Abstract
Multiple-point simulations have been introduced over the past decade to overcome the limitations of second-order stochastic simulations in dealing with geologic complexity, curvilinear patterns, and non-Gaussianity. However, a limitation is that they sometimes fail to generate results that comply with the statistics of the available data while maintaining the consistency of high-order spatial statistics. As an alternative, high-order stochastic simulations based on spatial cumulants or spatial moments have been proposed; however, they are also computationally demanding, which limits their applicability. The present work derives a new computational model to numerically approximate the conditional probability density function (cpdf) as a multivariate Legendre polynomial series based on the concept of spatial Legendre moments. The advantage of this method is that no explicit computations of moments (or cumulants) are needed in the model. The approximation of the cpdf is simplified to the computation of a unified empirical function. Moreover, the new computational model computes the cpdfs within a local neighborhood without storing the high-order spatial statistics through a predefined template. With this computational model, the algorithm for the estimation of the cpdf is developed in such a way that the conditional cumulative distribution function (ccdf) can be computed conveniently through another recursive algorithm. In addition to the significant reduction of computational cost, the new algorithm maintains higher numerical precision compared to the original version of the high-order simulation. A new method is also proposed to deal with the replicates in the simulation algorithm, reducing the impacts of conflicting statistics between the sample data and the training image (TI). A brief description of implementation is provided and, for comparison and verification, a set of case studies is conducted and compared with the results of the well-established multi-point simulation algorithm, filtersim. This comparison demonstrates that the proposed high-order simulation algorithm can generate spatially complex geological patterns while also reproducing the high-order spatial statistics from the sample data.
Highlights
For the past several decades, stochastic simulations have been used to quantify spatial uncertainty in earth science applications
Starting from the high-order simulation method based on Legendre polynomial series, a new computational model in the form of a unified empirical function is developed to approximate the conditional probability density function
It greatly reduces the computational requirements, but it provides a more accurate approximation of the cpdf through Legendre polynomial series in comparison to the previous high-order simulation algorithm based on Legendre cumulants
Summary
For the past several decades, stochastic simulations have been used to quantify spatial uncertainty in earth science applications. Research has been focused on various issues around mps algorithms, such as computational efficiency and various patch-based extensions (Zhang et al 2006; Arpat and Caers 2007; Wu et al 2008; Boucher 2009; Remy et al 2009; Honarkhah and Caers 2010; Mariethoz et al 2010; Parra and Ortiz 2011; Huang et al 2013; Boucher et al 2014; Strebelle and Cavelius 2014; Chatterjee et al 2016; Li et al 2016) These mps methods are TI-based, and their statistics are estimated from distributions of replicates of data events in the TI.
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