Introduction

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”.