Estimating the distribution of the field for Multiple Point Statistics

Multiple Point statistics typically provides a known distribution of the random field by means of the training image. Classical Geostatistics estimates the variogram, which is only an aspect of the distribution of the field. Both thus might use an inappropriate description of the distribution of the random field. The only exception are the high order cummulants methods and spline methods using a completly nonparametric approach. This contribution addresses the possibility to estimate the distributions for Nongaussian Random fields at the example of categorical random fields in a multiple point statistics setting. The core idea is to discribe possible characteristics of fields by using small training patches, which can be combinded to span a space of possible random field distribution models. The specific combination is selected by a distribution valued parameter, which can be estimated from an a sampled random fields using an estimation procedure based on observation likelihoods. Similarly to the difficulty in estimating the shape parameter of the Matern variogram there is little power in this procedure to estimate the roughness of boundaries. We will thus introduce a prior preweighting of the patches according to our physical assumptions about the boundaries. The same procedure allows to measure, how good the high order statistics of final simulation fit to the orignal observations. We will use this to check the conditional simulations for distributional consistency with the conditioning set.