Abstract: |
Most traditional and non-traditional geostatistical techniques rely on some form of stationarity in their approach to creating model realizations. This assumption of stationarity is inherent to any statistical modeling technique. Often, random fields are decomposed into the sum of a trend component and a stationary stochastic component. Trends can be simple changes in mean, but can also consist of changes in affinity or angular direction variations or any other feature of the local spatial field. However, such decomposition limits the extent of real world spatial phenomena that can be modeled since not all phenomena can be easily decomposed in this fashion. Moreover, the modeler has the arduous task of modeling both trend and stochastic component. In multiple-point geostatistics, the training image is central to the modeling of complex spatial distributions; however most techniques require the training image to model only the stationary component, which limits the application of such techniques to real 3D applications. In this paper, we propose a new technique for multiple-point geostatistical simulation that does not rely on any assumption of stationarity, nor decomposition and can directly create model realizations that depict the training image patterns, whatever their nature (trending or not). Our modeling approach relies on decomposing any 3D model into patterns and simulating these patterns directly. A distance between patterns is defined to make sure the simulated realizations exhibit the spatial continuity depicted in the training image. Based on this distance, patterns are classified for rapid pattern similarity searches during the conditional simulation. Starting from an empty grid, patterns are simulated one at a time; each time the pattern most similar to the current simulation grid is pasted. If the training image shows complexity that cannot be handled with stationary modeling approaches, then the vector representing the pattern values is augmented with the spatial location of the pattern. This simple addition to the distance calculations ensures that any feature, including those requiring non-stationary modeling approaches is maintained in the simulated realization. In this paper, we elaborate on the various implementation details that allow this simple idea to work for real complex 3D applications. |
