Estimation of regionalized compositions: A comparison of three methods

A regionalized composition is a random vector function whose components are positive and sum to a constant at every point of the sampling region. Consequently, the components of a regionalized composition are necessarily spatially correlated. This spatial dependence—induced by the constant sum constraint—is a spurious spatial correlation and may lead to misinterpretations of statistical analyses. Furthermore, the cross-covariance matrices of the regionalized composition are singular, as is the coefficient matrix of the cokriging system of equations. Three methods of performing estimation or prediction of a regionalized composition at unsampled points are discussed: (1) the direct approach of estimating each variable separately; (2) the basis method, which is applicable only when a random function is available that can he regarded as the size of the regionalized composition under study; (3) the logratio approach, using the additive-log-ratio transformation proposed by J. Aitchison, which allows statistical analysis of compositional data. We present a brief theoretical review of these three methods and compare them using compositional data from the Lyons West Oil Field in Kansas (USA). It is shown that, although there are no important numerical differences, the direct approach leads to invalid results, whereas the basis method and the additive-log-ratio approach are comparable.     

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.     

Factorial kriging of a geochemical dataset for heavy-metal spatial-variability characterization

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.     

Prediction of the collapsing risk of mining slopes based on geostatistical interpretation of geotechnical parameters

Almost all collapses of rock slopes especially in open pit mines are related to discontinuities such as bedding planes, faults and major joints.Geostatistical assessments can be used for understanding the distribution of regionalized variables in any spatial study. In this paper3D spatial dispersion of the fault planes in the Gole Gohar open pit iron mine, located in Kerman province, south east of Iran, is modeled. Then, regionalized variable theory is used to analyze and interpret spatial distribution of the following geotechnical parameters: Geological strength index (GSI), Rock quality designation (RQD), Cohesion (C) and angle of internal friction (f). In order to define regionalized variable distribution, variogram functions were determined for identifying the regional behavior. Structural analysis of variograms showed that the mentioned parameters have spatial structures that make it possible to set up a geostatistical model to predict their values for each non-sampled block on the pit wall. Results showed that there is a relation between the low values of geotechnical parameters and the existence of discontinuities around the pit area. The role of discontinuities in the occurrence of collapses in the area was clearly demonstrated by comparing the estimated parameters models and the model of discontinuities dispersion.     

岩土参数的空间变异性分析

岩土参数具有空间变异性和不确定性,本文将岩土参数视为具有随机性和结构性的区域化变量,利用区域化变量理论和地质统计方法分析上海地基土有代表性的抗剪强度指标的空间变异特性。结果表明该方法是可行的。     

Geochemical characterization of heavy metal contaminated area using multivariate factorial kriging

This paper describes a geostatistical method, known as factorial kriging analysis, which is well suited for analyzing multivariate spatial information. The method involves multivariate variogram modeling, principal component analysis, and cokriging. It uses several separate correlation structures, each corresponding to a specific spatial scale, and yields a set of regionalized factors summarizing the main features of the data for each spatial scale. This method is applied to an area of high manganese-ore mining activity in Amapá State, North Brazil. Two scales of spatial variation (0.33 and 2.0 km) are identified and interpreted. The results indicate that, for the short-range structure, manganese, arsenic, iron, and cadmium are associated with human activities due to the mining work, while for the long-range structure, the high aluminum, selenium, copper, and lead concentrations, seem to be related to the natural environment. At each scale, the correlation structure is analyzed, and regionalized factors are estimated by cokriging and then mapped.     

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.     

Singularity and Nonnormality in the Classification of Compositional Data

Geologists may want to classify compositional data and express the classification as a map. Regionalized classification is a tool that can be used for this purpose, but it incorporates discriminant analysis, which requires the computation and inversion of a covariance matrix. Covariance matrices of compositional data always will be singular (noninvertible) because of the unit-sum constraint. Fortunately, discriminant analyses can be calculated using a pseudo-inverse of the singular covariance matrix; this is done automatically by some statistical packages such as SAS. Granulometric data from the Darss Sill region of the Baltic Sea is used to explore how the pseudo-inversion procedure influences discriminant analysis results, comparing the algorithm used by SAS to the more conventional Moore–Penrose algorithm. Logratio transforms have been recommended to overcome problems associated with analysis of compositional data, including singularity. A regionalized classification of the Darss Sill data after logratio transformation is different only slightly from one based on raw granulometric data, suggesting that closure problems do not influence severely regionalized classification of compositional data.     

含水层渗透系数K的空间变异性研究

本文结合桥梁实例对公路双曲拱桥出现的一些主要病害进行了初步分析，并进而对桥梁状况提出评定意见和处理建议，仅供参考。     

Regionalized favorability theory for information synthesis in mineral exploration

This paper presents a regionalized method for the estimation of a favorability function through generalization of all relevant variables (explanatory and target) into random functions. The new method allows the use of cross-covariance functions in addition to ordinary covariances for extracting spatial joint information, which is virtually overlooked in the conventional analyses. The optimal weights for a favorability equation are derived from solving a generalized eigen-system established by the maximization of covariances between a favorability function and the principal components of a set of pre-selected target variables. Various correlation coefficients may be computed to assist in interpretation of the favorability estimates. Both favorability functions and correlation coefficients may be estimated for a point or a block. The regionalized favorability theory can be compared to cokriging in that both use the sample-sample covariances to account for the sample-sample relations and the point-sample covariances to account for the point-sample configurations. The new technique is demonstrated on a test case study, which involves the integration of geochemical, airborne-geophysical, and structural data sets for the target selection of hydrothermal gold-silver deposits.     

Spatial orthogonality of the principal components computed from coregionalized variables

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.     

生态地理建模中的多尺度问题

本文在分析生态地理建模内涵的基础上,讨论了生态地理建模中的尺度转换问题、跨尺度相互作用问题、空间尺度与时间尺度的关联问题和多尺度数据处理问题.由于生态地理问题的非线性、生态环境的异质性和随机事件,简单的线性尺度转换方法远不能满足生态地理建模的要求.为了从根本上解决生态地理建模中的时空尺度问题,除需要运用微分几何学和等级理论等经典方法外,还需要引入格点生成法和网格计算等现代理论和技术手段.     

基于GIS的岩石裂隙空间分布模式研究

文中以甘肃北山花岗岩中发育的构造裂隙(主要指节理尺度)为研究对象,应用地质统计分析理论,以裂隙面密度P21 作为地质统计分析的区域性变量,探索花岗岩岩体裂隙空间分布模式。结果表明:甘肃北山芨芨槽候选场址区域内花岗岩岩体裂隙空间分布具有明显的各向异性特征,变异主轴方向明显; 变异主轴区域内的裂隙分组地质统计特征显示,区域内第1分组裂隙并非是传统意义上的均匀随机分布,而是具有明显的空穴效应特征,应该在三维裂隙网络建模工作中予以考虑。本文在考虑本组裂隙空穴效应特征基础上,针对三维裂隙网络建模方法提出了新的建议。     

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。     

Two-Way Regionalized Classification of Multivariate Datasets and its Application to the Assessment of Hydrodynamic Dispersion

Zones of mixing between shallow groundwaters of different composition were unravelled by “two-way regionalized classification,” a technique based on correspondence analysis (CA), cluster analysis (ClA) and discriminant analysis (DA), aided by gridding, map-overlay and contouring tools. The shallow groundwaters are from a granitoid plutonite in the Fundão region (central Portugal). Correspondence analysis detected three natural clusters in the working dataset: 1, weathering; 2, domestic effluents; 3, fertilizers. Cluster analysis set an alternative distribution of the samples by the three clusters. Group memberships obtained by correspondence analysis and by cluster analysis were optimized by discriminant analysis, gridded over the entire Fundão region, and converted into “two-way regionalized classification” memberships as follows: codes 1, 2 or 3 were used when classification by correspondence analysis and cluster analysis produced the same results; code 0 when the grid node was first assigned to cluster 1 and then to cluster 2 or vice versa (mixing between weathering and effluents); code 4 in the other cases (mixing between agriculture and the other influences). Code-3 areas were systematically surrounded by code-4 areas, an observation attributed to hydrodynamic dispersion. Accordingly, the extent of code-4 areas in two orthogonal directions was assumed proportional to the longitudinal and transverse dispersivities of local soils. The results (0.7–16.8 and 0.4–4.3 m, respectively) are acceptable at the macroscopic scale. The ratios between longitudinal and transverse dispersivities (1.2–11.1) are also in agreement with results obtained by other studies.     

A Study of Spatial Variability on Aquifer Hydraulic Conductivity K

A structural analysis of K of an aquifer system in the study area is presented, and the main direction and degree of the variability of K are found by using the unstationary regionalized variable theory of geostatistics. Optimal estimation of K has been made by universal kriging method (U K M ). Both spatial variability distribution map and division map of K are given.     

Numerical analysis of hydrogeochemical data: a case study (Alto Guadalentı́n,southeast Spain)

This study examines the spatial variability of the factors obtained from the application of correspondence analysis to a hydrogeochemical data set. The goal was to synthesize the hydrogeochemical information using this multivariate statistical technique, by setting a series of factors which clarified the main properties of one aquifer. Then, a geostatistical framework to obtain a probabilistic assessment of groundwater quality was established. Experimental and theoretical semivariograms of the selected factors, considered as regionalized variables, were computed. These variographic information and factor values in the experimental sites were used in the ordinary kriging, which provides unbiased and linear estimates of the regionalized variables. These estimates were used to compile maps of the chosen factors, which explain their spatial distribution.The selected case study was the alluvial aquifer of Alto Guadalentı́n which is situated in southeast Spain, in the Internal Zones of Betic Cordilleras. These waters are chiefly SO₄ and Cl types, but HCO₃ facies are common in the central sector of the basin. High temperature, acid pH, problems of overexploitation and pollution by CO₂-gas characterise these waters. Available groundwater quality monitoring data were used to calibrate the numerical model. The present study focused on setting the main physical and chemical attributes and establishing the spatial pattern of groundwater quality and the temporal changes in this pattern.     

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.     

Sampling design optimization for spatial functions

A new procedure is presented for minimizing the sampling requirements necessary to estimate a mappable spatial function at a specified level of accuracy. The technique is based on universal kriging, an estimation method within the theory of regionalized variables. Neither actual implementation of the sampling nor universal kriging estimations are necessary to make an optimal design. The average standard errorand maximum standard error of estimationover the sampling domain are used as global indices of sampling efficiency. The procedure optimally selects those parameters controlling the magnitude of the indices, including the density and spatial pattern of the sample elements and the number of nearest sample elements used in the estimation. As an illustration, the network of observation wells used to monitor the water table in the Equus Beds of Kansas is analyzed and an improved sampling pattern suggested. This example demonstrates the practical utility of the procedure, which can be applied equally well to other spatial sampling problems, as the procedure is not limited by the nature of the spatial function.     

Variography of submarine morphology: Problems of deregularization, and cartographical implications

During the German Antarctic Expedition VI (leg 3, December 1987 to March 1988), bathymetric surveys were made in the Weddell Sea by the SEABEAM sonar system. For the first time geostatistical methods were applied in the SEABEAM-postprocessing. The investigations of variography that were necessary prior to the cartographical-geomorphological evaluation shed new light on classical geostatistical concerns. SEABEAM data provide a good example of a mean square, differentiable regionalized variable, where data are sampled over a two-dimensional support due to the technique of the sonar device. By deregularizations of the sample variograms, spatial continuity can be shown to be a property of seafloor depth as well as a point variable. The results are discussed in a sedimentological context. As an application of the regional variogram analyses, large-scale kriged bathymetric maps are presented.