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1.
Universal cokriging is used to obtain predictions when dealing with multivariate random functions. An important type of nonstationarity is defined in terms of multivariate random functions with increments which are stationary of orderk. The covariance between increments of different variables is modeled by means of the pseudo-cross-covariance function. Criteria are formulated to which the parameters of pseudo-cross-covariance functions must comply so as to ensure positive-definiteness. Cokriging equations and the induced cokriging equations are given. The study is illustrated by an example from soil science. 相似文献
2.
Multivariate Intrinsic Random Functions for Cokriging 总被引：2，自引：0，他引：2
In multivariate geostatistics, suppose that we relax the usual second-order-stationarity assumptions and assume that the component
processes are intrinsic random functions of general orders. In this article, we introduce a generalized cross-covariance function
to describe the spatial cross-dependencies in multivariate intrinsic random functions. A nonparametric method is then proposed
for its estimation. Based on this class of generalized cross-covariance functions, we give cokriging equations for multivariate
intrinsic random functions in the presence of measurement error. A simulation is presented that demonstrates the accuracy
of the proposed nonparametric estimation method. Finally, an application is given to a dataset of plutonium and americium
concentrations collected from a region of the Nevada Test Site used for atomic-bomb testing. 相似文献
3.
On the Equivalence of the Cokriging and Kriging Systems 总被引：2，自引：0，他引：2
Simple cokriging of components of a p-dimensional second-order stationary random process is considered. Necessary and sufficient conditions under which simple cokriging is equivalent to simple kriging are given. Essentially this condition requires that it should be possible to express the cross-covariance at any lag series h using the cross-covariance at |h|=0 and the auto-covariance at lag series h. The mosaic model, multicolocated kriging and the linear model of coregionalization are examined in this context. A data analytic method to examine whether simple kriging of components of a multivariate random process is equivalent to its cokriging is given 相似文献
4.
The variogram matrix function is an important measure for the dependence of a vector random field with second-order increments, and is a useful tool for linear predication or cokriging. This paper proposes an efficient approach to construct variogram matrix functions, based on three ingredients: a univariate variogram, a conditionally negative definite matrix, and a Bernstein function, and derives three classes of variogram matrix functions for vector elliptically contoured random fields. Moreover, various dependence structures among components can be derived through appropriate mixture procedures demonstrated in this paper. We also obtain covariance matrix functions for second-order vector random fields through the Schoenberg–Lévy kernels. 相似文献
5.
This paper is concerned with vector random fields on spheres with second-order increments, which are intrinsically stationary and mean square continuous and have isotropic variogram matrix functions. A characterization of the continuous and isotropic variogram matrix function on a sphere is derived, in terms of an infinite sum of the products of positive definite matrices and ultraspherical polynomials. It is valid for Gaussian or elliptically contoured vector random fields, but may not be valid for other non-Gaussian vector random fields on spheres such as a χ 2, log-Gaussian, or skew-Gaussian vector random field. Some parametric variogram matrix models are derived on spheres via different constructional approaches. A simulation study is conducted to illustrate the implementation of the proposed model in estimation and cokriging, whose performance is compared with that using the linear model of coregionalization. 相似文献
6.
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. 相似文献
7.
Noel Cressie 《Mathematical Geology》1987,19(5):425-449
Fitting trend and error covariance structure iteratively leads to bias in the estimated error variogram. Use of generalized increments overcomes this bias. Certain generalized increments yield difference equations in the variogram which permit graphical checking of the model. These equations extend to the case where errors are intrinsic random functions of order k, k=1, 2, ..., and an unbiased nonparametric graphical approach for investigating the generalized covariance function is developed. Hence, parametric models for the generalized covariance produced by BLUEPACK-3D or other methods may be assessed. Methods are illustrated on a set of coal ash data and a set of soil pH data. 相似文献
8.
时空多元协同克立格的理论研究 总被引：11，自引：0，他引：11
本文结合地质统计学的最新成果,在空间协同区域化理论的基础上,对时空域中多元信息的协同克立格（ＳＴＣＯＫ）理论进行了较为详细的研究。主要研究内容有：（１）ＳＴＣＯＫ中的互变异函数与互协方差函数;（２）ＳＴＣＯＫ方程组及求解估值权因子的三种方法：①传统普通协同克立格法（ＳＴＴＯＣＯＫ）,②标准普通协同克立格法（ＳＴＳＯＣＯＫ）,③简单协同克立格法（ＳＴＳＣＯＫ）;（３）排列协同克立格（ＳＴＣＯＬＣＯＫ）;（４）指示协同克立格（ＳＴＩＫＣＯＫ）。 相似文献
9.
Xavier Emery 《Mathematical Geology》2007,39(6):607-623
Conditioning realizations of stationary Gaussian random fields to a set of data is traditionally based on simple kriging.
In practice, this approach may be demanding as it does not account for the uncertainty in the spatial average of the random
field. In this paper, an alternative model is presented, in which the Gaussian field is decomposed into a random mean, constant
over space but variable over the realizations, and an independent residual. It is shown that, when the prior variance of the
random mean is infinitely large (reflecting prior ignorance on the actual spatial average), the realizations of the Gaussian
random field are made conditional by substituting ordinary kriging for simple kriging. The proposed approach can be extended
to models with random drifts that are polynomials in the spatial coordinates, by using universal or intrinsic kriging for
conditioning the realizations, and also to multivariate situations by using cokriging instead of kriging. 相似文献
10.
Large cokriging systems arise in many situations and are difficult to handle in practice. Simplifications such as simple kriging,
strictly collocated and multicollocated cokriging are often used and models under which such simplifications are, in fact,
equivalent to cokriging have recently received attention. In this paper, a two-dimensional second-order stationary random
process with known mean is considered and the redundancy of certain components of the data at certain locations vis-à-vis
the solution to the simple cokriging system is examined. Conditions for the simple cokriging weights of these components at
these locations are set to zero. The conditions generalise the notion of the autokrigeability coefficient and can, in principle,
be applied to any data configuration. In specific sampling situations such as the isotopic and certain heterotropic configurations,
models under which simple kriging, strictly collocated, multicollocated and dislocated cokriging are equivalent to simple
cokriging are readily identified and results already available in the literature are obtained. These are readily identified
and the results are already available in the literature. The advantage of the approach presented here is that it can be applied
to any data configuration for analysis of permissible simplifications in simple cokriging. 相似文献
11.
Comparative performance of indicator algorithms for modeling conditional probability distribution functions 总被引：1，自引：0，他引：1
P. Goovaerts 《Mathematical Geology》1994,26(3):389-411
This paper compares the performance of four algorithms (full indicator cokriging. adjacent cutoffs indicator cokriging, multiple indicator kriging, median indicator kriging) for modeling conditional cumulative distribution functions (ccdf).The latter three algorithms are approximations to the theoretically better full indicator cokriging in the sense that they disregard cross-covariances between some indicator variables or they consider that all covariances are proportional to the same function. Comparative performance is assessed using a reference soil data set that includes 2649 locations at which both topsoil copper and cobalt were measured. For all practical purposes, indicator cokriging does not perform better than the other simpler algorithms which involve less variogram modeling effort and smaller computational cost. Furthermore, the number of order relation deviations is found to be higher for cokriging algorithms, especially when constraints on the kriging weights are applied. 相似文献
12.
Ordinary Cokriging Revisited 总被引：12，自引：0，他引：12
P. Goovaerts 《Mathematical Geology》1998,30(1):21-42
This paper sets up the relations between simple cokriging and ordinary cokriging with one or several unbiasedness constraints. Differences between cokriging variants are related to differences between models adopted for the means of primary and secondary variables. Because it is not necessary for the secondary data weights to sum to zero, ordinary cokriging with a single unbiasedness constraint gives a larger weight to the secondary information while reducing the occurrence of negative weights. Also the weights provided by such cokriging systems written in terms of covariances or correlograms are not related linearly, hence the estimates are different. The prediction performances of cokriging estimators are assessed using an environmental dataset that includes concentrations of five heavy metals at 359 locations. Analysis of reestimation scores at 100 test locations shows that kriging and cokriging perform equally when the primary and secondary variables are sampled at the same locations. When the secondary information is available at the estimated location, one gains little by retaining other distant secondary data in the estimation. 相似文献
13.
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. 相似文献
14.
Giuseppe Raspa Massimiliano Moscatelli Francesco Stigliano Antonio Patera Fabrizio Marconi Daiane Folle Roberto Vallone Marco Mancini Gian Paolo Cavinato Salvatore Milli Joo Felipe Coimbra Leite Costa 《Engineering Geology》2008,101(3-4):251-268
We are presenting an attempt to evaluate the spatial variability of geotechnical parameters in the upper Pleistocene–Holocene alluvial deposits of Roma (Italy) by means of multivariate geostatistics.The upper Pleistocene–Holocene alluvial deposits of Roma are sensitive to high levels of geohazard. They occupy a sizable and significant part of the city, being the foundation for many monuments, historical neighborhoods, and archaeological areas, and the main host of the present and future subway lines. We have stored information from more than 2000 geotechnical boreholes crossing the alluvial deposits into a relational database. For the present study, only the boreholes with lithologic/textural interpretation and geotechnical information were selected. The set includes 283 boreholes and 719 samples, which have a set of geotechnical information comprising physical properties and mechanical parameters.Techniques of multivariate statistics and geostatistics were combined and compared to evaluate the estimation methods of the mechanical parameters, with special reference to the drained friction angle from direct shear test (φ′). Principal Component Analysis was applied to the dataset to highlight the relationships between the geotechnical parameters. Through cross-validation analysis, multiple linear regression, kriging, and cokriging were tested as estimators of φ′. Cross-validation demonstrates that the cokriging with granulometries as auxiliary variables is the most suitable method to estimate φ′. In addition to proving that cokriging is a good estimator of φ′, cross-validation demonstrates that input data are coherent and this allows us to use them for estimation of geotechnical parameters, although they come from different laboratories and different vintages.Nevertheless, to get the same good results of cross-validation in estimation, it is necessary for granulometries to be available at grid points. Since this information being not available at all grid points, it is expected that, in the future, textural information can be derived in an indirect way, i.e., from lithologic/textural spatial reconstructions. 相似文献
15.
Myers developed a matrix form of the cokriging equations, but one that entails the solution of a large system of linear equations.
Large systems are troublesome because of memory requirements and a general increase in the matrix condition number. We transform
Myers’s system into a set of smaller systems, whose solution gives the classical kriging results, and provides simultaneously
a nested set of lower dimensional cokriging results. In the course of developing the new formulation we make an interesting
link to the Cauchy-Schwarz condition for the invertibility of a system, and another to a simple situation of coregionalization.
In addition, we proceed from these new equations to a linear approximation to the cokriging results in the event that the
crossvariograms are small, allowing one to take advantage of a recent results of Xie and others which proceeds by diagonalizing
the variogram matrix function over the lag classes. 相似文献
16.
Environmental studies require multivariate data such as chemical concentrations with space-time coordinates. There are two
general conditions related to such data: the existence of correlations among the coregionalized variables and the differences
in numbers of data which occur because of insufficient data caused by measurement error or bad weather conditions. This study
proposes geostatistical techniques for space-time multivariate modeling that take into consideration these correlations and
data absences. These techniques consist of suitable modeling of semivariograms and cross-semivariograms for quantifying correlation
structures among multivariables and of extending standardized ordinary cokriging. The tensor product cubic smoothing surface
method is used for space-time semivariogram modeling. These methods are applied to the chemical component data of the Ariake
Sea, a typical closed sea in southwest Japan. In order to clarify environmental changes in the Ariake Sea, the concentration
data of four nutritive salts (NO2–N, NO3–N, NH4–N, and PO4–P) at 38 stations over 25 years are used as environmental indicators. For each of the kinds of data, there are spaces and
times for which there is no data available. The effectiveness of the modeling of space-time semivariograms and the high estimation
capability of the extended cokriging are demonstrated by cross-validation. Compared with ordinary kriging for a single variable,
multivariate space-time standardized ordinary cokriging can provide a more detailed concentration map of nutritive salts and
while elucidating their temporal changes over sparsely spaced data areas. In the space-time models by ordinary kriging, on
the other hand, smooth trends are obvious. 相似文献
17.
Xavier Emery 《Mathematical Geosciences》2008,40(1):83-99
This paper presents random field models with Gaussian or gamma univariate distributions and isofactorial bivariate distributions,
constructed by composing two independent random fields: a directing function with stationary Gaussian increments and a stationary
coding process with bivariate Gaussian or gamma distributions. Two variations are proposed, by considering a multivariate
directing function and a coding process with a separable covariance, or by including drift components in the directing function.
Iterative algorithms based on the Gibbs sampler allow one to condition the realizations of the substitution random fields
to a set of data, while the inference of the model parameters relies on simple tools such as indicator variograms and variograms
of different orders. A case study in polluted soil management is presented, for which a gamma model is used to quantify the
risk that pollutant concentrations over remediation units exceed a given toxicity level. Unlike the multivariate Gaussian
model, the proposed gamma model accounts for an asymmetry in the spatial correlation of the indicator functions around the
median and for a spatial clustering of high pollutant concentrations. 相似文献
18.
Multivariable spatial prediction 总被引：1，自引：0，他引：1
For spatial prediction, it has been usual to predict one variable at a time, with the predictor using data from the same type of variable (kriging) or using additional data from auxiliary variables (cokriging). Optimal predictors can be expressed in terms of covariance functions or variograms. In earth science applications, it is often desirable to predict the joint spatial abundance of variables. A review of cokriging shows that a new cross-variogram allows optimal prediction without any symmetry condition on the covariance function. A bivariate model shows that cokriging with previously used cross-variograms can result in inferior prediction. The simultaneous spatial prediction of several variables, based on the new cross-variogram, is then developed. Multivariable spatial prediction yields the mean-squared prediction error matrix, and so allows the construction of multivariate prediction regions. Relationships between cross-variograms, between single-variable and multivariable spatial prediction, and between generalized least squares estimation and spatial prediction are also given. 相似文献
19.
Spatial characterization of non-Gaussian attributes in earth sciences and engineering commonly requires the estimation of their conditional distribution. The indicator and probability kriging approaches of current nonparametric geostatistics provide approximations for estimating conditional distributions. They do not, however, provide results similar to those in the cumbersome implementation of simultaneous cokriging of indicators. This paper presents a new formulation termed successive cokriging of indicators that avoids the classic simultaneous solution and related computational problems, while obtaining equivalent results to the impractical simultaneous solution of cokriging of indicators. A successive minimization of the estimation variance of probability estimates is performed, as additional data are successively included into the estimation process. In addition, the approach leads to an efficient nonparametric simulation algorithm for non-Gaussian random functions based on residual probabilities. 相似文献
20.
Geochemical characterization of heavy metal contaminated area using multivariate factorial kriging 总被引：1，自引：0，他引：1
Joaquim C. B. Queiroz José R. Sturaro Augusto C. F. Saraiva Paulo M. Barbosa Landim 《Environmental Geology》2008,55(1):95-105
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. 相似文献