Simulation of an Unconditional Random Function

Suppose we have a random function $Z(/boma{x})$ with a know spatial law. More specific, we suppose stationarity of second order and $m = E Z(/boma{x})$ and $/gamma(/boma{h}) = E[(Z(/boma{x}) - Z(/boma{x}+/boma{h}))^{2}]$ (or $C(/boma{h}))$ are known. For reason of simplicity we also suppose that $Z$ is normally distributed. The goal is to find simulated values of $Z(/boma{x})$.

Many methods are available for simulating one-dimensional realizations of a stationary stochastic process with a given covariance. However, when these procedures are extended to the three-dimensional space, they are often inextricable or prohibitive in terms of computer time. The originality of the method known as the ``turning bands'' method, originated by G. Matheron, derives from reducing all three-dimensional (or more generally, $n$-dimensional) simulations to several independent one-dimensional simulations along lines which are then rotated in the three-dimensional space $R^3$ (or $R^n$). This turning band method provides multidimensional simulations for reasonable computer costs, equivalent, in fact, to the cost of one-dimensional simulations.