Multivariate Correlation in the Framework of Support and Spatial Scales of Variability

This paper extends the concept of dispersion variance to the multivariate case where the change of support affects dispersion covariances and the matrix of correlation between attributes. This leads to a concept of correlation between attributes as a function of sample supports and size of the physical domain. Decomposition of dispersion covariances into the spatial scales of variability provides a tool for computing the contribution to variability from different spatial components. Coregionalized dispersion covariances and elementary dispersion variances are defined for each multivariate spatial scale of variability. This allows the computation of dispersion covariances and correlation between attributes without integrating the cross-variograms. A correlation matrix, for a second-order stationary field with point support and infinite domain, converges toward constant correlation coefficients. The regionalized correlation coefficients for each spatial scale of variability, and the cases where the intrinsic correlation hypothesis holds are found independent of support and size of domain. This approach opens possibilities for multivariate geostatistics with data taken at different support. Two numerical examples from soil textural data demonstrate the change of correlation matrix with the size of the domain. In general, correlation between attributes is extended from the classic Pearson correlation coefficient based on independent samples to a most general approach for dependent samples taken with different support in a limited domain.     

Minimum-variance unbiased quadratic estimation of covariances of regionalized variables

The parameters of covariance functions (or variograms) of regionalized variables must be determined before linear unbiased estimation can be applied. This work examines the problem of minimum-variance unbiased quadratic estimation of the parameters of ordinary or generalized covariance functions of regionalized variables. Attention is limited to covariance functions that are linear in the parameters and the normality assumption is invoked when fourth moments of the data need to be calculated. The main contributions of this work are (1) it shows when and in what sense minimum-variance unbiased quadratic estimation can be achieved, and (2) it yields a well-founded, practicable, and easy-to-automate methodology for the estimation of parameters of covariance functions. Results of simulation studies are very encouraging.     

Derivatives of Spatial Variances of Growing Windows and the Variogram

Geostatistical analysis of spatial random functions frequently uses sample variograms computed from increments of samples of a regionalized random variable. This paper addresses the theory of computing variograms not from increments but from spatial variances. The objective is to extract information about the point support space from the average or larger support data. The variance is understood as a parametric and second moment average feature of a population. However, it is well known that when the population is for a stationary random function, spatial variance within a region is a function of the size and geometry of the region and not a function of location. Spatial variance is conceptualized as an estimation variance between two physical regions or a region and itself. If such a spatial variance could be measured within several sizes of windows, such variances allow the computation of the sample variogram. The approach is extended to covariances between attributes that lead to the cross-variogram. The case of nonpoint sample support of the blocks or elements composing each window is also included. A numerical example illustrates the application of this conceptualization.     

Exact formulas for approximating variogram function integrals

The application of regionalized variables requires the estimation of the variogram function and the evaluation of its integral. By representing the variogram by a polygonal function the integral may be easily approximated by closed form representations of polygonal integrals. This approach provides a basis for more extensive statistical evaluation not evident in existing approximation methods. This paper provides the closed form representations for two-dimensional variogram functions whose domain is represented by a finite collection of rectangles.     

General polygonal variogram functions: Evaluation of estimation variance integrals

The application of regionalized variables requires the estimation of the variogram function and the evaluation of its integral. By representing the variogram by a general polygonal function the requisite integrals may be easily computed by a closed form representation of simple integrals. This paper provides the integration formulas for two-dimensional variogram functions whose domain is represented as a finite collection of rectangles. The integration formulas essential for a fully developed polygonal approach to an extensive statistical evaluation of geostatistical quantities are presented.     

Spatial structure analysis of regionalized compositions

Like compositions in general, regionalized compositions present the problem of spurious spatial correlation. To avoid this problem, this paper uses the additive-logratio transformation of regionalized compositions, following techniques introduced over the last few years for the statistical analysis of compositional data. It leads to an appropriate definition of a spatial covariance structure to describe spatial dependence between regionalized variables subject to constant-sum constraints in the case of weak stationarity. To illustrate stated problems, simulated data are used.     

One-dimensional unsteady solute transport along unsteady flow through inhomogeneous medium

The one-dimensional linear advection–diffusion equation is solved analytically by using the Laplace integral transform. The solute transport as well as the flow field is considered to be unsteady, both of independent patterns. The solute dispersion occurs through an inhomogeneous semi-infinite medium. Hence, velocity is considered to be an increasing function of the space variable, linearly interpolated in a finite domain in which solute dispersion behaviour is studied. Dispersion is considered to be proportional to the square of the spatial linear function. Thus, the coefficients of the advection–diffusion equation are functions of both the independent variables, but the expression for each coefficient is considered in degenerate form. These coefficients are reduced into constant coefficients with the help of a new space variable, introduced in our earlier works, and new time variables. The source of the solute is considered to be a stationary uniform point source of pulse type.     

四川省白马钒钛磁铁矿床铁品位分布地质统计学研究

以白马钒钛磁铁矿区及及坪矿段为研究对象,通过对矿体主要区域化变量矿石铁品位半变异函数的分析,建立了相应的半变异函数的球状模型,从而求得主矿体在走向、倾向以及垂直3个方向上的块金值、变程等参数.根据这些参数,求得白马铁矿床及及坪矿段铁品位变化程度系数在走向、倾向与垂直方向均小于0.3,表明矿区成矿作用单一,矿化较均匀,有利于矿山的开采,并且统计得出铁品位属于正态分布,表明下一步用普通克立格法进行估值效果最好,为矿山规划设计和生产提供了理论依据.     

定向独立权模型及其在矿产资源预测中的应用

提出了一种矿产资源靶区定位预测统计方法－定向独立权法。该方法将统计单元关联度定义为诸地质变量的线性组合,即Ｙ＝Ｘｄ．通过研究地质变量作用于评价目标参考变量（例如,矿化强度、矿床和矿点数目或规模等）的方向和相对大小来确定诸地质变量的权系数ｄ;用统计单元之间的关联度（Ｙ）评价资源靶区的优劣。     

Principal components analysis using the hypothetical closed array

The application of R-mode principal components analysis to a set of closed chemical data is described using previously published chemical analyses of rocks from Gough Island. Different measures of similarity have been used and the results compared by calculating the correlation coefficients between each of the elements of the extracted eigenvectors and each of the original variables. These correlations provide a convenient measure of the contribution of each variable to each of the principal components. The choice of similarity measure (variance-covariance or correlation coefficient)should reflect the nature of the data and the view of the investigator as to which is the proper weighting of the variables—according to their sample variance or equally. If the data are appropriate for principal components analysis, then the Chayes and Kruskal concept of the hypothetical open and closed arrays and the expected closure correlations would seem to be useful in defining the structure to be expected in the absence of significant departures from randomness. If the data are not multivariate normally distributed, then it is possible that the principal components will not be independent. This may result in significant nonzero covariances between various pairs of principal components.     

南秦岭迭部地区碳硅泥岩型铀矿资源定量预测评价

通过对南秦岭迭部地区碳硅泥岩型铀矿成矿地质条件和成矿地质特征的研究,确定了铀矿床的关键控矿因素和成矿远景预测要素。在此基础上,利用MapGis软件对各要素进行了空间定位,并系统编制了成矿要素图和预测要素图;应用MRAS软件对预测要素进行配置,选取其最佳的预测要素组合,构置预测变量,并计算变量与铀成矿的相关性、最小单元成矿有利度,利用人机联合圈定、优选出最小靶区。最后,采用修正体积法和德尔菲法对资源进行定量和半定量估算,并对靶区进行可信度评价和分级。     

Nonparametric estimation of generalized covariances by modeling on-line data

Structural analysis of data displaying trends may be performed with the help of generalized increments, the variance of these increments being a function of a generalized covariance. Generalized covariances are estimated primarily by parametric methods (i. e., methods searching for the best coefficients of a predetermined function), but also may be computed by one known nonparametric alternative. In this paper, a new nonparametric method is proposed. It is founded on the following principles: (1) least-squares residues are generalized increments; and (2) the generalized covariance is not a unique function, but a family of functions (the system is indeterminate). The method is presented in a general context of a k order trend in R^d, although the full solution is given only fork = I in Rⁱ. In Rⁱ, higher order trends may be developed easily with the equations included in this paper. For higher dimensions in space, the problem is more complex, but a research approach is proposed. The method is tested on soil pH data and compared to a parametric and nonparametric method.     

Spatio-Temporal Covariance Functions Generated by Mixtures

Spatio-temporal covariance functions are introduced in this paper by using two approaches: (1) positive power mixture of purely spatial and purely temporal covariances, and (2) scale mixture of purely spatial and purely temporal covariances. Various parametric and nonparametric families of nonseparable spatio-temporal covariance functions are obtained with appropriate selections of the mixing function and covariances being mixed.     

Two Ordinary Kriging Approaches to Predicting Block Grade Distributions

Multigaussian kriging aims at estimating the local distributions of regionalized variables and functions of these variables (transfer or recovery functions) at unsampled locations. In this paper, we focus on the evaluation of the recoverable reserves in an ore deposit accounting for a change of support and information effect caused by ore/waste misclassifications. Two approaches are proposed: the multigaussian model with Monte Carlo integration and the discrete Gaussian model. The latter is simpler to use but requires stronger hypotheses than the former. In each model, ordinary multigaussian kriging gives unbiased estimates of the recoverable reserves that do not utilize the mean value of the normal score data. The concepts are illustrated through a case study on a copper deposit which shows that local estimates of the metal content based on ordinary multigaussian kriging are close to the optimal conditional expectation when the data are abundant and are not dominated by the global mean when the data are scarce. The two proposed approaches (Monte Carlo integration and discrete Gaussian model) lead to similar results when compared to two other geostatistical methods: service variables and ordinary indicator kriging, which show strong deviations from conditional expectation.     

Comparison of approaches to spatial estimation in a bivariate context

The problem of estimating a regionalized variable in the presence of other secondary variables is encountered in spatial investigations. Given a context in which the secondary variable is known everywhere (or can be estimated with great precision), different estimation methods are compared: regression, regression with residual simple kriging, kriging, simple kriging with a mean obtained by regression, kriging with an external drift, and cokriging. The study focuses on 19 pairs of regionalized variables from five different datasets representing different domains (geochemical, environmental, geotechnical). The methods are compared by cross-validation using the mean absolute error as criterion. For correlations between the principal and secondary variable under 0.4, similar results are obtained using kriging and cokriging, and these methods are superior slightly to the other approaches in terms of minimizing estimation error. For correlations greater than 0.4, cokriging generally performs better than other methods, with a reduction in mean absolute errors that can reach 46% when there is a high degree of correlation between the variables. Kriging with an external drift or kriging the residuals of a regression (SKR) are almost as precise as cokriging.     

Geostatistical Analysis of Inverse Problem Variables: Application to Seismic Tomography

In this article we present a geostatistical approach to the transmission tomographic inverse problem, which is based on consideration of the inverse problem variables (velocity and traveltime errors) as regionalized variables (R.V.). Their structural analysis provides us with a new method to study the geophysical anisotropy of the rock, an important source of a priori information in order to design the anisotropic corrections. The underlying idea is that the geophysical structure can be deduced from the spatial structure of the regionalized variables which result from solving the tomographic problem with an isotropic algorithm. Also, the application of the structural analysis technique to the anisotropic corrected velocity field allows us to characterize the reliability of these corrections (model quality analysis). Geostatistical formalism also provides us with different techniques (parametric and non-parametric) to estimate and even simulate the velocity in the areas where this field has been considered anomalous based on field studies and on geophysical and statistical criteria. The kriging acts as a low-pass smoothing filter for the anomalous model parameters (velocities), but is not a substitute for an adequate filtering of the outliers before the inversion. This methodology opens the possibility of considering the inverse problem variables as stochastic processes, an important feature in cases where the tomogram is to be used as a tool of assessment to quantify the rock heterogeneities.     

Geostatistical Simulation of Regionalized Pore-Size Distributions Using Min/Max Autocorrelation Factors

In many fields of the Earth Sciences, one is interested in the distribution of particle or void sizes within samples. Like many other geological attributes, size distributions exhibit spatial variability, and it is convenient to view them within a geostatistical framework, as regionalized functions or curves. Since they rarely conform to simple parametric models, size distributions are best characterized using their raw spectrum as determined experimentally in the form of a series of abundance measures corresponding to a series of discrete size classes. However, the number of classes may be large and the class abundances may be highly cross-correlated. In order to model the spatial variations of discretized size distributions using current geostatistical simulation methods, it is necessary to reduce the number of variables considered and to render them uncorrelated among one another. This is achieved using a principal components-based approach known as Min/Max Autocorrelation Factors (MAF). For a two-structure linear model of coregionalization, the approach has the attractive feature of producing orthogonal factors ranked in order of increasing spatial correlation. Factors consisting largely of noise and exhibiting pure nugget–effect correlation structures are isolated in the lower rankings, and these need not be simulated. The factors to be simulated are those capturing most of the spatial correlation in the data, and they are isolated in the highest rankings. Following a review of MAF theory, the approach is applied to the modeling of pore-size distributions in partially welded tuff. Results of the case study confirm the usefulness of the MAF approach for the simulation of large numbers of coregionalized variables.     

Application of disjunctive kriging for estimating economic grade distribution in an iron ore deposit: A case study of the Choghart North Anomaly,Iran

Most significant iron ore deposits in Iran are located in Central Iran Zone. These deposits belong to the Bafq mining district. The Bafq mining district is located in the Early Cambrian Kashmar-Kerman volcanic arc of Central Iran. Linear estimation of regionalized variables (for example by inverse distance weighting or ordinary Kriging) results in relatively high estimation variances, i.e. the estimates have very low precision. Assessment of project economics (or other critical decision making) based on linear estimation is therefore risky. Non-linear estimation methods like disjunctive kriging perform better and the lower estimation variance allows less risky economic decision-making. Another advantage of disjunctive kriging is that it allows estimation of functions of the primary variable, which here is the grade (Fe %) of the ore. In particular it permits estimation of indicator functions defined using thresholds on the primary variable. This paper is devoted to application of disjunctive kriging method in Choghart North Anomaly iron ore deposit in Central Iran, Yazd province, Iran. In this study, the Fe concentration of Choghart North Anomaly iron ore deposit was modelled and estimated. The exploration data consists of borehole samples measuring the Fe concentration. A Gaussian isofactorial model is fitted to these data and disjunctive kriging was used to estimate the regionalized variable (Fe %) at unsampled locations and to assess the probabilities that the actual concentrations exceed a threshold value at a given location. Consequently a three dimensional model of probability of exceeding a threshold value and the estimated value are provided by disjunctive kriging to divide the ore into an economic and uneconomic part on the basis of estimation of indicator functions using thresholds grades defined on point support. The tools and concepts are complemented by a set of computer programs that are applied to the case study. The study showed that disjunctive kriging can be applied successfully for modeling the grade of an ore deposit. Results showed that the correlation between the estimated value and real value at locations close to each other is 81.9%.     

Joint simulation of multiple variables with a Markov-type coregionalization model

Many applications are multivariate in character and call for stochastic images of the joint spatial variability of multiple variables conditioned by a prior model of covariances and cross- covariances. This paper presents an algorithm to perform cosimulation of such spatially intercorrelated variables. This new algorithm builds on a Markov-type hypothesis whereby collocated information screens further away data of the same type, allowing cosimulation without the burden of a full cokriging. The proposed algorithm is checked against a synthetic multi-Gaussian reference dataset, then against a multi-Gaussian cosimulation approach using full cokriging. The results indicate that the proposed algorithm perform as well as the full cokriging approach in reproducing the univariate and bivariate statistics of the reference set, yet at less cpu cost.     

Error bounds in cascading regressions

Cascading regressions is a technique for predicting a value of a dependent variable when no paired measurements exist to perform a standard regression analysis. Biases in coefficients of a cascaded-regression line as well as error variance of points about the line are functions of the correlation coefficient between dependent and independent variables. Although this correlation cannot be computed because of the lack of paired data, bounds can be placed on errors through the required properties of the correlation coefficient. The potential meansquared error of a cascaded-regression prediction can be large, as illustrated through an example using geomorphologic data.