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

Using third-order cumulants to investigate spatial variation: a case study on the porosity of the Bunter Sandstone

The multivariate cumulants characterize aspects of the spatial variability of a regionalized variable. A centred multivariate Gaussian random variable, for example, has zero third-order cumulants. In this paper it is shown how the third-order cumulants can be used to test the plausibility of the assumption of multivariate normality for the porosity of an important formation, the Bunter Sandstone in the North Sea. The results suggest that the spatial variability of this variable deviates from multivariate normality, and that this assumption may lead to misleading inferences about, for example, the uncertainty attached to kriging predictions

Temporal moments of a tracer pulse in a perfectly parallel flow system

Perfectly parallel groundwater transport models partition water flow into isolated one-dimensional stream tubes which maintain total spatial correlation of all properties in the direction of flow. The case is considered of the temporal moments of a conservative tracer pulse released simultaneously into N stream tubes with arbitrarily different advective–dispersive transport and steady flow speeds in each of the stream tubes. No assumptions are made about the form of the individual stream tube arrival-time distributions or about the nature of the between-stream tube variation of hydraulic conductivity and flow speeds. The tracer arrival-time distribution g(t,x) is an N-component finite-mixture distribution, with the mean and variance of each component distribution increasing in proportion to tracer travel distance x. By utilising moment relations of finite mixture distributions, it is shown (to r=4) that the rth central moment of g(t,x) is an rth order polynomial function of x or φ, where φ is mean arrival time. In particular, the variance of g(t,x) is a positive quadratic function of x or φ. This generalises the well-known quadratic variance increase for purely advective flow in parallel flow systems and allows a simple means of regression estimation of the large-distance coefficient of variation of g(t,x). The polynomial central moment relation extends to the purely advective transport case which arises as a large-distance limit of advective–dispersive transport in parallel flow models. The associated limit g(t,x) distributions are of N-modal form and maintain constant shapes independent of travel distance. The finite-mixture framework for moment evaluation is also a potentially useful device for forecasting g(t,x) distributions, which may include multimodal forms. A synthetic example illustrates g(t,x) forecasting using a mixture of normal distributions

Development of a Grade Control Technique Optimizing Dilution and Ore Loss Trade-off in Lateritic Bauxite Deposits

This thesis focusses on the development of new techniques to improve the resource estimation of laterite-type bauxite deposits. Contributions of the thesis include (1) a methodology to variogram-free modelling of the ore boundaries using multiple-point statistics, (2) an approach to automate the parameter tuning process for multiple-point statistical algorithms and (3) a grade control technique to minimise the economic losses due to dilution and ore loss

A Novel Pixel-based Multiple-Point Geostatistical Simulation Method for Stochastic Modeling of Earth Resources

Uncertainty is an integral part of modeling Earth\u27s resources and environmental processes. Geostatistical simulation technique is a well-established tool for uncertainty quantification of earth systems modeling. Multiple-point statistical (MPS) algorithms are specifically advantageous when dealing with the complexity and heterogeneity of geological data. MPS algorithms take advantage of using training images to mimic physical reality. This research presents a novel and efficient pixel-based multiple-point geostatistical simulation method for mineral resource modeling. Pixel-based simulation implies the sequential modeling of individual points on the simulation grid by borrowing spatial information from the training image and honoring conditioning data points. The developed method borrows information by integrating multiple machine learning algorithms, including Principal Component Analysis (PCA), t-Distributed Stochastic Neighbor Embedding (t-SNE), and Density-based Spatial Clustering of Applications with Noise (DBSCAN) algorithms. For automation and to ensure high-quality realizations, multiple optimizations, and parameter tuning strategies were introduced. The proposed methodology proved its applicability by accurate reproduction of complex geological features honoring conditioning data while maintaining reasonable computational time. The model is validated by simulating a variety of categorical and continuous variables for both two and three-dimensional cases and conditional and unconditional simulations. As a three-dimensional case study for categorical stochastic modeling, the proposed method is applied to a gold deposit for orebody modeling. The proposed algorithm can be applied to a variety of contexts, including but not limited to petroleum reservoir characterization, seismic inversion, mineral resources modeling, gap-filling in remote sensing, and climate modeling. The developed model can be extended for spatio-temporal modeling, multivariate simulation, non-stationary modeling, and super-resolution realizations

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

Developing New High-Order Sequential Simulation Methods Based on Learning-Oriented Kernels

RÉSUMÉ: La quantification de l’incertitude joue un rôle essentiel dans la gestion du risque technique de l’exploitation durable des ressources naturelles. Les modèles de champs aléatoires sont utilisés pour modéliser les attributs naturels d’intérêt, parmi lesquels les attributs à différents endroits sont représentés comme des variables aléatoires comprenant une distribution de probabilité conjointe. Les statistiques spatiales, qui varient selon différents modèles de champs aléatoires, décrivent mathématiquement les structures spatiales. Les méthodes de simulation stochastique génèrent des réalisations multiples basées sur certains modèles de champs aléatoires afin de représenter les résultats possibles des attributs naturels considérés. Elles visent à reproduire les statistiques spatiales des données perçues, fournissant ainsi des outils utiles pour quantifier l’incertitude spatiale des attributs cibles. Dans le contexte des applications minières, la reproduction de structures spatiales, à partir des donnéeséchantillons, a un impact significatif sur la gestion des risques liés aux décisions de planification minière. Plus précisément, la valeur actualisée nette (VAN) d’un gisement minéral, compte tenu d’un calendrier de planification minière donné, dépend des revenus générés par les séquences d’extraction des matériaux souterrains, les flux de trésorerie étant actualisés en fonction des périodes d’exploitation. Les séquences d’extraction des matériaux, à leur tour, sont déterminées par la distribution spatiale des teneurs en métaux, en particulier la continuité spatiale des éléments métalliques enrichis. Les méthodes de simulation stochastique d’ordre élevé ne présupposent aucune distribution de probabilité spécifique sur les modèles de champs aléatoires, évitant ainsi les limites des modèles de champs aléatoires gaussiens traditionnels. De plus, ces méthodes tiennent compte des statistiques spatiales d’ordre élevé qui caractérisent les interactions statistiques entre les attributs aléatoires en de multiples endroits et elles ont donc l’avantage de reproduire des structures spatiales complexes. Par conséquent, cette thèse développe de nouvelles méthodes de simulation stochastique d’ordre élevé basées sur un cadre proposé d’apprentissage statistique et de noyaux orientés sur l’apprentissage, visant à faire progresser les aspects théoriques des méthodes de simulation stochastique ainsi que les aspects pratiques des décisions minières sous incertitude. Le paradigme général de la simulation séquentielle est adopté dans cette thèse afin de générer des réalisations à partir de modèles de champs aléatoires, ce qui décompose les distributions de probabilités conjointes en une séquence de distributions de probabilités conditionnelles. La méthode originale de simulation d’ordre élevé utilise la série d’expansions du polynôme de Legendre pour l’approximation des distributions de probabilités conjointes des champs aléatoires. Les coefficients de la série d’expansion du polynôme sont dérivés du calcul de ce que l’on appelle les cumulants spatiaux qui doivent être stockés dans une structure arborescente en mémoire et certains termes de la série du polynôme sont abandonnés compte tenu de la complexité du calcul. À titre de première contribution, un nouveau modèle de calcul de simulation d’ordre élevé est ici proposé, évitant le calcul explicite des cumulants spatiaux et le stockage des résultats calculés. Une fonction unifiée est dérivée comme une forme d’équivalence à la série d’expansions du polynôme de Legendre sans abandonner aucun terme, tout en simplifiant les calculs en temps polynomial. La méthode de simulation proposée conduit à un algorithme récursif de dérivation de la distribution de probabilités conditionnelles. À titre de deuxième contribution, une nouvelle fonction du noyau, ce qu’on appelle le noyau spatial du moment de Legendre, est proposée pour intégrer des statistiques spatiales d’ordre élevé des données originales dans le nouvel espace du noyau. Un cadre d’apprentissage statistique est proposé pour découvrir la distribution de probabilité cible du champ aléatoire en la faisant correspondre aux statistiques spatiales d’ordre élevé observées dans les données disponibles grâce à un algorithme à noyau. Le nouveau cadre d’apprentissage statistique pour la simulation d’ordre élevé a la capacité de généralisation nécessaire pour atténuer les conflits statistiques entre les données-échantillons et l’image d’entraînement, comme le confirment les études de cas avec des données synthétiques. Un gisement d’or tridimensionnel est réalisé pour montrer ses aspects pratiques dans une mine réelle, en démontrant la reproduction de statistiques spatiales d’ordre élevé à partir des données-échantillons de forage. Pour éviter l’impact d’éventuels conflits statistiques avec les données-échantillons en utilisant une image d’entraînement, une méthode de simulation d’ordre élevé sans image d’entraînement est développée en se basant sur le cadre d’apprentissage statistique ci-dessus. Une nouvelle approche d’agrégation de noyaux est proposée afin de permettre la découverte de données éparses. Les événements de données, comme les données de conditionnement, correspondent aux valeurs d’attribut associées aux modèles spatiaux de diverses configurations géométriques. L’agrégation de noyaux combine l’ensemble des éléments dans différents sousespaces du noyau pour l’inférence statistique, en utilisant efficacement les informations incomplètes des répliques qui correspondent partiellement au modèle spatial d’un événement de données spécifique. L’étude de cas montre une bonne reproduction des statistiques spatiales d’ordre élevé des données-échantillons sans utiliser les images d’entraînement. Notre dernière contribution vise à atteindre la distribution de probabilité cible des modèles de champs aléatoires en apprenant des informations spatiales d’ordre élevé provenant de différentes sources à différentes échelles. Plus précisément, l’agrégation de noyaux est proposée pour incorporer les statistiques spatiales d’ordre élevé à une échelle grossière à partir des données-échantillons et pour compléter les statistiques spatiales d’ordre élevé à petite échelle à partir de l’image d’entraînement. De plus, un logiciel est développé et décrit pour faciliter les applications. Des études de cas, dans un gisement d’or et avec un ensemble de données synthétique, sont menées respectivement afin de tester la méthode et le programme développé.----------ABSTRACT:Uncertainty quantification plays a vital role in managing the technical risk of the sustainable exploitation of natural resources. Random field models are utilized to model the natural attributes of interest, within which the attributes at different locations are represented as random variables comprising a joint probability distribution. Spatial statistics, varied with different random field models, mathematically describe the spatial patterns. Stochastic simulation methods generate multiple realizations based on certain random field models to represent the possible outcomes of natural attributes under consideration. They aim to reproduce spatial statistics of the perceived data, thus providing useful tools to quantify the spatial uncertainty of the target attributes. In the context of mining applications, reproducing spatial patterns from the sample data has a significant impact on managing the risks of mine planning decisions. Specifically, the net present value (NPV) regarding a certain mine planning schedule of a mineral deposit depends on the revenue generated by the extraction sequences of the underground materials, as the cash flows are discounted by the mining periods. The extraction sequences of the materials, in turn, are driven by the spatial distributions of metal grades, especially the spatial continuity of enriched metal elements. High-order stochastic simulation methods make no assumption of any specific probability distribution on the random field models, avoiding the limitation of traditional Gaussian random field models. In addition, the methods account for the high-order spatial statistics that characterize the statistical interactions among random attributes at multiple locations and thus have the advantage of reproducing complex spatial patterns. Therefore, this thesis develops new high-order stochastic simulation methods based on a proposed framework of statistical learning and learning-oriented kernels, aiming to advance the theoretical aspects of the stochastic simulation methods, as well as the practical aspects of mining decisions under uncertainty. The general paradigm of sequential simulation is adopted in this thesis to generate realizations from the random field models, which decomposes the joint probability distributions into a sequence of conditional probability distributions. The original high-order simulation method uses the Legendre polynomial expansion series for the approximation of the joint probability distributions of the random fields. The coefficients of the polynomial expansion series are derived from the computation of so-called spatial cumulants, which have to be stored in a tree structure in memory. In addition, some terms from the polynomial series are dropped considering the computational complexity. As a first contribution, a new computational model of high-order simulation is proposed herein,which avoids the explicit computation of spatial cumulants and the storage of the computed results. A unified function is derived as the equivalency form to the Legendre polynomial expansion series without dropping out any terms, while simplifying the computations to polynomial time. The proposed simulation method leads to a recursive algorithm of deriving the conditional probability distribution. As a second contribution, a new kernel function, the so-termed spatial Legendre moment kernel, is proposed to embed high-order spatial statistics of the original data into the new kernel space. A statistical learning framework is proposed to learn the target probability distribution of the random field by matching the expected high-order spatial statistics with regard to the target distribution to the observed high-order spatial statistics of the available data through a kernelized algorithm. The new statistical learning framework for high-order simulation has the generalization capacity to mitigate the statistical conflicts between the sample data and the training image, as confirmed by the case studies with a synthetic data set. Case study at a three-dimensional gold deposit shows the practical aspects of the proposed method in a real-life mine, demonstrating the reproduction of high-order spatial statistics from the drill-hole sample data. To avoid the impact of potential statistical conflicts with the sample data by using a training image, a training-image free high-order simulation method is developed based on the above statistical learning framework. A new concept of aggregated kernel statistics is proposed to enable sparse data learning. The data events, as the conditioning data, correspond to the attribute values associated with the so-called spatial templates of various geometric configurations. The aggregated kernel statistics combine the ensemble of the elements in different kernel subspaces for statistical inference, efficiently utilizing the incomplete information from the replicates, which partially match to the spatial template of a given data event. The case study shows an effective reproduction of the high-order spatial statistics of the sample data without using the TI. Our last contribution aims to achieve the target probability distributions of the random field models by learning high-order spatial information from different sources at multiple scales. Specifically, the aggregated kernel statistics is proposed to incorporate the high-order spatial statistics at coarse-scale from the sample data and to complement the high-order spatial statistics at fine-scale from the TI. In addition, a software is developed and described to facilitate the applications. Case studies with a synthetic data set and at a gold deposit are conducted respectively to test the method and the developed program

Stochastic Inverse Methods to Identify non-Gaussian Model Parameters in Heterogeneous Aquifers

La modelación numérica del flujo de agua subterránea y del transporte de masa se está convirtiendo en un criterio de referencia en la actualidad para la evaluación de recursos hídricos y la protección del medio ambiente. Para que las predicciones de los modelos sean fiables, estos deben de estar lo más próximo a la realidad que sea posible. Esta proximidad se adquiere con los métodos inversos, que persiguen la integración de los parámetros medidos y de los estados del sistema observados en la caracterización del acuífero. Se han propuesto varios métodos para resolver el problema inverso en las últimas décadas que se discuten en la tesis. El punto principal de esta tesis es proponer dos métodos inversos estocásticos para la estimación de los parámetros del modelo, cuando estos no se puede describir con una distribución gausiana, por ejemplo, las conductividades hidráulicas mediante la integración de observaciones del estado del sistema, que, en general, tendrán una relación no lineal con los parámetros, por ejemplo, las alturas piezométricas. El primer método es el filtro de Kalman de conjuntos con transformación normal (NS-EnKF) construido sobre la base del filtro de Kalman de conjuntos estándar (EnKF). El EnKF es muy utilizado como una técnica de asimilación de datos en tiempo real debido a sus ventajas, como son la eficiencia y la capacidad de cómputo para evaluar la incertidumbre del modelo. Sin embargo, se sabe que este filtro sólo trabaja de manera óptima cuándo los parámetros del modelo y las variables de estado siguen distribuciones multigausianas. Para ampliar la aplicación del EnKF a vectores de estado no gausianos, tales como los de los acuíferos en formaciones fluvio-deltaicas, el NSEnKF propone aplicar una transformación gausiana univariada. El vector de estado aumentado formado por los parámetros del modelo y las variables de estado se transforman en variables con una distribución marginal gausiana.Zhou ., H. (2011). Stochastic Inverse Methods to Identify non-Gaussian Model Parameters in Heterogeneous Aquifers [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/12267Palanci