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.

The present work proposes a new high-order simulation framework based on statistical learning. The training data consist of the sample data together with a training image, and the learning target is the underlying random field model of spatial attributes of interest. The learning process attempts to find a model with expected high-order spatial statistics that coincide with those observed in the available data, while the learning problem is approached within the statistical learning framework in a reproducing kernel Hilbert space (RKHS). More specifically, the required RKHS is constructed via a spatial Legendre moment (SLM) reproducing kernel that systematically incorporates the high-order spatial statistics. The target distributions of the random field are mapped into the SLM-RKHS to start the learning process, where solutions of the random field model amount to solving a quadratic programming problem. Case studies with a known data set in different initial settings show that sequential simulation under the new framework reproduces the high-order spatial statistics of the available data and resolves the potential conflicts between the training image and the sample data. This is due to the characteristics of the spatial Legendre moment kernel and the generalization capability of the proposed statistical learning framework. A three-dimensional case study at a gold deposit shows practical aspects of the proposed method in real-life applications.

A training image free, high-order sequential simulation method is proposed herein, which is based on the efficient inference of high-order spatial statistics from the available sample data. A statistical learning framework in kernel space is adopted to develop the proposed simulation method. Specifically, a new concept of aggregated kernel statistics is proposed to enable sparse data learning. The conditioning data in the proposed high-order sequential simulation method appear as data events corresponding to the attribute values associated with the so-called spatial templates of various geometric configurations. The replicates of the data events act as the training data in the learning framework for inference of the conditional probability distribution and generation of simulated values. These replicates are mapped into spatial Legendre moment kernel spaces, and the kernel statistics are computed thereafter, encapsulating the high-order spatial statistics from the available data. To utilize the incomplete information from the replicates, which partially match the spatial template of a given data event, the aggregated kernel statistics combine the ensemble of the elements in different kernel subspaces for statistical inference, embedding the high-order spatial statistics of the replicates associated with various spatial templates into the same kernel subspace. The aggregated kernel statistics are incorporated into a learning algorithm to obtain the target probability distribution in the underlying random field, while preserving in the simulations the high-order spatial statistics from the available data. The proposed method is tested using a synthetic dataset, showing the reproduction of the high-order spatial statistics of the sample data. The comparison with the corresponding high-order simulation method using TIs emphasizes the generalization capacity of the proposed method for sparse data learning.

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A new non-stationary, high-order sequential simulation method is presented herein, aiming to accommodate complex curvilinear patterns when modelling non-Gaussian, spatially distributed and variant attributes of natural phenomena. The proposed approach employs spatial templates, training images and a set of sample data. At each step of a multi-grid approach, a template consisting of several data points and a simulation node located in the center of the grid is selected. To account for the non-stationarity exhibited in the samples, the data events decided by the conditioning data are utilized to calibrate the importance of the related replicates. Sliding the template over the training image generates a set of training patterns, and for each pattern a weight is calculated. The weight value of each training pattern is determined by a similarity measure defined herein, which is calculated between the data event of the training pattern and that of the simulation pattern. This results in a non-stationary spatial distribution of the weight values for the training patterns. The proposed new similarity measure is constructed from the high-order statistics of data events from the available data set, when compared to their corresponding training patterns. In addition, this new high-order statistics measure allows for the effective detection of similar patterns in different orientations, as these high-order statistics conform to the commutativity property. The proposed method is robust against the addition of more training images due to its non-stationary aspect; it only uses replicates from the pattern database with the most similar local high-order statistics to simulate each node. Examples demonstrate the key aspects of the method.

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