High-Order Spatial Simulation Using Legendre-Like Orthogonal Splines

Abstract

High-order sequential simulation techniques for complex non-Gaussian spatially distributed variables have been developed over the last few years. The highorder simulation approach does not require any transformation of initial data and makes no assumptions about any probability distribution function, while it introduces complex spatial relations to the simulated realizations via high-order spatial statistics. This paper presents a new extension where a conditional probability density function (cpdf) is approximated using Legendre-like orthogonal splines. The coefficients of spline approximation are estimated using high-order spatial statistics inferred from the available sample data, additionally complemented by a training image. The advantages of using orthogonal splines with respect to the previously used Legendre polynomials include their ability to better approximate a multidimensional probability density function, reproduce the high-order spatial statistics, and provide a generalization of high-order simulations using Legendre polynomials. The performance of the new method is first tested with a completely known image and compared to both the high-order simulation approach using Legendre polynomials and the conventional sequential Gaussian simulation method. Then, an application in a gold deposit demonstrates the advantages of the proposed method in terms of the reproduction of histograms, variograms, and high-order spatial statistics, including connectivity measures. The C++ course code of the high-order simulation implementation presented

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…”

“…where the elements t i are referred to as knots and m max represents the maximum number of knots; note that Minniakhmetov et al (2018) present a study on how to choose m max to obtain computationally stable polynomial approximations. The final Legendre-like splines are defined as…”

“…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.…”

“…Advantages of this method lie in the absence of assumptions on the distribution of the data and in being a data-driven approach. The Legendre polynomial was replaced by Legendre-like splines as the basis function for the estimation of conditional probabilities by Minniakhmetov et al (2018). Results show a more stable approximation of the related cpdf.…”

“…current paper uses Legendre-like splines(Wei et al 2013;Minniakhmetov et al 2018) as a means to obtain the basis function mentioned above. In short, those splines are a combination of Legendre polynomials (Lebedev 1965) up to order r and linear combinations of B-splines(de Boor 1978).…”