When one has observed a random field Z at some points and recorded its values (network observations), a natural problem is to estimate Z at points where there are no observations. This dissertation deals first with this problem in an abstract setting, in m dimensions; later, it considers the estimation of a spatial two-dimensional random field. The problem then is one of constructing an estimated map of Z over a geographic area. For a given network of stations the quality of a map depends on the method of estimation. But for the given method of estimation the quality of a map depends on the choice of locations for the stations. This is the problem of network design. Both the study of methods of estimation and the problem of network design are addressed. 16 figures. (RWR).

The theory of spatial models over lattices, or random fields as they are known, has developed significantly over recent years. This book provides a graduate-level introduction to the subject which assumes only a basic knowledge of probability and statistics, finite Markov chains, and the spectral theory of second-order processes. A particular strength of this book is its emphasis on examples - both to motivate the theory which is being developed, and to demonstrate the applications which range from statistical mechanics to image analysis and from statistics to stochastic algorithms.

This book contains a novel theory of random fields estimation of Wiener type, developed originally by the author and presented here. No assumption about the Gaussian or Markovian nature of the fields are made. The theory, constructed entirely within the framework of covariance theory, is based on a detailed analytical study of a new class of multidimensional integral equations basic in estimation theory.This book is suitable for graduate courses in random fields estimation. It can also be used in courses in functional analysis, numerical analysis, integral equations, and scattering theory.