Polynomial expansions in likelihoods for spatial data: A case study

This paper illustrates the computational benefits of polynomial representations for quantities in the likelihood function for the spatial linear model based on the power covariance scheme. These benefits include a comprehensive study of likelihoods and maximum likelihood estimators for data. For simplicity, we focus on a relatively simple covariance scheme and data observed at equal intervals along a transect; we briefly indicate how generalizations to more complicated covariance functions and higher dimensions will operate.     

Conditioning of coefficient matrices of Ordinary Kriging

The solution of a set of linear equations is central to Ordinary Kriging. Computers are commonly applied because of the amount of data and work involved. There has, until recently, been little attention devoted toward the conditioning of kriging matrices. This article considers implications of conditioning upon numerical stability, instead of on robustness which has been the main focus of past work. The effect of properties of the stationary covariance matrix on the conditioning of the kriging matrix is discussed. The relationship between the covariance and autocorrelation functions allows some conclusions about the conditioning of covariance matrices, based on past work in deconvolution. The conditioning of some coefficient matrices of stationary kriging, defined in terms of either the semivariogram or the covariance, is examined.     

Point-block covariance: A general solution for the two-dimensional stationary random functions

The application of the theory of random functions to problems of ore evaluation may involve computations of the covariance between the mean value of a given block and the functional value at a given point. However, an analytical solution for such a covariance does not exist for nonspherical blocks and for commonly applied models of covariance functions. Further, because this covariance is a function of the spatial arrangements of the block and the point, it has to be evaluated numerically each time for given point—block arrangements. This paper presents a readily available general solution to this problem in the form of a series of graduated curves which, together with some geometric manipulations, may be used to compute the covariance between a pointand a two-dimensional block for all possible point—block arrangements. The availability of the graph thus eliminates the necessity of using the time-absorbing programs on computers for such computations. Finally, many of the approximations that are made in order to avoid cumbersome covariance evaluations are no longer necessary due to the ease of such computations with the help of the graph provided.     

Combining sensitivities and prior information for covariance localization in the ensemble Kalman filter for petroleum reservoir applications

Sampling errors can severely degrade the reliability of estimates of conditional means and uncertainty quantification obtained by the application of the ensemble Kalman filter (EnKF) for data assimilation. A standard recommendation for reducing the spurious correlations and loss of variance due to sampling errors is to use covariance localization. In distance-based localization, the prior (forecast) covariance matrix at each data assimilation step is replaced with the Schur product of a correlation matrix with compact support and the forecast covariance matrix. The most important decision to be made in this localization procedure is the choice of the critical length(s) used to generate this correlation matrix. Here, we give a simple argument that the appropriate choice of critical length(s) should be based both on the underlying principal correlation length(s) of the geological model and the range of the sensitivity matrices. Based on this result, we implement a procedure for covariance localization and demonstrate with a set of distinctive reservoir history-matching examples that this procedure yields improved results over the standard EnKF implementation and over covariance localization with other choices of critical length.     

On the analysis of pollen-climate canonical transfer functions

Canonical correlation analysis, as described by Webb and Bryson, Quaternary Research 1972, provides a means of reconstructing past climates quantitatively from fossil pollen using a pollen-climate transfer function. This paper presents a method for analysis of variance of the transfer function model. This method is used to identify ecological relationships among the pollen and climate variables, to select climatically sensitive taxa, and to investigate the importance of site factors. Several criteria are presented, in addition to those used by Webb and Bryson, for choosing canonical variate pairs to include in the transfer function model, namely: the variate pair relationships should be ecologically meaningful; the transfer function model should yield stable paleoclimatic estimates; and, the variate pair relationships should be statistically meaningful. The application of these criteria to the set of variate pairs used in the transfer function model of Webb and Bryson is described and modifications are suggested.     

The Correlation Structure of Matheron's Classical Variogram Estimator Under Elliptically Contoured Distributions

The classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. When data have an elliptically contoured distribution with constant mean, the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix, the covariance matrix, and the kurtosis. Several specific cases are studied closely. A subclass of elliptically contoured distributions with a particular family of covariance matrices is shown to possess exactly the same correlation structure for the classical variogram estimator as the multivariate independent Gaussian distribution. The consequences on variogram fitting by generalized least squares are discussed.     

The Correlation Structure of Matheron's Classical Variogram Estimator Under Elliptically Contoured Distributions

The classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. When data have an elliptically contoured distribution with constant mean, the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix, the covariance matrix, and the kurtosis. Several specific cases are studied closely. A subclass of elliptically contoured distributions with a particular family of covariance matrices is shown to possess exactly the same correlation structure for the classical variogram estimator as the multivariate independent Gaussian distribution. The consequences on variogram fitting by generalized least squares are discussed.     

Generalized cross-covariances and their estimation

Generalized cross-covariances describe the linear relationships between spatial variables observed at different locations. They are invariant under translation of the locations for any intrinsic processes, they determine the cokriging predictors without additional assumptions and they are unique up to linear functions. If the model is stationary, that is if the variograms are bounded, they correspond to the stationary cross-covariances. Under some symmetry condition they are equal to minus the usual cross-variogram. We present a method to estimate these generalized cross-covariances from data observed at arbitrary sampling locations. In particular we do not require that all variables are observed at the same points. For fitting a linear coregionalization model we combine this new method with a standard algorithm which ensures positive definite coregionalization matrices. We study the behavior of the method both by computing variances exactly and by simulating from various models.     

1：5万登龙、热加幅的成矿预测研究

利用已知矿床（点）的地质特征，采用数学方法对变量进行筛选，提出最佳的变量组合，对工作区进行综合信息预测，最后，圈出成矿远景区．     

Evaluation and error analysis: Kalman gain regularization versus covariance regularization

Ensemble size is critical to the efficiency and performance of the ensemble Kalman filter, but when the ensemble size is small, the Kalman gain generally cannot be well estimated. To reduce the negative effect of spurious correlations, a regularization process applied on either the covariance or the Kalman gain seems to be necessary. In this paper, we evaluate and compare the estimation errors when two regularization methods including the distance-dependent localization and the bootstrap-based screening are applied on the covariance and on the Kalman gain. The investigations were carried out through two examples: 1D linear problem without dynamics but for which the true Kalman gain can be computed and a 2D highly nonlinear reservoir fluid flow problem. The investigation resulted in three primary conclusions. First, if localizations of two covariance matrices are not consistent, the estimate of the Kalman gain will generally be poor at the observation location. The consistency condition can be difficult to apply for nonlocal observations. Second, the estimate of the Kalman gain that results from covariance regularization is generally subject to greater errors than the estimate of the Kalman gain that results from Kalman gain regularization. Third, in terms of removing spurious correlations in the estimation of spatially correlated variables, the performance of screening Kalman gain is comparable as the performance of localization methods (applied on either covariance or Kalman gain), but screening Kalman gain outperforms the localization methods in terms of generality for application, as the screening method can be used for estimating both spatially correlated and uncorrelated variables, and moreover, no assumption about the prior covariance is required for the screening method.     

The U-WEDGE Transformation Method for Multivariate Geostatistical Simulation

To speed up multivariate geostatistical simulation it is common to transform the set of attributes into spatially uncorrelated factors that can be simulated independently. Spatial decorrelation methods are usually based on the diagonalisation of the variance/covariance and semivariogram matrices of the set of attributes for a chosen family of lag spacings. These matrices are symmetric and there are several efficient methods for the approximate joint diagonalisation of a family of symmetric matrices. One of these is the uniformly weighted exhaustive diagonalisation with Gauss iterations (U-WEDGE) method. In contrast to the method of minimum/maximum autocorrelation factors (MAF), where a two structure linear model of coregionalisation is assumed, U-WEDGE can be applied directly to the set of experimental semivariogram matrices without having to place restrictions on the number of structures in the linear model of coregionalisation, thus removing one of the restrictions placed on the subsequent modelling of the spatial structure of the factors. We use an iron-ore data set to illustrate the method and present a comparison between the simulated attributes obtained from U-WEDGE and MAF with the full co-simulation of the attributes.     

Generalized sequential kriging (GSK) for heterogeneous cell size: property modeling without matrices

Sequential kriging avoids the use of matrices and resolves the issue of unstable solutions. It allows for stepwise ways to get joint estimations and cosimulations that are equivalent to the simultaneous solution. The approach is proposed as the solution for geocellular modeling with variable cell size from heterogeneous structural properties (HSPs) as required for modeling with structural constraints. Rock properties are controlled by structural domains, regions, and structural geology parameters. In some cases, rock properties are cross-correlated to formation thickness, curvature of structures, and other structural attributes. Cell thickness may be proportional to formation thickness and may enter as a conditioning property in the estimation of rock property parameters for simulation. In addition, cell volume controls the upscaling of covariance structures (i.e., regularized variograms). Structural properties are priorly modeled. Perturbation response functions (PRFs) are computed for each cell vs all possible sample point locations to facilitate sequential kriging. Upscaled PRFs are modified following conditional updating after each new data value is included in the estimation of parameters. Generalized sequential kriging is expected to become the main tool for real-time spatial modeling of 3D cellular models with HSP. In addition, some new developments related to the sequential kriging algorithm are included. Sequential kriging can be used for the estimation of parameters for simulation in the so-called unstructured grids.     

Rate of Convergence of the Gibbs Sampler in the Gaussian Case

We show that the Gibbs Sampler in the Gaussian case is closely linked to linear fixed point iterations. In fact stochastic linear iterations converge toward a stationary distribution under the same conditions as the classical linear fixed point one. Furthermore the covariance matrices are shown to satisify a related fixed point iteration, and consequently the Gibbs Sampler in the gaussian case corresponds to the classical Gauss-Seidel iterations on the inverse of the covariance matrix, and the stochastic over-relaxed Gauss-Seidel has the same limiting distribution as the Gibbs Sampler. Then an efficient method to simulate a gaussian vector is proposed. Finally numerical investigations are performed to understand the effect of the different strategies such as the initial ordering, the blocking and the updating order for iterations. The results show that in a geostatistical context the rate of convergence can be improved significantly compared to the standard case.     

新疆塔河油田奥陶系碳酸盐岩溶洞、裂隙中方解石胶结物元素分析与成因

通过对新疆塔河油田奥陶系的溶洞 (穴 )和裂隙充填的 31件方解石胶结物元素分析及因子和聚类分析表明 :①方解石中的CaCO₃ 、MgCO₃ 、FeCO₃ 、MnCO₃ 相对平均摩尔百分含量分别为 98.38%、0.87%、0.6 8%和 0.0 7%,SiO₂ 、Al₂O₃ 平均分别为 1.95 %、0.73%,Sr、BaO平均分别为 131.4× 10-6、5 16.5× 10-6,F和Cl 平均分别为 0.0 14 %和0.0 13%,而Mn +Fe/Mg、Mn/Fe平均分别为 0.87、0.10;②方解石中Sr/Ca =0.0 0 6 18Mg/Ca + 0.0 0 0 5 33( R2 =0.197),主要显示出无机成因特点;③方解石常量与微量元素特征均可由六个因子可代表组成,具有一定的沉积与成岩作用意义;④据聚类分析将样品划分出四组 :其中,以S6 9 W 0 37等代表的可能为潜流带中的淡水淋滤的产物,并可能经历了相当强烈的水 岩作用或埋藏溶蚀作用;以S6 9 W 0 38等为代表的则为大气淋滤的产物;以S80 W 0 6 4等代表显示出埋藏作用或海水成岩及混合带的产物特征,但不排除埋藏热水作用的影响;S85 W 0 0 5等样品推测为海水浓缩背景下埋藏成岩产物,可能出现酸性还原条件.     

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.     

Variogram Fitting by Generalized Least Squares Using an Explicit Formula for the Covariance Structure

In the context of spatial statistics, the classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. If data are Gaussian with constant mean, then the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix and the variance matrix. When data are independent with unidimensional and regular support, an explicit formula for this correlation is available. The same is true for a multidimensional and regular support as can be shown by using Kronecker products of matrices. As variogram fitting is a crucial stage for correct spatial prediction, it is proposed to use a generalized least squares method with an explicit formula for the covariance structure (GLSE). A good approximation of the covariance structure is achieved by taking account of the explicit formula for the correlation in the independent situation. Simulations are carried out with several types of underlying variograms, as well as with outliers in the data. Results show that this technique (GLSE), combined with a robust estimator of the variogram, improves the fit significantly.     

On the condition number of covariance matrices in kriging, estimation, and simulation of random fields

The numerical stability of linear systems arising in kriging, estimation, and simulation of random fields, is studied analytically and numerically. In the state-space formulation of kriging, as developed here, the stability of the kriging system depends on the condition number of the prior, stationary covariance matrix. The same is true for conditional random field generation by the superposition method, which is based on kriging, and the multivariate Gaussian method, which requires factoring a covariance matrix. A large condition number corresponds to an ill-conditioned, numerically unstable system. In the case of stationary covariance matrices and uniform grids, as occurs in kriging of uniformly sampled data, the degree of ill-conditioning generally increases indefinitely with sampling density and, to a limit, with domain size. The precise behavior is, however, highly sensitive to the underlying covariance model. Detailed analytical and numerical results are given for five one-dimensional covariance models: (1) hole-exponential, (2) exponential, (3) linear-exponential, (4) hole-Gaussian, and (5) Gaussian. This list reflects an approximate ranking of the models, from best to worst conditioned. The methods developed in this work can be used to analyze other covariance models. Examples of such representative analyses, conducted in this work, include the spherical and periodic hole-effect (hole-sinusoidal) covariance models. The effect of small-scale variability (nugget) is addressed and extensions to irregular sampling schemes and higher dimensional spaces are discussed.     

Rapid solution of kriging equations,using a banded Gauss elimination algorithm

Summary A crucial concern when implementing computer algorithms for geostatistical analyses on microcomputers is speed of execution. Kriging, in particular, typically relies on a Gauss elimination algorithm to solve for weights. Because such an alogrithm is required for each estimate, the solution for weights can result in very slow program execution speed on a microcomputer. One approach to enhancing the efficiency of Gauss elimination is demonstrated herein. The upper triangle plus diagonal of the intersample covariance matrix is used in a modified banded Gauss elimination algorithm. Results show that such an algorithm yields approximately a two-fold reduction in execution time for kriging when the number of nearest neighbours used for estimation is large.     

Discriminant analysis of a Cretaceous foraminifer using shrunken estimators

Canonical variate analysis of living and fossil organisms, based on morphological characters, can sometimes greatly distort the biological interpretation (i.e., reification) of the coefficients of the eigenvectors forming the canonical variates through the inclusion of redundant within-group directions. Instability is associated with the smallest eigenvalues, particularly if these do not greatly differ from zero. In the present study of 46 borehole samples of the Late Cretaceous foraminifer Afrobolivina afraReyment, stability of the canonical variate coefficients is attained by removal of a near redundant direction of within-group variation. This leads to improved interpretability of the morphometric relationships in this species. on leave from Division of Mathematics and Statistics, CSIRO, Wembley, 6014, Western Australia.     

Means and Covariance Functions for Geostatistical Compositional Data: an Axiomatic Approach

This work focuses on the characterization of the central tendency of a sample of compositional data. It provides new results about theoretical properties of means and covariance functions for compositional data, with an axiomatic perspective. Original results that shed new light on geostatistical modeling of compositional data are presented. As a first result, it is shown that the weighted arithmetic mean is the only central tendency characteristic satisfying a small set of axioms, namely continuity, reflexivity, and marginal stability. Moreover, this set of axioms also implies that the weights must be identical for all parts of the composition. This result has deep consequences for spatial multivariate covariance modeling of compositional data. In a geostatistical setting, it is shown as a second result that the proportional model of covariance functions (i.e., the product of a covariance matrix and a single correlation function) is the only model that provides identical kriging weights for all components of the compositional data. As a consequence of these two results, the proportional model of covariance function is the only covariance model compatible with reflexivity and marginal stability.