Higher-order Spatial Statistics Research Articles

High-order simulation of geological domains and effects on stochastic long-term planning of mining complexes

Stochastic long-term mine planning has evolved to account for different sources of uncertainty. Typically, the uncertainty and local variability of boundaries in geological domains have been overlooked by experts through their deterministic interpretation of available data. Categorical attributes are used to model geological domains, and their stochastic simulation accounts for the mentioned issues. The ability of two-points simulation methods to reproduce complex patterns or the requirement of a training image in multiple-points simulation methods has limited their implementation in mining environments. The high-order simulation of categorical attributes presents a mathematically consistent framework that overcomes these limitations by using high-order spatial statistics from sample data. The case study at a gold mining complex shows two stochastic mine plans based on two sets of geological realisations: geological domains in the first set are modelled using conventional wireframes, while, in the second, they are simulated through the high-order method. The resulting mine plans are substantially different; while both plans present a similar quantity of metal recovered and lifespan, risk profiles are up to 40% wider, and the expected NPV is 20% higher for the case of simulated geological domains, given the decrease of waste handling costs and the corresponding reduction in environmental footprint.

Updating geostatistically simulated models of mineral deposits in real-time with incoming new information using actor-critic reinforcement learning

The existing technologies that update geostatistically simulated models of mineral deposits cannot self-learn from incoming new information generated in operating mines and do not account for high-order spatial statistics. This work proposes a novel self-learning artificial intelligence algorithm that learns from incoming new information and accounts for high-order spatial statistics, in order to update the geostatistically simulated models of mineral deposits in real-time. The proposed algorithm uses deep policy gradient reinforcement learning with an actor and a critic agent. The grid nodes of the geostatistically simulated model are visited sequentially in a random path, the environment generates the states for each grid node, and feeds the state to the actor and critic agents that respectively predict and evaluate the updated property of the grid node The data is stored in a replay memory, which is sampled at regular intervals to train the agents. The trained agents are then used for further rounds of self-learning. An application of the proposed algorithm at a copper mining operation with incoming drilling machine sensor data (collected spatially), and processing mill sensor data (collected over time), demonstrates its applied aspects in updating the geostatistically simulated models of copper grades of the mineral deposit in real-time, while also reproducing spatial patterns and high-order spatial statistics.

A Truly Spatial Random Forests Algorithm for Geoscience Data Analysis and Modelling

Spatial data mining helps to find hidden but potentially informative patterns from large and high-dimensional geoscience data. Non-spatial learners generally look at the observations based on their relationships in the feature space, which means that they cannot consider spatial relationships between regionalised variables. This study introduces a novel spatial random forests technique based on higher-order spatial statistics for analysis and modelling of spatial data. Unlike the classical random forests algorithm that uses pixelwise spectral information as predictors, the proposed spatial random forests algorithm uses the local spatial-spectral information (i.e., vectorised spatial patterns) to learn intrinsic heterogeneity, spatial dependencies, and complex spatial patterns. Algorithms for supervised (i.e., regression and classification) and unsupervised (i.e., dimension reduction and clustering) learning are presented. Approaches to deal with big data, multi-resolution data, and missing values are discussed. The superior performance and usefulness of the proposed algorithm over the classical random forests method are illustrated via synthetic and real cases, where the remotely sensed geophysical covariates in North West Minerals Province of Queensland, Australia, are used as input spatial data for geology mapping, geochemical prediction, and process discovery analysis.

High-Order Data-Driven Spatial Simulation of Categorical Variables

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.

Training Image Free High-Order Stochastic Simulation Based on Aggregated Kernel Statistics

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.

Learning high-order spatial statistics at multiple scales: A kernel-based stochastic simulation algorithm and its implementation

This paper presents a learning-based stochastic simulation method that incorporates high-order spatial statistics at multiple scales from sources with different resolutions. Regarding the simulation of a certain spatial attribute, the high-order spatial information from different sources is encapsulated as aggregated kernel statistics in a spatial Legendre moment kernel space, and the probability distribution of the underlying random field model is derived by a statistical learning algorithm, which matches the high-order spatial statistics of the target model to the observed ones. In addition, a related software is developed as the SGeMS plugin. Case studies are conducted with a known data set and a gold deposit, demonstrating reproduction of high-order spatial statistics from the available data, as well as practical aspects in mining applications.

Towards Geostatistical Learning for the Geosciences: A Case Study in Improving the Spatial Awareness of Spectral Clustering

The particularities of geosystems and geoscience data must be understood before any development or implementation of statistical learning algorithms. Without such knowledge, the predictions and inferences may not be accurate and physically consistent. Accuracy, transparency and interpretability, credibility, and physical realism are minimum criteria for statistical learning algorithms when applied to the geosciences. This study briefly reviews several characteristics of geoscience data and challenges for novel statistical learning algorithms. A novel spatial spectral clustering approach is introduced to illustrate how statistical learners can be adapted for modelling geoscience data. The spatial awareness and physical realism of the spectral clustering are improved by utilising a dissimilarity matrix based on nonparametric higher-order spatial statistics. The proposed model-free technique can identify meaningful spatial clusters (i.e. meaningful geographical subregions) from multivariate spatial data at different scales without the need to define a model of co-dependence. Several mixed (e.g. continuous and categorical) variables can be used as inputs to the proposed clustering technique. The proposed technique is illustrated using synthetic and real mining datasets. The results of the case studies confirm the usefulness of the proposed method for modelling spatial data.

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

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.

Graph Database Solution for Higher-order Spatial Statistics in the Era of Big Data

Abstract We present an algorithm for the fast computation of the general N-point spatial correlation functions of any discrete point set embedded within an Euclidean space of . Utilizing the concepts of kd-trees and graph databases, we describe how to count all possible N-tuples in binned configurations within a given length scale, e.g., all pairs of points or all triplets of points with side lengths < r MAX. Through benchmarking, we show the computational advantage of our new graph-based algorithm over more traditional methods. We show measurements of the three-point correlation function up to scales of ∼200 Mpc (beyond the baryon acoustic oscillation scale in physical units) using current Sloan Digital Sky Survey (SDSS) data. Finally, we present a preliminary exploration of the small-scale four-point correlation function of 568,776 SDSS Constant (stellar) Mass (CMASS) galaxies in the northern Galactic cap over the redshift range of 0.43 < z < 0.7. We present the publicly available code GRAMSCI (GRAph Made Statistics for Cosmological Information; bitbucket.org/csabiu/gramsci), under a Gnu is Not Unix (GNU) General Public License.

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

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.

High-Order Spatial Simulation Using Legendre-Like Orthogonal Splines

High-order sequential simulation techniques for complex non-Gaussian spatially distributed variables have been developed over the last few years. The high-order 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 herein, along with an example demonstrating its utilization, are provided online as supplementary material.

Pattern-based calibration of complex subsurface flow models against dynamic response data

Abstract Calibration of heterogeneous subsurface flow models usually leads to ill-posed nonlinear inverse problems, where too many unknown parameters are estimated from limited response measurements. When the underlying parameters form complex (non-Gaussian) structured spatial connectivity patterns, classical variogram-based geostatistical techniques cannot describe the underlying distributions. Modern pattern-based geostatistical methods that incorporate higher-order spatial statistics are more suitable for describing such complex spatial patterns. Moreover, when the unknown parameters are discrete (e.g., geologic facies distribution), conventional model calibration techniques that are designed for continuous parameters cannot be applied directly. In this paper, we introduce a novel pattern-based model calibration method to reconstruct spatially complex facies distributions from dynamic flow response data. To reproduce complex connectivity patterns during model calibration, we impose a geologic feasibility constraint that ensures the solution honors the expected higher-order spatial statistics. For model calibration, we adopt a regularized least-squares formulation, involving (i) data mismatch, (ii) pattern connectivity, and (iii) feasibility constraint terms. Using an alternating directions optimization algorithm, the regularized objective function is divided into a parameterized model calibration sub-problem, which is solved via gradient-based optimization. The resulting parameterized solution is then mapped onto the feasible set, using the k-nearest neighbors (k-NN) as a supervised machine learning approach, to honor the expected spatial statistics. The two steps of the model calibration formulation are repeated until the convergence criterion is met. Several numerical examples are used to evaluate the performance of the developed method.

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

Joint geostatistical simulation techniques are used to quantify uncertainty for spatially correlated attributes, including mineral deposits, petroleum reservoirs, hydrogeological horizons, environmental contaminants. Existing joint simulation methods consider only second-order spatial statistics and Gaussian processes. Motivated by the presence of relatively large datasets for multiple correlated variables that typically are available from mineral deposits and the effects of complex spatial connectivity between grades on the subsequent use of simulated realizations, this paper presents a new approach for the joint high-order simulation of spatially correlated random fields. First, a vector random function is orthogonalized with a new decorrelation algorithm into independent factors using the so-termed diagonal domination condition of high-order cumulants. Each of the factors is then simulated independently using a high-order univariate simulation method on the basis of high-order spatial cumulants and Legendre polynomials. Finally, attributes of interest are reconstructed through the back-transformation of the simulated factors. In contrast to state-of-the-art methods, the decorrelation step of the proposed approach not only considers the covariance matrix, but also high-order statistics to obtain independent non-Gaussian factors. The intricacies of the application of the proposed method are shown with a dataset from a multi-element iron ore deposit. The application shows the reproduction of high-order spatial statistics of available data by the jointly simulated attributes.

Testing geological heterogeneity representations for enhanced oil recovery techniques

This paper analyzes the effects of geological heterogeneity representation in a producing reservoir, when different stochastic simulation methods are used, so as to assess the consequent effects on flow responses for different enhanced oil recovery (EOR) techniques employed. First, the spatial heterogeneity of a fluvial reservoir is simulated using three different stochastic methods: (1) the well-known two-point sequential Gaussian simulation (SGS), (2) a multiple-point filter-based algorithm (FILTERSIM), and (3) a new alternative high-order simulation method that uses high-order spatial statistics (HOSIM). Numerical results show that SGS suffers from the inability of describing the highly permeable channel network whereas FILTERSIM better reproduces this connectivity. By means of the recent HOSIM, a more appropriate description of the curvilinear high-permeability channels is obtained. Second, the realizations generated above represent permeability fields in EOR numerical simulations. In particular, four different methods are considered, namely: (1) surfactant, (2) polymer, (3) alkaline-surfactant-polymer and (4) foam flooding processes. The numerical results show that properly reproducing the main geological features of the reference images has a higher impact if surfactant or alkaline chemicals are injected rather than polymer or foam. This is due to the fact that these latter chemicals act by mitigating the effects of heterogeneities.

A new approach to spatial data interpolation using higher-order statistics

Interpolation techniques for spatial data have been applied frequently in various fields of geosciences. Although most conventional interpolation methods assume that it is sufficient to use first- and second-order statistics to characterize random fields, researchers have now realized that these methods cannot always provide reliable interpolation results, since geological and environmental phenomena tend to be very complex, presenting non-Gaussian distribution and/or non-linear inter-variable relationship. This paper proposes a new approach to the interpolation of spatial data, which can be applied with great flexibility. Suitable cross-variable higher-order spatial statistics are developed to measure the spatial relationship between the random variable at an unsampled location and those in its neighbourhood. Given the computed cross-variable higher-order spatial statistics, the conditional probability density function is approximated via polynomial expansions, which is then utilized to determine the interpolated value at the unsampled location as an expectation. In addition, the uncertainty associated with the interpolation is quantified by constructing prediction intervals of interpolated values. The proposed method is applied to a mineral deposit dataset, and the results demonstrate that it outperforms kriging methods in uncertainty quantification. The introduction of the cross-variable higher-order spatial statistics noticeably improves the quality of the interpolation since it enriches the information that can be extracted from the observed data, and this benefit is substantial when working with data that are sparse or have non-trivial dependence structures.

A Multiple Training Image Approach for Spatial Modeling of Geologic Domains

Characterization of complex geological features and patterns remains one of the most challenging tasks in geostatistics. Multiple point statistics (MPS) simulation offers an alternative to accomplish this aim by going beyond classical two-point statistics. Reproduction of features in the final realizations is achieved by borrowing high-order spatial statistics from a training image. Most MPS algorithms use one training image at a time chosen by the geomodeler. This paper proposes the use of multiple training images simultaneously for spatial modeling through a scheme of data integration for conditional probabilities known as a linear opinion pool. The training images (TIs) are based on the available information and not on conceptual geological models; one image comes from modeling the categories by a deterministic approach and another comes from the application of conventional sequential indicator simulation. The first is too continuous and the second too random. The mixing of TIs requires weights for each of them. A methodology for calibrating the weights based on the available drillholes is proposed. A measure of multipoint entropy along the drillholes is matched by the combination of the two TIs. The proposed methodology reproduces geologic features from both TIs with the correct amount of continuity and variability. There is no need for a conceptual training image from another modeling technique; the data-driven TIs permit a robust inference of spatial structure from reasonably spaced drillhole data.

Accelerating experimental high-order spatial statistics calculations using GPUs

High-order spatial statistics have been widely used to describe the spatial phenomena in the field of geology science. Spatial statistics are subject to extremely heavy computational burden for large geostatistical models. To improve the computational efficiency, a parallel approach based on GPU (Graphics Processing Unit) is proposed for the calculation of high-order spatial statistics. The parallel scheme is achieved by utilizing a two-stage method to calculate the replicate of a moment for a given template simultaneously termed as the node-stage parallelism, and transform the spatial moments to cumulants for all lags of a template simultaneously termed as the template-stage parallelism. Also, a series of optimization strategies are proposed to take full advantage of the computational capabilities of GPUs, including the appropriate task allocation to the CUDA (Compute Unified Device Architecture) threads, proper organization of the GPU physical memory, and optimal improvement of the existed parallel routines. Tests are carried out on two training images to compare the performance of the GPU-based method with that of the serial implementation. Error analysis results indicate that the proposed parallel method can generate accurate cumulant maps, and the performance comparisons on various examples show that all the speedups for third-order, fourth-order and fifth-order cumulants calculation are over 17 times.

Large-aperture approximation for not-so-large apertures

In-flight wavefront measurements around a flat-window turret at subsonic Mach numbers are analyzed in instantaneous and time-averaged sense. In addition to the root-mean-squared levels of aero-optical distortions, higher-order spatial statistics are calculated, and their dependence as a function of the viewing angle is discussed. Given the optical data obtained, the applicability of the commonly used large aperture approximation (LAA) is revisited. We show that, for all angles, the LAA consistently underestimates the time-averaged Strehl ratio, so the LAA should be used very cautiously. Some reasons for these discrepancies are traced to non-Gaussian spatial distribution of the optical wavefronts. A different approximation for computing time-averaged Strehl ratios is proposed, and the results are discussed.

An Algorithm for 3D Pore Space Reconstruction from a 2D Image Using Sequential Simulation and Gradual Deformation with the Probability Perturbation Sampler

The construction of a faithful 3D pore space model of a porous medium that could reproduce the macroscopic behavior of that medium is of great interest in various fields including medicine, material science, hydrology and petroleum engineering. A computationally efficient algorithm is developed that uses the probability perturbation method and sequential multiple-point statistics simulations to generate 3D stochastic and equiprobable representations of random porous media when only a 2D thin section image is available. By employing the probability perturbation method as a gradual deformation technique, the pore patterns of a single 2D image are deformed to generate a series of 2D stochastically simulated images. The 3D pore structure is then generated by simply stacking the 2D-simulated images. The quality of the 3D reconstruction is critically dependent on the rate of deformation and a simple general procedure for choosing this parameter is presented. Various criteria such as porosity, two-point auto-correlation function, multiple-point connectivity function, local percolation probability, absolute permeability obtained by lattice-Boltzmann method (LBM), formation factor and two-phase relative permeability calculations are used to validate the results. The method is tested on two random porous solids; Berea Sandstone and synthetic Silica, for which directly measured 3D micro-CT images are available. The stochastically reconstructed 3D pore space preserves the low- and high-order spatial statistics, the macroscopic flow properties and the microstructure of the 3D micro-CT images.

High-Order Correlations in Internal Protein Motions and Energetics

Background: Despite originating as a linear sequence of amino acids, folded proteins display a remarkable diversity and specificity of motions, or dynamics, which constitute the building blocks of cellular metabolism. A growing body of evidence also suggests that these motions are hierarchical, involving a multitude of spatial and temporal scales. A key task in biology is thus to elucidate the relationship between hierarchical nature of protein dynamics and function. Characterizing these spatial/ temporal fluctuations in molecular dynamics (MD) simulations is critical to understanding enzyme catalysis, ligand binding, and allosteric signaling - all therapeutically exploitable processes.Results: Using 0.5us simulation for ubiquitin, a 76 residue protein that labels other proteins for degradation, we show that positional deviations exhibit non-Gaussian behavior (kurtosis >> 3), at functionally important regions of the protein. To analyze the spatial deviations we propose a general and statistically rigorous method Quasi Anharmonic Analysis (QAA) that meaningfully captures non-Gaussian behaviors overlooked with established methods. QAA, which is an extension of independent component analysis (ICA) techniques, is a more realistic encoding of protein fluctuations and atomic coupling since its basis vectors capture, in addition to variance, higher-order spatial statistics. QAA benefits from relaxing the constraint of orthogonality in basis vectors (e.g., PCA) or assumptions of Gaussian deviations. This coupling between the basis vectors from QAA allows one to elucidate how ‘fast’ and ‘slow’ motions in ubiquitin allow it to bind to different substrates with high specificity.Conclusions: QAA is a novel approach to organize and visualize conformational landscape spanned by a protein. QAA naturally characterizes long-tailed distributions and separates conformational clusters with exceptional clarity when projected onto the novel representation space. The transitions we observe in ubiquitin signal biologically important structural shifts and highlight meaningful energetic barriers in the underlying energy landscape.