A New Computational Model of High-Order Stochastic Simulation Based on Spatial Legendre Moments

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 Geosci (2018) 50:929-960 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.

High-Order Block Support Spatial Simulation Method and Its Application at a Gold Deposit

“…be a set of orthogonal functions defined in the same space Ω. Then, a fixed number ω of those orthogonal functions can approximate f (z) (Lebedev 1965;Mustapha and Dimitrakopoulos 2010a;Minniakhmetov et al 2018;Yao et al 2018), when multiplied by the coefficients L i…”

“…Previous studies have shown resulting realizations that comply with the TI used but do not necessarily reproduce the spatial statistics inferred from the data (Osterholt and Dimitrakopoulos 2007;Goodfellow et al 2012). As an alternative, to address the above limitations, a high-order simulation (HOSIM) framework has been proposed as a natural generalization of the second-order-based random field paradigm (Dimitrakopoulos et al 2010;Dimitrakopoulos 2010a, b, 2011;Minniakhmetov and Dimitrakopoulos 2017a, b;Minniakhmetov et al 2018;Yao et al 2018). The HOSIM framework does not make any assumptions about the data distribution, and the resulting realizations reproduce the high-order spatial statistics of the data.…”

“…Results show a more stable approximation of the related cpdf. Improving upon the computational performance, Yao et al (2018) proposed a new approach, where the calculation of the cpdf is simplified and no explicit calculation of cumulants is required. Although effective, the methods described above are performed at point support scale.…”

High-Order Sequential Simulation via Statistical Learning in Reproducing Kernel Hilbert Space

“…. , ζ t,N ) corresponding to T , the EPDF f emp embedded in the SLM-RKHS is identical to the density estimator in Yao et al (2018) in the kernel form as…”

“…Further developments of the high-order simulation paradigm include the simulation of spatially correlated variables (Minniakhmetov and Dimitrakopoulos 2017 ) and the direct simulation at the block scale (de Carvalho et al 2019 ). Most recently, Yao et al ( 2018 ) proposed a new computational model of high-order simulation as a unified empirical function, which avoids CPU-demanding computations of expansion coefficients. Furthermore, a kernel function can be derived from this model and will be used in the present work.…”