Multiple-point statistics and non-colocational soft data integration

Abstract

In geostatistics, available conditional information is typically categorized as either hard (with no uncertainty) or soft (associated with an uncertainty) data. 2-point based Gaussian (kriging) simulation methods have since their inception been able to account for hard and soft data. In contrast, multiple-point statistical (MPS) simulation methods, that allow quantifying more complex phenomena than Gaussian methods, have focused mostly on conditioning to hard data. The most widely used method for accounting for soft data is to consider only co-located soft data. Here we define conditional information data through a probability distribution and show how a general probabilistic solution, in the form of a posterior probability distribution, can be obtained, combining conditional categorical information with information from a categorical training image. We demonstrate how to compute the conditional distribution, as needed by most MPS-based sequential simulation algorithms, that allows accounting for both co- and non-co-located hard and soft data. Examples are provided comparing the direct computation of conditional statistics to using more computational demanding rejection sampling. In order to reduce the significant computational demands, a distance-based measure is proposed to increase the entropy of conditional data, based on the distance to the point being estimated and simulated.