Spatial vs ensemble statistics for spatial simulation algorithms

Spatial simulation algorithms are designed to generate realizations of random functions and to reproduce statistical characteristics of the random function. In practice only a small number of realizations are generated and hence the characteristics of the realizations are more important than those of the random function. Most algorithms reproduce the statistical characteristics, such as the covariance function, mean and univariate distribution, only in an average sense. At least second order stationarity is an implicit assumption in using the algorithms. Empirical and theoretical results for convergence are given for several of the commonly used algorithms such as LU, Sequential Gaussian and Turning Bands. Simulated Annealing is an algorithm where each realization reproduces the statistical properties with large probability.