Stepwise Conditional Transformation for Simulation of Multiple Variables

Most geostatistical studies consider multiple-related variables. These relationships often show complex features such as nonlinearity, heteroscedasticity, and mineralogical or other constraints. These features are not handled by the well-established Gaussian simulation techniques. Earth science variables are rarely Gaussian. Transformation or anamorphosis techniques make each variable univariate Gaussian, but do not enforce bivariate or higher order Gaussianity. The stepwise conditional transformation technique is proposed to transform multiple variables to be univariate Gaussian and multivariate Gaussian with no cross correlation. This makes it remarkably easy to simulate multiple variables with arbitrarily complex relationships: (1) transform the multiple variables, (2) perform independent Gaussian simulation on the transformed variables, and (3) back transform to the original variables. The back transformation enforces reproduction of the original complex features. The methodology and underlying assumptions are explained. Several petroleum and mining examples are used to show features of the transformation and implementation details.     

Multivariable variogram and its application to the linear model of coregionalization

In this article, we present the multivariable variogram, which is defined in a way similar to that of the traditional variogram, by the expected value of a distance, squared, in a space withp dimensions. Combined with the linear model of coregionalization, this tool provides a way for finding the elementary variograms that characterize the different spatial scales contained in a set of data withp variables. In the case in which the number of elementary components is less than or equal to the number of variables, it is possible, by means of nonlinear regression of variograms and cross-variograms, to estimate the coregionalization parameters directly in order to obtain the elementary variables themselves, either by cokriging or by direct matrix inversion. This new tool greatly simplifies the procedure proposed by Matheron (1982) and Wackernagel (1985). The search for the elementary variograms is carried out using only one variogram (multivariable), as opposed to thep(p + 1)/2 required by the Matheron approach. Direct estimation of the linear coregionalization model parameters involves the creation of semipositive definite coregionalization matrices of rank 1.     

The Linear Coregionalization Model and the Product–Sum Space–Time Variogram

The product covariance model, the product–sum covariance model, and the integrated product and integrated product–sum models have the advantage of being easily fitted by the use of marginal variograms. These models and the use of the marginals are described in a series of papers by De Iaco, Myers, and Posa. Such models allow not only estimating values at nondata locations but also prediction in future times, hence, they are useful for analyzing air pollution data, meteorological data, or ground water data. These three kinds of data are nearly always multivariate and because the processes determining the deposition or dynamics will affect all variates, a multivariate approach is desirable. It is shown that the use of marginal variograms for space–time modeling can be extended to the multivariate case and in particular to the use of the Linear Coregionalization Model (LCM) for cokriging in space–time. An application to an environmental data set is given.     

Extensions to Spatial Factor Methods with an Illustration in Geochemistry

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.     

Deriving Constraints on Small-Scale Variograms due to Variograms of Large-Scale Data

The application of kriging-based geostatistical algorithms to integrate large-scale seismic data calls for direct and cross variograms of the seismic variable and primary variable (e.g., porosity) at the modeling scale, which is typically much smaller than the seismic data resolution. In order to ensure positive definiteness of the cokriging matrix, a licit small-scale coregionalization model has to be built. Since there are no small-scale secondary data, an analytical method is presented to infer small-scale seismic variograms. The method is applied to estimate the 3-D porosity distribution of a West Texas oil field given seismic data and porosity data at 62 wells.     

Markov Models for Cross-Covariances

Markov models based on various data screening hypotheses are often used because they reduce the statistical inference burden. In the case of co-located cokriging, the commonly used Markov model results in the cross-covariance being proportional to the primary covariance. Such model is inappropriate in the presence of a smoothly varying secondary variable defined on a much larger volume support than the primary variable. For such cases, an alternative Markov screening hypothesis is proposed that results in a more continuous cross-covariance proportional to the secondary covariance model. A parallel development of both Markov models is presented. A companion paper provides a comparative application to a real data set.     

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.     

Identifying spatial heterogeneity of coal resource quality in a multilayer coal deposit by multivariate geostatistics

This study presents a new geostatistical approach to characterization of the geometry and quality of a multilayer coal deposit using the data of seam thickness as a geometric property and the contents of ash, sodium, total sulphur, and the heating value as quality properties. A coal deposit in East Kalimantan (Borneo), Indonesia, which has a synclinal geological structure, was chosen as the study site. Semivariogram analysis clarified the strong dependence of heating value on ash content in the top and bottom parts of each seam and the existence of a strong correlation with sodium content over the sub-seams in the same location. The correlations between the geometry and quality of the seams were generally weak. A linear coregionalization model was used to derive the spatial correlation coefficients of two variables at each scale component from the single- and cross-semivariogram matrices. Because the data were correlated spatially in the same seam or over different seams, multivariate techniques (ordinary cokriging and factorial cokriging) were mainly used and the resultant spatial estimates were compared to those derived using a univariate technique (ordinary kriging). A factorial cokriging was effective to decompose the spatial correlation structures with different scales. Another important characteristic was that the sodium content shows distinct segregation: the low zones are concentrated near the boundary of the sedimentary basin, while the high zones are concentrated in the central part. The main component of sodium originates from the abundance of saline water. Therefore, it can be inferred that seawater had stronger effects on the coal depositional process in the central basin than in the border part. The geostatistical modeling results suggest that the thicknesses of all the major seams were controlled by the syncline structure, while the coal qualities chiefly were originated from the coal depositional and diagenetic processes.     

因子克立格分析的理论综述及应用展望

   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。