Modeling complex geological structures with elementary training images and transform-invariant distances

We present a new framework for multiple‐point simulation involving small and simple training images. The use of transform‐invariant distances (by applying random transformations) expands the range of structures available in the simple patterns of the training image. The training image is no longer regarded as a global conceptual geological model, but rather a basic structural element of the subsurface. Complex geological structures are obtained whose spatial structure can be parameterized by adjusting the statistics of the random transformations, on the basis of field data or geological context. In most cases, such parameterization is possible by adjusting two numbers only. This method allows us to build models that (1) reproduce shapes corresponding to a desired prior geological concept and (2) are in phase with different types of field observations such as orientation, hydrofacies, or geophysical measurements. The main advantage is that the training images are so simple that they can be easily built even in 3‐D. We apply the method on a synthetic example involving seismic data where the transformation parameters are data‐driven. We also show examples where realistic 2‐ and 3‐D structures are built from simplistic training images, with transformation parameters inferred using a small number of orientation data. Key Points The possibility to parameterize the model of spatial continuity Practical advantages for training image inference and construction Facilitated data integration