Comparing training-image based algorithms using an analysis of distance

As additional multiple-point statistical (MPS) algorithms are developed, there is an increased need for scientific ways for comparison beyond the usual visual comparison or simple metrics, such as connectivity measures. In this paper we start from the general observation that any (not just MPS) geostatistical simulation algorithm represents two types of variability: 1) the within-realization variability, namely, that realizations reproduce a spatial continuity model (variogram, Boolean or training-image based), 2) the between-realization variability representing a model of spatial uncertainty. In this paper, we argue that any comparison of algorithms needs, at a minimum, to be based on these two randomizations. In fact, for the MPS algorithms, we illustrate, with examples, that there is often a trade-off: increased pattern reproduction entails reduced spatial uncertainty. In this paper we make the subjective choice that the best algorithm maximizes pattern reproduction while at the same time maximizes spatial uncertainty. In order to render these fundamental principles quantitative, we rely on a distance-based measure for both within-realization variability (pattern reproduction) and between-realization variability (spatial uncertainty). To compare any two (or more) algorithms, we first generate a set of realizations with each algorithm. Each realization is up-gridded into a set of successively coarser grids; a single realization is turned into a multi-resolution pyramid of grids. The same operation is performed on the training image. We calculate for each realization the Jensen-Shannon divergence (a distance) between the pattern histograms of each multiresolution grid of the realizations generated with both algorithms and the training image. A weighted average of the Jensen-Shannon divergence of all multi-resolution grids represents a single quantitative measure of the within-realization variability (pattern reproduction) of a given algorithm. The same distance, but now calculated and averaged between sets of realizations of both algorithms is then a single quantitative measure of the between-realization variability. We illustrate in this paper that this method is efficient and effective for 2D, 3D, continuous and discrete training images. Multiple-point statistics, spatial uncertainty, training images, pattern modeling