Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling

The advent of multiple-point geostatistics (MPS) gave rise to the integration of complex subsurface geological structures and features into the model by the concept of training images. Initial algorithms generate geologically realistic realizations by using these training images to obtain conditional probabilities needed in a stochastic simulation framework. More recent pattern-based geostatistical algorithms attempt to improve the accuracy of the training image pattern reproduction. In these approaches, the training image is used to construct a pattern database. Consequently, sequential simulation will be carried out by selecting a pattern from the database and pasting it onto the simulation grid. One of the shortcomings of the present algorithms is the lack of a unifying framework for classifying and modeling the patterns from the training image. In this paper, an entirely different approach will be taken toward geostatistical modeling. A novel, principled and unified technique for pattern analysis and generation that ensures computational efficiency and enables a straightforward incorporation of domain knowledge will be presented. In the developed methodology, patterns scanned from the training image are represented as points in a Cartesian space using multidimensional scaling. The idea behind this mapping is to use distance functions as a tool for analyzing variability between all the patterns in a training image. These distance functions can be tailored to the application at hand. Next, by significantly reducing the dimensionality of the problem and using kernel space mapping, an improved pattern classification algorithm is obtained. This paper discusses the various implementation details to accomplish these ideas. Several examples are presented and a qualitative comparison is made with previous methods. An improved pattern continuity and data-conditioning capability is observed in the generated realizations for both continuous and categorical variables. We show how the proposed methodology is much less sensitive to the user-provided parameters, and at the same time has the potential to reduce computational time significantly.