Which Path to Choose in Sequential Gaussian Simulation

TLDR: A comprehensive analysis of the influence of the path on the simulation errors is presented, based on which guidelines for choosing an optimal path were developed, and indicates that the optimal path is defined as the one minimizing the information lost by the omission of neighbors.

Abstract: Sequential Gaussian Simulation is a commonly used geostatistical method for populating a grid with a Gaussian random field. The theoretical foundation of this method implies that all previously simulated nodes, referred to as neighbors, should be included in the kriging system of each newly simulated node. This would, however, require solving a large number of linear systems of increasing size as the simulation progresses, which, for computational reasons, is generally not feasible. Traditionally, this problem is addressed by limiting the number of neighbors to the ones closest to the simulated node. This does, however, result in artifacts in the realization. The simulation path, that is, the order in which nodes are visited, is known to influence the location and magnitude of these artifacts. So far, few rigorous studies linking the simulation path to the associated biases are available and, correspondingly, recommendations regarding the choice of the simulation path are largely based on empirical evidence. In this study, a comprehensive analysis of the influence of the path on the simulation errors is presented, based on which guidelines for choosing an optimal path were developed. The most common path types are systematically assessed based on the comparison of the simulation covariance matrices with the covariance of the underlying spatial model. Our analysis indicates that the optimal path is defined as the one minimizing the information lost by the omission of neighbors. Classification into clustering paths, that is, paths simulating consecutively close nodes, and declustering paths, that is, paths simulating consecutively distant nodes, was found to be an efficient way of determining path performance. Common examples of the latter are multi-grid, mid-point, and quasi-random paths, while the former include row-by-row and spiral paths. Indeed, clustering paths tend to inadequately approximate covariances at intermediate and large lag distances, because their neighborhood is only composed of nearby nodes. On the other hand, declustering paths minimize the correlation among nodes, thus ensuring that the neighbors are more diverse, and that only weakly correlated neighbors are omitted.