Beyond Stationary Simulation; Modern Approaches to Stochastic Modelling

Abstract

Stochastic and conditional simulation methods have been effective towards producing realistic realizations and simulations of spatial numerical models that share equal probability of occurrence. Application of these methods are valuable throughout the domain of earth science for their ability to simulate sampled study data. Such stochastic methods have also been adopted into other fields outside of geostatistics domains, especially within the computing and data science community. Classical techniques for stochastic simulation have primarily consisted of stationary methods due to their brisk simulation speed and mathematical simplicity. However, advances in modern computing now allow for the implementation of more advanced non-stationary simulation methods, consisting of multiple varying structures, and allowing for much more accurate and realistic simulations. As some of these calculations may still be slow, the application of machine learning techniques, namely the Generative Adversarial Network (GAN) can be used to surpass previous simulation generation speeds and allows for greater parameterization. This work presents three stochastic simulation methods: stochastic simulation using non-stationary covariance, multipoint simulation, and conditional GANs. An SPDE method was used as a benchmark comparison. Experiments using synthetic data are used to showcase the effectiveness of each of these methods at maintaining non-stationary structures and conditioned data. A case study implementing non-stationary covariances is also presented on real geochemical samples coming from La Ceinture de roches vertes de la Haute-Eastmain, located in the Superior Province.