Scaling multiple-point statistics to different univariate proportions

Multiple-point statistics are used in geostatistical simulation to improve forecasting of responses that are highly dependent on the reproduction of complex features of the phenomenon. Often, complex features cannot be captured by conventional two-point simulation methods, based on the variogram. Inference of multiple-point statistics requires a training image that depicts the geological features of the geological setting being modelled. The proportions of facies in the training image may not match the target statistics of the final model. This is a problem because taking multiple point statistics from a training image also takes the univariate proportions, that is, the multiple point statistics contain all lower order statistics. There is a need to scale multiple-point statistics to different target univariate proportions. In other cases, locally varying facies proportions must be honored, but a single training image is available. The multiple-point statistics from the training image are scaled to the appropriate target univariate proportions of facies. An iterative scaling approach based on the expression for scaling multiple-point statistics in a purely random case is proposed. The implementation is illustrated through an example where it is shown that the proposed method lies between two extreme cases for a Boolean simulation, namely, the change in size of the objects and the change in their number of occurrences. A second example is presented to illustrate the potential use of this scaling procedure for nonstationary multiple-point geostatistical simulation.