Stochastic Visibility in Random Fields

The present monograph is a comprehensive summary of the research on visibility in random fields, which I have conducted with the late Professor Micha Yadin for over ten years. This research, which resulted in several published papers and technical reports (see bibliography), was motivated by some military problems, which were brought to our attention by Mr. Pete Shugart of the US Army TRADOC Systems Analysis Activity, presently called US Army TRADOC Analysis Command. The Director ofTRASANA at the time, the late Dr. Wilbur Payne, identified the problems and encouraged the support and funding of this research by the US Army. Research contracts were first administered through the Office of Naval Research, and subsequently by the Army Research Office. We are most grateful to all involved for this support and encouragement. In 1986 I administered a three-day workshop on problem solving in the area of sto­ chastic visibility. This workshop was held at the White Sands Missile Range facility. A set of notes with some software were written for this workshop. This workshop led to the incorporation of some of the methods discussed in the present book into the Army simulation package CASTFOREM. Several people encouraged me to extend those notes and write the present monograph on the level of those notes, so that the material will be more widely available for applications.

0. Introduction.- 0.1. Aims and Objectives.- 0.2. Some Military Applications.- 0.3. Synopsis.- 1. Probability Models.- 1.1. Probability Models For Obscuring Elements.- 1.2. Glossary of Distributions.- 1.2.1. Some Discrete Distributions.- 1.2.1.1. Binomial Distributions.- 1.2.1.2. Poisson Distributions.- 1.2.1.3. Multinomial Distributions.- 1.2.2. Some Continuous Distributions.- 1.2.2.1. Uniform Distributions.- 1.2.2.2. Beta Distributions.- 1.2.2.3. Gamma Distributions.- 1.2.2.4. Normal Distributions.- 1.2.2.5. Bivariate Normal Distributions.- 1.3. Random Fields.- 2. Geometrical Probability, Coverage and Visibility in Random Fields.- 2.1. Intersection of Lines By Random Segments.- 2.2. Random Lines Intersecting Circles.- 2.3. Random Disks Intersecting Lines.- 2.4. The Coverage of a Circle By Random Arcs.- 2.5. Vacancies On The Circle.- 2.6. Vacancies On The Plane.- 2.6.1. Vacancy of a Point under Bivariate Normal Dispersion.- 2.6.2. Complete Vacancy of a Triangle.- 2.7. Visitiblity of Points on a Circle In a Poisson Field.- 2.8. Distribution of Clump Size In a Poisson Field on The Line.- 3. Visibility Probabilities.- 3.1. Geometric Methods: Standard Poisson Fields.- 3.1.1. The Target(s) and Observation Point Are Within the Scattering Region.- 3.1.2. The Targets and Observation Points Are Outside a Rectangular Scattering Region.- 3.2. Analytic Methods: General Poisson Fields.- 3.2.1. The General Theoretical Framework.- 3.2.2. Standard Poisson Fields with Uniform Distribution of Radii.- 3.2.2.1. Annular Scattering Regions.- 3.2.2.2. Trapezoidal Scattering Regions.- 3.3. An Alternative Geometric-Analytic Method.- 3.3.1. Computing the Probability of B+(r) in the Bivariate Normal Case.- 3.4. The Visibility of Windows.- 4. Visibility Probabilities II.- 4.1. The Multi-Observer Multi-Target Shadowing Model and Simultaneous Visibility Probabilities.- 4.2. General Formulae of mk(n,n?) for the Standard Poisson Field.- 4.3. Determination of mk(n,n?) in Cases of Non-Standard Poisson Fields.- 4.4. Joint Visibility of Windows.- 4.5. Visibility of Points in Space.- 4.5.1. Single Target.- 4.5.2. Several Target Points on a Line.- 4.5.3. Uniform Distribution of Sphere Radius.- 4.5.4. Derivation of K(s,t).- 5. Distributions of Visibility Measures.- 5.1. The Distribution of The Number of Visible Targets.- 5.1.1. Introductory Examples With One Observation Point.- 5.1.2. General Method For Computing Probabilities of Elementary Events.- 5.1.3. Joint Distributions of Counting Variables.- 5.2. An Integrated Measure of Visibility on a Star-Shaped Curve.- 5.3. The Moments of W.- 5.4. Approximations to the Distribution of W.- 5.4.1. A Beta Approximation.- 5.4.2. Discrete Approximation.- 5.4.3. Recursive Determination of h(N)(k; ?).- 6. Distributions of The Visible and Invisible Segments.- 6.1. The Distribution of The Length of A Visible Segment.- 6.2. The Functions K+*(x,t) in the Standard-Uniform Case.- 6.3. Distribution of The Right-Hand Limit of A Shadow Cast by a Single Disk.- 6.4. Distribution of The Right Hand Limit of a Shadow Starting at a Given Point.- 6.5. Discrete Approximation.- 6.6. Distribution of the Number of Shadows.- 6.7. Survival Probability Functions.- 7. Problems and Solutions.- 7.1.1. Problems For Chapter 1.- 7.1.2. Solutions For Chapter 1.- 7.2.1. Problems For Chapter 2.- 7.2.2. Solutions For Chapter 2.- 7.3.1. Problems For Chapter 3.- 7.3.2. Solutions For Chapter 3.- 7.4.1. Problems For Chapter 4.- 7.4.2. Solutions For Chapter 4.- 7.5.1. Problems For Chapter 5.- 7.5.2. Solutions For Chapter 5.- 7.6.1. Problems For Chapter 6.- 7.6.2. Solutions For Chapter 6.- References.- Computer Programs.