Estimation using multiple-point statistics

In the last two decades, several geostatistical simulation techniques have appeared that allow taking into account complex structural information, by quantifying a multiple-point statistical (MPS) model. The higher-order statistics are typically informed from a training image, sample model, or geological exposure. Such MPS models can represent the subsurface and are then used to simulate subsurface models with high(er) order of realism than is possible using, for example, the widely used 2-point statistical models (such as sequential Gaussian simulation). All available methods dealing with MPS models are simulation methods, that can be used to generate multiple realizations of the underlying statistical model conditioned to hard and, to some extent, soft data. Such realizations can be very useful but are also computationally expensive to obtain. Here we consider the case when the end-user may not be interested in the set of produced realizations themselves, but rather in parameter-wise marginal statistical properties of a single model parameter derived from the simulated realizations. To obtain these, one would typically generate a larger number of independent realizations, and then compute some marginal statistics of these. Here we propose an MPS estimation algorithm, a variant of the widely used sequential simulation algorithm, that can be used to directly compute and store parameter-wise conditional statistics. This allows for a potentially faster and more accurate estimation than using sequential simulation. The method is demonstrated on both ENESIM-and SNESIM-type MPS algorithms and results compared using both sequential simulation and estimation. As an example, the method is applied for estimating the existence of near-surface buried valley systems in Kasted, Denmark.