Model based multiple point statistics and training model estimation

We would like to put the workflows of two point geostats and multi point geostatistics in a common framework of model estimation, model selection, algorithmic parameter selection and than finally simulation and prediction. This workflow is well established in to point geostats. In this contribution we extend it to multi point geostatistics. In this way we can improve the performance by better choices for the all method parameters.
Two point geostatistics typically puts three decisions ahead of each estimation or simulation: Based on the observations we choose a variogram model, estimate its parameters, and select a kriging neighbourhood to weigh between computational speed and algorithmic accuracy.
For Multiple Point statistics simulation methods we typically provide all knowledge about the spatial dependence by a fixed training image. Three different ways for generating training images have been proposed: Real maps or 3D models of the phenomenon, constructed images capturing our knowledge, and realisations of random field models. The random field model used for the simulation could be understood as our model of the distribution of the random field we would like to interpolate. The simulation of the training image and the following multiple point simulation is an imperfect numerical algorithm computing the conditional distribution based on that model. Just like two point geostatistics it again has the patterns as algorithmic parameters describing a geostatistical neighbourhood.
Like with variogram models, for any more complicated random fields models, we have model parameters. Depending on them different sets of observations will have different likelihoods. I.e. like with classical variogram estimation we can estimate the model parameters from the observations. As the likelihoods are typically not computable and variogram based methods can only capture two point dependences, we propose a quasilikelihood based multi point approach for the estimation of these parameters. Analog to variogram model comparison we also propose cross validation based methods to check the fit of the model and the performance of the simulation algorithm on the model. In analogy to the selection of search neighbourhood the simulation performance of the algorithm can be checked against the model.