共查询到20条相似文献，搜索用时 46 毫秒
1.
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
2.
Eulogio Pardo-Igúzquiza 《Mathematical Geology》1998,30(1):95-108
In this paper, the maximum likelihood method for inferring the parameters of spatial covariances is examined. The advantages of the maximum likelihood estimation are discussed and it is shown that this method, derived assuming a multivariate Gaussian distribution for the data, gives a sound criterion of fitting covariance models irrespective of the multivariate distribution of the data. However, this distribution is impossible to verify in practice when only one realization of the random function is available. Then, the maximum entropy method is the only sound criterion of assigning probabilities in absence of information. Because the multivariate Gaussian distribution has the maximum entropy property for a fixed vector of means and covariance matrix, the multinormal distribution is the most logical choice as a default distribution for the experimental data. Nevertheless, it should be clear that the assumption of a multivariate Gaussian distribution is maintained only for the inference of spatial covariance parameters and not necessarily for other operations such as spatial interpolation, simulation or estimation of spatial distributions. Various results from simulations are presented to support the claim that the simultaneous use of maximum likelihood method and the classical nonparametric method of moments can considerably improve results in the estimation of geostatistical parameters. 相似文献
3.
The problem to predict a direction, axis, or orientation (rotation) from corresponding geocoded data is discussed and a general solution by virtue of embedding a sphere/hemisphere in a real vector space is presented. Its explicit justification in terms of mathematical assumptions concerning stationarity/homogeneity and isotropy is included. The data are modelled by a stationary random field, and the spatial correlation is represented by modified multivariate variograms and covariance functions. Various types of isotropy assumptions concerning invariance under translation/rotation of the data locations, the measurements, or a combination of both, can be distinguished and lead to different simplifications of the general cross-covariance function. Beyond spatial prediction a measure of confidence in the estimates is provided. 相似文献
4.
This paper applies the relationship between the matrix multivariate covariance and the covariance of a linear combination of a single attribute to analyze modeling with nested structures. This analysis for modeling of covariances is introduced to the multivariate case for nonorthogonal vector spatial components. Results validate the classic linear model of coregionalization for a more general case of nonorthogonality, that produces additional terms including cross-covariance in the nested structures. Linear combinations of nested structures have been applied in the frequency domain to a more general case where the coefficients are nonconstant but valid transfer functions. This allows for a tool for the production of cross-covariance and covariance models that are convolutions of valid models. An example for modeling of the hole effect is illustrated. 相似文献
5.
Multivariate Spatial Modeling for Geostatistical Data Using Convolved Covariance Functions 总被引：1，自引：0，他引：1
Soil pollution data collection typically studies multivariate measurements at sampling locations, e.g., lead, zinc, copper
or cadmium levels. With increased collection of such multivariate geostatistical spatial data, there arises the need for flexible
explanatory stochastic models. Here, we propose a general constructive approach for building suitable models based upon convolution
of covariance functions. We begin with a general theorem which asserts that, under weak conditions, cross convolution of covariance
functions provides a valid cross covariance function. We also obtain a result on dependence induced by such convolution. Since,
in general, convolution does not provide closed-form integration, we discuss efficient computation.
We then suggest introducing such specification through a Gaussian process to model multivariate spatial random effects within
a hierarchical model. We note that modeling spatial random effects in this way is parsimonious relative to say, the linear
model of coregionalization. Through a limited simulation, we informally demonstrate that performance for these two specifications
appears to be indistinguishable, encouraging the parsimonious choice. Finally, we use the convolved covariance model to analyze
a trivariate pollution dataset from California. 相似文献
6.
Matrix formulation of co-kriging 总被引：11，自引：0，他引：11
Donald E. Myers 《Mathematical Geology》1982,14(3):249-257
The matrix form of the general co-kriging problem is presented. Matrix solutions are given for SRFs with covariance functions, for IRFs of order zero using variograms and for universal co-kriging. General methods for obtaining cross-covariance or cross-variogram models are given. The relationship of the general co-kriging problem to the problem of one under sampled variable is presented. 相似文献
7.
Geostatistical analysis of spatial random functions frequently uses sample variograms computed from increments of samples of a regionalized random variable. This paper addresses the theory of computing variograms not from increments but from spatial variances. The objective is to extract information about the point support space from the average or larger support data. The variance is understood as a parametric and second moment average feature of a population. However, it is well known that when the population is for a stationary random function, spatial variance within a region is a function of the size and geometry of the region and not a function of location. Spatial variance is conceptualized as an estimation variance between two physical regions or a region and itself. If such a spatial variance could be measured within several sizes of windows, such variances allow the computation of the sample variogram. The approach is extended to covariances between attributes that lead to the cross-variogram. The case of nonpoint sample support of the blocks or elements composing each window is also included. A numerical example illustrates the application of this conceptualization. 相似文献
8.
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
9.
Non-stationary Geostatistical Modeling Based on Distance Weighted Statistics and Distributions 总被引：3，自引：0，他引：3
A common assumption in geostatistics is that the underlying joint distribution of possible values of a geological attribute at different locations is stationary within a homogeneous domain. This joint distribution is commonly modeled as multi-Gaussian, with correlations defined by a stationary covariance function. This results in attribute maps that fail to reproduce local changes in the mean, in the variance and, particularly, in the spatial continuity. The proposed alternative is to build local distributions, variograms, and correlograms. These are inferred by weighting the samples depending on their distance to selected locations. The local distributions are locally transformed into Gaussian distributions embedding information on the local histogram. The distance weighted experimental variograms and correlograms are able to adapt to local changes in the direction and range of spatial continuity. The automatically fitted local variogram models and the local Gaussian transformation parameters are used in spatial estimation algorithms assuming local stationarity. The resulting maps are rich in nonstationary spatial features. The proposed process implies a higher computational effort than traditional stationary techniques, but if data availability allows for a reliable inference of the local distributions and statistics, a higher accuracy of estimates can be achieved. 相似文献
10.
The aim of this short article is to stress the importance of using only positive-definite functions as models for covariance functions and variograms.The two examples presented show that a negative variance can easily be obtained when a nonadmissible function is chosen for the variogram model. 相似文献
11.
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
12.
The product covariance model, the product–sum covariance model, and the integrated product and integrated product–sum models have the advantage of being easily fitted by the use of marginal variograms. These models and the use of the marginals are described in a series of papers by De Iaco, Myers, and Posa. Such models allow not only estimating values at nondata locations but also prediction in future times, hence, they are useful for analyzing air pollution data, meteorological data, or ground water data. These three kinds of data are nearly always multivariate and because the processes determining the deposition or dynamics will affect all variates, a multivariate approach is desirable. It is shown that the use of marginal variograms for space–time modeling can be extended to the multivariate case and in particular to the use of the Linear Coregionalization Model (LCM) for cokriging in space–time. An application to an environmental data set is given. 相似文献
13.
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
14.
T. R. P. Singh 《Mathematical Geology》1982,14(2):107-123
This paper presents results of some investigations on the statistical properties of accumulation of discrete as well as continuous values when a two-dimensional stationary random function is reduced to a one-dimensional function. Based on the results of these investigations, an indirect technique of estimating covariances or variograms of point support grade values is proposed. The efficiency of the technique, particularly in relation to the conventional procedure of covariance estimation, has been demonstrated by an actual case study of a pre-Cambrian high-grade iron ore deposit in eastern India. 相似文献
15.
Spatio-Temporal Covariance Functions Generated by Mixtures 总被引：2，自引：0，他引：2
Chunsheng Ma 《Mathematical Geology》2002,34(8):965-975
Spatio-temporal covariance functions are introduced in this paper by using two approaches: (1) positive power mixture of purely spatial and purely temporal covariances, and (2) scale mixture of purely spatial and purely temporal covariances. Various parametric and nonparametric families of nonseparable spatio-temporal covariance functions are obtained with appropriate selections of the mixing function and covariances being mixed. 相似文献
16.
In order to determine to what extent a spatial random field can be characterized by its low-order distributions, we consider
four models (specifically, random spatial tessellations) with exactly the same univariate and bivariate distributions and
we compare the statistics associated with various multiple-point configurations and the responses to specific transfer functions.
The three- and four-point statistics are found to be the same or experimentally hardly distinguishable because of ergodic
fluctuations, whereas change of support and flow simulation produce very different outcomes. This example indicates that low-order
distributions may not discriminate between contending random field models, that simulation algorithms based on such distributions
may not reproduce the spatial properties of a given model or training image, and that the inference of high-order distribution
may require very large training images. 相似文献
17.
James Gunning 《Mathematical Geology》2002,34(1):43-62
Increasing attention has been paid to the use of non-Gaussian distributions as models of heterogeneity in sedimentary formations in recent years. In particular, the Lévy-stable distribution has been shown to be a useful model of the distribution of the increments of data measured in well logs. Frequently, the width of this distribution follows a power–law type scaling with increment lag, thus suggesting a nonstationary, fractal, multivariate Lévy distribution as a useful random field model. However, in this paper we show that it is very difficult to formulate a multivariate Lévy distribution with any nontrivial spatial correlations that can be sampled from rigorously in large models. Conventional sequential simulation techniques require two properties to hold of a multivariate distribution in order to work: (1) the marginal distributions must be of relatively simple form, and (2) in the uncorrelated limit, the multivariate distribution must factor into a product of independent distributions. At least one of these properties will break down in a multivariate Lévy distribution, depending on how it is formulated. This makes a rigorous derivation of a sequential simulation algorithm impossible. Nonetheless, many of the original observations that spurred the original interest in multivariate Lévy distributions can be reproduced with a conventional normal scoring procedure. Secondly, an approximate formulation of a sequential simulation algorithm can adequately reproduce the Lévy distributions of increments and fractal scaling frequently seen in real data. 相似文献
18.
The majority of geostatistical estimation and simulation algorithms rely on a covariance model as the sole characteristic of the spatial distribution of the attribute under study. The limitation to a single covariance implicitly calls for a multivariate Gaussian model for either the attribute itself or for its normal scores transform. The Gaussian model could be justified on the basis that it is both analytically simple and it is a maximum entropy model, i.e., a model that minimizes unwarranted structural properties. As a consequence, the Gaussian model also maximizes spatial disorder (beyond the imposed covariance) which can cause flow simulation results performed on multiple stochastic images to be very similar; thus, the space of response uncertainty could be too narrow entailing a misleading sense of safety. The ability of the sole covariance to adequately describe spatial distributions for flow studies, and the assumption that maximum spatial disorder amounts to either no additional information or a safe prior hypothesis are questioned. This paper attempts to clarify the link between entropy and spatial disorder and to provide, through a detailed case study, an appreciation for the impact of entropy of prior random function models on the resulting response distributions. 相似文献
19.
This paper extends the concept of dispersion variance to the multivariate case where the change of support affects dispersion covariances and the matrix of correlation between attributes. This leads to a concept of correlation between attributes as a function of sample supports and size of the physical domain. Decomposition of dispersion covariances into the spatial scales of variability provides a tool for computing the contribution to variability from different spatial components. Coregionalized dispersion covariances and elementary dispersion variances are defined for each multivariate spatial scale of variability. This allows the computation of dispersion covariances and correlation between attributes without integrating the cross-variograms. A correlation matrix, for a second-order stationary field with point support and infinite domain, converges toward constant correlation coefficients. The regionalized correlation coefficients for each spatial scale of variability, and the cases where the intrinsic correlation hypothesis holds are found independent of support and size of domain. This approach opens possibilities for multivariate geostatistics with data taken at different support. Two numerical examples from soil textural data demonstrate the change of correlation matrix with the size of the domain. In general, correlation between attributes is extended from the classic Pearson correlation coefficient based on independent samples to a most general approach for dependent samples taken with different support in a limited domain. 相似文献
20.
Xavier Emery 《Mathematical Geology》2005,37(2):163-181
The multigaussian model is used in mining geostatistics to simulate the spatial distribution of grades or to estimate the recoverable reserves of an ore deposit. Checking the suitability of such model to the available data often constitutes a critical step of the geostatistical study. In general, the marginal distribution is not a problem because the data can be transformed to normal scores, so the check is usually restricted to the bivariate distributions. In this work, several tests for diagnosing the two-point normality of a set of Gaussian data are reviewed and commented. An additional criterion is proposed, based on the comparison between the usual variogram and the variograms of lower order: the latter are defined as half the mean absolute increments of the attribute raised to a power between 0 and 2. This criterion is then extended to other bivariate models, namely the bigamma, Hermitian and Laguerrian models. The concepts are illustrated on two real data-sets. Finally, some conditions to ensure the internal consistency of the variogram under a given model are given. 相似文献