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

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## 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.

## Keywords

High-order stochastic simulation Multi-point statistics Spatial moments Legendre polynomials## 1 Introduction

For the past several decades, stochastic simulations have been used to quantify spatial uncertainty in earth science applications. Traditionally, stochastic models are built on the basis of the Gaussian distribution and two-point statistics, where covariance or variograms are used to capture the spatial correlations (David 1988; Deutsch and Journel 1992; Journel 1994; Goovaerts 1997). The limitations of the existing two-point simulation methods have been reported in various publications (Guardiano and Srivastava 1993; Xu 1996; Journel 1997, 2003; De Iaco and Maggio 2011), which are mostly related to the poor reproduction of spatial distributions while dealing with the complex spatial patterns, spatial connectivity of extreme values, and non-Gaussianity. To reflect the complex geological patterns, multi-point statistics have to be introduced instead of conventional two-point statistics. Guardiano and Srivastava (1993) propose a multiple-point simulation (mps) framework and the concept of the training image (TI). The primary difference between mps and two-point simulations is that the conditional cumulative distribution functions (ccdfs) are built on empirical estimations of conditional probabilities with multiple-point configurations, which is equivalent to solving a normal equation according to the Bayes’ rule. Strebelle (2002) formalizes the method and developed the first computationally efficient implementation. For over a decade, 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). In general, these mps methods are TI-based, and their statistics are estimated from distributions of replicates of data events in the TI. Their main drawbacks are: (1) the high-order statistics are partially and indirectly considered; (2) the methods are not driven by a consistent mathematical framework; and (3) since they are TI-driven, they may not generate results that comply with the statistics of actual available data. The latter shortcoming becomes distinctly clear in mining applications, where dense data sets are used (Osterholt and Dimitrakopoulos 2007; Goodfellow et al. 2012).

As an alternative, a high-order simulation framework with mathematical consistency is proposed with the introduction of a new concept of spatial cumulants (Dimitrakopoulos et al. 2010). The so-called high-order simulation algorithm (hosim) and its implementation are developed by Mustapha and Dimitrakopoulos (2010b, 2011). 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. In Mustapha and Dimitrakopoulos (2011), some terms of the expansion series have to be discarded to obtain computational feasibility, which compromises the accuracy of the approximated cpdf. In addition, the computational cost limits the approach for larger-scale applications.

To take full advantage of the high-order simulation, that is, its data-driven aspect and no presumption of data distribution, and address the computational difficulties, a new stochastic simulation algorithm based on high-order spatial Legendre moments is presented herein. Rather than just a mathematical equivalency of the previous model of the high-order simulation, the approximation of the cpdf by Legendre polynomial series is reformulated under the framework of the sequential simulation, leading to a much more concise form of the computational model. In this new method, all explicit calculations of moments are encapsulated in a unified function to derive the cpdf, cutting down the previous complex computations into a few iterations of simple operations with polynomial time. Moreover, there is no predefined template configuration in the new algorithm, as required for the normal mps methods and the previous hosim model. The spatial configuration of the template will, instead, depend on the local neighborhood of the node to be simulated; note that there is no need to store the intermediate results in a tree as in most of the mps methods, including the previous hosim model. The variable template also has the advantage of simultaneously capturing the spatial patterns either on a local scale or a global scale.

The remainder of the paper continues with Sect. 2, which describes the stochastic model based on the concepts of high-order spatial Legendre moments. Section 3 develops the computational model as a statistical function. Section 4 describes the new proposed high-order simulation algorithm and analyzes the computational complexity. Section 5 explores the implementation of the new high-order simulation algorithm. Section 6 shows examples to assess the new method and compare it with filtersim. Finally, conclusions and future research are presented in Sect. 7.

## 2 Stochastic Model of High-Order Simulation with Spatial Legendre Moments

### 2.1 Sequential Simulation

The basic idea of sequential simulation is to sequentially draw random values from the decomposed univariate cpdfs through a random path that visits all the nodes to be simulated. Irrespective of the node’s location corresponding to the sequence number, there is no difference in the sampling procedures. Without loss of generality, the cpdf in every single sampling procedure can be symbolized uniformly as \( f_{{Z_{0} }} (z_{0} |{\Lambda }) \), where \( Z_{0} \) means the current simulating node and \( {\Lambda } \) means the set of conditioning data around \( Z_{0} \)’s location \( \varvec{u}_{0} \). Considering the computational intensity and the statistical relevancy, the conditioning data are usually confined to a neighborhood closest to the simulation node instead of taking account of all available data on the whole domain of the random field. For more details on this screen-effect approximation, the reader is referred to Dimitrakopoulos and Luo (2004).

- (1)
Draw a random path to visit all the \( N \) nodes to be simulated.

- (2)
Starting from \( i = 1 \) and for each node \( Z\left( {\varvec{u}_{i} } \right) \), derive the conditional probability cumulative distribution \( F_{{Z_{i} }} (z_{i} |{\Lambda }_{i - 1} ) \) or the density function \( f_{{Z_{i} }} (z_{i} |{\Lambda }_{i - 1} ) \).

- (3)
Draw a random value \( \zeta \left( {\varvec{u}_{i} } \right) \) from the conditional probability distribution in step (2) and update the conditioning data by adding the node value \( \zeta \left( {\varvec{u}_{i} } \right) \) into the current data set \( {\Lambda }_{i} \).

- (4)
Repeat from step (2) until all the nodes are visited.

### 2.2 High-Order Spatial Legendre Moments

*P*is a continuous function \( f_{Z} \left( z \right) \). The moment of order \( w \) is defined as

*i*th element of vector

**Z**. The spatial moments of a discrete random field \( {\mathbf{Z}} = \left[ {Z\left( {\varvec{u}_{0} } \right), \ldots ,Z\left( {\varvec{u}_{n} } \right)} \right] \) are functions of spatial location variables \( \varvec{u}_{0} , \ldots ,\varvec{u}_{n} \). Assuming the random field \( {\mathbf{Z}}\left( \varvec{u} \right) \) is stationary and ergodic, the spatial moments of \( {\mathbf{Z}}\left( \varvec{u} \right) \) can be expressed as functions of distance vectors, and, thus, they are independent of the locations. These distance vectors, which keep the spatial configuration of a center node and nodes within its neighborhood, can be expressed using a spatial template

**T**(Fig. 1). The terminologies of the spatial template

**T**and data events (Strebelle 2002; Dimitrakopoulos et al. 2010) are as follows:

- (i)
Spatial template

**T**: geometry defined by*N*distance vectors \( \left( {\varvec{h}_{1} , \ldots ,\varvec{h}_{N} } \right) \) from the center node \( \varvec{u}_{0} \), \( {\mathbf{T}} = \left\{ {\varvec{u}_{0} ,\varvec{u}_{0} + \varvec{h}_{1} , \ldots ,\varvec{u}_{0} + \varvec{h}_{N} } \right\} \). - (ii)
Data events: outcomes of the random field in the spatial template

**T**. Specifically, the data events are conditioning data set \( {\Lambda } \) in the present work.

**Z**in a template

**T**can be expressed element-wise as

**T**, \( \left( {\varvec{h}_{1} , \ldots ,\varvec{h}_{N} } \right) \) are the distance vectors to represent the geometry of

**T**, and \( w_{i} \) are the orders of the moments with each random variable \( Z\left( {\varvec{u}_{i} } \right) \)(\( i = 1, \ldots ,N) \).

*m*th-degree Legendre polynomial. The infinite sequence of polynomials forms a complete orthogonal basis set on the domain D = [− 1, 1]. The orthogonal property of the Legendre polynomials can be expressed as

**T**; the extra coefficient \( \left( {w_{i} + \frac{1}{2}} \right) \) on the right-hand side of the equation is intentionally introduced as a normalization term for the convenience of the later computation (see the Appendix for details).

### 2.3 Multivariate Expansion Series of a Joint pdf

*N*+ 1)-dimensional domain in the same way. Specifically, suppose that the multivariate function is a density function related to the joint distribution of random variables on a spatial template

**T**. The density function can be expanded into Legendre polynomial series in terms of Legendre spatial moments and Legendre polynomials as (see the Appendix for details)

*M*replicates of data events associated with template

**T**found in the TI, the spatial Legendre moments can be calculated as

**T**, \( t \) is the sequence number of replicates, and \( i \) is the sequence number of random variables.

## 3 Computational Model

Equation (14) gives a unified computational model of empirical estimation of the density function on the spatial template **T**, noticing that, on the right-hand side of the equation, the subscript \( i \) of \( w_{i} \) is dropped because of the symmetry of computation.

Now let’s consider the cpdf \( f_{{Z_{0} }} \left( {z_{0} | {\Lambda }} \right) \) of a single sampling step in sequential simulation (ref. Sect. 2.1). The joint pdf can be marginalized from Eq. (14) to get the marginal pdf of conditioning random variables. To specify the difference between the empirical models and theoretical models in Eqs. (10) and (11), \( \tilde{f} \) and \( \tilde{f}_{W} \) specifically denote the experimental function corresponding to pdf \( f \) and its Legendre polynomial series truncated at order \( W \), respectively.

In fact, Eqs. (16) and (17) ensure that the integral of the approximated pdf to be 1, a necessary property of probability density.

The above development provides a theoretical equivalency of the approximation of the cpdf by a truncated Legendre series, which was proposed by Mustapha and Dimitrakopoulos (2010b, 2011). However, the new reformulated model in the current paper leads to a different stochastic simulation method in view of the related computational aspects. The advantage of the new model represented by Eq. (19) is that no explicit computations of moments or cumulants are needed. In addition, the new model is computationally more accurate than the hosim program in Mustapha and Dimitrakopoulos (2011), in which some terms have to be dropped from the full expansion of the Legendre series in the form of spatial cumulants to gain computational efficiency.

## 4 Algorithm Description and Computational Analysis

### 4.1 Algorithm for Computing a cpdf

By the property of Legendre polynomials that \( P_{0} \left( z \right) = 1,\forall z \in \left[ { - 1,1} \right] \), combined with Eqs. (15) and (21), the computation of coefficients \( c_{w} \left( {w = 1, \ldots ,W} \right) \) can be divided into the computation of functions \( X_{t} \left( {z_{i} } \right) \) over the nodes of each replicate. Especially, the first term of \( c_{w} \) is always fixed as \( c_{0} = \frac{1}{2} \).

### 4.2 Recursive Algorithm for Computing a ccdf

The coefficient \( c_{0} = \frac{1}{2} \) is taken out from the summation in Eq. (22) so that the Bonnet’s recursion relation of Legendre polynomials can be smoothly applied in the followed derivation.

As can be seen from Eq. (25), the ccdf is also expressed as the summation of the univariate Legendre polynomials, with the order of the Legendre polynomials increasing by one because of the integration. Furthermore, the new coefficients \( d_{w} \left( {w = 0, \ldots ,W,W + 1} \right) \) can now be computed through Eq. (25) in an iterative way, as shown in Algorithm 2.

### 4.3 Computational Complexity

The most computationally demanding part of the high-order simulation algorithm is to calculate the Legendre series coefficients, which is the basis for estimating the cpdfs. Considering that the cpdfs are approximated by Legendre series truncated to a certain order *W*, as Eq. (11) shows, the number of the different coefficients is \( \left( {W + 1} \right)^{N + 1} \), where *N* is the number of data points. Even the Legendre series is approximated by truncated series, where the sum of orders of different variables is not greater than *W*, which is the form adopted by Mustapha and Dimitrakopoulos (2011). The number of the different coefficients is still as big as \( \mathop \sum \limits_{w = 0}^{W} \left( {\begin{array}{*{20}c} {N + w} \\ w \\ \end{array} } \right) \) for a single data event. Although this computational complexity can be reduced by discarding some terms which are regarded as negligible, it should be noted that this simplification may lead to a loss of accuracy.

From Eqs. (15) and (19), it can be seen that all of the different coefficients introduced by the explicit expansion of Legendre series are reduced to a calculation of the function \( \mathop \prod \nolimits_{i = 1}^{N} X_{t} \left( {z_{i} } \right) \). There are only \( NW \) computations of Legendre polynomials and a few products and additions included in the calculation of the function \( \mathop \prod \nolimits_{i = 1}^{N} X_{t} \left( {\zeta_{t,i} } \right) \) for each replicate of the data event encountered in the TI. It should be noted that the computational time still depends on the number of the replicates encountered in the TI, as well as the maximal order of Legendre polynomials and the number of conditionings in the neighborhood. However, the computational cost regarding the above-mentioned parameters is significantly reduced, as opposed to computing the large number of coefficients in the previous version of high-order simulation.

## 5 Implementation

The implementation is relatively straightforward in terms of the above algorithms estimating the cpdf and ccdf according to the framework of sequential simulation. However, a method is proposed in this section to deal with the replicates, aiming to reduce the conflicts of spatial statistics between the sample data and the TI. The main idea of the method is to deliberately select replicates which are similar to the conditioning data within a certain range according to some measure of similarity. The reason for this is that the conditional probability distribution is a one-dimensional intercept from the multivariate joint probability distribution and, therefore, the replicates that are close to the conditioning data are more relevant to estimate this one-dimensional local probability distribution.

- (1)
Read the sample data and TI into memory. In order to apply the multivariate expansion of Legendre polynomials, the property values of the samples or TI are scaled to the interval [− 1, 1] through a linear transformation.

- (2)
Specify dimensions of a certain neighborhood for searching the conditional data and other parameters, such as the minimum or maximum number of the conditional data. The geometry of the local template totally depends on the locations of the conditional data. In the present work, a rectangular shape neighborhood was used and a searching policy was applied to find the closest points to the center. Nevertheless, the shape of the neighborhood and the searching policy can be manipulated to further control the spatial configuration of the template.

- (3)
Set the lag tolerance, angle tolerance, and bandwidth tolerance to enable searching approximate replicates from the TI (see Fig. 2).

- (4)
Generate a random sequence on the indices of the simulation grid to create a random visiting path.

- (5)
According to the predefined visiting path, sequentially pick one node at a time for the simulation. If the property value is already known (copied from the hard data), then continue to choose another single node until the property value is not assigned. The conditioning data are searched inside the neighborhood centered on the chosen node by the previously specified searching policy from both the hard data and the simulated nodes.

- (6)
A local spatial template is determined by the data and the center node for later simulation. This spatial template is then used to find similar replicates from the TI according to the parameters set in steps (2) and (3). If the number of approximated replicates is not adequate for statistical inference, then drop the furthest node to the center node and repeat until the minimum number of conditioning data is reached.

- (7)
The local ccdf is estimated from the replicates using the algorithms elaborated in Sect. 3. A random value is drawn from the local ccdf using the Monte Carlo method and set as the property value of the node to be simulated.

- (8)
Repeat from step (5) until all the nodes in the random path are visited.

## 6 Examples and Comparisons

### 6.1 Example 1

### 6.2 Example 2

### 6.3 Parameter Sensitivity Testing

Most parameters in the current implementation of the high-order stochastic simulation method are experimental choices. Amongst all the parameters encountered in the current implementation, some follow common practices in the parameter selection for conventional geostatistical simulations, such as the size of the search window, the lag, and angle tolerance. Additionally, in the high-order simulation method presented here, the number of conditioning data corresponding to a certain template needs more consideration, as it determines the dimension of the local probability distribution. In the current implementation, the number of the conditioning data is limited for two important reasons. First, the limited number of conditioning data reduces the computational time needed to estimate the cpdf. Second, the method resembles the so-called multiple grid strategy (Strebelle 2002) applied in many multi-point simulation methods in order to maintain both large- and small-scale spatial structures. In the early stage of the simulation process, the neighborhoods are more likely to capture large-scale patterns, since the known data are sparse. The neighborhoods gradually correspond to finer-scale patterns as the simulation continues and more known data are generated. A similar search strategy has also been applied and discussed by Mariethoz et al. (2010).

The maximum order of the polynomials is another parameter of importance in the high-order simulation, since it affects the precision of the approximation of a cpdf by a truncated Legendre polynomial series. Theoretically, the coefficients in the Legendre polynomial series decay exponentially, and, in general, much faster than in Taylor series (Cohen and Tan 2012; Wang and Xiang 2012). The numerical results of Cohen and Tan (2012) show that Legendre polynomial series with six non-zero coefficients (orders 10 and 11 in their examples) are highly accurate approximations to the targets. The numerical test to approximate a probability distribution regarding the order of Legendre polynomial series has also been investigated by Mustapha and Dimitrakopoulos (2010b) and led to similar results. However, it should be noted that the above tests are conducted for the approximation of a determined function, whereas for the approximation of the pdf, there is also the impact from the limitation of the number of replicates. Depending on different data sets, Legendre polynomial series with an order from 6 to 20 should be a reasonable range to select.

## 7 Conclusions

The main contributions of this paper are as follows. Firstly, 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 (cpdf). The computational model leads to an estimation of the cpdf without calculating the high-order spatial cumulants or moments term by term. As a consequence, it not only greatly reduces the computational requirements, but it also 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. Secondly, two new algorithms to derive the cpdf and conditional cumulative distribution function (ccdf) based on the above computational model are developed; they both use the properties of Legendre polynomials to simplify the computation and avoid an explicit expansion of a multivariate Legendre series. Lastly, the spatial template used in the current high-order simulation method is dynamically changing with the computation of the probability distribution in real time, without storing data events. In addition, a flexible strategy to search replicates from the training image (TI) is proposed and implemented to deal with the conflicts between the statistics of the sample data and the TI.

Tests show the capacity of the proposed algorithm to reproduce complex geological patterns, and, in addition, that both the overall distribution and the high-order spatial statistics of the data are reproduced by the high-order simulations. Comparing the results of the high-order simulation in different cases with those of filtersim, the high-order simulation outperforms in the reproduction of high-order spatial statistics. This result becomes more notable in cases where there are conflicts in the spatial statistics between the sample data and the TI. This demonstrates that the high-order simulation has a more data-driven nature, whereas the filtersim is more TI-driven. Although the computational cost is significantly reduced (depending on the size of the training image, the number of neighborhoods, and the maximum order of Legendre polynomial series), the simulation is still slower than the filtersim method. However, since the computations of the cpdf are carried out on each replicate with the same type of calculation, the procedure could be parallelized so that the simulation can be further accelerated through parallelization techniques, such as GPU programming. It should also be noted that the approximation of cpdfs by Legendre series or any kind of polynomial series may generate problems of non-positive probability densities; further research is needed to address this issue.

## Notes

### Acknowledgements

This work was funded by Fonds de recherche du Québec - Nature et technologies, FQRNT Grant “Développement d’une stratégie globale d’optimisation de sites miniers avec incertitude: Amélioration de la viabilité économique et de la gestion environnementale des résidus miniers d’une mine de fer dans le nord”, with New Millennium Iron Corporation being the industry collaborator, and the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant 239019. Thanks go to Dr. Ilnur Minniakhmetov and Dr. Elena Tamayo-Mas for their technical comments on an earlier version of the manuscript.

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