The SIM2D Procedure

Introduction to Spatial Simulation

The purpose of spatial simulation is to produce a set of partial realizations of a spatial random field (SRF) $Z(\bm {s}), \bm {s} \in D \subset \mathcal{R}^2$ in a way that preserves a specified mean $\mu (\bm {s})=E\left[Z(\bm {s})\right]$ and covariance structure $C_ z(\bm {s}_1-\bm {s}_2)=\mr{Cov}\left(Z(\bm {s}_1),Z(\bm {s}_2)\right)$. The realizations are partial in the sense that they occur only at a finite set of locations $(\bm {s}_1, \bm {s}_2,\cdots ,\bm {s}_ n)$. These locations are typically on a regular grid, but they can be arbitrary locations in the plane.

PROC SIM2D produces simulations for continuous processes in two dimensions by using the lower-upper (LU) decomposition method. In these simulations the possible values of the measured quantity $Z(\bm {s}_0)$ at location $\bm {s}_0=(x_0,y_0)$ can vary continuously over a certain range. An additional assumption, needed for computational purposes, is that the spatial random field $Z(\bm {s})$ is Gaussian. The section Details: SIM2D Procedure provides more information about different types of spatial simulation and associated computational methods.

Spatial simulation is different from spatial prediction, where the emphasis is on predicting a point value at a given grid location. In this sense, spatial prediction is local. In contrast, spatial simulation is global; the emphasis is on the entire realization $(Z(\bm {s}_1), Z(\bm {s}_2),\cdots ,Z(\bm {s}_ n))$.

Given the correct mean $\mu (\bm {s})$ and covariance structure $C_ z(\bm {s}_1-\bm {s}_2)$, SRF quantities that are difficult or impossible to calculate in a spatial prediction context can easily be approximated by functions of multiple simulations.