Complex-Valued Random Fields for Vectorial Data: Estimating and Modeling Aspects

Complex-valued random fields represent a natural extension of real-valued random fields and can be useful for modeling vectorial data in two dimensions (i.e., a wind field). In such a case, some theoretical issues arise concerning generating and fitting complex covariance functions to be used for prediction purposes. In this paper, some general aspects and properties of complex-valued random fields are summarized and a procedure to fit complex stationary covariance functions is proposed. A case study for analyzing wind speed data is presented.     

Modeling Ocean Currents Through Complex Random Fields Indexed in Time

Surface ocean currents are often of interest in environmental monitoring. These vectorial data can be reasonably treated as a finite realization of a complex-valued random field, where the decomposition in modulus (current speed) and direction (current direction) of the current field is natural. Moreover, when observations are also available for different time points (other than at several locations), it is useful to evaluate the evolution of their complex correlation over time (rather than in space) and the corresponding modeling which is required for estimation purposes. This paper illustrates a first approach where the temporal profile of surface ocean currents is considered. After introducing the fundamental aspects of the complex formalism of a random field indexed in time, a new class of models suitable for including the temporal component is proposed and applied to describe the time-varying complex covariance function of current data. The analysis concerns ocean current observations, taken hourly on 30 April 2016 through high frequency radar systems at some stations located in the Northeastern Caribbean Sea. The selected complex covariance model indexed in time is used for estimation purposes and its reliability is confirmed by a numerical analysis.

     

Conditional Simulation of Random Fields by Successive Residuals

This paper presents a new approach to the LU decomposition method for the simulation of stationary and ergodic random fields. The approach overcomes the size limitations of LU and is suitable for any size simulation. The proposed approach can facilitate fast updating of generated realizations with new data, when appropriate, without repeating the full simulation process. Based on a novel column partitioning of the L matrix, expressed in terms of successive conditional covariance matrices, the approach presented here demonstrates that LU simulation is equivalent to the successive solution of kriging residual estimates plus random terms. Consequently, it can be used for the LU decomposition of matrices of any size. The simulation approach is termed conditional simulation by successive residuals as at each step, a small set (group) of random variables is simulated with a LU decomposition of a matrix of updated conditional covariance of residuals. The simulated group is then used to estimate residuals without the need to solve large systems of equations.     

Conditioning Simulations of Gaussian Random Fields by Ordinary Kriging

Conditioning realizations of stationary Gaussian random fields to a set of data is traditionally based on simple kriging. In practice, this approach may be demanding as it does not account for the uncertainty in the spatial average of the random field. In this paper, an alternative model is presented, in which the Gaussian field is decomposed into a random mean, constant over space but variable over the realizations, and an independent residual. It is shown that, when the prior variance of the random mean is infinitely large (reflecting prior ignorance on the actual spatial average), the realizations of the Gaussian random field are made conditional by substituting ordinary kriging for simple kriging. The proposed approach can be extended to models with random drifts that are polynomials in the spatial coordinates, by using universal or intrinsic kriging for conditioning the realizations, and also to multivariate situations by using cokriging instead of kriging.     

Atomic-Scale Study of Mineral Luminescent Materials and Their Microstructures

The aim of this thesis is to study the luminescence of fluorite: Eu3+ fluorite, Eu2+ and wollastonite: Ce3+, Tba+,to discuss crystal structures and defects using solid atomic images with high resolution transmission electron microscope,and to deal with the luminescent effect of mineral microscopic structure.In general, because of Eu3+ belongs to electric dipole transition, it is forbidden to free ion. But in fluorite, Eu3+system, Eu3+ is located at the nonsymmetrical center, and theparity selection rules are broken.     

Conditioned Simulations of Random Velocity Fields

In petroleum exploration and production, it is essential to have good estimations of the uncertainties on the reserves. Uncertainties on the velocity model used during the data processing are of major importance in this estimation. The generation of several velocity fields gives access to a quantified estimation of the uncertainties due to the velocity model inversion. The use of statistical methods helps in generating several important, equiprobable velocity fields, matching all the available velocity information. This paper presents an efficient simulation algorithm to generate instantaneous velocity fields, constrained by the distribution of values measured at the wells, and calibrated by the stacking velocities, taken as root-mean-square velocities. The simulations also match the covariance model given for the instantaneous velocity fields. The method is developed in a simple one-layer case with constant velocity, and then extended to more realistic situations. Finally, a real data application is shown, using data provided by ENI–Agip Division, and the efficiency of the proposed simulation method is discussed.     

High-order Statistics of Spatial Random Fields: Exploring Spatial Cumulants for Modeling Complex Non-Gaussian and Non-linear Phenomena

The spatial distributions of earth science and engineering phenomena under study are currently predicted from finite measurements and second-order geostatistical models. The latter models can be limiting, as geological systems are highly complex, non-Gaussian, and exhibit non-linear patterns of spatial connectivity. Non-linear and non-Gaussian high-order geostatistics based on spatial connectivity measures, namely spatial cumulants, are proposed as a new alternative modeling framework for spatial data. This framework has two parts. The first part is the definition, properties, and inference of spatial cumulants—including understanding the interrelation of cumulant characteristics with the in-situ behavior of geological entities or processes, as examined in this paper. The second part is the research on a random field model for simulation based on its high-order spatial cumulants. Mathematical definitions of non-Gaussian spatial random functions and their high-order spatial statistics are presented herein, stressing the notion of spatial cumulants. The calculation of spatial cumulants with spatial templates follows, including anisotropic experimental cumulants. Several examples of two- and three-dimensional images, including a diamond bearing kimberlite pipe from the Ekati Mine in Canada, are analyzed to assess the relations between cumulants and the spatial behavior of geological processes. Spatial cumulants of orders three to five are shown to capture directional multiple-point periodicity, connectivity including connectivity of extreme values, and spatial architecture. In addition, they provide substantial information on geometric characteristics and anisotropy of geological patterns. It is further shown that effects of complex spatial patterns are seen even if only subsets of all cumulant templates are computed. Compared to second-order statistics, cumulant maps are found to include a wealth of additional information from underlying geological patterns. Further work seeks to integrate this information in the predictive capabilities of a random field model.     

Stationary and Isotropic Vector Random Fields on Spheres

This paper presents the characterization of the covariance matrix function of a Gaussian or second-order elliptically contoured vector random field on the sphere which is stationary, isotropic, and mean square continuous. This characterization involves an infinite sum of the products of positive definite matrices and Gegenbauer??s polynomials, and may not be available for other non-Gaussian vector random fields on spheres such as a ?? ² or log-Gaussian vector random field. We also offer two simple but efficient constructing approaches, and derive some parametric covariance matrix structures on spheres.     

Numerical Method for Conditional Simulation of Levy Random Fields

Stochastic simulations of subsurface heterogeneity require accurate statistical models for spatial fluctuations. Incremental values in subsurface properties were shown previously to be approximated accurately by Levy distributions in the center and in the start of the tails of the distribution. New simulation methods utilizing these observations have been developed. Multivariate Levy distributions are used to model the multipoint joint probability density. Explicit bounds on the simulated variables prevent nonphysical extreme values and introduce a cutoff in the tails of the distribution of increments. Long-range spatial dependence is introduced through off-diagonal terms in the Levy association matrix, which is decomposed to yield a maximum likelihood type estimate at unobserved locations. This procedure reduces to a known interpolation formula developed for Gaussian fractal fields in the situation of two control points. The conditional density is not univariate Levy and is not available in closed form, but can be constructed numerically. Sequential simulation algorithms utilizing the numerically constructed conditional density successfully reproduce the desired statistical properties in simulations.     

Spectral Simulation of Isotropic Gaussian Random Fields on a Sphere

A spectral algorithm is proposed to simulate an isotropic Gaussian random field on a sphere equipped with a geodesic metric. This algorithm supposes that the angular power spectrum of the covariance function is explicitly known. Direct analytic calculations are performed for exponential and linear covariance functions. In addition, three families of covariance functions are presented where the calculation of the angular power spectrum is simplified (shot-noise random fields, Yadrenko covariance functions and solutions of certain stochastic partial differential equations). Numerous illustrative examples are given.

     

Conditional Simulation of Random Fields with Bivariate Gamma Isofactorial Distributions

This work focuses on a random function model with gamma marginal and bivariate isofactorial distributions, which has been applied in mining geostatistics for estimating recoverable reserves by disjunctive kriging. The objective is to widen its use to conditional simulation and further its application to the modeling of continuous attributes in geosciences. First, the main properties of the bivariate gamma isofactorial distributions are analyzed, with emphasis in the destructuring of the extreme values, the presence of a proportional effect (higher variability in high-valued areas), and the asymmetry in the spatial correlation of the indicator variables with respect to the median threshold. Then, we provide examples of stationary random functions with such bivariate distributions, for which the shape parameter of the marginal distribution is half an integer. These are defined as the sum of squared independent Gaussian random fields. An iterative algorithm based on the Gibbs sampler is proposed to perform the simulation conditional to a set of existing data. Such ‘multivariate chi-square’ model generalizes the well-known multigaussian model and is more flexible, since it allows defining a shape parameter which controls the asymmetry of the marginal and bivariate distributions.     

Markov Chain Random Fields for Estimation of Categorical Variables

Multi-dimensional Markov chain conditional simulation (or interpolation) models have potential for predicting and simulating categorical variables more accurately from sample data because they can incorporate interclass relationships. This paper introduces a Markov chain random field (MCRF) theory for building one to multi-dimensional Markov chain models for conditional simulation (or interpolation). A MCRF is defined as a single spatial Markov chain that moves (or jumps) in a space, with its conditional probability distribution at each location entirely depending on its nearest known neighbors in different directions. A general solution for conditional probability distribution of a random variable in a MCRF is derived explicitly based on the Bayes’ theorem and conditional independence assumption. One to multi-dimensional Markov chain models for prediction and conditional simulation of categorical variables can be drawn from the general solution and MCRF-based multi-dimensional Markov chain models are nonlinear.     

Markov Chain Random Fields for Estimation of?Categorical Variables

Multi-dimensional Markov chain conditional simulation (or interpolation) models have potential for predicting and simulating categorical variables more accurately from sample data because they can incorporate interclass relationships. This paper introduces a Markov chain random field (MCRF) theory for building one to multi-dimensional Markov chain models for conditional simulation (or interpolation). A MCRF is defined as a single spatial Markov chain that moves (or jumps) in a space, with its conditional probability distribution at each location entirely depending on its nearest known neighbors in different directions. A general solution for conditional probability distribution of a random variable in a MCRF is derived explicitly based on the Bayes’ theorem and conditional independence assumption. One to multi-dimensional Markov chain models for prediction and conditional simulation of categorical variables can be drawn from the general solution and MCRF-based multi-dimensional Markov chain models are nonlinear.     

三维可视化动态地质建模系统研发与应用

地质体三维可视化动态建模软件系统的研发涉及地质数据三维可视化、地质体三维动态构模和地质专业空间决策分析等一系列方法、技术问题。采用基于非均匀有理样条(non uniform rational B-spline,简称NURBS)等多种函数的空间插值法,基于钻孔、剖面、散点集和多源数据的构模法,以及基于单体塌陷的通用网格简化算法(general mesh simplification,简称GSM)和Grid+CSR-Tree三维空间混合索引法,在QuantyView软件平台上研发了一个具有快速动态重构能力的三维动态地质建模系统。在多个地区的实际应用表明,开展"玻璃国土"建设对于推进地质勘查与开发工作信息化具有重要的意义。     

Investigation of Soil Conditioning Tests with Three-Dimensional Numerical Modeling

Due to the diverse and complex structure of soil and the variety of foam-modifier materials that are used, it is difficult to provide a model to predict the laboratory behavior of modified soils. For example, several studies have shown independently that the amount of the foam-modified soil depends on several factors, such as the internal friction angle and normal stiffness. Of late, modeling by numerical methods has become popular in engineering sciences and the modeling of complex material behavior is possible with the help of numerical methods. In this research, the performance and efficiency of the numerical method in the modeling of laboratory tests such as the slump test and the uniaxial compressive strength test were investigated and it was found that numerical modeling performs very well in predicting the results of these tests for foam-modified sand samples. In order to achieve this goal, the slump test and the uniaxial compressive strength test were performed in the laboratory on several modified sand samples in order to obtain the laboratory results for these samples. Then, numerical simulation of these experiments was carried out using PFC^3D software. The results of numerical modeling were compared with the experimental results, and good agreement was observed. Finally, after calibration of the numerical model using the experimental results, the effect of changes in the internal friction angle and the normal stiffness of the modified sand in the amount of the slump was investigated. According to the results of this sensitivity analysis, it was determined that by increasing both effective parameters the amount of the slump of foam-modified sand decreases and that the parameters are the most important factors in controlling the slump value.     

寒区高等级公路宽幅路基的随机温度场

寒区高等级公路宽幅路面比普通路面吸热面积更大,增加了保证路基稳定性的难度.在对其进行路基温度场的稳定性分析时,传统的有限元方法都是进行确定性分析,没有考虑到边界条件等因素的随机性、变异性,因此不能得知以上随机因素引起温度场的变幅.采用随机有限元法,由温度场的变分原理,通过摄动法建立随机温度场,从而得到随机温度场的有限元格式,理论公式可广泛适用于其他工程随机温度场问题. 将宽幅路基气温边界条件参数设为随机变量,考虑升温效应,对寒区宽幅路基温度场进行计算,分析了温度场均值和方差的分布规律.结果表明：宽幅路基中部聚热效应明显,路基下部出现厚度不均匀的融化盘,可能造成路基不均匀沉降;边界条件的随机性造成温度场方差分布呈明显的边界效应,即越靠近路基上边界温度场方差越大,在一定深度以下趋于零.此外,不论是路基下出现的融化盘,还是温度场方差分布下边界都将随时间增长而扩大.     

Conditional Latin Hypercube Simulation of (Log)Gaussian Random Fields

In earth and environmental sciences applications, uncertainty analysis regarding the outputs of models whose parameters are spatially varying (or spatially distributed) is often performed in a Monte Carlo framework. In this context, alternative realizations of the spatial distribution of model inputs, typically conditioned to reproduce attribute values at locations where measurements are obtained, are generated via geostatistical simulation using simple random (SR) sampling. The environmental model under consideration is then evaluated using each of these realizations as a plausible input, in order to construct a distribution of plausible model outputs for uncertainty analysis purposes. In hydrogeological investigations, for example, conditional simulations of saturated hydraulic conductivity are used as input to physically-based simulators of flow and transport to evaluate the associated uncertainty in the spatial distribution of solute concentration. Realistic uncertainty analysis via SR sampling, however, requires a large number of simulated attribute realizations for the model inputs in order to yield a representative distribution of model outputs; this often hinders the application of uncertainty analysis due to the computational expense of evaluating complex environmental models. Stratified sampling methods, including variants of Latin hypercube sampling, constitute more efficient sampling aternatives, often resulting in a more representative distribution of model outputs (e.g., solute concentration) with fewer model input realizations (e.g., hydraulic conductivity), thus reducing the computational cost of uncertainty analysis. The application of stratified and Latin hypercube sampling in a geostatistical simulation context, however, is not widespread, and, apart from a few exceptions, has been limited to the unconditional simulation case. This paper proposes methodological modifications for adopting existing methods for stratified sampling (including Latin hypercube sampling), employed to date in an unconditional geostatistical simulation context, for the purpose of efficient conditional simulation of Gaussian random fields. The proposed conditional simulation methods are compared to traditional geostatistical simulation, based on SR sampling, in the context of a hydrogeological flow and transport model via a synthetic case study. The results indicate that stratified sampling methods (including Latin hypercube sampling) are more efficient than SR, overall reproducing to a similar extent statistics of the conductivity (and subsequently concentration) fields, yet with smaller sampling variability. These findings suggest that the proposed efficient conditional sampling methods could contribute to the wider application of uncertainty analysis in spatially distributed environmental models using geostatistical simulation.     

计算机辅助矿产资源储量动态估算与管理模型

针对我国矿产资源储量监管水平落后,矿山资源储量动态估算、管理及图件编制信息化程度低的现状,以矿山日常勘探与开发工作流程为主线,以地矿点源数据库为基础,研究了一种新的可对不断勘探与开采的矿山提供有效支持的计算机辅助矿产资源储量动态估算与管理模型。该模型可实现复杂多金属矿山资源储量二维及三维空间一体化与可视化动态估算,创新性地解决了复杂多金属矿山勘探与开发所面临的资源储量动态核减与管理问题。以基于该模型实现的矿产资源勘查与开发软件--QuantyMine传统方法矿产资源储量动态估算子系统为工具,以福建紫金矿业集团所属的多个矿山为实际应用对象进行模型检验。经与核实报告对比分析,所选取的某典型多金属矿床矿石量、金属量动态估算结果与报告结果相差分别为-1.78％～2.86％和-5.05％～4.45％,资源储量各类别结果接近。     

垂直有限线源三维地电场有限差分正演研究

利用套管供电,通过地表观测电位研究油田注水分布和剩余油分布在油田开发中的应用。针对垂直有限线源,首先推导了线电流源的边界条件,利用有限差分实现了三维地电场的正演。正演结果表明：在全井套管供电时,地表观测电位主要受浅层电阻率不均匀性的影响;在射孔段供电时,地表观测电位主要反映射孔段的电阻率分布。