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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.
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
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.
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
6.
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
7.
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
8.
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
9.
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
10.
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
11.
Three approaches for estimating the hydraulic conductivity (K) of the Trifa aquifer, Morocco were investigated: (1) kriging of the K values obtained from pumping tests, (2) cokriging of the pumping test data with electrical resistivity data as a secondary variable, and (3) cokriging of the pumping test data with the slope of the water table. Gauss-transformed values of the variables are used because they provide more robust variograms and transformed values of the primary and secondary variables show correlations higher than the raw values, which is beneficial in cokriging. In cokriging with electrical resistivity, two zones are considered since the geological deposits are different from the north to the south of the aquifer, which is reflected in different correlations between the variables. Comparison of the three approaches is based mainly on the estimation errors, and to a lesser degree on the cross-validations of the corresponding variogram models and general considerations, like the measurements’ reliability and aquifer make-up. The best-estimated K is given by cokriging with the slope of the water table and is therefore preferred for further use in groundwater flow modeling. Thus, electrical resistivity or the slope of the water table can both be used as secondary variables to estimate K, especially in heterogeneous aquifers with lateral variations in lithology, as is the case of the Trifa aquifer. 相似文献
12.
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
13.
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
14.
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
15.
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
16.
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
17.
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
18.
Spatial and temporal study of nitrate concentration in groundwater by means of coregionalization 总被引：1，自引：0，他引：1
Spatial and temporal behavior of hydrochemical parameters in groundwater can be studied using tools provided by geostatistics.
The cross-variogram can be used to measure the spatial increments between observations at two given times as a function of
distance (spatial structure). Taking into account the existence of such a spatial structure, two different data sets (sampled
at two different times), representing concentrations of the same hydrochemical parameter, can be analyzed by cokriging in
order to reduce the uncertainty of the estimation. In particular, if one of the two data sets is a subset of the other (that
is, an undersampled set), cokriging allows us to study the spatial distribution of the hydrochemical parameter at that time,
while also considering the statistical characteristics of the full data set established at a different time.
This paper presents an application of cokriging by using temporal subsets to study the spatial distribution of nitrate concentration
in the aquifer of the Lucca Plain, central Italy. Three data sets of nitrate concentration in groundwater were collected during
three different periods in 1991. The first set was from 47 wells, but the second and the third are undersampled and represent
28 and 27 wells, respectively. Comparing the result of cokriging with ordinary kriging showed an improvement of the uncertainty
in terms of reducing the estimation variance. The application of cokriging to the undersampled data sets reduced the uncertainty
in estimating nitrate concentration and at the same time decreased the cost of the field sampling and laboratory analysis.
Received: 23 July 1997 · Accepted: 31 March 1998 相似文献
19.
Xiao-Lin Sun Yun-Jin Wu Hui-Li Wang Yu-Guo Zhao Gan-Lin Zhang 《Mathematical Geosciences》2014,46(4):429-443
Information on the spatial distribution of soil particle-size fractions (psf) is required for a wide range of applications. Geostatistics is often used to map spatial distribution from point observations; however, for compositional data such as soil psf, conventional multivariate geostatistics are not optimal. Several solutions have been proposed, including compositional kriging and transformation to a composition followed by cokriging. These have been shown to perform differently in different situations, so that there is no procedure to choose an optimal method. To address this, two case studies of soil psf mapping were carried out using compositional kriging, log-ratio cokriging, cokriging, and additive log-ratio cokriging; and the performance of Mahalanobis distance as a criterion for choosing an optimal mapping method was tested. All methods generated very similar results. However, the compositional kriging and cokriging results were slightly more similar to each other than to the other pair, as were log-ratio cokriging and additive log-ratio cokriging. The similar results of the two methods within each pair were due to similarities of the methods themselves, for example, the same variogram models and prediction techniques, and the similar results between the two pairs were due to the mathematical relationship between original and log-ratio transformed data. Mahalanobis distance did not prove to be a good indicator for selecting an optimal method to map soil psf. 相似文献
20.
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