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1.
Generalized covariance functions in estimation 总被引：3，自引：0，他引：3
Peter K. Kitanidis 《Mathematical Geology》1993,25(5):525-540
I discuss the role of generalized covariance functions in best linear unbiased estimation and methods for their selection. It is shown that the experimental variogram (or covariance function) of the detrended data can be used to obtain a preliminary estimate of the generalized covariance function without iterations and I discuss the advantages of other parameter estimation methods. 相似文献
2.
When estimating the mean value of a variable, or the total amount of a resource, within a specified region it is desirable to report an estimated standard error for the resulting estimate. If the sample sites are selected according to a probability sampling design, it usually is possible to construct an appropriate design-based standard error estimate. One exception is systematic sampling for which no such standard error estimator exists. However, a slight modification of systematic sampling, termed 2-step tessellation stratified (2TS) sampling, does permit the estimation of design-based standard errors. This paper develops a design-based standard error estimator for 2TS sampling. It is shown that the Taylor series approximation to the variance of the sample mean under 2TS sampling may be expressed in terms of either a deterministic variogram or a deterministic covariance function. Variance estimation then can be approached through the estimation of a variogram or a covariance function. The resulting standard error estimators are compared to some more traditional variance estimators through a simulation study. The simulation results show that estimators based on the new approach may perform better than traditional variance estimators. 相似文献
3.
Peter K. Kitanidis 《Mathematical Geology》1991,23(5):741-758
In linear geostatistics, models for the mean function (drift) and the variogram or generalized covariance function are selected on the basis of the modeler's understanding of the phenomenon studied as well as data. One can seldom be assured that the most appropriate model has been selected; however, analysis of residuals is helpful in diagnosing whether some important characteristic of the data has been neglected and, ultimately, in providing a reasonable degree of assurance that the selected model is consistent with the available information. The orthonormal residuals presented in this work are kriging errors constructed so that, when the correct model is used, they are uncorrelated and have zero mean and unit variance. It is suggested that testing of orthonormal residuals is a practical way for evaluating the agreement of the model with the data and for diagnosing model deficiencies. Their advantages over the usually employed standardized residuals are discussed. A set of tests are presented. Orthonormal residuals can also be useful in the estimation of the covariance (or variogram) parameters for a model that is considered correct. 相似文献
4.
Olivier Dubrule 《Mathematical Geosciences》2017,49(4):441-465
Many variogram (or covariance) models that are valid—or realizable—models of Gaussian random functions are not realizable indicator variogram (or covariance) models. Unfortunately there is no known necessary and sufficient condition for a function to be the indicator variogram of a random set. Necessary conditions can be easily obtained for the behavior at the origin or at large distance. The power, Gaussian, cubic or cardinal-sine models do not fulfill these conditions and are therefore not realizable. These considerations are illustrated by a Monte Carlo simulation demonstrating nonrealizability over some very simple three-point configurations in two or three dimensions. No definitive result has been obtained about the spherical model. Among the commonly used models for Gaussian variables, only the exponential appears to be a realizable indicator variogram model in all dimensions. It can be associated with a mosaic, a Boolean or a truncated Gaussian random set. In one dimension, the exponential indicator model is closely associated with continuous-time Markov chains, which can also lead to more variogram models such as the damped oscillation model. One-dimensional random sets can also be derived from renewal processes, or mosaic models associated with such processes. This provides an interesting link between the geostatistical formalism, focused mostly on two-point statistics, and the approach of quantitative sedimentologists who compute the probability distribution function of the thickness of different geological facies. The last part of the paper presents three approaches for obtaining new realizable indicator variogram models in three dimensions. One approach consists of combining existing realizable models. Other approaches are based on the formalism of Boolean random sets and truncated Gaussian functions. 相似文献
5.
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
6.
Assessment of the sampling variance of the experimental variogram is an important topic in geostatistics as it gives the uncertainty of the variogram estimates. This assessment, however, is repeatedly overlooked in most applications mainly, perhaps, because a general approach has not been implemented in the most commonly used software packages for variogram analysis. In this paper the authors propose a solution that can be implemented easily in a computer program, and which, subject to certain assumptions, is exact. These assumptions are not very restrictive: second-order stationarity (the process has a finite variance and the variogram has a sill) and, solely for the purpose of evaluating fourth-order moments, a Gaussian distribution for the random function. The approach described here gives the variance–covariance matrix of the experimental variogram, which takes into account not only the correlation among the experiemental values but also the multiple use of data in the variogram computation. Among other applications, standard errors may be attached to the variogram estimates and the variance–covariance matrix may be used for fitting a theoretical model by weighted, or by generalized, least squares. Confidence regions that hold a given confidence level for all the variogram lag estimates simultaneously have been calculated using the Bonferroni method for rectangular intervals, and using the multivariate Gaussian assumption for K-dimensional elliptical intervals (where K is the number of experimental variogram estimates). A general approach for incorporating the uncertainty of the experimental variogram into the uncertainty of the variogram model parameters is also shown. A case study with rainfall data is used to illustrate the proposed approach. 相似文献
7.
Sample schemes used in geostatistical surveys must be suitable for both variogram estimation and kriging. Previously schemes
have been optimized for one of these steps in isolation. Ordinary kriging generally requires the sampling locations to be
evenly dispersed over the region. Variogram estimation requires a more irregular pattern of sampling locations since comparisons
must be made between measurements separated by all lags up to and beyond the range of spatial correlation. Previous studies
have not considered how to combine these optimized schemes into a single survey and how to decide what proportion of sampling
effort should be devoted to variogram estimation and what proportion devoted to kriging
An expression for the total error in a geostatistical survey accounting for uncertainty due to both ordinary kriging and variogram
uncertainty is derived. In the same manner as the kriging variance, this expression is a function of the variogram but not
of the sampled response data. If a particular variogram is assumed the total error in a geostatistical survey may be estimated
prior to sampling. We can therefore design an optimal sample scheme for the combined processes of variogram estimation and
ordinary kriging by minimizing this expression. The minimization is achieved by spatial simulated annealing. The resulting
sample schemes ensure that the region is fairly evenly covered but include some close pairs to analyse the spatial correlation
over short distances. The form of these optimal sample schemes is sensitive to the assumed variogram. Therefore a Bayesian
approach is adopted where, rather than assuming a single variogram, we minimize the expected total error over a distribution
of plausible variograms. This is computationally expensive so a strategy is suggested to reduce the number of computations
required 相似文献
8.
On the Use of Non-Euclidean Distance Measures in Geostatistics 总被引：4，自引：0，他引：4
Frank C. Curriero 《Mathematical Geology》2006,38(8):907-926
In many scientific disciplines, straight line, Euclidean distances may not accurately describe proximity relationships among
spatial data. However, non-Euclidean distance measures must be used with caution in geostatistical applications. A simple
example is provided to demonstrate there are no guarantees that existing covariance and variogram functions remain valid (i.e.
positive definite or conditionally negative definite) when used with a non-Euclidean distance measure. There are certain distance
measures that when used with existing covariance and variogram functions remain valid, an issue that is explored. The concept
of isometric embedding is introduced and linked to the concepts of positive and conditionally negative definiteness to demonstrate
classes of valid norm dependent isotropic covariance and variogram functions, results many of which have yet to appear in
the mainstream geostatistical literature or application. These classes of functions extend the well known classes by adding
a parameter to define the distance norm. In practice, this distance parameter can be set a priori to represent, for example,
the Euclidean distance, or kept as a parameter to allow the data to choose the metric. A simulated application of the latter
is provided for demonstration. Simulation results are also presented comparing kriged predictions based on Euclidean distance
to those based on using a water metric. 相似文献
9.
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. 相似文献
10.
11.
Nonstationarity of the Mean and Unbiased Variogram Estimation: Extension of the Weighted Least-Squares Method 总被引：1，自引：0，他引：1
When concerned with spatial data, it is not unusual to observe a nonstationarity of the mean. This nonstationarity may be modeled through linear models and the fitting of variograms or covariance functions performed on residuals. Although it usually is accepted by authors that a bias is present if residuals are used, its importance is rarely assessed. In this paper, an expression of the variogram and the covariance function is developed to determine the expected bias. It is shown that the magnitude of the bias depends on the sampling configuration, the importance of the dependence between observations, the number of parameters used to model the mean, and the number of data. The applications of the expression are twofold. The first one is to evaluate a priori the importance of the bias which is expected when a residuals-based variogram model is used for a given configuration and a hypothetical data dependence. The second one is to extend the weighted least-squares method to fit the variogram and to obtain an unbiased estimate of the variogram. Two case studies show that the bias can be negligible or larger than 20%. The residual-based sample variogram underestimates the total variance of the process but the nugget variance may be overestimated. 相似文献
12.
Geostatistics has traditionally used a probabilistic framework, one in which expected values or ensemble averages are of primary importance. The less familiar deterministic framework views geostatistical problems in terms of spatial integrals. This paper outlines the two frameworks and examines the issue of which spatial continuity measure, the covarianceC (h) or the variogram (h), is appropriate for each framework. AlthoughC (h) and (h) were defined originally in terms of spatial integrals, the convenience of probabilistic notation made the expected value definitions more common. These now classical expected value definitions entail a linear relationship betweenC (h) and (h); the spatial integral definitions do not. In a probabilistic framework, where available sample information is extrapolated to domains other than the one which was sampled, the expected value definitions are appropriate; furthermore, within a probabilistic framework, reasons exist for preferring the variogram to the covariance function. In a deterministic framework, where available sample information is interpolated within the same domain, the spatial integral definitions are appropriate and no reasons are known for preferring the variogram. A case study on a Wiener-Levy process demonstrates differences between the two frameworks and shows that, for most estimation problems, the deterministic viewpoint is more appropriate. Several case studies on real data sets reveal that the sample covariance function reflects the character of spatial continuity better than the sample variogram. From both theoretical and practical considerations, clearly for most geostatistical problems, direct estimation of the covariance is better than the traditional variogram approach.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987. 相似文献
13.
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
14.
Quadratic estimators of components of a nested spatial covariance function are presented. Estimators are unbiased and possess a minimum norm property. Inversion of a covariance matrix is required but, by assuming that spatial correlation is absent, a priori, matrix inversion can be avoided. The loss of efficiency that results from this assumption is discussed. Methods can be generalized to include estimation of components of a generalized polynomial covariance assuming the underlying process to be an intrinsic random function. Particular attention is given to the special case where just two components of spatial covariance exist, one of which represents a nugget effect. 相似文献
15.
Changik Han Jiyang Wang Mingguo Zheng Ende Wang Jianming Xia GwangSu Li Sunchol Choe 《Earth Science Informatics》2016,9(2):197-213
A critical step for kriging in geostatistics is estimation of the variogram. Traditional variogram modeling comprise of the experimental variogram calculation, appropriate variogram model selection and model parameter determination. Selecting of the variogram model and fitting of model parameters is the most controversial aspect of geostatistics. Shapes of valid variogram models are finite, and sometimes, the optimal shape of the model can not be fitted, leading to reduced estimation accuracy. In this paper, a new method is presented to automatically construct a model shape and fit model parameters to experimental variograms using Support Vector Regression (SVR) and Multi-Gene Genetic Programming (MGGP). The proposed method does not require the selection of a variogram model and can directly provide the model shape and parameters of the optimal variogram. The validity of the proposed method is demonstrated in a number of cases. 相似文献
16.
The classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. When data have an elliptically contoured distribution with constant mean, the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix, the covariance matrix, and the kurtosis. Several specific cases are studied closely. A subclass of elliptically contoured distributions with a particular family of covariance matrices is shown to possess exactly the same correlation structure for the classical variogram estimator as the multivariate independent Gaussian distribution. The consequences on variogram fitting by generalized least squares are discussed. 相似文献
17.
Marc G. Genton 《Mathematical Geology》2000,32(1):127-137
The classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. When data have an elliptically contoured distribution with constant mean, the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix, the covariance matrix, and the kurtosis. Several specific cases are studied closely. A subclass of elliptically contoured distributions with a particular family of covariance matrices is shown to possess exactly the same correlation structure for the classical variogram estimator as the multivariate independent Gaussian distribution. The consequences on variogram fitting by generalized least squares are discussed. 相似文献
18.
Michael L. Stein 《Mathematical Geology》1986,18(7):625-633
Marshall and Mardia (1985) and Kitanidis (1985) have suggested using minimum norm quadratic estimation as a method to estimate parameters of a generalized covariance function. Unfortunately, this method is difficult to use with large data sets as it requires inversion of an n × n matrix, where n is number of observations. These authors suggest replacing the matrix to be inverted by the identity matrix, which eliminates the computational burden, although with a considerable loss of efficiency. As an alternative, the data set can be broken into subsets, and minimum norm quadratic estimates of parameters of the generalized covariance function can be obtained within each subset. These local estimates can be averaged to obtain global estimates. This procedure also avoids large matrix inversions, but with less loss in efficiency. 相似文献
19.
Geostatistical analyses require an estimation of the covariance structure of a random field and its parameters jointly from
noisy data. Whereas in some cases (as in that of a Matérn variogram) a range of structural models can be captured with one
or a few parameters, in many other cases it is necessary to consider a discrete set of structural model alternatives, such
as drifts and variograms. Ranking these alternatives and identifying the best among them has traditionally been done with
the aid of information theoretic or Bayesian model selection criteria. There is an ongoing debate in the literature about
the relative merits of these various criteria. We contribute to this discussion by using synthetic data to compare the abilities
of two common Bayesian criteria, BIC and KIC, to discriminate between alternative models of drift as a function of sample size when drift and variogram parameters are
unknown. Adopting the results of Markov Chain Monte Carlo simulations as reference we confirm that KIC reduces asymptotically to BIC and provides consistently more reliable indications of model quality than does BIC for samples of all sizes. Practical considerations often cause analysts to replace the observed Fisher information matrix
entering into KIC with its expected value. Our results show that this causes the performance of KIC to deteriorate with diminishing sample size. These results are equally valid for one and multiple realizations of uncertain
data entering into our analysis. Bayesian theory indicates that, in the case of statistically independent and identically
distributed data, posterior model probabilities become asymptotically insensitive to prior probabilities as sample size increases.
We do not find this to be the case when working with samples taken from an autocorrelated random field. 相似文献
20.
A stationary specification of anisotropy does not always capture the complexities of a geologic site. In this situation, the
anisotropy can be varied locally. Directions of continuity and the range of the variogram can change depending on location
within the domain being modeled. Kriging equations have been developed to use a local anisotropy specification within kriging
neighborhoods; however, this approach does not account for variation in anisotropy within the kriging neighborhood. This paper
presents an algorithm to determine the optimum path between points that results in the highest covariance in the presence
of locally varying anisotropy. Using optimum paths increases covariance, results in lower estimation variance and leads to
results that reflect important curvilinear structures. Although CPU intensive, the complex curvilinear structures of the kriged
maps are important for process evaluation. Examples highlight the ability of this methodology to reproduce complex features
that could not be generated with traditional kriging. 相似文献