Synopsis
1 Introduction.- 1.1 The Nature of the Problem.- 1.2 Examples of Spatially Random Material.- 1.3 Notes.- 2 Deterministic Design I: Conceptual Formulation.- 2.1 Relative Mass Resolution.- 2.2 Response Functions.- 2.3 Point-Source Response Sets.- 2.4 The Convexity Theorem: Complete Response Sets.- 2.5 Relative Mass Resolution and the Concept of Expansion.- 2.6 Example: Pu Assay With One Detector.- 2.7 Example: Pu Assay With Two Detectors.- 2.8 Example: Coincidence Measurements.- 2.8.1 Formulation of the Assay Problem.- 2.8.2 One-Detector Assay Systems.- 2.8.3 Two-Detector Assay Systems.- 2.8.4 Multi-Detector Assay Systems: High-Order Coincidences.- 2.9 Computation of the Relative Mass Resolution.- 2.9.1 Intuitive Derivation.- 2.9.2 Exploiting Symmetry of the Response Set.- 2.9.3 Summary of the Algorithm for Evaluating the Relative Mass Resolution.- 2.10 Example: Pu Assay With Four Detectors.- 2.11 The Inclusion of Statistical Uncertainty.- 2.11.1 Single-Detector systems.- 2.11.2 Multi-Detector Systems.- 2.12 Example: Radioactive Waste Assay.- 2.13 Summary.- 2.13.1 The Deterministic Measure of Performance.- 2.13.2 Statistical Uncertainty.- 2.14 Notes.- 3 Deterministic Design II: General Formulation.- 3.1 Motivation.- 3.2 Unconstrained Spatial Distributions.- 3.3 Constrained Spatial Distributions.- 3.3.1 Evaluation of the Relative Resolution.- 3.3.2 Fundamental Response Sets.- 3.3.3 Inclusion of Statistical Uncertainty.- 3.3.4 Convexity and Compactness of $$\tilde C$$(h, u).- 3.4 Example: Simple Constrained Distributions.- 3.5 Example: Constrained Normal Distributions.- 3.6 Example: Meteorological Measurements.- 3.7 Example: Assay of a Pulmonary Aerosol.- 3.7.1 Formulation.- 3.7.2 Unconstrained Spatial Distributions.- 3.7.3 Constrained Spatial Distributions.- 3.8 Example: Thickness Measurement.- 3.8.1 Formulation.- 3.8.2 A Minimization Problem: Choosing Q.- 3.8.3 Resolution Without Statistical Uncertainty.- 3.8.4 Including Statistical Uncertainty.- 3.9 Example: Enrichment Assay.- 3.10 Auxiliary Parameters.- 3.11 Example: Variable Matrix Structure.- 3.12 Variable Spatial Distributions and Auxiliary Parameters.- 3.13 Constrained Time-Varying Distributions.- 3.14 Example: Flow-Rate Measurement.- 3.15 Notes.- 4 Probabilistic Interpretation of Measurement.- 4.1 Probability Density of the Measurement.- 4.1.1 Single Measurement of a Single Source Particle.- 4.1.2 Single Measurement of Identical Source Particles.- 4.1.3 Multiple Measurements of Identical Source Particles.- 4.1.4 Multiple Measurements of Nonidentical Source Particles.- 4.1.5 Example: Medical Whole-Body Assay.- 4.2 Probability Density of the Total Source Mass.- 4.2.1 Bayes' Rule.- 4.2.2 The Poisson Distribution.- 4.2.3 The Likelihood Function.- 4.2.4 Statistical Uncertainty.- 4.2.5 Example: Pu Assay With One Detector.- 4.2.6 Example: Assay of Aerosol Particles in the Lungs.- 4.3 Bayes' Decision Theory.- 4.3.1 General Formulation.- 4.3.2 Binary Decisions.- 4.3.3 Minimum Probability of Error.- 4.3.4 Quadratic Penalty.- 4.3.5 Biased Quadratic Penalty.- 4.3.6 Example: Plutonium Assay With One Detector.- 4.3.7 Estimating Continuous Parameters.- 4.4 Neyman-Pearson Decision Theory.- 4.4.1 Threshold Detection.- 4.4.2 Maximum Likelihood Estimation.- 4.5 Direct Probabilistic Calibration.- 4.5.1 General Formulation - One Detector.- 4.5.2 General Formulation - Multiple Detectors.- 4.5.3 Example: U Prospecting - Assay of a Thick Deposit.- 4.6 Summary.- 4.7 Notes.- 5 Probabilistic Design.- 5.1 Motivation.- 5.2 Relative Error Criterion.- 5.2.1 General Considerations.- 5.2.2 Example: U Prospecting - Assay of a Thin Deposit.- 5.3 Minimum Variance Criterion.- 5.3.1 Rao-Cramer Inequality.- 5.3.2 Examples.- 5.4 Probabilistic Expansion.- 5.4.1 Definition of the Overlap Function.- 5.4.2 Example: Overlap Functions.- 5.4.3 Reduction Theorems: Inequalities for the Overlap Function.- 5.5 Notes.- 6 Adaptive Assay.- 6.1 Motivation.- 6.2 Sequential Analysis.- 6.2.1 Formulation.- 6.2.2 Example: Batch
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