Conditioning Multiple-Point Statistics Simulation to Inequality Data

Stochastic modeling is often employed in environmental sciences for the analysis and understanding of complex systems. For example, random fields are key components in uncertainty analysis or Bayesian inverse modeling. Multiple-point statistics (MPS) provides efficient simulation tools for simulating fields reproducing the spatial statistics depicted in a training image (TI), while accounting for local or block conditioning data. Among MPS methods, the direct sampling algorithm is a flexible pixel-based technique that consists in first assigning the conditioning data values (so-called hard data) in the simulation grid, and then in populating the rest of the simulation domain in a random order by successively pasting a value from a TI cell sharing a similar pattern. In this study, an extension of the direct sampling method is proposed to account for inequality data, that is, constraints in given cells consisting of lower and/or upper bounds for the simulated values. Indeed, inequality data are often available in practice. The new approach involves the adaptation of the distance used to compare and evaluate the match between two patterns to account for such constraints. The proposed method, implemented in the DeeSse code, allows generating random fields both reflecting the spatial statistics of the TI and honoring the inequality constraints. Finally examples of topography simulations illustrate and show the capabilities of the proposed method.