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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 nonnegative 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 
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DEVELOPMENT OF OPTIMIZED SOLUTION METHODS FOR INVERSE RADIATION TRANSPORT PROBLEMS INVOLVING THE CHARACTERIZATION OF DISTRIBUTED RADIONUCLIDES IN THE LARGE WASTE CONTAINER By CHINJEN 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 ChinJen 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 FourDetector Positions ............ 11 The Concept of Using Multiple Gamma Energy Peaks .......................................... 11 Computer Modeling and Verification Procedure ................................................... 14 4 CONJUGATE GRADIENT WITH NONNEGATIVE 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 NoiseFree Condition .............................................................. 64 Comparisons with the Segmented Gamma Scanning Method ................................ 66 Heterogeneous and NoiseFree Conditions ............................................................ 68 Heterogeneous and NoisePerturbed 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 41. Results of the convergence test for different zero criteria of CGNN by using 10,000 random source distribution cases of homogeneous density .............. 23 51. Results of the convergence test for different iteration number of MLEM by using 10,000 random source distribution cases of homogeneous density ......... 27 61. Characteristics of check sources and related efficiency calibrations ................ 33 62. Activity estimation errors for 18 homogeneous density experiments by using multiple energy peaks in the CGNN and MLEM reconstruction algorithms .... 43 63. Summary of activity estimation errors for 18 homogeneous density experiments by using multiple energy peaks in the CGNN and MLEM algorithms ...... 44 64. Location of different material channels inside the barrel for the study of Type A and Type B density heterogeneity ................................................................. 46 65. Activity estimation errors for the heterogeneous density experiments by using multiple energy peaks in the CGNN and MLEM algorithms ....................... 50 66. Summary of activity estimation errors for the heterogeneous density experiments by using multiple energy peaks in the CGNN and MLEM algorithms ...... 51 67. Summary of activity prediction for 36 experiments .......................................... 53 68. Comparisons of ideal and observed detector responses for experiment case H1 and case H 3 .................................................................................................. 57 71. Confidence levels for total activity estimation by different optimization algorithms under the homogeneous and noisefree condition ...................... 65 72. Comparisons of confidence levels between the SGS and the 2energy CGNN and MLEM reconstruction algorithms under the homogeneous and noisefree condition ..................................................................................... 69 73. Confidence levels for total activity estimation by different optimization algorithms under the heterogeneous and noisefree condition ...................... 71 74. Confidence levels for total activity estimation by different optimization algorithms under the 20% heterogeneity and noiseperturbed condition .......... 74 75. Confidence levels for total activity estimation by different optimization algorithms under the 50% heterogeneity and noiseperturbed condition ......... 78 76. Comparison of the confidence levels of 10,000 random source distribution cases and 100 biased source distribution cases ..................................................... 82 77. Comparisons of confidence levels for activity reconstruction using perturbed kernel table and base density kernel table under 20% heterogeneity condition ..................................................................................................... . 86 78. Comparisons of confidence levels for activity reconstruction using perturbed kernel table and base density kernel table under 50% heterogeneity condition ..................................................................................................... . 87 79. Confidence levels for activity estimation using different reconstruction algorithms and eight detector positions ....................................................... 89 710. Confidence levels for activity estimation using the base density kernel table and eight detector positions ......................................................................... 94 711. Confidence levels of activity estimation for the full length of a 208 e waste barrel under 50% heterogeneity and different noiseperturbed conditions ...... 101 LIST OF FIGURES Figure page 11. A generic configuration of a distributed source S in a selfabsorbing medium of volume V measured by several external detectors .......................................... 3 31. Configuration of a fourdetector positions measuring system in a vertical segment of a 208 e nuclear waste barrel ....................................................... 12 32. ir'92 gam m a spectrum .......................................................................................... 15 61. Experimental setup for the 208 e waste barrel characterization ......................... 30 62. The acrylicglass matrix container inside the waste barrel ............................... 31 63. The efficiency calibration curve for HPGe detector .......................................... 34 64. The ROI for a 316 keV energy peak of the Ir'92 gamma spectrum .................... 36 65. The ROI for the 468 keV and 588 keV energy peaks of the Ir192 spectrum ..... 38 66. Six materials of different density in the insertion channels and an experimental heterogeneous configuration inside the waste barrel ................................... 40 67. Configurations of Ir'92 source distribution for 18 experiments under the hom ogeneous density condition ................................................................... 42 68. Density distribution and configurations of Ir192 location for 9 experiments under the Type A heterogeneity condition ................................................... 47 69. Density distribution and configurations of Ir'92 location for 9 experiments under the Type B heterogeneity condition ................................................... 48 71. Distributions of relative prediction error for the 2energy CGNN and MLEM algorithms under the homogeneous and noisefree condition ..................... 67 72. Distributions of relative prediction error for the SGS method using 4 and 64 detector positions under the homogeneous and noisefree condition ........... 70 73. Distributions of relative prediction error for 2energy CGNN and MLEM methods under the heterogeneous and noisefree condition ......................... 73 74. Distributions of relative prediction error for 2energy CGNN and MLEM methods under the 20% heterogeneity and different noiseperturbed condition ..................................................................................................... . 76 75. Distributions of relative prediction error for 2energy CGNN and MLEM methods under the 50% heterogeneity and different noiseperturbed condition ..................................................................................................... . 79 76. Representative configurations for six types of biased source distribution ...... 81 77. Distributions of relative prediction error for 2energy CGNN method using 100 biased source distribution cases under heterogeneity and noiseperturbed conditions ................................................................................................... . 84 78. Distributions of relative prediction error for 2energy CGNN and MLEM methods (8 detector positions) under the 20% heterogeneity and different noiseperturbed conditions ............................................................................ 91 79. Distributions of relative prediction error for 2energy CGNN and MLEM methods (8 detector positions) under the 50% heterogeneity and different noiseperturbed conditions ............................................................................ 92 710. Confidence levels of activity estimations by using perturbed kernel table and base density kernel table in the 2energy CGNN method under different base density conditions ......................................................................................... 97 711. 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 ChinJen 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 selfabsorbing 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 eggcrate 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 selfabsorbing medium. Figure 11 shows a generic configuration of a distributed source S in a selfabsorbing 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 (11) 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 (12) 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 11 substituted into Equation 12 for the calculation of total source strength Stot. In matrix form, the problem can be represented as follows: D=K.S (13) lJ D. x Figure 11. A generic configuration of a distributed source S in a selfabsorbing 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 13 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 13 is either highly singular or the response kernel K is so illconditioned 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 selfabsorbing 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 eggcrate 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.89 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..011 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 photofission and photoneutron reactions. Neutron detection methods usually need a coincidence counting system and time of flight detection method to distinguish between timecorrelated 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 FourDetector 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 31 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 31. 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 13 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 (31) 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 fourdetector 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 31 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 fourdetector positions and two energy peaks detection system, Equation 13 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) K22) 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 wellseparated gamma peaks ranging from 296 keV to 884 keV (Figure 32), 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 fourdetector 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 32) 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 32. 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 noisefree 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 NONNEGATIVE 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 mid1970s.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 13 can be found by minimizing the following squared error (see the derivation in Appendix A): e= DKS =(DKS)t(DK9) (41) 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" (42) 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 (43) with r=K' (D KSJ) (44) Equation 43 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 (45) To minimize e, we need to take the partial derivative of Equation 41 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 GramSchmidt 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 (47) and the recursion relation (see the derivation in Appendix C) is dn = rn 3,,_ld._ or d,, = r+, 6,d (48) where /6. is the orthogonality factor. Taking the inner product of Equation 48 with K' Kdn_1 and applying the property of Equation 47, 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 46, calculate S,+ based on Equation 42, calculate r.1 based on Equation 45, calculate /3n based on Equation 49, and then calculate d,,,, based on Equation 48. 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 41 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 44 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 (410) 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 410 are checked. If the criteria in Equation 410 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 410 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 410 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 44. That is 37 ZrPm ZERO = = (411) 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 41 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 41. 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 41. 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 6,674 104 10,000 10,000 10,000 9,993 9,341 105 10,000 10,000 10,000 10,000 9,783 106 10,000 10,000 10,000 10,000 9,901 107 10,000 10,000 10,000 10,000 9,823 10 10,000 10,000 10,000 10,000 9,824 109 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 13 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(51) 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()) (52) 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 (53) i=1 j=1 XO ! where m is the total number of detector response. Its logarithm is ln f (X,S) = ly (X, + X, lnY, lnX !) i j = + X, ln(KSj) lnX,!) (54) i j The conditional expectation of X, in Equation 54 with respective to Di and S(") is K S(n D U= E(X D (55) N K ,S (n) Because the third term in Equation 54 has no dependence on the iteration of S, the conditional expectation of Equation 54 becomes E(lnf (X,S) D,S (') =I[K Sj + N, ln(KySj)1+C (56) 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. (57) The partial derivative of the above equation is 92 E(Inf(X,S) D,S(n)) 1 SNU (58) Equation 58 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 ZKuZK,,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 59, 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 51 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 106. 51. 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 61 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 GEM10195P) has a transistorreset 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 62) with 37 eggcrate type compartments made of acrylicglass (1.5 mm in thickness) was built and cut to fit the round shape of the barrel at the periphery. This acrylicglass 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 PCAT 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 MAESTROII controlled Figure 61. Experimental setup for the 208 e waste barrel characterization Figure 62. The acrylicglass 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 61 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 63 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 104 9.88770 x 107 E + 9.60628 x 10 E2 3.22253 x 103 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 61. 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) 2.151Ox104 122.3 Co57 (0.8560) 3.4730x1O4 136.8 Co5" (0.1068) 3.2070x104 276.3 Ba33 (0.0715) 2.7140x104 302.8 Ba'33 (0.1830) 2.2530x104 356.0 Ba133 (0.6194) 1.8405x104 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 * Halflife: Ba'33 10.57 years, Na222.60 years, Co57271.8 c Cs13730.17 years, Mn54312 days, 4 E1 11 ,,Iii II 3.50E4 Check Source Caibrabon Points  Fit 2: 3rd Order Polynomial Fitting . 3.00E4 X Fitted Poins CU " 2.50E4 I 0 Cl) 316 keV L 2.00E4 468 keV U  588 kev o 1.50E4 58V 1.00E4 5.00E5 'I I i Iiii 0 300 600 900 1200 1500 Gamma Energy (keV) Figure 63. The efficiency calibration curve for HPGe detector Region of Interest of Ir92 Gamma Spectrum MAESTROII 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 64 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 straightline background. The background is given by the following equation: B =+2C + Ci)hl+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 (62). 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: h3 Aag =j C' (63) i=1+3 The net area is the adjusted gross area minus the adjusted background as follows: B. (h l 5) (64) An=Ao (hI+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 64. 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 64, 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 FullWidthHalfMaximum (FWHM) of this peak is 1.77 keV. The ROI encompasses a region three times that of the FWHM. Figure 65 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 66 shows that the acrylicglass 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 65. 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 66 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 67 shows the positions of Ir'92 sources for 18 experiments under the homogeneous density condition. These positions were chosen to represent different Figure 66. 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 13, 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 62. Using the CGNN method, estimation errors ranged from 33.30% to 27.52% for 2energy reconstruction and from 34.38% to 28.67% for 3energy reconstruction. Using the MLEM method, estimation errors ranged from 29.22% to 23.79% for 2energy reconstruction and from 28.30% to 24.62% for 3energy reconstruction. Table 63 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 67. Configurations of Ir92 source distribution for 18 experiments under the homogeneous density condition (0 1 Ir192 piece ;@: 3 Ir92 pieces) Table 62. Activity estimation errors for 18 homogeneous density experiments by using multiple energy peaks in the CGNN and MLEM reconstruction algorithms CGNN Reconstruction Error (%) Case 2Energy 3Energy 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 (%) 2Energy 3Energy 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 63. 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% 2Energy 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%) 3Energy 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 2energy reconstruction, and 18 out of 18 cases (100%) by using 3energy reconstruction. For 20% error, the best prediction of the four reconstruction schemes was the MLEM using 3energy 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 eggcrate 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 64 was the average value from these four measurements. Table 64 also lists the positions of different material channels inserted into the barrel for both Type A and Type B heterogeneity conditions. Figure 68 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 69. As in the homogeneous density Table 64. 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 68. 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 69. 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 65. 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 66 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 2energy reconstruction, and MLEM predicted 8 out of 9 cases (88.9%). By using 3energy 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 2energy or 3energy reconstruction, and MLEM predicted 9 out of 9 cases (100%) by using 2energy or 3energy reconstruction. All the 9 cases predicted by MLEM using 3energy 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 65. Activity estimation errors for the heterogeneous density experiments by using multiple energy peaks in the CGNN and MLEM algorithms CGNN Reconstruction Error (%) Case 2Energy 3Energy MLEM Reconstruction Error (%) 2Energy 3Energy Number of Ir'92 Pieces in the Barrel Cases of Type A Heterogeneity: (A1A9) 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 66. 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 2Energy 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%) 3Energy 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 2Energy 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%) 3Energy 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 67 summarizes all results. All experiments were within 50% accuracy. Using 2energy reconstruction, the CGNN predicted 32 cases (88.9%) and the MLEM predicted 34 cases (94.4%) with 30% accuracy. Using 3energy 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 3energy 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 67. Summary of activity prediction for 36 experiments Error <50% <30% <25% <20% <10% 2Energy 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%) 3Energy 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 acrylicglass 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 61 and Figure 63). The efficiency of the 302.8 keV gammaray 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 31. The 316 keV kernel responses for grid 19 or grid 36 with respect to detector position 1 were minK,,19 = 2.9166x106 maxK,,,9 = 1.0995x10' aveK,19 = 6.0387x106; minK,36 = 1.9622x104 maxKI,36 = 1.2040x103 aveK,36 = 5.3356x104; 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, 36v )= 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 68 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 68. 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 68. 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 = VD/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 68. 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)] (71) 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 gammarays 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 31, based on calculated path length and density distribution calculated from Equation 71. 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 32. 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)] (72) 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 68, 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 72 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(_(X2)(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 73, 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 noisefree, (ii) heterogeneous with 20% density perturbation and noisefree, (iii) heterogeneous with 50% density perturbation and noisefree, (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 ' (74) '( 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 1energy, 2energy and 3energy 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 1energy, 2energy, and 3energy 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 NoiseFree Condition Table 71 shows simulation results of different reconstruction schemes under the homogeneous and noisefree condition (simulation category (i)). In this simulation category, 1energy, 2energy, and 3energy reconstructions were used for both the CGNN and MLEM algorithms. Confidence levels of 1energy 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 3energy CGNN reconstruction. The confidence level for 10% error was 92.1% for Table 71. Confidence levels for total activity estimation by different optimization algorithms under the homogeneous and noisefree condition Category (i) : Homogeneous and NoiseFree 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: 1energy; E2: 2energy; E3: 3energy. 3energy MLEM reconstruction. In this simulation condition (category (i)), prediction ability of the CGNN proved better than that of the MLEM. Figure 71 shows the distributions of relative prediction error for 2energy 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 70605040302010 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 70605040302010 0 10 20 30 40 50 60 70 Relative Error of Total Activity Estimation (%) Figure 71. Distributions of relative prediction error for the 2energy CGNN and MLEM algorithms under the homogeneous and noisefree condition Table 72 shows the results of the SGS method using various multiple detector positions and compares them with the results of the 2energy 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 72. The maximum error for 4 detector positions occurred at +170%. For 64 detector positions, the maximum error occurred at + 100%. Compared with the 2energy CGNN or MLEM method, the SGS's prediction was very poor. Heterogeneous and NoiseFree Conditions Simulation results of 2energy and 3energy CGNN and MLEM algorithms under heterogeneous and noisefree conditions (simulation categories (ii) and (iii) ) are shown in Table 73. For 20% heterogeneity, 2energy and 3energy CGNN reconstruction schemes predicted 10% error at 97% and 98% confidence levels respectively. Under the same conditions for the MLEM, 2energy and 3energy 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, 3energy reconstructions performed better than 2energy reconstructions, and the CGNN's prediction was better than that of MLEM. Table 72. Comparisons of confidence levels between the SGS and the 2energy CGNN and MLEM reconstruction algorithms under the homogeneous and noisefree 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 2energy CGNN reconstruction. Homogeneous density and 2energy MLEM reconstruction. 0.20 0.15 0.10 0.05 0.00 200150100 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  200150100 50 0 50 100 150 200 Relative Error of Total Activity Estirration (%) Figure 72. Distributions of relative prediction error for the SGS method using 4 and 64 detector positions under the homogeneous and noisefree condition Table 73. Confidence levels for total activity estimation by different optimization algorithms under the heterogeneous and noisefree condition Category (ii) : 20% Heterogeneity and NoiseFree 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 NoiseFree 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: noisefree; E2: 2energy; E3: 3energy; K: perturbed density kernel table. Distribution of relative prediction error for 2energy CGNN and MLEM algorithms are shown in Figure 73. Maximum relative error for the 2energy MLEM method was +30% and +60% for 20% and 50% heterogeneity, respectively. For the 2energy CGNN method, maximum relative errors was 20% and 30% for 20% and 50% heterogeneity, respectively. Heterogeneous and NoisePerturbed 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 noiseperturbed detector response. Simulation Results of the 20% Heterogeneity Level Table 74 shows simulation results for different noise levels under 20% density heterogeneity (simulation category (iv)). The results of 1energy, 2energy, and 3energy CGNN and MLEM reconstruction schemes are compared at 5% noise level. It is evident again that the total activity predictions of 1energy CGNN and MLEM methods were very poor. A surprising finding is that CGNN results deteriorated by changing from a 2 energy to a 3energy 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 3energy MLEM results at this noise level. Table 74 also clearly shows influences of noise at this heterogeneity level. For 2energy CGNN reconstruction, the confidence level for 30% accuracy decreased from  IIi I 111 ii_ C2NOOKE2 70605040 30 20 10 0 10 20 30 40 50 60 70 Relative Error of Total Activity Estimation (%) 70605040302010 0 10 20 30 40 50 60 70 Relative Error of Total Activity Estimation (%) 70605040302010 0 10 20 30 40 50 60 70 Relative Error of Total Activity Estimation (%) 70605040302010 0 10 20 30 40 50 60 70 Relative Error of Total Activity Estimation (%) Figure 73. Distributions of relative prediction error for 2energy CGNN and MLEM methods under the heterogeneous and noisefree condition Table 74. Confidence levels for total activity estimation by different optimization algorithms under the 20% heterogeneity and noiseperturbed 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: 2energy; E3: 3energy 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 2energy 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 noisefree 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 noiseperturbed condition. For 50% accuracy, the confidence level was 97.9% for the 2energy CGNN method and 99.0% for the 2energy MLEM method. A most likely accuracy for both the CGNN and MLEM methods under 20% heterogeneity condition can be determined from Table 74. It is defined as the highest accuracy with more than 95% confidence level which can be found from the table. For both 2energy CGNN and MLEM results shown in Table 74, 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 2energy CGNN method and 96.5% for the 2energy 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 2energy CGNN and MLEM methods under 5%, 20% and 50% noise level are shown in Figure 74. 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 : 70605040302010 0 10 20 30 40 50 60 70 70605040302010 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 70605040302010 0 10 20 30 40 50 60 70 70 60 50 40 30 2010 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  8070605040302010 0 10 20 30 40 50 60 70 80 8070605040302010 0 10 20 30 40 50 60 70 80 Relative Error of Total Activity Estimation (%) Relative Error of Total Activity Estimation (%) Figure 74. Distributions of relative prediction eiTor for 2energy CGNN and MLEM methods under the 20% heterogeneity and different noiseperturbed 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 75 shows simulation results for different noise levels under 50% density heterogeneity (simulation category (v)). Figure 75 shows the distributions of relative prediction error for 2energy CGNN and MLEM methods under 5%, 20% and 50% noise levels. The results of 2energy and 3energy CGNN and MLEM reconstruction schemes are compared at the 50% noise level in Table 75. Again, it shows that for the CGNN, 3energy reconstruction deteriorated the results of 2energy 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 3energy reconstruction was comparable to that of 2energy 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 3energy 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 2energy reconstruction scheme was the optimal choice. For 50% accuracy, both 2energy CGNN and 2energy 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 75. Confidence levels for total activity estimation by different optimization algorithms under the 50% heterogeneity and noiseperturbed 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: 2energy; E3: 3energy K: perturbed density kernel table; NO 1: 1% noise; N05: 5% noise; N10: 10% noise; N20: 20% noise; N50: 50% noise. 0.20 70605040302010 0 10 20 30 40 50 60 70 Relative Error of Total Activity Estimation (%) 0.00 70605040302010 0 10 20 30 40 50 60 70 Relative Error of Total Activity Estimation (%) 0.15 0.00 8070 0504030 2010 0 10 20 30 40 50 60 70 80 Relative Error of Total Activity Estimation (%) 7060504030 2010 0 10 20 30 40 50 60 70 Relative Error of Total Activity Estimation (%) 70605040302010 0 10 20 30 40 50 60 70 Relative Error of Total Activity Estimation (%) 8070 605040302010 0 10 20 30 40 50 60 70 80 Relative Error of Total Activity Estimation (%) Figure 75. Distributions of relative prediction error for 2energy CGNN and MLEM methods under the 50% heterogeneity and different noiseperturbed condition within 50% error. Using the 2energy CGNN reconstruction scheme, the most likely accuracy was 30% error with a noise level up to 10%. However, using 2energy 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 75 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 2energy 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 76 shows representative configurations for these six types of source distribution. Table 76 shows that resulting confidence levels based on these 100 cases were comparable to the results of the 2energy 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 76. Representative configurations for six types of biased source distribution (b) Clustered Sources in Peripheral Grids Table 76. 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 C2NO5KE2BS 1.000 0.960 0.920 0.860 0.560 C2N20KE2 0.998 0.951 0.896 0.805 0.495 C2N20KE2BS 1.000 0.940 0.880 0.810 0.500 C2N50KE2 0.979 0.821 0.743 0.630 0.314 C2N50KE2BS 0.970 0.810 0.730 0.640 0.320 C5N05KE2 0.998 0.954 0.918 0.844 0.531 C5NO5KE2BS 1.000 0.960 0.930 0.870 0.570 C5N20KE2 0.998 0.942 0.892 0.789 0.458 C5N20KE2BS 1.000 0.940 0.910 0.810 0.410 C5N50KE2 0.978 0.818 0.719 0.605 0.315 C5N50KE2BS 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: 2energy; 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 77. 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 71 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 70605040302010 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 70605040302010 0 10 20 30 40 50 60 70 Relative Error of Total Activity Estimation (%) 706050403020 10 0 10 20 30 40 50 60 70 Relative Error of Total Activity Estimation (%) CSN05KE2 I Based on 100 Biased Source Distributions 7060504030 2010 0 10 20 30 40 50 60 70 Relative Error of Total Activity Estimation (%) 70605040302010 0 10 20 30 40 50 60 70 Relative Error of Total Activity Estimation (%) 7060504030 2010 0 10 20 30 40 50 60 70 Relative Error of Total Activity Estimation (%) Figure 77. Distributions of relative prediction error for 2energy CGNN method using 100 biased source distribution cases under heterogeneity and noiseperturbed 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 77 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 78 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 77), 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 78), 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 77), 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 78), 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 2energy CGNN reconstruction scheme, the most likely accuracy was 50% error with a noise level as high as 20%. Using the 2energy MLEM Table 77. 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: 2energy; K: perturbed density kernel table; B: base density kernel table; N05: 5% noise; N20: 20% noise; N50: 50% noise. Table 78. 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: 2energy; K: perturbed density kernel table; B: base density kernel table; N05: 5% noise; N20: 20% noise; N50: 50% noise. 