Bayesian Inference of Covariance Parameters in Spectral Approach to Geostatistical Simulation

Summary The spectral simulation approach (described in Ismagilov and Lifshits (ECMOR XVI)) is a relatively new geostatistical method of stochastic reservoir property simulation. It is based on Fourier analysis of well log data and simulation of Fourier expansion coefficients in the interwell space. The key advantage of this method is its ability to automatically recognize and reproduce vertical non-stationarities observed in well data (Ismagilov et al. (ATCE 2019)). This comes at a price of having many parameters: while usual geostatistical approaches like kriging or sequential Gaussian simulation require estimating one covariance function or variogram (in practice, estimated parameters are variogram model type and ranges in three directions), the spectral approach requires estimating a lot of them (typically, 100—200 covariance functions). Obviously, automatic covariance estimation becomes crucial in this setting. While assuming parametric models for the aforementioned covariance functions and estimating their parameters by maximizing the likelihood works reasonably well in practice, this strategy has some drawbacks. First, in cases when likelihood surface turns out to be multi-modal or flat, the point estimation of parameters may lead to many problems such as incorrect uncertainty estimation or even choosing a wrong model. Second, maximum likelihood estimation usually does not provide a way to incorporate prior knowledge about parameters --- a typical example is constraining the resulting variogram range to be in geologically reasonable limits. We argue that Bayesian inference of parameters is a way to overcome both these challenges. Treating covariance parameters as random variables avoids limitations of deterministic point estimations while introducing prior distributions for parameters is the most natural way of incorporating the prior knowledge. We develop and implement in software a version of spectral approach where covariance parameters are treated in a Bayesian way. We show via computations on practical examples that Bayesian inference enables to build better models in cases with complex likelihood surfaces and to account for prior knowledge about covariance parameters.