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, it is argued that any comparison of algorithms needs, at a minimum, to be based on these two randomizations. In fact, for certain MPS algorithms, it is illustrated with different examples that there is often a trade-off: Increased pattern reproduction entails reduced spatial uncertainty. In this paper, the subjective choice that the best algorithm maximizes pattern reproduction is made while at the same time maximizes spatial uncertainty. The discussion is also limited to fairly standard multiple-point algorithms and that our method does not necessarily apply to more recent or possibly future developments. In order to render these fundamental principles quantitative, this paper relies on a distance-based measure for both within-realization variability (pattern reproduction) and between-realization variability (spatial uncertainty). It is illustrated in this paper that this method is efficient and effective for two-dimensional, three-dimensional, continuous, and discrete training images.