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## Über dieses Buch

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.

## Inhaltsverzeichnis

### 0. Introduction

Abstract
Visibility problems are of much interest in military applications of operations research, in communications and many other subject areas. Imagine that you are standing in a forest not far from a vehicle. You see the vehicle completely. The vehicle starts to move away from you, driving among the trees. After a short while you see only some parts of the vehicle, and shortly after that you lose sight of the vehicle completely. Trees are randomly dispersed between you and the vehicle and interfere with the lines of sight. This is a typical visibility problem. What is the stochastic or random elements of the problem? If you are told exactly where the trees are located, the width of their trunks and all other pertinent information, and asked what fraction of the vehicle you would be able to see after it drove in a certain direction 100 meters away from you, you will be able in principle to figure this out. The problem is deterministic. On the other hand, if we do not have all the pertinent information we could figure only, assuming that the trees are randomly dispersed, their sizes are random, etc., certain probabilities that the fractions of the vehicle that could be seen are of certain size. We turn the problem from a deterministic problem, which requires a lot of information, that is often unavailable, to a stochastic problem whose solution depends on the assumed model of randomness. The present book provides the reader the methods of determining visibility probabilities and related distributions, assuming that the objects which obscure the visibility are randomly distributed in certain regions according to a model called “Poisson random field”.
Shelemyahu Zacks

### 1. Probability Models

Abstract
In the present chapter we present the probability models which are basic to the problems discussed in the monograph. We provide also a short glossary to the distribution functions used in the following chapters. It is assumed that the reader is familiar with probability theory. The glossary is provided for establishing notation. The reader who needs additional studying of this material is referred to the books of William Feller (1968) or Sheldon Ross(1976).
Shelemyahu Zacks

### 2. Geometrical Probability, Coverage and Visibility In Random Fields

Abstract
The present chapter introduces some classical problems of geometrical probability coverage theory and random fields. The emphasis is, however, on practical aspects of the theory. The examples, which illustrate the general theory, focus on computational aspects. The objective in this chapter is to introduce basic concepts and techniques, which are encountered in later chapters.
Shelemyahu Zacks

### 3. Visibility Probabilities

Abstract
In the previous chapter we discussed various coverage problems. In the present chapter we start the development of the theory of stochastic visibility in random fields.
Shelemyahu Zacks

### 4. Visibility Probabilities II

Abstract
In the present chapter we consider problems associated with the visibility of targets from several observation points, subject to interference by a random field of obscuring elements, and visibility probabilities in three dimensional spaces. This chapter presents extensions of the results of Chapter 3. We consider a random field of obscuring elements, which are centered in a region located between the observation points and the targets. The targets are stationary points in the plane, or in space.
Shelemyahu Zacks

### 5. Distributions of Visibility Measures

Abstract
In the present chapter we develop methods for determining the distributions of some measures of visibility for a given field of obstacles. We start first with the distribution of the number of targets which are simultaneously visible, out of m specified target points, either from one or from several observation points. This distribution is not the frequently encountered binomial distribution, since the visibility for specified targets are generally not independent, and the visibility probabilities are generally not the same. The other measure of visibility studied in this chapter is an integrated measure of the total length, of a specified star-shaped curve, that can be observed from one or several observation points. These measures have various applications.
Shelemyahu Zacks

### 6. Distributions of Visible and Invisible Segments

Abstract
In the present chapter we develop formulae for the distributions of the length of visible and invisible (shadowed) segments of the target curve C in the plane. We will focus attention on trapezoidal fields, in which the strip S is bounded by two parallel lines U and W, and the target curve C is a straight line parallel to S. As before, the distances of U, W, and C from O are u, w and r, respectively, where 0 < u < w < r. We will assume that the Poisson field is standard and the distribution of the radius of a random disk centered in S is uniform on (a, b), and 2b < u. The trapezoidal region is a subset of S, C*, bounded by U, W and the rays $${_{x_L^ * }}$$ and where $${_{x_U^ * }}$$, where x*L and x*U are the rectangular coordinates of points on C specified below. Let $$\bar C$$ be an interval on C of interest. The rectangular x-coordinates of the points on $$\bar C$$ are bounded by x L and x U , x L > x U , and, as before,
$$x_L^* = {x_L} - b\frac{r}{u}{(1 + {(\frac{{{x_L}}}{r})^2})^{1/2}}$$
$$x_U^* = {x_U} - b\frac{r}{u}{(1 + {(\frac{{{x_U}}}{r})^2})^{1/2}}$$
.
Shelemyahu Zacks

### 7. Problems and Solutions

Abstract
In the present chapter we present problems for solution for each one of the previous six chapters. The problems are followed by the solution of the problem. Often the solutions are detailed. This is done in order to further explain the methodology. The furnished software is being utilized extensively in solving the problems. In solving problems of Chapter 2 we often use numerical integration. For this we have used either MATHCAD 4.0® or MATHEMATICA®.
Shelemyahu Zacks

### Backmatter

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