共查询到20条相似文献，搜索用时 62 毫秒
1.
Chunsheng Ma 《Mathematical Geosciences》2012,44(6):765-778
This paper presents the characterization of the covariance matrix function of a Gaussian or second-order elliptically contoured vector random field on the sphere which is stationary, isotropic, and mean square continuous. This characterization involves an infinite sum of the products of positive definite matrices and Gegenbauer??s polynomials, and may not be available for other non-Gaussian vector random fields on spheres such as a ?? 2 or log-Gaussian vector random field. We also offer two simple but efficient constructing approaches, and derive some parametric covariance matrix structures on spheres. 相似文献
2.
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
3.
4.
Chunsheng Ma 《Mathematical Geosciences》2011,43(2):229-242
This paper studies vector (multivariate, multiple, or multidimensional) random fields in space and/or time with second-order
increments, for which the variogram matrix is an important tool to measure the dependence within each component and between
each pair of distinct components. We introduce an efficient approach to construct Gaussian or non-Gaussian vector random fields
from the univariate random field with higher dimensional index domain, and particularly to generate a class of variogram matrices. 相似文献
5.
Gilles Bourgault 《Mathematical Geosciences》2012,44(8):1005-1037
The likelihood of Gaussian realizations, as generated by the Cholesky simulation method, is analyzed in terms of Mahalanobis distances and fluctuations in the variogram reproduction. For random sampling, the probability to observe a Gaussian realization vector can be expressed as a function of its Mahalanobis distance, and the maximum likelihood depends only on the vector size. The Mahalanobis distances are themselves distributed as a Chi-square distribution and they can be used to describe the likelihood of Gaussian realizations. Their expected value and variance are only determined by the size of the vector of independent random normal scores used to generate the realizations. When the vector size is small, the distribution of Mahalanobis distances is highly skewed and most realizations are close to the vector mean in agreement with the multi-Gaussian density model. As the vector size increases, the realizations sample a region increasingly far out on the tail of the multi-Gaussian distribution, due to the large increase in the size of the uncertainty space largely compensating for the low probability density. For a large vector size, realizations close to the vector mean are not observed anymore. Instead, Gaussian vectors with Mahalanobis distance in the neighborhood of the expected Mahalanobis distance have the maximum probability to be observed. The distribution of Mahalanobis distances becomes Gaussian shaped and the bulk of realizations appear more equiprobable. However, the ratio of their probabilities indicates that they still remain far from being equiprobable. On the other hand, it is observed that equiprobable realizations still display important fluctuations in their variogram reproduction. The variance level that is expected in the variogram reproduction, as well as the variance of the variogram fluctuations, is dependent on the Mahalanobis distance. Realizations with smaller Mahalanobis distances are, on average, smoother than realizations with larger Mahalanobis distances. Poor ergodic conditions tend to generate higher proportions of flatter variograms relative to the variogram model. Only equiprobable realizations with a Mahalanobis distance equal to the expected Mahalanobis distance have an expected variogram matching the variogram model. For large vector sizes, Cholesky simulated Gaussian vectors cannot be used to explore uncertainty in the neighborhood of the vector mean. Instead uncertainty is explored around the n-dimensional elliptical envelop corresponding to the expected Mahalanobis distance. 相似文献
6.
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. 相似文献
7.
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
8.
Before optimal linear prediction can be performed on spatial data sets, the variogram is usually estimated at various lags and a parametric model is fitted to those estimates. Apart from possible a priori knowledge about the process and the user's subjectivity, there is no standard methodology for choosing among valid variogram models like the spherical or the exponential ones. This paper discusses the nonparametric estimation of the variogram and its derivative, based on the spectral representation of positive definite functions. The use of the estimated derivative to help choose among valid parametric variogram models is presented. Once a model is selected, its parameters can be estimated—for example, by generalized least squares. A small simulation study is performed that demonstrates the usefulness of estimating the derivative to help model selection and illustrates the issue of aliasing. MATLAB software for nonparametric variogram derivative estimation is available at http://www-math.mit.edu/~gorsich/derivative.html. An application to the Walker Lake data set is also presented. 相似文献
9.
Covariance and variogram functions have been extensively studied in Euclidean space. In this article, we investigate the validity
of commonly used covariance and variogram functions on the sphere. In particular, we show that the spherical and exponential
models, as well as power variograms with 0<α≤1, are valid on the sphere. However, two Radon transforms of the exponential model, Cauchy model, the hole-effect model and
power variograms with 1<α≤2 are not valid on the sphere. A table that summarizes the validity of commonly used covariance and variogram functions on
the sphere is provided. 相似文献
10.
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
11.
Simulation of Non-Gaussian Transmissivity Fields Honoring Piezometric Data and Integrating Soft and Secondary Information 总被引：1，自引：0，他引：1
The conditional probabilities (CP) method implements a new procedure for the generation of transmissivity fields conditional to piezometric head data capable to sample nonmulti-Gaussian random functions and to integrate soft and secondary information. The CP method combines the advantages of the self-calibrated (SC) method with probability fields to circumvent some of the drawbacks of the SC method—namely, its difficulty to integrate soft and secondary information or to generate non-Gaussian fields. The SC method is based on the perturbation of a seed transmissivity field already conditional to transmissivity and secondary data, with the perturbation being function of the transmissivity variogram. The CP method is also based on the perturbation of a seed field; however, the perturbation is made function of the full transmissivity bivariate distribution and of the correlation to the secondary data. The two methods are applied to a sample of an exhaustive non-Gaussian data set of natural origin to demonstrate the interest of using a simulation method that is capable to model the spatial patterns of transmissivity variability beyond the variogram. A comparison of the probabilistic predictions of convective transport derived from a Monte Carlo exercise using both methods demonstrates the superiority of the CP method when the underlying spatial variability is non-Gaussian. 相似文献
12.
Estimation of non-ergodic variograms and their sampling variance by design-based sampling strategies
Design-based sampling strategies based on classical sampling theory offer unprecedented potentials for estimation of non-ergodic variograms. Unbiased and uncorrelated estimates of the semivariance at the selected lags and of its sampling variance can be simply obtained. These estimates are robust against deviations from an assumed spatial autocorrelation model. The same holds for the variogram model parameters and their sampling (co)variances. Moreover, an objective measure for lack of fit of the fitted model can simply be derived. The estimators for two basic sampling designs, simple random sampling and stratified simple random sampling of pairs of points, are presented. The first has been tested in real world for estimating the non-ergodic variograms of three soil properties. The parameters of variogram models and their sampling (co)variances were estimated with 72 pairs of points distributed over six lags. 相似文献
13.
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
14.
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
15.
Dae S. Young 《Mathematical Geology》1987,19(6):467-479
Geostatistics is extended to the spatial analysis of vector variables by defining the estimation variance and vector variogram in terms of the magnitude of difference vectors. Many random variables in geotechnology are in vectorial terms rather than scalars, and its structural analysis requires those sample variable interpolations to construct and characterize structural models. A better local estimator will result in greater quality of input models; geostatistics can provide such estimators: kriging estimators. The efficiency of geostatistics for vector variables is demonstrated in a case study of rock joint orientations in geological formations. The positive cross-validation encourages application of geostatistics to spatial analysis of random vectors in geoscience as well as various geotechnical fields including optimum site characterization, rock mechanics for mining and civil structures, cavability analysis of block cavings, petroleum engineering, and hydrologic and hydraulic modelings. 相似文献
16.
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. 相似文献
17.
用变差函数研究重磁场的区域变化特征.变差函数的变程反映重磁场的相干范围, 块金效应反映随机干扰, 基台值反映变异程度.重磁场的理论模拟说明: 重力场的相干范围大于磁场, 重磁场变程主要取决于场源深度, 浅源重磁场变差函数近似为球状模型或指数模型, 深源重磁场近似为连续性更好的高斯模型.磁场场源深度近似等于变程的一半, 重力场场源深度近似等于变程的四分之一.湖北大冶铁矿垂直分量磁异常具有几何各向异性, 北西-南东走向, 变差函数推测磁铁矿平均深度为250m.磁异常小波多尺度分解细节和逼近部分磁场具有协调几何各向异性, 变差函数的各阶场源深度估计结果与功率谱估计结果吻合. 相似文献
18.
Estimating concentrations or flow rates along a stream network requires specific models. Two classes of models, recently proposed in the literature, are generalized, to the intrinsic case in particular. We present a global construction by ‘streams’, i.e. on the whole set of paths between sources and outlet. Combining stationary or intrinsic one-dimensional random functions leads to stationary or intrinsic models on segments, with discontinuities at the forks. A construction from outlet to sources, leads to stationary or intrinsic models on each stream, without any discontinuity at the forks. The linear variogram is found as a particular case. The extension to the linear model of coregionalization is immediate, allowing a multivariate modelling of concentrations. To cite this article: C. de Fouquet, C. Bernard-Michel, C. R. Geoscience 338 (2006). 相似文献
19.
Variograms of hydrologic characteristics are usually obtained by estimating the experimental variogram for distinct lag classes
by commonly used estimators and fitting a suitable function to these estimates. However, these estimators may fail the conditionally
positive-definite property and the better results for the statistics of cross-validation, which are two essential conditions
for choosing a valid variogram model. To satisfy these two conditions, a multi-objective bilevel programming estimator (MOBLP)
which is based on the process of cross-validation has been developed for better estimate of variogram parameters. This model
is illustrated with some rainfall data from Luan River Basin in China. The case study demonstrated that MOBLP is an effective
way to achieve a valid variogram model. 相似文献
20.
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