-correlated fission neutrons emitted

by the fission process and random neutrons from the (ct,n) reaction and the background.

The passive neutron assay method often requires a long measurement time (typically,

several hours to one day for measuring low level waste) because of the low spontaneous

fission probability. The measurement time can be shortened by using the active neutron

assay, but the accurate determination of thermal neutron flux within the moderator matrix

causes another measurement difficulty. Determination of the inventory of actinides

within the waste drum is a challenge due to the nature of nonhomogeneous distribution of

actinides. This challenge exists whether using passive or active neutron assays and is the

same challenge faced in the gamma assay method. Therefore, no significant

announcement of accurate estimation of actinide activity inside a nuclear waste drum has

been found for the neutron assay method.













CHAPTER 3
THE MULTIPLE ENERGY PEAKS GAMMA DETECTION SYSTEM

The Model of a Gamma Scanning System Using Four-Detector Positions


This research presents a calculation model that uses four orthogonal detector

positions to measure the emerging gamma flux from the distributed source in a vertical

segment of a 208 e waste barrel. Figure 3-1 shows the configuration of a vertical segment

of the waste barrel. Each vertical segment is modeled to have 37 computational grids.

Each grid is assigned a number as indicated in Figure 3-1. The volume of each cubic grid

is 8x8x8 cm3. The barrel is a 0.16 cm thickness steel drum and has 56.84 cm inner

diameter. The gamma signal is measured at a distance of 42.85 cm from the center of the

barrel. The calculation model assumes that the HPGe detector (7.5 cm in diameter) is

well collimated and detects only those gamma rays coming from the same vertical

segment.

The Concept of Using Multiple Gamma Energy Peaks


Using the grid system as described above, the matrix element KiJ in Equation 1-3

corresponds to the counting rate of a specific energy peak contributed by thejth voxel to

the detector at the ith position and is given by

Kij = exp(- Z (0---)pq tq).cos(O). "-.,8 (3-1)
qeRay(ij) P0 'O
where tq is the path length of the ray(ij) in the qth voxel, (u/p) is the linear mass





12








, I, I ,, 1 1 II ,I,,I ,,,,I,,,,I,,,,I,,,,,11111


40


30


20


10-


0


-10-


-20-


-30-


-40-


-40 -30 -20 -10 0 10 20 30
Relative Dimension (cm)


lll0l


Figure 3- 1. Configuration of a four-detector positions measuring system in a vertical
segment of a 208 1 nuclear waste barrel


Detector 1







30 31 32 .33 34


23 24 25 26 27 28 29

16 17 18 9 .20 21 .22

Detector 4 4 10 11 12 13 14 15 Detector 2










Detector 3


- - - - - - - - - - - - -








attenuation coefficient of the gamma energy considered, Pq is the density in the qth

voxel, Q j is the solid angle extended from a source point in thejth voxel to the detector

at the ith position, c is the detector efficiency at a calibration position with solid angle

QO, and f8is the branching ratio of the emitted gamma ray. The cos() in Equation 3-1

accounts for correction of the inclination if the gamma ray is not normal to the detector

surface. In real implementation, D is measured, K is precalculated based on a known

geometric model and detector efficiency, and S can be solved by using a numerical

optimization algorithm such as CGNN or MLEM. Total activity can be obtained by

summing over all the elements of S. If few detector positions are used, this is an

underdetermined linear algebra problem. If single peak energy is used to analyze this

problem, it becomes a 4x37 linear system for which a reasonable optimized solution is

difficult to obtain. Whenever multiple energy peaks are possible for analyzing a given

radionuclide, it becomes a 4mx37 linear algebra system (m being the number of identified

energy peaks). If the number of detectors is increased to eight, this system becomes

8mx37. For the four-detector positions and two energy peaks detection system, Equation

1-3 in its detailed form is

D ~ 1) re -( I ) K ~( IN
K K K37' S1

1,2 2,2 3,2 37,2
D 2)I (2) K2-2) K(2) K32) $
2 K1,2 22 32 2 S4 (3 2)



S36
D42) K(') K(2) K(2) K(2) ) 37
1, ,4 3K "37,4 37








where D(") is the mth energy peak response at ith detector position, K() is the point

kernel response contributed by thejth voxel to the ith detector position, and Sj is the

volumetric source in thejth voxel.

This research uses Ir'92 radionuclide to simulate the distributed source. Ir'9' has

several prominent and well-separated gamma peaks ranging from 296 keV to 884 keV

(Figure 3-2), and is appropriate for simulating the gamma spectrum of plutonium which

originates from reprocessing plants and is of great interest for nuclear waste management.

Three energy peaks of Ir'92 are selected for this research: 316.6 keV (/6= 0.8285), 468.1

keV (f6= 0.4810), and 588.6 keV (68= 0.0457). This research illustrates that from a two

dimensional analysis, using a model with four-detector positions and multiple energy

peaks to evaluate the activity of randomly distributed sources inside the waste drum,

dramatic improvement of the accuracy can be obtained. In the numerical simulation,

source strengths are randomly generated and are assigned to a randomly selected grid. If

a source is assigned to one of the grids, it is assumed that the source strength is uniformly

distributed within that grid.

Computer Modeling and Verification Procedure


The following steps are taken for the computer modeling and its verification:

1. A computer program based on the first flight kernel method is written. For

every measurement or numerical simulation, a kernel table comprised of the values of the

elements of matrix K (shown in Equation 3-2) is generated. Element Ki, I is a point

kernel response which is an averaged value of the counting rate (for the mth energy peak










































Figure 3-2. Ir192 gamma spectrum


Counts


10000

1000

100


250


200
10000

1000

100

10


450 500 550 600 650 700
Gamma Energy (keV)








at the ith detector position) contributed by the Monte Carlo sampling of 5,000 point

sources within gridj. The computer program can generate a kernel table based on the

consideration of heterogeneous density distribution.

2. Two numerical optimization algorithms, CGNN and MLEM, are built into the

model to solve the linear algebra system. The convergence criteria are selected based on

the study of the optimum result from 10,000 random source distribution cases.

3. The computer model is verified by various physical experiments described in

Chapter 6. These experiments involve the cases of nonuniform distribution of Ir'192

sources under heterogeneous density conditions.

4. Numerical experiments, described in detail in Chapter 7, are performed to

obtain the error bound and the associated confidence level of this model under the

influences of density perturbation and signal noise perturbation. First, the homogeneous

condition with density 1.0 g/cm3 (called base density) is analyzed for a total of 10,000

cases. For each case, total source activity is randomly distributed in 37 grids and each

detector response for each energy peak is calculated. The Monte Carlo procedure is used

to simulate random distribution of the source within the barrel. A random number

generator is used to randomly choose one of 37 grids. A randomly chosen fraction of

total source activity is assigned to the randomly selected grid. Then, a second grid is

chosen and a randomly selected fraction of the remaining source activity is assigned to

this grid. The process continues until all remaining source activity is reduced to a very

small value (10A of original activity). The same calculation is done by considering the

effect of density heterogeneity. Density perturbation is simulated by selecting a

perturbation range which is some fraction (positive or negative) deviated from the base








density. Again, the Monte Carlo procedure is used, grid by grid, to randomly choose any

value within the range and assign this perturbed density for the grid. The detector

responses thus calculated are called the noise-free signal. The noise perturbation effect is

analyzed by considering the possible uncertainties of detector signal in a real

measurement, which may be caused by detector calibration uncertainties, counting

statistics, and the locally nonuniform distribution of the source within a computational

grid. The ideal signal is perturbed, first by a selected fraction and then by a statistical

sampling process. Several noise perturbation levels are tested and analyzed.













CHAPTER 4
CONJUGATE GRADIENT WITH NON-NEGATIVE CONSTRAINT METHOD

The conjugate gradient (CG) algorithm is an iterative algebraic solution scheme

for solving linear systems of equations and represents an important computational

innovation of the early 1950s. This method came into wide use only in the mid-1970s.4

Due to the rapid improvement of computer technology, iterative algebra algorithms,

including the conjugate gradient method and expectation maximum (EM, described in the

next chapter) method, have become more and more important in the fields of image

reconstruction and remote sensing which usually require a lot of matrix operations and

computer memory. An application similar to nuclear waste drum characterization is the

single photon emission computed tomography (SPECT) for medical imaging in clinical

nuclear medicine. There have been some discussions of using CG in SPECT,..-17 but

wide applications have not been seen. SPECT reconstructs images of body organs which

are much smaller than the typical nuclear waste drum. The total activity of nuclear

medicine injected into the human body is always known and is usually much larger than

the total activity in waste drum characterization, which is usually unknown. Therefore,

applying CG in the nuclear waste drum characterization has different considerations from

that in the SPECT.

From the optimization theorem, the least squared solution S of Equation 1-3 can

be found by minimizing the following squared error (see the derivation in Appendix A):









e= D-KS =(D-KS)t(D-K9) (4-1)

where 2 means the L2 norm, ( )t means the transpose of the matrix and S means the

least squared solution of S. In the conjugate gradient method, solution S is found by

using an iterative algorithm:
A
S'+1 = Sn + and" (4-2)

where S is the estimate of S at the nth iteration, a,, is a scalar search factor and d,, is a

search vector which is required to be orthogonal to the modified residual vector r,, by the

following relation:

r,ld,, = 0 (4-3)

with

r=K' (D KSJ) (4-4)


Equation 4-3 means that to approach the minimum point of e the search vector at a

certain iteration step must be orthogonal to the residual vector at the next step. From the

relation

S'1+1 = Sn + andn KS,+ KSn + a,,Kd,,


> K' (KSn+l D) = Kt (KS, D) + a KKd,

and from equation (6), we get r+, = r, a,,KtKdn (4-5)

To minimize e, we need to take the partial derivative of Equation 4-1 with respect

to an and set the result at zero. This leads to (see the derivation in Appendix B):








^9DK 2dt
K11) _c(D- K (S + a,,d) -Od(46
,a 0 d K 'K d ,,

The search vector dn of CG must satisfy the general Gram-Schmidt conjugate properties

and is related to the residual vector r through a recursion formula.8 The conjugate

property is

d,Kt Kd=O for n# m (4-7)

and the recursion relation (see the derivation in Appendix C) is

dn = rn -3,,_ld._- or d,, = r+, 6,d (4-8)

where /6. is the orthogonality factor.

Taking the inner product of Equation 4-8 with K' Kdn_1 and applying the property

of Equation 4-7, we have

r,,Kt Kd,,- r K'
d_ = K' Kd or Kd,-9)
dK' Kd (4n

The algorithm of CG without constraint is:

Step 1: Set an initial guess of S, and calculate d, = r K' (D KS )

Step 2: Set the loop index starting from n=l, calculate an based on

Equation 4-6, calculate S,+ based on Equation 4-2, calculate r.1

based on Equation 4-5, calculate /3n based on Equation 4-9, and

then calculate d,,,, based on Equation 4-8. The loop ends when a

specified criterion of residual difference is reached or a specified

iteration number is reached.








The algorithm of CGNN is more complicated than the above algorithm. Suppose

there is a hypersurface H which is the boundary of the CGNN solution S, which

minimizes the quadratic function e of Equation 4-1 and has at least m components which

are zero. Then, at S, 1, any slight increase of component S,,+, at boundary H will

increase the e value. Therefore, if I is a set of indices i < m such that S = 0, we have

the relations for the optimum point S,

_0 for /inI and 0 otherwise


Since the residual is defined as the negative value of the gradient (see Equation 4-4 and

Appendix A), the criteria for terminating the search and finding a CGNN solution S,,+1 is

equivalent to have

r, + 0 for i in I and r1 = 0 otherwise (4-10)

The search method is conducted by a restart algorithm. Within any iteration, if there

exists a search vector dn that will produce negative components of S,+, the most

negative component is determined and the scale of the search length is calculated to force

this component to be zero. Then, the I set is redefined and the criteria in Equation 4-10

are checked. If the criteria in Equation 4-10 are not matched, S,, is reset to be S, and

the whole algorithm is restarted.

The most important issue for CGNN algorithm is the termination point. The zero

criterion in Equation 4-10 is customarily set at a very small value which is near to the

precision limit of the computer digit. However, because the source activity level covers a








very wide range, a fixed value for zero may sometimes stall the algorithm and may also

not be an optimal converged point. Zero criterion in Equation 4-10 for this study is

dynamically defined to be a small fraction of the average value of the initial guess of r

which is defined in Equation 4-4. That is

37
ZrPm

ZERO = = (4-11)
37

where r,' is the mth component of initial guess of r and is varied case by case with respect

to the measured or calculated detector responses. Table 4-1 shows the convergence test

on a Pentium/120 computer by studying 10,000 random source distribution cases under

homogeneous density condition. A total activity of 106 Bq is input for every case.

Detector responses are calculated using forward projection. Then, the source distribution

is reconstructed by CGNN algorithm using four detector positions and two energy peaks.

The reconstructed total activity is then compared with the input total activity to determine

the percent error for each case. Five categories of percent error are defined as shown in

Table 4-1. The cases which fall into a certain category are counted. The value for 6 is

varied from 10-' to 101. It is evident that the optimal value of e among them is 10.6 which

predicts 9,901 out of 10,000 cases with 10% accuracy.












Table 4-1. Results of the convergence test for different zero criteria of CGNN by using
10,000 random source distribution cases of homogeneous density


s Error* <50% <30% <25% <20% <10%


10"1 4,573 2,957 2,455 2,057 931
10"2 8,356 4,243 3,418 2,555 1,207
103 10,000 9,929 9,708 9,213 6,674
10-4 10,000 10,000 10,000 9,993 9,341
10-5 10,000 10,000 10,000 10,000 9,783
10-6 10,000 10,000 10,000 10,000 9,901
10-7 10,000 10,000 10,000 10,000 9,823
10- 10,000 10,000 10,000 10,000 9,824
10-9 10,000 10,000 10,000 10,000 9,823
-10 10,000 10,000 10,000 10,000 9,820
* Error means the percent estimation error of the total activity and
the values in the 2nd to 5th column are the number of cases which
fall into that error category.













CHAPTER 5
MAXIMUM LIKELIHOOD EXPECTATION MAXIMUM METHOD

Maximum likelihood (ML) method is an approach to maximize the likelihood

ftmction, which is the probability that the source strength S produces the measured

detector responses. As is CG, ML is an iterative algebraic solution scheme for solving

linear systems of equations. It was introduced to emission tomography by Rockmore and

Macovski in 1976."9 Lange and Carson incorporated the expectation maximum (EM)

method in ML to compute maximum likelihood estimates.3 MLEM then became a

popular application on SPECT. It assumes that both D and S in Equation 1-3 are Poisson

distributions. Thus, the expectation value of the ith detector response for a waste barrel

with 37 computational grids is

37 37
1 Y Usj I Yu(5-1)
j=j j=

where Sj is the source strength at voxelj and Kij is the probability that a photon leaving

voxelj is counted by the ith detector. X,1 is the expectation value of Xij, which is the

statistical number of photons that are emitted from voxelj and contributes to the counts

for detector response i. In the EM algorithm, a conditional expectation function is

defined as

E(lnf (X,S) D,S(-)) (5-2)

where In denotes the natural logarithm and S(") is the estimation of S at the nth iteration.








This equation defines an expectation function of a logarithm likelihood function f(X,S)

under the condition of observed detector response D and source distribution S at the nth

iteration step. f(X, S) has the Poisson distribution form of


f(XS) =HH (5-3)
i=1 j=1 XO !

where m is the total number of detector response. Its logarithm is

ln f (X,S) = ly (-X, + X, ln-Y, lnX !)
i j

= + X, ln(KSj) lnX,!) (5-4)
i j

The conditional expectation of X, in Equation 5-4 with respective to Di and S(") is

K S(n D
U= E(X D (5-5)
N K ,S (n)

Because the third term in Equation 5-4 has no dependence on the iteration of S, the

conditional expectation of Equation 5-4 becomes

E(lnf (X,S) D,S (') =-I[-K Sj + N, ln(KySj)1+C (5-6)
j J

where C is the term not dependent on the iteration of S. To find the maximum of this

expectation, we need

6 E(lnf(X,S)ID,S("))
=O=-XKY +EN.- (5-7)


The partial derivative of the above equation is








92 E(Inf(X,S) D,S(n)) 1
SNU- (5-8)



Equation 5-8 yields a negative value and this assures that E(lnf(X,S)D,S(n ) is a

concave function of S and has a maximum at

IN S(n) K.Di = 1...,m
S(n+l) $Y
Sj~~ ~ I u ZKu-ZK,,S ") j,l1 = 1,...,3 7(59
i i 1

This iteration algorithm will always assure nonnegative results if the initial guess of the

source is positive.

To determine the optimal termination point for the iterative algorithm in Equation

5-9, the same procedure as described in the last paragraph of Chapter 4 is adopted. The

termination point is set to be the optimal number of iterations which will give the most

satisfactory results from the process of studying 10,000 random source distribution cases.

Table 5-1 shows the results of the convergence test by varying the iteration number from

100 to 20,000. The optimal prediction among them is 10,000 iterations. MLEM predicts

9,350 out of 10,000 cases with 10% accuracy. The computation time of MLEM using

10,000 iterations is about 10 times longer than that of CGNN using zero criterion 10-6.











5-1. Results of the convergence test for different iteration number of
10,000 random source distribution cases of homogeneous density


It.# Error* <50% <30% <25% <20% <10%


100 9,906 6,918 5,832 4,572 2,821
500 10,000 9,963 9,872 9,582 6,547
1,000 10,000 10,000 9,991 9,831 7,800
2,000 10,000 10,000 10,000 9,906 8,573
5,000 10,000 10,000 10,000 9,942 9,122
10,000 10,000 10,000 10,000 9,943 9,350
20,000 10,000 10,000 10,000 9,943 9,163
* Error means the percent estimation error of the total activity and
the values in the 2nd to 5th column are the number of cases which
fall into that error category.


Table
using


MLEM by














CHAPTER 6
EXPERIMENTS AND MODEL VERIFICATIONS

A series of experiments were performed to verify the computer model and to

evaluate the properties of the concept of using multiple energy peaks in numerical

reconstruction algorithms. These experiments were designed to simulate the random

source distributions and the heterogeneous density conditions which are usually met in

real waste barrel measurements. To simulate the distributed source, several pieces of Ir' 92

sources were prepared. Each piece of Ir"' source was 3 mm long and 0.5 mm in

diameter and had activity of 0.0726 5% mCi (2.6869x 106 5% Bq) on October 16,

1996. Three Regions of Interest (ROI) in the Ir'92 gamma spectrum were defined for the

peak area calculation. These ROIs were centered at 316.6 keV (fl= 0.8285), 468.1 keV

(93= 0.4810), and 588.6 keV (fl= 0.0457) respectively. For reconstruction using one

energy peak, the 316.6 keV peak was used. For reconstruction using two energy peaks,

the 316.6 keV and 588.6 keV peaks were used; and for reconstruction using three energy

peaks, all three peaks were used.

Experimental Setup


Figure 6-1 is a photograph of the experimental setup including a 208 e waste

barrel, lead collimators, a HPGe gamma detector, electronic modules, and a data access

computer. The HPGe gamma detector (EG&G ORTEC model GEM-10195-P) has a

transistor-reset type preamplifier and has 10% relative efficiency and 1.95 keV resolution








measured at 1.33 MeV. The electronic modules include an amplifier (EG&G ORTEC

model 572), a bias voltage supply (EG&G ORTEC model 459), and a multichannel

analyzer (MCA) interface card (EG&G ORTEC 920 MCB).

The waste barrel had a steel wall 0.16 cm thick. Its inner diameter was measured

56.84 cm and the height was 88 cm. The waste barrel was seated on top of a rotation

device which was capable of 360 degree manual rotation. A cylindrical wooden stand

25.4 cm high and with about the same inner diameter as the barrel was built and placed

inside and at the bottom of the barrel. To simulate the 37 computational grids for a

vertical segment of the barrel, a matrix container (see Figure 6-2) with 37 egg-crate type

compartments made of acrylic-glass (1.5 mm in thickness) was built and cut to fit the

round shape of the barrel at the periphery. This acrylic-glass matrix container was seated

on top of the wooden stand. Each of the 37 compartments was 8x8 cm2 in area and 25.4

cm high. The HPGe detector was located 42.85 cm away from the center of the barrel

(14.31 cm away from the periphery of the barrel) with its center line at half of the height

of the matrix container, and collimated by three lead bricks. The front brick had an open

hole about the same diameter as the face of the detector and allowed the detector to

penetrate through it. Two other lead bricks were positioned behind the front one and

seated on both sides of the detector to ensure coverage of the whole collimating system.

The data access computer was a PC-AT with a MCA interface card installed.

The gamma ray signal was processed by electronic modules and signal counts were

stored on the buffer of the MCA card. EG&G ORTEC software MAESTRO-II controlled



































Figure 6-1. Experimental setup for the 208 e waste barrel characterization




































Figure 6-2. The acrylic-glass matrix container inside the waste barrel







the timing of the signal processing and interpreted the information stored on the MCA

buffer as pulse height data (often called an energy spectrum).

Detector Efficiency Calibration


Five check sources were used for HPGe detector efficiency calibration. The

source activities on September 15, 1996, were 1.037 gCi, 1.025 4Ci, 0.9173 tCi, 0.9084

piCi, and 0.8978 [tCi for Ba'33, Cs137, Mn54, Na22, and Co57 respectively. Each check

source was placed along the center line of the detector and at 38.1 cm (15 inches) away

from the detector front face. Electronic settings for the calibration were bias voltage

2300V, amplifier gain 24, and shaping time 2 isec. Table 6-1 shows the characteristics

of these five check sources and the measured gamma energy efficiencies. The energy

range extended from 81.2 kev to 1331.0 kev. Figure 6-3 shows the measured efficiencies

and the resulting calibration curve based on least squares fitting. Two steps of least

squares fitting were used to establish the calibration curve. A second order polynomial

fitting was used from 81.2 to 136.8 keV and a third order polynomial fitting was used

from 136.8 to 1331 keV. The equation for the third order polynomial fitted curve is

s(E) = 4.44942 x 10-4 9.88770 x 10-7 E + 9.60628 x 10- E2 3.22253 x 10-3 E3

where E is the point source efficiency at a distance of 15" from the detector, and E is the

gamma energy in keV. Efficiencies based on this calibration curve for 316.68 keV,

468.09 keV, and 588.09 keV energy peaks of Ir'2 source, were found to be 2.1792lxl 04,

1.59539x104, and 1.30146x104 respectively.














Table 6-1. Characteristics of check sources and related efficiency calibrations


Gamma Energy for the Efficiency Calibrated at
Check Sources (key) Nuclide* (Branching Ratio) 15" from the Detector

81.2 Ba'33 (0.3297) 2.151Ox10-4
122.3 Co57 (0.8560) 3.4730x1O-4
136.8 Co5" (0.1068) 3.2070x10-4
276.3 Ba33 (0.0715) 2.7140x104
302.8 Ba'33 (0.1830) 2.2530x10-4
356.0 Ba133 (0.6194) 1.8405x10-4
383.9 Ba133 (0.0891) 1.6799x104
662.0 Cs'37 (0.8510) 1.2885x104
835.1 Mn4 (0.9998) 1.0325x104
1173.0 Co6 (1.0000) 7.9700x105
1331.0 Co60 (1.0000) 7.4400x10"5


* Half-life: Ba'33 -10.57 years,
Na22-2.60 years, Co57-271.8 c


Cs137-30.17 years, Mn54-312 days,














4 E-1 11 ,,Iii II


3.50E-4 Check Source Caibrabon Points
-- Fit 2: 3rd Order Polynomial Fitting
. 3.00E-4 X Fitted Poins
CU
" 2.50E-4 I
0
Cl) -316 keV

L 2.00E-4
--468 keV
U
-- 588 kev-
o 1.50E-4 58V


1.00E-4


5.00E-5 'I I i Iiii

0 300 600 900 1200 1500
Gamma Energy (keV)


Figure 6-3. The efficiency calibration curve for HPGe detector








Region of Interest of Ir92 Gamma Spectrum


MAESTRO-II calculates total counts of a measured energy peak by integrating

over the peak area defined by ROI.20 Precise definition of the ROI is very important to

accurate peak area calculation. Figure 6-4 shows the configuration for defining a ROI of

a 316 kev energy peak. The ROI is a group of MCA channels bounded by a low channel

number I and a high channel number h. The background on the low channel side of the

peak is the average of the first three channels of the ROI. The background on the high

channel side of the peak is the average of the last three channels of the ROI. These two

pairs of channels define the end points of the straight-line background. The background

is given by the following equation:

B =+2C + Ci)h-l+l(61
=1 i~h- 2 6

where B is the total background area; I is the ROI low limit; h is the ROI high limit; and

Ci is the number of counts of channel i. A gross area Ag is defined as

h
Ag= Ci (6-2).
i=1

The adjusted gross area is the sum of all the channels within the ROI but not used in the

background, and is represented by the following relation:

h-3
Aag =j C' (6-3)
i=1+3

The net area is the adjusted gross area minus the adjusted background as follows:

B. (h -l -5) (6-4)
An=Ao- (h-I+1)












ROI





An









B B
1









760 770 780 790
MCA Channel Number

I I
311.42 316.68 318.70
Gamma Energy (keV)


Figure 6-4. The ROI for a 316 keV energy peak of the Ir"' gamma spectrum








This research uses the net area counts An as the total uncollided photons measured by the

detector for a specific gamma energy peak.

The ROIs of Ir192 were defined at the time that the energies were calibrated. As

shown in Figure 6-4, the energy calibration set the center of 316 keV peak at channel 775,

with a lower ROI limit set at channel 766 (313.52 keV) and a higher ROI limit set at

channel 782 (319.13 keV). The Full-Width-Half-Maximum (FWHM) of this peak is 1.77

keV. The ROI encompasses a region three times that of the FWHM. Figure 6-5 shows

the configurations of the ROI for the 468 keV and 588 keV energy peaks based on the

calibrated energy spectrum. The 468 keV energy peak was centered at channel 1150 and

had a FWHM of 1.85 keV, a low ROI limit of 1141, and a high ROI limit of 1157. The

588 keV energy peak was centered at channel 1448 and had a FWHM of 1.87 kev, a low

ROI limit of 1440, and a high ROI limit of 1456.

Data Collection Process


To simulate a four fixed detector system, detector readings were taken at each 90

degree position by rotating the barrel counterclockwise. For the homogeneous condition,

the barrel was flooded with water and 18 nonuniform source distribution cases were

studied. For heterogeneous conditions, in addition to water, six other materials were

used: sand (p = 1.45 g/cm3), small stones (p = 1.33 g/cm3), salt (p = 1.14 g/cm3), flour (p

= 0.83 g/cm3), bird seed (p = 0.78 g/cm3), and pine ashes (p = 0.27 g/cm3). Figure 6-6

shows that the acrylic-glass insertion channels were specifically designed to hold the

above mentioned materials and to exactly fit into the 37 compartments. A thin string was













ROI for 468 kev Peak


0h
/0Q









1135 1140 1145 1150 1155 1160 1165
MCA Channel Nuter



462.83 468.09 473.35
Gamma Energy (kev)


ROI for 588 kev Peak


1435 1440 1445 1450 1455 1460 1465
MCA Channel Number




584.03 588.09 594,55
Gamma Energy (kev)
Figure 6-5. The ROI for the 468 keV and 588 keV energy peaks of the Ir192 spectrum








attached to the top of every insertion channel to facilitate its insertion into and removal

from the compartment. Figure 6-6 also shows a configuration of an experimental

heterogeneous condition inside the waste barrel. A number of insertion channels are used

in this configuration. Plastic bags were used for water channels to avoid the possibility of

leakage. Different density distributions were modeled by shuffling the insertion

channels. Two heterogeneous conditions (Type A and Type B) were simulated by using

combinations of the seven materials. Density Type A simulated low density perturbation

condition and its density distribution varied from 0.83 to 1.14 g/cm3. Density Type B

simulated high density perturbation condition and its density distribution varied from

0.27 to 1.45 g/cm3. Both Type A and Type B conditions had 9 nonuniform source

distribution cases.

For each source distribution case, Ir192 sources were placed at the center of the

height of some insertion channels. In order to avoid any bias resulting from orientation

of Ir192 sources, and to obtain reasonable statistical results, four 300 second measurements

were made at each detector position. Insertion channels containing Ir'92 sources were

removed, rotated 90 degree clockwise, and reinserted into their compartments for each

successive measurements. Detector responses were then averaged for these four

measurements.

Experiments of Distributed Ir'92 Sources under the Homogeneous Density Condition


Figure 6-7 shows the positions of Ir'92 sources for 18 experiments under the

homogeneous density condition. These positions were chosen to represent different




















































Figure 6-6. Six materials of different density in the insertion channels and an
experimental heterogeneous configuration inside the waste barrel








source distribution behaviors. Cases Hi, H2, H3, and H13 represented single isolated

source distribution. Cases H7, H8, H9, H11, H14, H15, H16, and H17 represented

clustered source distributions. All other cases belonged to multiple but scattered source

distributions. In the figure, the activity marked on top of each configuration was the total

activity for that configuration. Case H14 had three Ir192 pieces located at grid 20 and

another piece at grid 21. Case H15 had three Ir'92 pieces located at grid 25 and three other

pieces scattered at grid 6, 14, and 33 respectively.

Four measurements were made at each detector position. For each measurement,

total counts of 316 keV, 468 keV, and 588 keV energy peaks were recorded respectively,

and then these values were divided by counting time to get detector responses (counting

rates). Final detector responses were averaged from these four measurement values.

Four orthogonal detector positions were measured with a total of 16 measurements made

for each experiment. These measured detector responses formed the elements of

response matrix D in Equation 1-3, and when combined with the precalculated kernel

matrix K, a numerical reconstruction algorithm was used to solve for source distribution.

When two energies were used for the reconstruction, the linear system was 8x37. When

three energies were used, the linear system was 12x37. Results of total activity

estimation for these 18 experiments are shown in Table 6-2. Using the CGNN method,

estimation errors ranged from -33.30% to 27.52% for 2-energy reconstruction and from

-34.38% to 28.67% for 3-energy reconstruction. Using the MLEM method, estimation

errors ranged from -29.22% to 23.79% for 2-energy reconstruction and from -28.30% to

24.62% for 3-energy reconstruction. Table 6-3 shows the summary of these results. All

of the 18 experiments were within 50% error for the four reconstruction schemes used.









(HI) 2.6869x106 Bq (H2) 2.6869x106 Bq (H3) 2.6869x106 Bq (H4) 5.3738x106 Bq


L~~~ ~~~ L~ L1,I, L! G,,kII,



(H5) 5.3738x106 Bq (H6) 5.2730x106 Bq (H7) 5.2730x106 Bq (H8) 7.9095x106 Bq






















(H17) 8.4168x106 Bq (H18) 8.4168x106 Eq







Figure 6-7. Configurations of Ir92 source distribution for 18 experiments under the
homogeneous density condition (0 1 Ir192 piece ;@: 3 Ir92 pieces)













Table 6-2. Activity estimation errors for 18 homogeneous density experiments by using
multiple energy peaks in the CGNN and MLEM reconstruction algorithms


CGNN
Reconstruction
Error (%)
Case 2-Energy 3-Energy


H1
H2
H3
H4
H5
H6
H7
H8
H9
H10
Hil
H12
H13
H14
H15
H16
H17
H18


-3.90
-9.54
27.52
4.91
4.21
-14.29
-1.14
3.24
15.07
21.32
3.58
-6.58
-21.10
-25.50
-33.49
-30.63
-17.56
-29.51


-2.09
-8.38
28.67
2.84
3.52
-11.76
-0.16
3.44
13.92
23.65
4.55
-6.35
-18.51
-34.38
-31.55
-29.70
-14.17
-29.39


MLEM
Reconstruction
Error (%)
2-Energy 3-Energy


-3.98
-9.65
-1.37
5.11
5.21
-14.92
-2.40
1.36
-1.26
23.79
1.72
-8.41
-20.59
-28.15
-29.22
-26.55
-19.12
-26.43


Number of
ir192 Pieces
in the Barrel


-1.85
-8.55
-1.40
6.56
4.13
-12.29
-1.03
2.49
-0.61
24.62
3.53
-7.86
-15.58
-25.52
-28.30
-25.44
-18.42
-24.77












Table 6-3. Summary of activity estimation errors for 18 homogeneous density
experiments by using multiple energy peaks in the CGNN and MLEM algorithms



Error <50% <30% <25% <20% <10%


2-Energy Reconstruction

CGNN 18" 17 13 11 8
(100%)" (94.4%) (76.5%) (61.1%) (44.4%)

MLEM 18 17 13 12 8
(100%) (94.4%) (76.5%) (66.7%) (44.4%)

3-Energy Reconstruction

CGNN 18 18 13 12 8
(100%) (100%) (76.5%) (66.7%) (44.4%)

MLEM 18 18 15 13 10
(100%) (100%) (83.3%) (76.5%) (55.6%)

Number of cases within the specified error bound.
Percentage of the 18 cases within the specified error bound.








For 30% error, both the CGNN and MLEM predicted 17 out of 18 cases (94.4%) by using

2-energy reconstruction, and 18 out of 18 cases (100%) by using 3-energy reconstruction.

For 20% error, the best prediction of the four reconstruction schemes was the MLEM

using 3-energy reconstruction which predicted 13 out of 18 cases (76.5%).

Experiments of Distributed Ir'92 Sources under Heterogeneous Density Conditions


Two types of density heterogeneity were studied. Type A heterogeneity

simulated a low density perturbation condition by filling insertion channels with salt,

water, or flour and then inserting them into the egg-crate compartments in the waste

barrel. Type B heterogeneity simulated a high density perturbation condition by using

sand, small stones, salt, water, flour, bird seed and pine ashes. Before placing these

materials into the barrel, their densities were measured by transmission density

measurement. An Ir'92 piece was used as the transmission source and was placed 15"

away from the detector. Counting rates of 588 keV peak were recorded for measurements

both with and without an insertion channel in front of the detector. Density then was

calculated based on the exponential attenuation relation. For each material, a

transmission density measurement was made for each of the four faces of the rectangular

channel. The density for each material listed in Table 6-4 was the average value from

these four measurements. Table 6-4 also lists the positions of different material channels

inserted into the barrel for both Type A and Type B heterogeneity conditions. Figure 6-8

shows the configuration of Ir"' locations for the nine experiments under the Type A

heterogeneity condition. Source configurations for the other nine experiments under the

Type B heterogeneity condition are shown in Figure 6-9. As in the homogeneous density












Table 6-4. Location of different material channels inside the barrel for the study of Type
A and Type B density heterogeneity


Type A Heterogeneity (Density : 0.83 1.14 g/cm3)


Grid Number
Salt (p =1.14 g/cm3) 17, 19, 21, 25, 27, 28, 31, 33, 34, 36

Water (p =1.00 g/cm3) 1, 2, 3, 9, 12, 14, 15, 16, 18, 20,
22, 23, 26, 29, 32, 35, 37
Flour (p =0.83 g/cm3) 4, 5, 6, 7, 8, 10, 11, 13, 24, 30

Type B Heterogeneity (Density: 0.27 1.45 g/cm3)


Sand (p =1.45 g/cm3)

Stones (p =1.33 g/cm3)

Salt (p =1.14 g/cm3)

Water (p =1.00 g/cm3)

Flour (p =0.83 g/cm3)

Bird Seed (p =0.78 g/cm3)

Pine Ashes (p =0.27 g/cm3)


Grid Number
2, 4, 5, 6,11

10, 17, 18

8, 25, 26, 31, 32, 36

1, 3, 9, 15, 16,22,23,29,35,37

7, 12, 13, 14, 20, 33, 34

19, 24, 30

21, 27, 28

























(Al) 4.1693x106Bq


(A4) 2.0652x106 Bq


(A7) 6.0240x106 Bq


(A2) 2.0652x106 Bq


(A5) 2.0652x106Bq


(A8) 6.0240x106 Bq


(A3) 4.1304x106Bq


(A6) 4.0160x106 Bq


(A9) 8.2608x106 Bq


Figure 6-8. Density distribution and configurations of Ir'92 location for 9 experiments
under the Type A heterogeneity condition


4.0 0.83 1.14 1.0 1.14 1.14 1.0)
1.0 1.14 1.0 1.14 1.0 1.14 1.0

1.01 0.83 0.83 1.0 0.83 1.0 1.0


























(B1) 2.0460x106 Bq


(B4) 4.0540x106 Bq


(B7) 6.0240x106 Bq


(B2) 2.0460x106 Bq


(B5) 4.0160x106 Bq


(B8) 8.0320x106 Bq


(B3) 4.0540x106 Bq


(B6) 6.0240x106 Bq








(B9) 8.0320x106 Bq


Figure 6-9. Density distribution and configurations of Ir192 location for 9 experiments
under the Type B heterogeneity condition


1.0 0.78 1.14 1.14 0.27 0.27 1.0
1.0 1.33 1.3 0.7 0.8 0.27 1.0

1.0 133 145 .83 0.82 0.8








cases, the Ir92 source locations (shown in these two figure) simulate single, clustered,

and/or multiple but scattered source distributions.

The results of total activity estimation for these experiments appear in Table 6-5.

For all of the reconstruction schemes used, the activity of most of the cases is

underestimated. The underestimation may be attributed to the biases of the average

density measurements of the insertion channels, or to the biases of the choice of Ir...

distribution. Since the number of experiments is limited, the exact bias tendency of the

estimation can be assured only after performing more experiments through the numerical

simulations.

Table 6-6 summarizes experimental results. All of the experiments were within

50% error for the four reconstruction schemes used. In Type A heterogeneity

experiments with 30% error, CGNN predicted 7 out of 9 cases (77.8%) by using

2-energy reconstruction, and MLEM predicted 8 out of 9 cases (88.9%). By using

3-energy reconstruction, both CGNN and MLEM predicted 8 out of 9 cases (88.9%). In

Type B heterogeneity experiments with 30% error, CGNN predicted 8 out of 9 cases

(88.9%) by using 2-energy or 3-energy reconstruction, and MLEM predicted 9 out of 9

cases (100%) by using 2-energy or 3-energy reconstruction. All the 9 cases predicted by

MLEM using 3-energy reconstruction were within 20% error.

Discussion of the Results of the Experiments


Two important findings were observed from the above 36 experiments: (a) the

computer model was able to accurately estimate total activity, and (b) the locally













Table 6-5. Activity estimation errors for the heterogeneous density experiments by using
multiple energy peaks in the CGNN and MLEM algorithms


CGNN
Reconstruction
Error (%)
Case 2-Energy 3-Energy


MLEM
Reconstruction
Error (%)
2-Energy 3-Energy


Number of
Ir'92 Pieces
in the Barrel


Cases of Type A Heterogeneity: (A1-A9)


-38.77
-18.36
-36.86
1.36
3.74
-23.75
-24.42
-9.40
-19.01


-34.18
-15.34
-16.61
4.49
3.13
-14.72
-22.12
-12.67
-20.66


Cases of Type B Heterogeneity: (B 1 -B9)


-24.74
0.86
-38.27
-14.90
-3.83
-23.47
-19.20
-24.55
-19.29


-22.82
0.67
-36.95
-14.97
-0.34
-23.36
-19.87
-23.15
-21.31


-31.66
-19.56
-23.74
2.16
0.95
-25.09
-19.99
1.96
-17.34


-30.15
-16.44
-12.32
5.78
1.23
-17.23
-18.25
1.55
-15.38


-23.92
-0.86
-6.39
-15.85
-6.14
-21.35
-9.50
-18.52
-18.84


-18.58
-0.50
-2.23
-13.56
-1.09
-18.62
-8.82
-17.14
-14.66










Table 6-6. Summary of activity estimation errors for the heterogeneous density
experiments by using multiple energy peaks in the CGNN and MLEM algorithms



Error <50% <30% <25% <20% <10%


Type A Heterogeneity : total 9 cases

2-Energy Reconstruction
CGNN 9' 7 7 5 3
(100%)** (77.8%) (77.8%) (55.6%) (33.3%)
MLEM 9 8 7 6 3
(100%) (88.9%) (77.8%) (66.7%) (33.3%)

3-Energy Reconstruction
CGNN 9 8 8 6 2
(100%) (88.9%) (88.9%) (66.7%) (22.2%)
MLEM 9 8 8 8 3
(100%) (88.9%) (88.9%) (88.9%) (33.3%)


Error <50% <30% <25% <20% <10%


Type B Heterogeneity : total 9 cases

2-Energy Reconstruction
CGNN 9 8 8 5 2
(100%) (88.9%) (88.9%) (55.6%) (22.2%)
MLEM 9 9 9 7 4
(100%) (100%) (100%) (77.8%) (44.4%)

3-Energy Reconstruction
CGNN 9 8 8 4 2
(100%) (88.9%) (88.9%) (44.4%) (22.2%)
MLEM 9 9 9 9 4
(100%) (100%) (100%) (100%) (44.4%)

Number of cases within the specified error bound.
Percentage of the 9 cases within the specified error bound.










nonuniform distribution of an Ir'92 source in a computational grid caused observed

detector responses to deviate from calculated ideal detector responses. Such local

nonuniformity of source distribution was especially important to the modeling of

numerical simulations described in the next chapter. The following paragraphs discuss

these two findings.

Results of the 36 experiments described in the last two sections reveal that the

computational model using multiple energy peaks in the CGNN or MLEM algorithms

was able to predict the total activity of randomly distributed sources inside a waste barrel

at a high accuracy level. Table 6-7 summarizes all results. All experiments were within

50% accuracy. Using 2-energy reconstruction, the CGNN predicted 32 cases (88.9%)

and the MLEM predicted 34 cases (94.4%) with 30% accuracy. Using 3-energy

reconstruction, the CGNN predicted 34 cases (94.4%) and the MLEM predicted 35 cases

(97.2%). With an accuracy level as high as 20%, the MLEM predicted 30 cases (83.3%)

by using 3-energy reconstruction. These experimental facts verified the prediction ability

of the computer model developed in this research. However, the number of experiments

was still too limited to be able to define a statistically reasonable confidence level for the

computer model. Further and more extensive verifications were done by using the

numerical experiments described in the next chapter.

An important fact needs to be carefully examined in regard to the locally

nonuniform distribution of Ir'92 sources in a computational grid for these 36 experiments.

The calculation model used 37 computational grids for a vertical segment of the waste

barrel and assumed that once a source was in a grid, the source was uniformly distributed












Table 6-7. Summary of activity prediction for 36 experiments



Error <50% <30% <25% <20% <10%



2-Energy Reconstruction

CGNN 36* 32 28 21 13
(100%)* (88.9%) (77.8%) (58.3%) (36.1%)

MLEM 36 34 29 25 15
(100%) (94.4%) (80.6%) (69.4%) (41.7%)

3-Energy Reconstruction

CGNN 36 34 29 22 12
(100%) (94.4%) (80.6%) (61.6%) (33.3%)

MLEM 36 35 32 30 17
(100%) (97.2%) (88.9%) (83.3%) (47.2%)

Number of cases within the specified error bound.
** Percentage of the 36 cases within the specified error bound.








within that grid. It is difficult, however, to simulate a fully uniform source distribution

condition within a computational grid by using a Ir'92 source of only 3 mm in length and

0.5 mm in diameter. Also, in the experiments, no specific bias and no attempt were made

to place Ir'92 pieces in acrylic-glass compartments such that a uniform distribution

condition could be simulated. Therefore, the experimental source distribution for a

computational grid deviated somewhat from the uniform assumption. Because of this

local nonuniformity, observed detector responses must have deviations from the ideal

calculation. This deviation is modeled as a nonuniformity noise in this research. In the

real world, radioactive sources inside the waste drum are usually clustered and rarely

have a point source inside a computational grid. Local nonuniformity noise is more

severe in a point source case than in a clustered source case. The reason for this is that

for a clustered source, observed counts of a detector result from photon emissions

averaged over a certain volume which is often larger than the size of the Ir'92 source used

in the above experiments. Source emissions averaged over a larger volume in a

computational grid have a greater tendency toward a uniform emission, meaning that the

above experiments had the inherent noise caused by the local nonuniformity of the small

piece of Ir'92 inside a grid. The magnitude of this noise may compete with that of other

noise caused by statistics of the detector reading or the uncertainties of the calibration

parameters. Differences of magnitude between these noises are analyzed in the following

paragraphs.

First, the efficiency calibration was very trivial and is easy to control its

uncertainty below a few percent. For example, the efficiency of 316 keV energy peaks of

Ir'92 were found to be 2.17921x10- and this energy was very close to one of the








calibration points of the 302.8 keV of Ba'33 check source (see Table 6-1 and Figure 6-3).

The efficiency of the 302.8 keV gamma-ray was measured as 2.2530x10' and the fitted

efficiency based on the equation of the calibration curve for this energy was 2.2467x 104.

The relative deviation was only 0.28%. The uncertainty (i.e. the noise) of efficiency

calibration for the 316 keV energy peak was expected to be close to this value.

Therefore, the contribution of noise of the ideal detector response from the efficiency

calibration was very small.

Second, the magnitude of local nonuniformity noise can be examined by

comparing minimum, maximum, and average values of calculated detector kernel

responses. Detector kernel responses for the homogeneous condition (listed in Appendix

D) were calculated by the computer model according to Equation 3-1. The 316 keV

kernel responses for grid 19 or grid 36 with respect to detector position 1 were

minK,,19 = 2.9166x10-6 maxK,,,9 = 1.0995x10-' aveK,19 = 6.0387x10-6;

minK,36 = 1.9622x10-4 maxKI,36 = 1.2040x10-3 aveK,36 = 5.3356x10-4;

where aveKij was the average kernel response for detector position i contributed by

5,000 Monte Carlo samplings of point sources in gridj; and minKij was the minimum

and maxKij was the maximum point kernel response among the 5,000 points,

respectively. Deviations from the average for these two grids were

(min K, 36-v )= -63.22% (max K,36 aveK,36)
=a_63.22%= 125.65%
aveK,136 aveK,,36


(min K,,19 aveK,.,9 )
= -51.70%
aveK,,19


(max Kl,,9 aveK,,,9)
= 82.08%.
aveK,,,9








A similar induction used for grid 36 with respect to detector position 4 was

(min K4,36 aveK4,36) (max K4,36 aveK4,36)
avK,6 = -57.54%; avK,6 = 105.10%
aveK4,36 aveK4,36

These calculations reveal that from the point of view of detector position 1, a point source

at the very far corner of grid 36 contributed to a detector response which was 63.22%

lower than that contributed by a uniformly distributed source in the grid, and a point

source at the very near comer contributed to a detector response which is 125.65% higher

than that contributed by the uniformly distributed source. These two values defined the

low and high limits of the local nonuniformity noise of grid 36 with respect to detector

position 1. For grid 19, the low and high limits of noise became -51.70% and 82.08%.

For grid 36 with respect to detector position 4, the low and high limits of noise were

-57.54% and 105.10%. From these comparisons, it is evident that local nonuniformity of

a point source caused a wide range of perturbation of the ideal detector response which

was calculated based on the average kernel response of each computational grid. This

situation was more severe in a peripheral grid than in a central grid. Table 6-8 shows the
comparisons of ideal and observed counting rates for experiment case H1 (1 Ir"9 piece in


central grid 19) and case H3 (1 Ir92 piece in peripheral grid 36). These two cases

represent two extremes of source distribution inside the waste barrel. Obviously, case H3

was much more perturbed than case HI, as shown in Table 6-8. Case Hi was a fully

symmetric case and all four detector positions were ideally supposed to have the same

readings. Case H3 was a symmetric case for detector 2 and detector 4, and ideally these












Table 6-8. Comparisons of ideal and observed detector responses for experiment case HI
and case H3



Case H1
316 keV 468 keV 588 keV
Ideal Obs. %Dev.* Ideal Obs. %Dev. Ideal Obs. %Dev.

Det 1" 13.60 15.09 10.96 9.41 10.31 9.56 0.97 1.06 9.28
Det 2 13.60 13.95 2.57 9.41 10.10 7.33 0.97 0.99 2.06
Det 3 13.60 14.22 4.56 9.41 9.84 4.57 0.97 0.93 -4.12
Det 4 13.60 13.39 -1.54 9.41 9.47 0.64 0.97 0.96 -1.03


Case H3
316 keV 468 keV 588 keV
Ideal Obs. %Dev. Ideal Obs. %Dev. Ideal Obs. %Dev.

Det 1 1190.00 1024.00 -13.95 544.60 481.10 -11.66 43.93 39.01 -11.20
Det2 6.48 7.19 10.96 4.71 5.21 10.62 0.50 0.65 30.00
Det 3 0.34 0.30 -11.76 0.35 0.32 -8.57 0.04 0.02 -50.00
Det 4 6.48 9.78 50.93 4.71 6.76 43.52 0.50 0.68 36.00

Percent of deviation of observed response from ideal responses
Detector position 1








two detectors were supposed to have the same readings. However, no obvious symmetry

was observed from either of these two experiments.

The standard deviation of the counting rate for the counting statistics was

o- = V-D/t (D being counting rate and t being total counting time). For a 300 second

counting time, the standard deviation of the 316 kev counting rate for detector position 1

was 0.22 in experiment HI and was 1.85 in experiment H2. These were only 1.62% and

0.16% of the ideal responses for these two cases respectively far below the percentage

deviations of 10.96% and -13.95% as shown in Table 6-8. Therefore, counting statistics

was not the major factor causing the observed response deviated from the ideal response

for these two experiments. Perturbations caused by local nonuniform source distributions

were the important contributors to the detector response noise in the experiments.













CHAPTER 7
NUMERICAL SIMULATIONS

In addition to 36 experiments described in Chapter 6, this research also performed

numerical simulations to determine the error bound and its associated confidence level for

the computer model in a reasonable statistical sense. In real experimentation, the

computer model uses backward reconstruction algorithms CGNN and/or MLEM to

estimate total activity based on observed external detector responses. In numerical

experimentation, total activity was obtained from backward reconstruction based on ideal

detector responses which were calculated by a forward projection algorithm. However,

as discussed in the last section of Chapter 6, observed responses are always different from

ideal responses. These differences were treated as "noises" in this research. To simulate

real measurements by using numerical experiments, "noise" needed to be modeled and

incorporated in ideal responses. This research modeled two types of noise local

nonuniformity noise and counting statistical noise. Since waste barrel contents are

mostly unknown, density heterogeneity has a certain degree of influence on the accuracy

of computed result. This research used a random density perturbation model to simulate

different types of density heterogeneity and combined these density perturbations with

noise perturbations to form a wide spectrum of numerical simulations.








The Density Perturbation Model


Density heterogeneity was modeled by specifying a relative density perturbation

level with respect to a homogeneous base density. The perturbed density for each

computational grid was assigned by the Monte Carlo process:

P, = p' [I + f (217j 1)] (7-1)


where p' was the perturbed density for the computational gridj, P0 was the base

density, f was the relative density perturbation level, and 77j was the random number

generated for the computational gridj.

In order to have a consistent basis of comparison with experiments described in

Chapter 6, 1.0 g/cm3 was used as the base density in the simulation. Two relative density

perturbation levels, 20% (low) and 50% (high), were studied. Uncollided gamma-rays

emitted from any randomly sampled source point needed to pass one or several grids to

reach the detector. Path length within each grid was accurately calculated for density

perturbation conditions. For each simulated case, a perturbed kernel table was obtained

by using Equation 3-1, based on calculated path length and density distribution calculated

from Equation 7-1. Then, the kernel table was incorporated with the assignment of

random source fraction of each grid, to give the forward projection of ideal detector

responses by using Equation 3-2.

The Model of Local Nonuniformity Noise


Ideal detector responses were generated using the assumption that the source was

uniformly distributed within a computational grid. As found in Chapter 6, local








nonuniformity of source distribution within a computational grid was an important

contributor of signal noise and was modeled by specifying a relative perturbation level

for detector responses. This level was the perturbation range of ideal detector responses,

which was assumed to occur in real experiments. A Monte Carlo process was used to

select the nonuniformity noise from this range. That is, for each detector position i, the

perturbed response (counting rate) was

D'(Ej) = DO (Ej).[1 + p.(2. r7 1)] (7-2)

where D'(Ej) was the perturbed response for energy peak j of detector position i;

D (Ej ) was the ideal detector response; p was the relative perturbation level contributed

from local nonuniformity noise; and 77, was the random number generated for detector

position i. Perturbation levels from 0% to 50% were studied. As shown in Table 6-8, the

50% perturbation case was equivalent to experiment case H3 which was a highly biased

case with an Ir192 source very close to one of the detector positions and having a

maximum deviation around 50% between observed and ideal responses.

The Model of Counting Statistics


Any radiation measurement is subject to the uncertainty of counting statistics. A

perturbed detector response D,(Ej) calculated from Equation 7-2 was considered an

average value of the measurement of randomly distributed sources nonuniformly

distributed inside any computational grid. To simulate a real response with counting

statistical behavior, distribution of perturbed detector response D'(Ej) was modeled by

using a Gaussian distribution form








1 exp(_(X-2)(-3
P(X) = 2- 20(73)



where X = D'(Ej), Uo = and Twas the counting time. Using the rejection


technique, each detector response in the numerical simulation was obtained by sampling

from Equation 7-3, with a sampling range from X 3o- to X + 3o-.21

Error Bound and Confidence Level


Five numerical experiment categories were designed with the following

simulation conditions:

(i) homogeneous and noise-free,

(ii) heterogeneous with 20% density perturbation and noise-free,

(iii) heterogeneous with 50% density perturbation and noise-free,

(iv) heterogeneous with 20% density perturbation and different noise levels, and

(v) heterogeneous with 50% density perturbation and different noise levels.

For each category, evaluations of error bound and its associated confidence level

for activity estimation were carried out by using different backward reconstruction

schemes under different simulation conditions. Each simulation is performed with

respect to 10,000 random source distribution cases. Confidence level for the simulation

was defined as


C= PrJA<- A0 '- (7-4)
'( Ao


with Ai being the estimated activity for any single case, A0 the reference activity, 6 the

required accuracy, and N (=10,000) the total number of cases. Pr is the frequency of








occurrence for relative error within c For example, if the error bound s is specified as

10%, and 95% of the predicted cases fell within 10% of the actual source strength, the

prediction error would be said to be 10% at the 95% confidence level.

In the simulation, counting time was 300 seconds and reference activity was

106 Bq. This activity level was about the same order of magnitude as those activities used

in the experiments described in Chapter 6.

The Naming System for the Numerical Experiment


In order to distinguish the results of a large number of simulation cases, a naming

system was designed. For category (i), six evaluations were studied and were named as

HCE1, HCE2, HCE3, HME1, HME2, and HME3. "H" stands for homogeneous, "C"

stands for the CGNN method, "M" stands for the MLEM method, and "El", "E2", and

"E3" denote 1-energy, 2-energy and 3-energy reconstruction, respectively. For categories

(ii) to (v), the naming system used the notation "XnNmmYEp" explained by the

following keys:

"X" is "C" or "M" and represents the CGNN or the MLEM method, respectively;

"n" is "2" or "5" and means 20% or 50% density perturbation, respectively;

"N" denotes local nonuniformity noise;

"mm" is the relative perturbation level of local nonuniformity noise;

"Y" is "B" or "K" and represent the reconstruction is based on the base density

kernel table or the reconstruction is based on the perturbed kernel table, respectively;

"E" means energy; and








"p" is "1", "2" or "3" and represents 1-energy, 2-energy, and 3-energy

reconstruction, respectively.

For example, a simulation result denoted by C2NO5KE2 would mean that the

result was based on using 2 energies and the perturbed kernel table in the CGNN

reconstruction algorithm under a 20% density perturbation and a 5% noise condition. A

result denoted by M5N50BE2 would mean that the result was based on using 2 energies

and the base density kernel table in the MLEM reconstruction algorithm under a 50%

density perturbation and a 50% noise condition.

Homogeneous and Noise-Free Condition


Table 7-1 shows simulation results of different reconstruction schemes under the

homogeneous and noise-free condition (simulation category (i)). In this simulation

category, 1-energy, 2-energy, and 3-energy reconstructions were used for both the CGNN

and MLEM algorithms. Confidence levels of 1-energy reconstruction were very low.

However, when two energies were used, confidence levels dramatically increased to a

very satisfactory result for both CGNN and MLEM reconstructions. All the predictions

of the CGNN were within 20% error and the confidence level reached 98.1% for an

accuracy as high as 10%. All the predictions of the MLEM were within 25% error and

the confidence level for 10% error was 93.5%. Confidence level increased even more

when three energies were used, but not as dramatically as when increasing from one

energy to two energies. Almost all the predictions (99.7%) were within 10% error for

3-energy CGNN reconstruction. The confidence level for 10% error was 92.1% for












Table 7-1. Confidence levels for total activity estimation by different optimization
algorithms under the homogeneous and noise-free condition

Category (i) : Homogeneous and Noise-Free Condition

Error <50% <30% <25% <20% <10%

HCEI* 0.438 0.301 0.267 0.229 0.120
HMEI** 0.630 0.372 0.323 0.277 0.158

HCE2 1.000 1.000 1.000 1.000 0.981
HME2 1.000 1.000 1.000 0.994 0.935

HCE3 1.000 1.000 1.000 1.000 0.997
HME3 1.000 1.000 1.000 1.000 0.921
H: homogeneous density; C: CGNN; M: MLEM;
El: 1-energy; E2: 2-energy; E3: 3-energy.








3-energy MLEM reconstruction. In this simulation condition (category (i)), prediction

ability of the CGNN proved better than that of the MLEM.

Figure 7-1 shows the distributions of relative prediction error for 2-energy CGNN

and MLEM methods in simulation category (i). It is even more evident that the CGNN's

predictions were all within 15% error and had no significant bias toward underestimation

or overestimation. For the MLEM, a tendency toward overestimation can be seen, but the

error is not more than 25%.

Comparisons with the Segmented Gamma Scanning Method


The traditional Segmented Gamma Scanning (SGS) method assumes that the

material and radionuclides are distributed homogeneously and uniformly within the waste

drum. Data collection is done by measuring the gamma signal emitted from the drum as

it is rotated stepwisely or continuously. The rotation is used to compensate for activity

distribution asymmetries. Under this assumption, the total detector response DT is the

sum of all detector responses and is proportional to total activity:

DT = "D c S,.


For 1 Bq homogeneous source, the total response DThom can be easily calculated by

doing an analytical integration or a Monte Carlo integration over the drum volume.

Therefore, for any measured detector response DTnon which is essentially

nonhomogeneous in nature, total activity is estimated to be
Slot = DT" (Bq).

DThom












0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00


0.50
0.45

0.40

0.35

0.30
0.25

0.20

0.15

0.10

0.05

0.00


-65 -55 -45 -35 -25 -15 -5 5 15 25 35 45 55 65
-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%)


-65 -55 -45 -35 -25 -15 -5 5 15 25 35 45 55 65
-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%)


Figure 7-1. Distributions of relative prediction error for the 2-energy CGNN and MLEM
algorithms under the homogeneous and noise-free condition








Table 7-2 shows the results of the SGS method using various multiple detector

positions and compares them with the results of the 2-energy CGNN and MLEM

methods. It is evident that increasing the number of detector positions in SGS did not

improve the accuracy of the measurement. For prediction within 30% error, the

confidence level was 34.3% at best. Even for an error bound as large as 50%, the

confidence level in the SGS measurement was very low and less than 66%. Distributions

of relative prediction error for 4 and 64 detector positions are shown in Figure 7-2. The

maximum error for 4 detector positions occurred at +170%. For 64 detector positions,

the maximum error occurred at + 100%. Compared with the 2-energy CGNN or MLEM

method, the SGS's prediction was very poor.

Heterogeneous and Noise-Free Conditions


Simulation results of 2-energy and 3-energy CGNN and MLEM algorithms under

heterogeneous and noise-free conditions (simulation categories (ii) and (iii) ) are shown in

Table 7-3. For 20% heterogeneity, 2-energy and 3-energy CGNN reconstruction schemes

predicted 10% error at 97% and 98% confidence levels respectively. Under the same

conditions for the MLEM, 2-energy and 3-energy reconstructions predicted 10% error at

91.2% and 91.5% confidence levels respectively. For 50% heterogeneity, the CGNN

predicted 10% error with 92.9% confidence by using 2 energies and 10% error with

96.2% confidence by using 3 energies. The MLEM predicted 20% error at a confidence

level of less than 90%, but, for 25% error, the predictions had a confidence level of more

than 92%. In these two categories, 3-energy reconstructions performed better than

2-energy reconstructions, and the CGNN's prediction was better than that of MLEM.












Table 7-2. Comparisons of confidence levels between the SGS and the 2-energy CGNN
and MLEM reconstruction algorithms under the homogeneous and noise-free condition


Error <50% <30% <25% <20% <10%


SGS04* 0.589 0.277 0.226 0.174 0.093
SGS08* 0.653 0.343 0.266 0.212 0.100
SGS16* 0.656 0.336 0.278 0.218 0.105
SGS32' 0.657 0.335 0.278 0.217 0.105
SGS64* 0.657 0.336 0.279 0.217 0.105

HCE2** 1.000 1.000 1.000 1.000 0.981
HME2*" 1.000 1.000 1.000 0.994 0.935
* Digits indicate the number of detector positions for SGS method.
Homogeneous density and 2-energy CGNN reconstruction.
Homogeneous density and 2-energy MLEM reconstruction.











0.20


0.15


0.10


0.05


0.00


-200-150-100 -50 0 50 100 150 200
Relative Error of Total Activity Estirmtion (%)


0.20


0.15


0.10


0.05


0.00 '"'1'" h1h"' '"l FTll11T'" jl -
-200-150-100 -50 0 50 100 150 200
Relative Error of Total Activity Estirration (%)

Figure 7-2. Distributions of relative prediction error for the SGS method using 4 and 64
detector positions under the homogeneous and noise-free condition











Table 7-3. Confidence levels for total activity estimation by different optimization
algorithms under the heterogeneous and noise-free condition

Category (ii) : 20% Heterogeneity and Noise-Free Condition


Error <50% <30% <25% <20% <10%

C2NOOKE2* 1.000 1.000 1.000 1.000 0.970
M2N0OKE2 1.000 1.000 0.999 0.995 0.912

C2NOOKE3 1.000 1.000 1.000 1.000 0.980
M2NOOKE3 1.000 1.000 1.000 0.998 0.915

Category (iii) : 50% Heterogeneity and Noise-Free Condition


Error <50% <30% <25% <20% <10%

C5NOOKE2 1.000 1.000 0.999 0.996 0.929
M5NOOKE2 0.997 0.953 0.923 0.851 0.599

C5NOOKE3 1.000 1.000 0.998 0.998 0.962
M5NOOKE3 0.999 0.967 0.939 0.886 0.616
Notation for the 1st column- C2: CGNN and 20% heterogeneity;
M2: MLEM and 20% heterogeneity; C5: CGNN and 50% heterogeneity;
M5: MLEM and 50% heterogeneity; NO: noise-free; E2: 2-energy;
E3: 3-energy; K: perturbed density kernel table.








Distribution of relative prediction error for 2-energy CGNN and MLEM

algorithms are shown in Figure 7-3. Maximum relative error for the 2-energy MLEM

method was +30% and +60% for 20% and 50% heterogeneity, respectively. For the

2-energy CGNN method, maximum relative errors was -20% and -30% for 20% and 50%

heterogeneity, respectively.

Heterogeneous and Noise-Perturbed Conditions


Local nonuniformity noise and counting statistics were both simulated under

heterogeneous conditions. Five relative perturbation levels of local nonuniformity (1%,

5%, 10%, 20% and 50%) were simulated. Gaussian statistics were applied for every

noise-perturbed detector response.

Simulation Results of the 20% Heterogeneity Level

Table 7-4 shows simulation results for different noise levels under 20% density

heterogeneity (simulation category (iv)). The results of 1-energy, 2-energy, and 3-energy

CGNN and MLEM reconstruction schemes are compared at 5% noise level. It is evident

again that the total activity predictions of 1-energy CGNN and MLEM methods were

very poor. A surprising finding is that CGNN results deteriorated by changing from a 2-

energy to a 3-energy reconstruction scheme. Confidence levels for 30% accuracy

decreased from 96.4% to 93.7%, and for 25% accuracy, decreased from 92.8% to 89.7%.

This deterioration phenomenon was not found in the 3-energy MLEM results at this noise

level.

Table 7-4 also clearly shows influences of noise at this heterogeneity level. For

2-energy CGNN reconstruction, the confidence level for 30% accuracy decreased from



















- IIi -I 111 ii_

C2NOOKE2












70-60-50-40 -30 -20 -10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%)


-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%)


-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%)


-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%)


Figure 7-3. Distributions of relative prediction error for 2-energy CGNN and MLEM
methods under the heterogeneous and noise-free condition












Table 7-4. Confidence levels for total activity estimation by different optimization
algorithms under the 20% heterogeneity and noise-perturbed condition

Category (iv) : 20% Heterogeneity with Local
Nonuniformity and Statistical Noise


Error <50% <30% <25% <20% <10%

C2NO1KE2' 0.998 0.969 0.932 0.860 0.563
M2N01KE2 0.998 0.968 0.944 0.877 0.600

C2NO5KE1 0.468 0.319 0.278 0.237 0.141
M2NO5KE1 0.663 0.395 0.337 0.282 0.160
C2NO5KE2 0.998 0.964 0.928 0.857 0.557
M2N05KE2 0.998 0.970 0.944 0.866 0.588
C2NO5KE3 0.993 0.937 0.897 0.831 0.553
M2NO5KE3 0.999 0.973 0.951 0.894 0.638

C2N1OKE2 0.999 0.963 0.919 0.850 0.529
M2N1OKE2 0.999 0.968 0.940 0.865 0.573

C2N2OKE2 0.998 0.951 0.896 0.805 0.495
M2N2OKE2 0.999 0.965 0.912 0.826 0.524

C2N5OKE2 0.979 0.821 0.743 0.630 0.314
M2N5OKE2 0.990 0.840 0.757 0.639 0.328

Notation for the 1 st column- C2: CGNN and 20% heterogeneity;
M2: MLEM and 20% heterogeneity; E2: 2-energy; E3: 3-energy
K: perturbed density kernel table; NO 1: 1% noise; N05: 5% noise;
N10: 10% noise; N20: 20% noise; N50: 50% noise.








96.9% to 82.1% when the noise level increased from 1% to 50%. For 2-energy MLEM

reconstruction, the confidence level for 30% accuracy decreased from 96.8% to 84.0%

when the noise level increased from 1% to 50%. Compared with the noise-free cases

described in the last two sections, the performance of the CGNN is reversed with respect

to the MLEM in the noisy condition. The CGNN confidence level is degraded faster than

that of the MLEM when noise increased. Even in only a 1% noise condition, the MLEM

outperformed the CGNN. This phenomenon can be explained by the fact that the MLEM

is theoretically based on a statistical derivation and the noise is also modeled statistically

by this research. Therefore, the MLEM's performance is somehow more regulative in the

noise-perturbed condition.

For 50% accuracy, the confidence level was 97.9% for the 2-energy CGNN

method and 99.0% for the 2-energy MLEM method. A most likely accuracy for both the

CGNN and MLEM methods under 20% heterogeneity condition can be determined from

Table 7-4. It is defined as the highest accuracy with more than 95% confidence level

which can be found from the table. For both 2-energy CGNN and MLEM results shown

in Table 7-4, no confidence level greater than 95% could be found for accuracy higher

than 30%. The confidence level for 30% accuracy was 95.1% for the 2-energy CGNN

method and 96.5% for the 2-energy MLEM method at 20% noise level. Therefore, the

most likely accuracy for both of them was said to be 30% error with a noise level up to

20%.

Distributions of relative prediction error for 2-energy CGNN and MLEM methods

under 5%, 20% and 50% noise level are shown in Figure 7-4. Maximum relative error







76








0.20 I0I.I, I I 1, o20 | I I I I I, I I I-


C2NO5KE2 M2NO5KE2
0.15 0.15



0.10 0.10
-,


2 005 0.05



000 000 :-

-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70 -70-60-50-40-30-20-10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%) Relative Error of Total Activity Estimation (%)


0.15 I 1 1 1 1 1 1 1 1 1 1 , 1 0.15 IIII I 1 1 1 1 1 1 1 1 1 ,


C2N2OKE2 M2N20KE2

0.10 0.10

0N


0.05 0.05

12
LL


0.00 -T -1 000 -TT

-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70 -70 -60 -50 -40 -30 -20-10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%) Relative Error of Total Activity Estimation (%)


015I 05 -


C2N50KE2 M2N50KE2

0.10 0.10

0


Q 0.05 -- 0.05 -







-80-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70 80 -80-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70 80
Relative Error of Total Activity Estimation (%) Relative Error of Total Activity Estimation (%)



Figure 7-4. Distributions of relative prediction eiTor for 2-energy CGNN and MLEM

methods under the 20% heterogeneity and different noise-perturbed condition








for the CGNN in these three noise levels was +60%, +60%, and +70%, respectively. For

the MLEM, it was +55%, +60%, and +65%, respectively. The frequency of occurrence

for all these maximum relative errors was below 0.5%.

Simulation Results of the 50% Heterogeneity Level

Table 7-5 shows simulation results for different noise levels under 50% density

heterogeneity (simulation category (v)). Figure 7-5 shows the distributions of relative

prediction error for 2-energy CGNN and MLEM methods under 5%, 20% and 50% noise

levels. The results of 2-energy and 3-energy CGNN and MLEM reconstruction schemes

are compared at the 50% noise level in Table 7-5. Again, it shows that for the CGNN,

3-energy reconstruction deteriorated the results of 2-energy reconstruction. The

confidence level for 50% accuracy decreased from 97.8% to 96.2%. For 30% accuracy,

the confidence level decreased from 81.8% to 78.6%. For the MLEM, the prediction

ability of 3-energy reconstruction was comparable to that of 2-energy reconstruction.

Confidence levels for 30% and 25% accuracy deteriorated, but for 20% and 10%

accuracy, the confidence level is improved. This means that the overall performance of

3-energy MLEM reconstruction was limited when the noise level was as high as 50%.

Therefore, for both CGNN and MLEM application in the high noise condition, the

2-energy reconstruction scheme was the optimal choice.

For 50% accuracy, both 2-energy CGNN and 2-energy MLEM reconstruction

schemes provided more than a 95% confidence level under a noise level as high as 50%.

These results verified the fact that 36 experimental results described in Chapter 6 were all










Table 7-5. Confidence levels for total activity estimation by different optimization
algorithms under the 50% heterogeneity and noise-perturbed condition


Category (v) : 50% Heterogeneity with Local
Nonuniformity and Statistical Noise


Error <50% <30% <25% <20% <10%

C5N01KE2* 0.996 0.953 0.922 0.851 0.550
M5NO1KE2 0.997 0.953 0.919 0.853 0.585

C5NO5KE2 0.998 0.954 0.918 0.844 0.531
M5NO5KE2 0.996 0.953 0.916 0.850 0.573

C5NlOKE2 0.998 0.955 0.910 0.834 0.515
M5N1OKE2 0.996 0.956 0.910 0.840 0.554

C5N20KE2 0.998 0.942 0.892 0.789 0.458
M5N20KE2 0.997 0.951 0.893 0.814 0.498

C5N5OKE2 0.978 0.818 0.719 0.605 0.315
M5N5OKE2 0.986 0.828 0.738 0.628 0.319
C5N5OKE3 0.962 0.786 0.706 0.582 0.296
M5N5OKE3 0.986 0.825 0.730 0.636 0.335
Notation for the 1st column- C5: CGNN and 50% heterogeneity;
M5: MLEM and 50% heterogeneity; E2: 2-energy; E3: 3-energy
K: perturbed density kernel table; NO 1: 1% noise; N05: 5% noise;
N10: 10% noise; N20: 20% noise; N50: 50% noise.
















0.20


-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%)


0.00


-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%)


0.15


0.00


-80-70 -0-50-40-30 -20-10 0 10 20 30 40 50 60 70 80
Relative Error of Total Activity Estimation (%)


-70-60-50-40-30 -20-10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%)


-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%)


-80-70 -60-50-40-30-20-10 0 10 20 30 40 50 60 70 80
Relative Error of Total Activity Estimation (%)


Figure 7-5. Distributions of relative prediction error for 2-energy CGNN and MLEM

methods under the 50% heterogeneity and different noise-perturbed condition








within 50% error. Using the 2-energy CGNN reconstruction scheme, the most likely

accuracy was 30% error with a noise level up to 10%. However, using 2-energy MLEM

reconstruction scheme, the most likely accuracy was found to be 30% error with a noise

level up to 20%.

For the CGNN, the maximum relative error of the three noise levels shown in

Figure 7-5 was +60%, +70%, and +75%, respectively. For the MLEM, it was +65%,

+65%, and +70%, respectively. The frequency of occurrence for all these maximum

relative errors was below 0.5%.

Simulations of Biased Source Distribution Cases


Using the Monte Carlo process in the computer model, a source fraction for each

computational grid was randomly assigned. However, this process was not be able to

generate biased source distribution cases. In order to check if these source distribution

cases could produce biased total activity estimation, a total of 100 biased source

distribution cases were simulated and the total activity of each of them were

reconstructed by using 2-energy CGNN method. These 100 cases encompassed six types

of biased source distribution: (a) single source concentrated in a grid, (b) clustered source

concentrated in a few peripheral grids, (c) clustered source concentrated in a few central

grids, (d) sources distributed symmetrically in peripheral grids, (e) sources scattered in a

quadrant or half of the 37 grids, and (f) sources distributed symmetrically in central grids.

Figure 7-6 shows representative configurations for these six types of source distribution.

Table 7-6 shows that resulting confidence levels based on these 100 cases were

comparable to the results of the 2-energy CGNN method which were based on 10,000



















(a) Single Source Concentrated in a Grid


(c) Clustered Sources in Central Grids (d) Sources Distributed Symmetrically in Peripheral Grids


(e) Sources Scattered in a Quadrant (f) Sources Distributed Symmetrically in Central Grids

Figure 7-6. Representative configurations for six types of biased source distribution


(b) Clustered Sources in Peripheral Grids












Table 7-6. Comparison of the confidence levels of 10,000 random source distribution
cases and 100 biased source distribution case

Error <50% <30% <25% <20% <10%


C2NO5KE2* 0.998 0.964 0.928 0.857 0.557
C2NO5KE2-BS 1.000 0.960 0.920 0.860 0.560

C2N20KE2 0.998 0.951 0.896 0.805 0.495
C2N20KE2-BS 1.000 0.940 0.880 0.810 0.500

C2N50KE2 0.979 0.821 0.743 0.630 0.314
C2N50KE2-BS 0.970 0.810 0.730 0.640 0.320

C5N05KE2 0.998 0.954 0.918 0.844 0.531
C5NO5KE2-BS 1.000 0.960 0.930 0.870 0.570

C5N20KE2 0.998 0.942 0.892 0.789 0.458
C5N20KE2-BS 1.000 0.940 0.910 0.810 0.410

C5N50KE2 0.978 0.818 0.719 0.605 0.315
C5N50KE2-BS 0.990 0.850 0.760 0.610 0.290

Notation for the 1 st column- C2: CGNN and 20% heterogeneity;
C5: CGNN and 50% heterogeneity; E2: 2-energy;
K: perturbed density kernel table; BS: biased source distribution cases;
N05: 5% noise; N20: 20% noise; N50: 50% noise.








random source distribution cases. Distributions of relative prediction error are shown in

Figure 7-7. It is evident that these 100 highly biased source distribution cases do not

deteriorate the results or cause significant biased effects with respect to those presented in

the previous section.

Simulations Using the Base Density Kernel Table


The numerical simulations of the heterogeneous condition presented above were

performed by simulating a perturbed density distribution and constructing a perturbed

kernel table. Using the perturbed kernel table and a random source distribution, the

forward projection of ideal detector responses was obtained. Ideal detector responses

were perturbed by the noise effect to simulate observed detector responses. Total activity

was then reconstructed based on the known perturbed kernel table and the simulated

detector responses. This simulates a situation in which transmission tomography or

radiography is performed and detailed density information is provided. In a situation in

which detailed density information can not be provided, for instance when application of

transmission tomography is impossible, backward reconstruction of total activity can only

be implemented by using the appropriate base density kernel table. A base density kernel

table is constructed based on the average density of the waste barrel. A prior knowledge

of the weight and volume of the barrel or a transmission density measurement can help on

the determination of the average density. However, resulting confidence levels will be

different from results based on a known density distribution. In this research, perturbed

density was calculated from Equation 7-1 in which the base density used is 1.0 g/cm3.

Naturally, the simulation system had an average density close to 1.0 g/cm3 as well. This
































-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%)

'l'lil'l'l'l1l,1'l'l'l11111

C2N20KE2 Based on 100
Biased Source Distributions















-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%)


-70-60-50-40-30-20 -10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%)


CSN05KE2 I Based on 100
Biased Source Distributions















-70-60-50-40-30 -20-10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%)


-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%)


-70-60-50-40-30 -20-10 0 10 20 30 40 50 60 70
Relative Error of Total Activity Estimation (%)


Figure 7-7. Distributions of relative prediction error for 2-energy CGNN method using
100 biased source distribution cases under heterogeneity and noise-perturbed conditions


0.30


0.00


0.20


0.00


0.20








section presents numerical simulations for the 20% and 50% heterogeneity conditions in

which detailed density information was unknown and therefore, the base density kernel

table was used for backward reconstruction.

Table 7-7 shows the comparisons of confidence levels for the prediction of total

activity using perturbed kernel and base density kernel tables under the 20%

heterogeneity condition. Table 7-8 shows the same comparisons under the 50%

heterogeneity condition. From these two tables, it is evident that accuracy was degraded

when using the base density kernel table as compared to when using the perturbed kernel

table. This is because density heterogeneity became an extra noise component added to

ideal detector responses which were calculated using the perturbed kernel table. Under

20% heterogeneity (Table 7-7), the confidence levels of 30% accuracy for the CGNN

were 91.1% for the 5% noise level, 87.7% for the 20% noise level, and 76.1% for the

50% noise level. For the MLEM, these confidence levels were 93.9%, 90.0%, and

79.4%. For 30% accuracy at the 50% heterogeneity level (Table 7-8), the CGNN had the

confidence levels of 88.5% for the 5% noise level, 85.6% for the 20% noise level, and

74.4% for the 50% noise level. For the MLEM, these confidence level were 91.1%,

88.8%, and 76.8%. At the 20% heterogeneity and 50% noise level (Table 7-7), the

confidence level of 50% accuracy for the CGNN was 93.8% and for the MLEM was

96.8%. At the 50% heterogeneity and 50% noise level (Table 7-8), the confidence level

of 50% accuracy for the CGNN was 93.5% and for the MLEM was 95.5%.

When detailed density information was not available and only an average density

was known, using the 2-energy CGNN reconstruction scheme, the most likely accuracy

was 50% error with a noise level as high as 20%. Using the 2-energy MLEM












Table 7-7. Comparisons of confidence levels for activity reconstruction using perturbed
kernel table and base density kernel table under 20% heterogeneity condition

Category (iv) : 20% Heterogeneity with Local
Nonuniformity and Statistical Noise

Error <50% <30% <25% <20% <10%

C2NO5KE2* 0.998 0.964 0.928 0.857 0.557
C2N05BE2 0.990 0.911 0.839 0.753 0.468

C2N20KE2 0.998 0.951 0.896 0.805 0.495
C2N20BE2 0.984 0.877 0.810 0.718 0.425

C2N50KE2 0.979 0.821 0.743 0.630 0.314
C2N50BE2 0.938 0.761 0.661 0.543 0.284

M2NO5KE2 0.998 0.970 0.944 0.866 0.588
M2NO5BE2 0.996 0.939 0.884 0.802 0.499

M2N20KE2 0.999 0.965 0.912 0.826 0.524
M2N20BE2 0.995 0.900 0.846 0.752 0.455

M2N50KE2 0.990 0.840 0.757 0.639 0.328
M2N50BE2 0.968 0.794 0.687 0.571 0.295
*Notation for the 1 st column- C2: CGNN and 20% heterogeneity;
M2: MLEM and 20% heterogeneity; E2: 2-energy;
K: perturbed density kernel table; B: base density kernel table;
N05: 5% noise; N20: 20% noise; N50: 50% noise.












Table 7-8. Comparisons of confidence levels for activity reconstruction using perturbed
kernel table and base density kernel table under 50% heterogeneity condition

Category (v): 50% Heterogeneity with Local
Nonuniformity and Statistical Noise

Error <50% <30% <25% <20% <10%

C5N05KE2* 0.998 0.954 0.918 0.844 0.531
C5NO5BE2 0.984 0.885 0.830 0.741 0.449

C5N20KE2 0.998 0.942 0.892 0.789 0.458
C5N20BE2 0.977 0.856 0.785 0.688 0.406

C5N50KE2 0.978 0.818 0.719 0.605 0.315
C5N50BE2 0.935 0.744 0.669 0.528 0.286

M5NO5KE2 0.996 0.953 0.916 0.850 0.573
M5NO5BE2 0.993 0.911 0.853 0.761 0.462

M5N20KE2 0.997 0.951 0.893 0.814 0.498
M5N20BE2 0.992 0.888 0.806 0.713 0.408

M5N50KE2 0.986 0.828 0.738 0.628 0.319
M5N50BE2 0.955 0.768 0.675 0.548 0.290
*Notation for the 1 st column- C5: CGNN and 50% heterogeneity;
M5: MLEM and 50% heterogeneity; E2: 2-energy;
K: perturbed density kernel table; B: base density kernel table;
N05: 5% noise; N20: 20% noise; N50: 50% noise.