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.     

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.     

Joint Consistent Mapping of High-Dimensional Geochemical Surveys

Geochemical surveys often contain several tens of components, obtained from different horizons and with different analytical techniques. These are used either to obtain elemental concentration maps or to explore links between the variables. The first task involves interpolation, the second task principal component analysis (PCA) or a related technique. Interpolation of all geochemical variables (in wt% or ppm) should guarantee consistent results: At any location, all variables must be positive and sum up to 100 %. This is not ensured by any conventional geostatistical technique. Moreover, the maps should ideally preserve any link present in the data. PCA also presents some problems, derived from the spatial dependence between the observations, and the compositional nature of the data. Log-ratio geostatistical techniques offer a consistent solution to all these problems. Variation-variograms are introduced to capture the spatial dependence structure: These are direct variograms of all possible log ratios of two components. They can be modeled with a function analogous to the linear model of coregionalization (LMC), where for each spatial structure there is an associated variation matrix describing the links between the components. Eigenvalue decompositions of these matrices provide a PCA of that particular spatial scale. The whole data set can then be interpolated by cokriging. Factorial cokriging can also be used to map a certain spatial structure, eventually projected onto those principal components (PCs) of that structure with relevant contribution to the spatial variability. If only one PC is used for a certain structure, the maps obtained represent the spatial variability of a geochemical link between the variables. These procedures and their advantages are illustrated with the horizon C Kola data set, with 25 components and 605 samples covering most of the Kola peninsula (Finland, Norway, Russia).     

Using Simultaneous Diagonalization to Identify a Space–Time Linear Coregionalization Model

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.     

Tests of Significance for Structural Correlations in the Linear Model of Coregionalization

In the linear model of coregionalization (LMC), when applicable to the experimental direct variograms and the experimental cross variogram computed for two random functions, the variability of and relationships between the random functions are modeled with the same basis functions. In particular, structural correlations can be defined from entries of sill matrices (coregionalization matrices) under second-order stationarity. In this article, modified t-tests are proposed for assessing the statistical significance of estimated structural correlations. Their specific aspects and fundamental differences, compared with an existing modified t-test for global correlation analysis with spatial data, are discussed via estimated effective sample sizes, in relation to the superimposition of random structural components, the range of autocorrelation, the presence of correlation at another structure, and the sampling scheme. Accordingly, simulation results are presented for one structure versus two structures (one without and the other with autocorrelation). The performance of tests is shown to be related to the uncertainty associated with the estimation of variogram model parameters (range, sill matrix entries), because these are involved in the test statistic and the degrees of freedom of the associated t-distribution through the estimated effective sample size. Under the second-order stationarity and LMC assumptions, the proposed tests are generally valid.     

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.     

Designing an optimal multivariate geostatistical groundwater quality monitoring network using factorial kriging and genetic algorithms

The optimal selection of monitoring wells is a major task in designing an information-effective groundwater quality monitoring network which can provide sufficient and not redundant information of monitoring variables for delineating spatial distribution or variations of monitoring variables. This study develops a design approach for an optimal multivariate geostatistical groundwater quality network by proposing a network system to identify groundwater quality spatial variations by using factorial kriging with genetic algorithm. The proposed approach is applied in designing a groundwater quality monitoring network for nine variables (EC, TDS, Cl⁻, Na, Ca, Mg, SO ₄ ²⁻ , Mn and Fe) in the Pingtung Plain in Taiwan. The spatial structure results show that the variograms and cross-variograms of the nine variables can be modeled in two spatial structures: a Gaussian model with ranges 28.5 km and a spherical model with 40 km for short and long spatial scale variations, respectively. Moreover, the nine variables can be grouped into two major components for both short and long scales. The proposed optimal monitoring design model successfully obtains different optimal network systems for delineating spatial variations of the nine groundwater quality variables by using 20, 25 and 30 monitoring wells in both short scale (28.5 km) and long scale (40 km). Finally, the study confirms that the proposed model can design an optimal groundwater monitoring network that not only considers multiple groundwater quality variables but also monitors variations of monitoring variables at various spatial scales in the study area.     

On a controversial method for modeling a coregionalization

This paper reviews two alternative approaches for modeling the (cross) variograms in a coregionalization: (1) fitting the traditional linear model of coregionalization. or (2) deducing the crossvariogram model as a linear combination of prior direct (auto) variogram models while checking the Cauchy-Schwarz inequalities. We show that the second approach has no practical advantage over the traditional one and may not be valid if more than two variables are involved. In such case. Cauchy-Schwarz inequalities are necessary but not sufficient conditions for validity of a coregionalization model.     

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.     

Scale Effect on Principal Component Analysis for Vector Random Functions

Principal component analysis (PCA) is commonly applied without looking at the spatial support (size and shape, of the samples and the field), and the cross-covariance structure of the explored attributes. This paper shows that PCA can depend on such spatial features. If the spatial random functions for attributes correspond to largely dissimilar variograms and cross-variograms, the scale effect will increase as well. On the other hand, under conditions of proportional shape of the variograms and cross-variograms (i.e., intrinsic coregionalization), no scale effect may occur. The theoretical analysis leads to eigenvalue and eigenvector functions of the size of the domain and sample supports. We termed this analysis growing scale PCA, where spatial (or time) scale refers to the size and shape of the domain and samples. An example of silt, sand, and clay attributes for a second-order stationary vector random function shows the correlation matrix asymptotically approaches constants at two or three times the largest range of the spherical variogram used in the nested model. This is contrary to the common belief that the correlation structure between attributes become constant at the range value. Results of growing scale PCA illustrate the rotation of the orthogonal space of the eigenvectors as the size of the domain grows. PCA results are strongly controlled by the multivariate matrix variogram model. This approach is useful for exploratory data analysis of spatially autocorrelated vector random functions.     

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.     

On the pseudo cross-variogram

Normal cross-variograms cannot be estimated from data in the usual way when there are only a few points where both variables have been measured. But the experimental pseudo cross-variogram can be computed even where there are no matching sampling points, and this appears as its principal advantage. The pseudo cross-variogram may be unbounded, though for its existence the intrinsic hypothesis alone is not a sufficient stationarity condition. In addition the differences between the two random processes must be second order stationary. Modeling the function by linear coregionalization reflects the more restrictive stationarity condition: the pseudo cross-variogram can be unbounded only if the unbounded correlation structures are the same in all variograms. As an alternative to using the pseudo cross-variogram a new method is presented that allows estimating the normal cross variogram from data where only one variable has been measured at a point.     

On the pseudo cross-variogram

Normal cross-variograms cannot be estimated from data in the usual way when there are only a few points where both variables have been measured. But the experimental pseudo cross-variogram can be computed even where there are no matching sampling points, and this appears as its principal advantage. The pseudo cross-variogram may be unbounded, though for its existence the intrinsic hypothesis alone is not a sufficient stationarity condition. In addition the differences between the two random processes must be second order stationary. Modeling the function by linear coregionalization reflects the more restrictive stationarity condition: the pseudo cross-variogram can be unbounded only if the unbounded correlation structures are the same in all variograms. As an alternative to using the pseudo cross-variogram a new method is presented that allows estimating the normal cross variogram from data where only one variable has been measured at a point.     

黑河下游荒漠河岸林典型样带植被空间异质性

应用地统计学的理论与方法,分析了黑河流域下游荒漠河岸林主要种群,即老林胡杨(Popu luseuphratica)、幼林胡杨、柽柳(Tamarix chinensis)和苦豆子(Sophora alopecuroides)的空间异质性程度、异质性组成及尺度依赖问题.结果表明:黑河下游荒漠河岸林种群斑块格局明显,斑块内部异质性较小,斑块之间异质性增强,出现空间异质性变化较大的尺度为430m.而斑块内部因不同种群而变化,苦豆子和柽柳具有较大的空间异质性尺度,分别为43m和55m.老林和幼林胡杨种群的空间异质性尺度(8m和13m)较小,但异质性存在多尺度变化.采用30m分辨率的遥感数据能够较好地分析种群斑块格局,但是对种群内的空间异质性分析需要更高的分辨率.     

Characterization and Quantification of Uncertainty in?Coregionalization Analysis

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.     

Mapping and assessment-based modeling of soil fertility differences in the central and eastern parts of the Black Sea region using GIS and geostatistical approaches

This study was carried out on arable lands of the central and eastern Black Sea regions including eight provinces (Artvin, Giresun, Gümü?hane, Ordu, Rize, Samsun, Sinop, and Trabzon). The present study aims to generate a soil fertility map for agricultural lands in the central and eastern parts of the Black Sea region. The main objective of this research is to quantify soil fertility by developing a soil fertility index (SFI) model at the regional level. The related objectives were to map the spatial distribution of soil fertility by using auxiliary variables and to model soil fertility within the study region. To accomplish this, a data set for soil fertility differences was collected and a model was developed to predict the spatial distribution of differences across the region. The study area was divided into 2.5 × 2.5-km grid squares. A total of 3400 soil samples were collected from the surface (0–20 cm) of each grid intersection point. The geostatistical method was used to generate the SFI distribution map of the study area for surface soils. Of the total study area, 93.76% had good (S1) or moderately fertile (S2) soil while 6.15% of the area had marginally fertile (S3) soil. Only a very small area (N) had low-fertility soil.     

Characterization and Quantification of Uncertainty in Coregionalization Analysis

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.     

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.     

Geostatistical analysis of soil moisture measurements and remotely sensed data at different spatial scales

The European remote sensing satellite (ERS-2) synthetic aperture radar (SAR) data was used for temporal monitoring of soil moisture at Sukhothai, Thailand. Higher correlations were found between the observed soil moisture and the radar backscattering coefficient. The soil moisture distribution shows great variation in space and time due to its stochastic nature. In order to obtain a better understanding of the nature and causes of spatial variation of soil moisture, the extensive soil moisture measurements observed in Thailand and also remotely sensed ERS-2 SAR data were used for geostatistical analysis. The observed soil moisture shows seasonal variations with mean varying from 3.33 %v/v (dry season) to 33.44 %v/v (wet season). The spatial geostatistical structure also shows clear seasonal variations in the geostatistical characteristics such as range and sill. The sills vary from 1.00 (%v/v)² for the driest day to 107.57 (%v/v)² for one of the wet days. The range or the correlation lengths varies between 46.5 and 149.8 m for the wettest and driest periods. The nugget effect does not show strong seasonal pattern or trend but the dry periods usually have a smaller nugget effect than the wet periods. The spherical variogram model fits the sample variograms very well in the case of soil moisture observations while the exponential model fits those of the remotely sensed data. The ranges observed from the observed soil moisture data and remotely sensed data at the same resolution are very similar. Resolution degradation affects the geostatistical structure of the data by reducing the sills, and increasing the ranges.     

储层物性的空间结构特征分析与预测—以青海尕斯库勒油田E3^1油藏为例

储层物性受诸多地质因素的影响，经常表现出强烈的非均质性。这种非均质性在一定的空间尺度上往往具有明显的结构特点。本文以青海尕斯库勒油田E3^1油藏为例，讨论了地质统计学方法在油田储层特性空间结构特征分析与预测中的应用。采用变异函数定量描述了孔隙度和渗透率的空间分布结构特征，并在此基础上利用克立格法进行了最成插值预测。结果表明，研究区储层物性（孔隙度和渗透率）具有显著的空间结构性特点，变程一般在800－2000m之间；不同小层的储层物性具有不同的空间结构方向性。这种特征主要沉积相带空间展布的影响，各小层孔隙度和渗透率的实验半变异函数均可用具有块金效应的球状模型来拟合并进行预测。作为验证，本文还采用“多重趋势面”预测模型对储层的孔隙度和渗透率进行了预测分析。