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1.
The geostatistical analysis of multivariate data involves choosing and fitting theoretical models to the empirical matrix. This paper considers the specific case of the model of linear coregionalization, and describes an automated procedure for fitting models, that are adequate in the mathematical sense, using a least-squares like technique. It also describes how to decide whether the number of parameters of the cross-variogram matrix model should be reduced to improve stability of fit. The procedure is illustrated with an analysis of the spatial relations among the physical properties of an alluvial soil. The results show the main influence of the scale and the shape of the basic models on the goodness of fit. The choice of the number of basic models appears of secondary importance, though it greatly influences the resulting interpretation of the coregionalization analysis.  相似文献   

2.
Although there are multiple methods for modeling matrix covariance functions and matrix variograms in the geostatistical literature, the linear coregionalization model is still widely used. In particular it is easy to check to ensure whether the matrix covariance function is positive definite or that the matrix variogram is conditionally negative definite. One of the difficulties in using a linear coregionalization model is in determining the number of basic structures and the corresponding covariance functions or variograms. In this paper, a new procedure is given for identifying the basic structures of the space–time linear coregionalization model and modeling the matrix variogram. This procedure is based on the near simultaneous diagonalization of the sample matrix variograms computed for a set of spatiotemporal lags. A case study using a multivariate spatiotemporal data set provided by the Environmental Protection Agency of Lombardy, Italy, illustrates how nearly simultaneous diagonalization of the empirical matrix variograms simplifies modeling of the matrix variograms. The new methodology is compared with a previous one by analyzing various indices and statistics.  相似文献   

3.
The study described herein concerns the application of geostatistical methods to data soil from Montemor-O-Novo area (Southern Portugal). In the area, the gold mineralised zones (Banhos, Caeiras, Falés, Gamela, Malaca and Monfurado) are characterised by different geological settings and mineralogical assemblages. A total of 1211 soil samples were collected in Montemor-O-Novo area and analysed for Cu, Pb, Zn, As, Ba and Au by atomic absorption spectrometry.To account for spatial structure, simple and cross variograms were computed for the main directions of the grid sampling. From the experimental variograms a linear model of coregionalization composed of three structures, a nugget effect and two anisotropic spherical structures, was fitted to each of the six variables. The coregionalization matrices deduced from the theoretical model show the relationships between the variables at different scales. These matrices were compared with those obtained by principal component analysis (PCA).This methodology was the basis for estimating the corresponding spatial components (Y0, Y1 and Y2) using factorial kriging analysis (FKA). Maps of raw data, Y0, Y1 and Y2 were made for each variable.The use of multivariate analysis permit the study of the spatial structure intrinsic to geochemical data and the identification and refinement of significant anomalies related to Au-bearing mineral deposits.  相似文献   

4.
Characterizing the spatial patterns of variability is a fundamental aspect when investigating what could be the causes behind the spatial spreading of a set of variables. In this paper, a large multivariate dataset from the southeast of Belgium has been analyzed using factorial kriging. The purpose of the study is to explore and retrieve possible scales of spatial variability of heavy metals. This is achieved by decomposing the variance-covariance matrix of the multivariate sample into coregionalization matrices, which are, in turn, decomposed into transformation matrices, which serve to decompose each regionalized variable as a sum of independent factors. Then, factorial cokriging is used to produce maps of the factors explaining most of the variance, which can be compared with maps of the underlying lithology. For the dataset analyzes, this comparison identifies a few point scale concentrations that may reflect anthropogenic contamination, and it also identifies local and regional scale anomalies clearly correlated to the underlying geology and to known mineralizations. The results from this analysis could serve to guide the authorities in identifying those areas which need remediation.  相似文献   

5.
   因子克立格分析是研究多元地质统计学的基拙，是由因子克立格法和协区域化分析两部分组成，方法上包括了区域化变量(集)的分解和分解后每一空间分童的估值;本文阐述了因子克立格分析从产生到应用的理论发展、研究成果及应用展望。  相似文献   

6.
Coregionalization analysis has been presented as a method of multi-scale analysis for multivariate spatial data. Despite an increasing use of this method in environmental and earth sciences, the uncertainty associated with the estimation of parameters in coregionalization analysis (e.g., sills and functions of sills) is potentially high and has not yet been characterized. This article aims to discuss the theory underlying coregionalization analysis and assess the robustness and limits of the method. A theoretical framework is developed to calculate the ergodic and fluctuation variance-covariance matrices of least-squares estimators of sills in the linear model of coregionalization. To adjust for the positive semidefiniteness constraint on estimated coregionalization matrices, a confidence interval estimation procedure for sills and functions of sills is presented. Thereafter, the relative importance of uncertainty measures (bias and variance) for sills and structural coefficients of correlation and determination is assessed under different scenarios to identify factors controlling their uncertainty. Our results show that the sampling grid density, the choice of the least-squares estimator of sills, the positive semidefiniteness constraint, the presence of scale dependence in the correlations, and the number and range of variogram models, all affect the level of uncertainty, sometimes through multiple interactions. The asymptotic properties of variogram model parameter estimators in a bounded sampling domain impose a theoretical limit to their accuracy and precision. Because of this limit, the uncertainty was found to be high for several scenarios, especially with three variogram models, and was often more dependent on the ratio of variogram range to domain extent than on the sampling grid density. In practice, in the coregionalization analysis of a real dataset, the circular requirement for sill estimates in the calculation of uncertainty measures makes the quantification of uncertainty very problematic, if not impossible. The use of coregionalization analysis must be made with due knowledge of the uncertainty levels and limits of the method.  相似文献   

7.
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.  相似文献   

8.
Within the frame of the linear model of coregionalization, this paper sets up equations relating the variogram matrix of the principal components extracted from the variance-covariance matrix to the diagonal variogram matrices of the regionalized factors. The spatial orthogonality of the principal components is investigated in three situations: the intrinsic correlation, two basic structures with independent nugget components, three basic structures with independent nugget components and uncorrelated subsets of variables. Two examples point out that the correlation between the principal components may be nonnegligible at short distances, especially if the correlation structure changes according to the spatial scale considered. For one of the two case studies, an orthogonal varimax rotation of the first principal components is found to greatly reduce the spatial correlation between some of them.  相似文献   

9.
Coregionalization by Linear Combination of Nonorthogonal Components   总被引：1，自引：0，他引：1  
This paper applies the relationship between the matrix multivariate covariance and the covariance of a linear combination of a single attribute to analyze modeling with nested structures. This analysis for modeling of covariances is introduced to the multivariate case for nonorthogonal vector spatial components. Results validate the classic linear model of coregionalization for a more general case of nonorthogonality, that produces additional terms including cross-covariance in the nested structures. Linear combinations of nested structures have been applied in the frequency domain to a more general case where the coefficients are nonconstant but valid transfer functions. This allows for a tool for the production of cross-covariance and covariance models that are convolutions of valid models. An example for modeling of the hole effect is illustrated.  相似文献   

10.
Many applications involving spatial data require several layers of information to be simultaneously analyzed in relation to underlying geography and topographic detail. This in turn generates a need for forms of multivariate analysis particularly oriented to spatial problems and designed to handle spatial structure and dependency both within and between spatially indexed multivariate responses. In this paper we focus on one group of such methods sometimes referred to as spatial factor analysis. Use of these techniques has so far been mostly restricted to applications in the geosciences and in some forms of image processing, but the methods have potential for wider use outside these fields. They are concerned with identifying components of a multivariate data set with a spatial covariance structure that predominantly acts over a particular spatial range or zone of influence. We review the various forms of spatial factor analysis that have been proposed and emphasize links between them and with the linear model of coregionalization employed in geostatistics. We then introduce extensions to such methods that may prove useful in exploratory spatial analysis, both generally and more specifically in the context of multivariate spatial prediction. Application of our proposed exploratory techniques is demonstrated on a small but illustrative geochemical data set involving multielement measurements from stream sediments.  相似文献   

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