Joint High-Order Simulation of Spatially Correlated Variables Using High-Order Spatial Statistics

Minniakhmetov, Ilnur; Dimitrakopoulos, Roussos

doi:10.1007/s11004-016-9662-x

Cited by 25 publications

(8 citation statements)

References 38 publications

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“…In this algorithm, the conditional probability density function (cpdf) is approximated by a multivariate expansion with coefficients expressed in terms of spatial cumulants. The hosim algorithm has been extended mostly recently to deal with the joint simulation of multiple variables, as well as the simulation of categorical data (Minniakhmetov and Dimitrakopoulos 2017a , b ); other extensions are approximating the cpdf with different types of orthogonal polynomial bases, such as expansion series with Laguerre polynomials and Legendre-like spline polynomials (Mustapha and Dimitrakopoulos 2010a ; Minniakhmetov and Dimitrakopoulos 2018 ). However, the related calculations are computationally demanding, since the number of spatial cumulants involved in the series increases exponentially either as the order of cumulants or the quantity of conditioning data increases.…”

Section: Introductionmentioning

confidence: 99%

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

2018

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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.

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Section: Introductionmentioning

confidence: 99%

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

2018

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“…Nowadays, point and block kriging or simulation for grade variables and indicator-based techniques (indicator kriging, sequential indicator simulation, plurigaussian simulation) for categorical variables are accepted standard techniques. Beyond the framework of Gaussian random fields, cumulant based (Dimitrakopoulos et al 2010;Minniakhmetov and Dimitrakopoulos 2017) and Copula based (Musafer et al 2013(Musafer et al , 2017 proposals, as well as multiple point geostatistics (MPS) can be found in scientific papers, though their penetration and acceptance in the industry is yet negligible. Multivariate issues are also seldom considered, though compositions (mineral or chemical) are geometallurgically relevant primary variables, and techniques do exist to predict or simulate them at both point (Pawlowsky 1989;Pawlowsky-Glahn and Burger 1992;Pawlowsky-Glahn and Olea 2004;Tolosana-Delgado 2006;Tolosana-Delgado et al 2011;Mueller et al 2014) and block support in a fashion consistent with their scale, namely delivering positive and constant-sum predictions/simulations abiding to a relative scale.…”

Section: Orebody Modellingmentioning

confidence: 99%

Predictive Geometallurgy: An Interdisciplinary Key Challenge for Mathematical Geosciences

Boogaart

Tolosana‐Delgado

2018

Handbook of Mathematical Geosciences

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Predictive geometallurgy tries to optimize the mineral value chain based on a precise and quantitative understanding of: the geology and mineralogy of the ores, the minerals processing, and the economics of mineral commodities. This chapter describes the state of the art and the mathematical building blocks of a possible solution to this problem. This solution heavily relies on all classical fields of mathematical geosciences and geoinformatics, but requires new mathematical and computational developments. Geometallurgy can thus become a new defining challenge for mathematical geosciences, in the same fashion as geostatistics has been in the first 50 years of the IAMG.Geometallurgy, from the Greek words for earth (geia), metal (metallo) and work (ergon), can be understood as the exploitation of a metallic ore based on a precise understanding of its geoscientific characteristics. Geometallurgy is hence a cooperation field for geoscientists and mineral processing engineers, something which has occurred in virtually all mining operations. A modern understanding of geometallurgy, what we could call predictive geometallurgy, proposes a quantitative approach to the subject. In rough terms, that requires optimizing the ore processing based on automated mineralogy and microstructure characterisation of the ore, coupled with geometallurgical tests. These are tests conducted at several scales (from lab to plant) along which the actual ore is processed in realistic conditions in order to study the differential behaviour of the several ore and waste mineral phases, and thus the enriching potential of the ore through the processes considered.

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“…The proposed high-order simulation approach estimates the third-and fourth-order spatial statistics from data and complements them with higher-order statistics from the TI. Further developments in algorithmic performance (Yao et al 2018(Yao et al , 2020, generalization using splines (Minniakhmetov et al 2018), a high-order decorrelation method (Minniakhmetov and Dimitrakopoulos 2017a), and efficient block simulations (de Carvalho et al 2019), and training-image-free simulations (Yao et al 2021) have made the approach more practical. These approaches are based on the approximation of a conditional distribution using Legendre polynomials, which are smooth functions and are incapable of an adequate approximation of the discrete distribution of categorical variables.…”

Section: Introductionmentioning

confidence: 99%

High-Order Data-Driven Spatial Simulation of Categorical Variables

Minniakhmetov

Dimitrakopoulos

2021

Math Geosci

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Modern approaches for the spatial simulation of categorical variables are largely based on multi-point statistical methods, where a training image is used to derive complex spatial relationships using relevant patterns. In these approaches, simulated realizations are driven by the training image utilized, while the spatial statistics of the actual sample data are ignored. This paper presents a data-driven, high-order simulation approach based on the approximation of high-order spatial indicator moments. The high-order spatial statistics are expressed as functions of spatial distances that are similar to variogram models for two-point methods, while higher-order statistics are connected with lower-orders via boundary conditions. Using an advanced recursive B-spline approximation algorithm, the high-order statistics are reconstructed from the available data and are subsequently used for the construction of conditional distributions using Bayes’ rule. Random values are subsequently simulated for all unsampled grid nodes. The main advantages of the proposed technique are its ability to (a) simulate without a training image to reproduce the high-order statistics of the data, and (b) adapt the model’s complexity to the information available in the data. The practical intricacies and effectiveness of the proposed approach are demonstrated through applications at two copper deposits.

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Joint High-Order Simulation of Spatially Correlated Variables Using High-Order Spatial Statistics

Cited by 25 publications

References 38 publications

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

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

Predictive Geometallurgy: An Interdisciplinary Key Challenge for Mathematical Geosciences

High-Order Data-Driven Spatial Simulation of Categorical Variables

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