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1.
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
2.
Spectral simulation of multivariable stationary random functions using covariance fourier transforms 总被引：1，自引：0，他引：1
A coregionalization simulation consists of the generation of realizations of a group of spatially related random variables. The Fourier integral method is presented, modified to carry out such a multivariable simulation. This method allows the simulation of realizations with any specified symmetrical covariance matrix and it is not limited to the classic linear model of coregionalization. The results of gaussian nonconditinal simulations from a case study modeling the spatial characteristics of a layer of coal are given. 相似文献
3.
The application of kriging-based geostatistical algorithms to integrate large-scale seismic data calls for direct and cross variograms of the seismic variable and primary variable (e.g., porosity) at the modeling scale, which is typically much smaller than the seismic data resolution. In order to ensure positive definiteness of the cokriging matrix, a licit small-scale coregionalization model has to be built. Since there are no small-scale secondary data, an analytical method is presented to infer small-scale seismic variograms. The method is applied to estimate the 3-D porosity distribution of a West Texas oil field given seismic data and porosity data at 62 wells. 相似文献
4.
On a controversial method for modeling a coregionalization 总被引：2，自引：0，他引：2
P. Goovaerts 《Mathematical Geology》1994,26(2):197-204
This paper reviews two alternative approaches for modeling the (cross) variograms in a coregionalization: (1) fitting the traditional linear model of coregionalization. or (2) deducing the crossvariogram model as a linear combination of prior direct (auto) variogram models while checking the Cauchy-Schwarz inequalities. We show that the second approach has no practical advantage over the traditional one and may not be valid if more than two variables are involved. In such case. Cauchy-Schwarz inequalities are necessary but not sufficient conditions for validity of a coregionalization model. 相似文献
5.
Multivariable variogram and its application to the linear model of coregionalization 总被引：1，自引：0，他引：1
In this article, we present the multivariable variogram, which is defined in a way similar to that of the traditional variogram, by the expected value of a distance, squared, in a space withp dimensions. Combined with the linear model of coregionalization, this tool provides a way for finding the elementary variograms that characterize the different spatial scales contained in a set of data withp variables. In the case in which the number of elementary components is less than or equal to the number of variables, it is possible, by means of nonlinear regression of variograms and cross-variograms, to estimate the coregionalization parameters directly in order to obtain the elementary variables themselves, either by cokriging or by direct matrix inversion. This new tool greatly simplifies the procedure proposed by Matheron (1982) and Wackernagel (1985). The search for the elementary variograms is carried out using only one variogram (multivariable), as opposed to thep(p + 1)/2 required by the Matheron approach. Direct estimation of the linear coregionalization model parameters involves the creation of semipositive definite coregionalization matrices of rank 1. 相似文献
6.
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
7.
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
8.
《Mathematical Geology》1997,29(6):779-799
Generalized cross-covariances describe the linear relationships between spatial variables observed at different locations.
They are invariant under translation of the locations for any intrinsic processes, they determine the cokriging predictors
without additional assumptions and they are unique up to linear functions. If the model is stationary, that is if the variograms
are bounded, they correspond to the stationary cross-covariances. Under some symmetry condition they are equal to minus the
usual cross-variogram. We present a method to estimate these generalized cross-covariances from data observed at arbitrary
sampling locations. In particular we do not require that all variables are observed at the same points. For fitting a linear
coregionalization model we combine this new method with a standard algorithm which ensures positive definite coregionalization
matrices. We study the behavior of the method both by computing variances exactly and by simulating from various models. 相似文献
9.
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
10.
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
11.
We evaluate the performance and statistical accuracy of the fast Fourier transform method for unconditional and conditional simulation. The method is applied under difficult but realistic circumstances of a large field (1001 by 1001 points) with abundant conditioning criteria and a band limited, anisotropic, fractal-based statistical characterization (the von Kármán model). The simple Fourier unconditional simulation is conducted by Fourier transform of the amplitude spectrum model, sampled on a discrete grid, multiplied by a random phase spectrum. Although computationally efficient, this method failed to adequately match the intended statistical model at small scales because of sinc-function convolution. Attempts to alleviate this problem through the covariance method (computing the amplitude spectrum by taking the square root of the discrete Fourier transform of the covariance function) created artifacts and spurious high wavenumber content. A modified Fourier method, consisting of pre-aliasing the wavenumber spectrum, satisfactorily remedies sinc smoothing. Conditional simulations using Fourier-based methods require several processing stages, including a smooth interpolation of the differential between conditioning data and an unconditional simulation. Although kriging is the ideal method for this step, it can take prohibitively long where the number of conditions is large. Here we develop a fast, approximate kriging methodology, consisting of coarse kriging followed by faster methods of interpolation. Though less accurate than full kriging, this fast kriging does not produce visually evident artifacts or adversely affect the a posteriori statistics of the Fourier conditional simulation. 相似文献
12.
Conditional Spectral Simulation with Phase Identification 总被引：2，自引：0，他引：2
Tingting Yao 《Mathematical Geology》1998,30(3):285-308
Spectral simulation is used widely in electrical engineering to generate random fields with a given covariance spectrum. The algorithms used are fast particularly when based on Fast Fourier Transform (FFT). However, because of lack of phase identification, spectral simulation only generates unconditional realizations. Local data conditioning is obtained typically by adding a simulated kriging residual. This conditioning process requires an additional kriging at each simulated node thus forfeiting the speed advantage of FFT. A new algorithm for conditioning is proposed whereby the phase values are determined iteratively to ensure approximative data reproduction while reproducing the frequency spectrum, that is, the covariance model. A case study is presented to demonstrate the algorithm. 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.
Fitting semivariograms with analytical models can be tedious and restrictive. There are many smooth functions that could be
used for the semivariogram; however, arbitrary interpolation of the semivariogram will almost certainly create an invalid
function. A spectral correction, that is, taking the Fourier transform of the corresponding covariance values, resetting all
negative terms to zero, standardizing the spectrum to sum to the sill, and inverse transforming is a valuable method for constructing
valid discrete semivariogram models. This paper addresses some important implementation details and provides a methodology
to working with spectrally corrected semivariograms. 相似文献
15.
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
16.
Guillaume Larocque Pierre Dutilleul Bernard Pelletier James W. Fyles 《Mathematical Geology》2007,39(3):263-288
Coregionalization analysis has been presented as a method of multi-scale analysis for multivariate spatial data. Despite an
increasing use of this method in environmental and earth sciences, the uncertainty associated with the estimation of parameters
in coregionalization analysis (e.g., sills and functions of sills) is potentially high and has not yet been characterized.
This article aims to discuss the theory underlying coregionalization analysis and assess the robustness and limits of the
method. A theoretical framework is developed to calculate the ergodic and fluctuation variance-covariance matrices of least-squares
estimators of sills in the linear model of coregionalization. To adjust for the positive semidefiniteness constraint on estimated
coregionalization matrices, a confidence interval estimation procedure for sills and functions of sills is presented. Thereafter,
the relative importance of uncertainty measures (bias and variance) for sills and structural coefficients of correlation and
determination is assessed under different scenarios to identify factors controlling their uncertainty. Our results show that
the sampling grid density, the choice of the least-squares estimator of sills, the positive semidefiniteness constraint, the
presence of scale dependence in the correlations, and the number and range of variogram models, all affect the level of uncertainty,
sometimes through multiple interactions. The asymptotic properties of variogram model parameter estimators in a bounded sampling
domain impose a theoretical limit to their accuracy and precision. Because of this limit, the uncertainty was found to be
high for several scenarios, especially with three variogram models, and was often more dependent on the ratio of variogram
range to domain extent than on the sampling grid density. In practice, in the coregionalization analysis of a real dataset,
the circular requirement for sill estimates in the calculation of uncertainty measures makes the quantification of uncertainty
very problematic, if not impossible. The use of coregionalization analysis must be made with due knowledge of the uncertainty
levels and limits of the method. 相似文献
17.
Analytical solution to 1D coupled water infiltration and deformation in unsaturated soils 总被引：1，自引：0，他引：1
An analytical solution to 1D coupled water infiltration and deformation is derived using a Fourier integral transform. Exponential functional forms are used to represent the hydraulic conductivity–pore‐water pressure relationship and the soil‐water characteristic curve. Fredlund's incremental‐linear constitutive model for unsaturated soils is adopted. The analytical solution considers arbitrary initial pore‐water pressure distributions and flux and pressure boundary conditions. The corresponding analytical solutions to coupled steady‐state problems are also obtained. The analytical solutions demonstrate that the coupling of seepage and deformation plays an important role in water infiltration in unsaturated soils. In the early stages of infiltration, the difference between uncoupled and coupled conditions becomes marked over time, and in late stages, the difference caused by the coupling effects diminishes toward the steady state. The difference between the uncoupled and coupled conditions increases with decreasing desaturation coefficient (α). Pore‐water pressure or deformation changes caused by the coupling effects are mainly controlled by the degree of soil volume change due to a change in soil suction (H). The smaller the absolute value of H, the greater the effect of coupling on the infiltration and deformation. The ratio of rainfall intensity to saturated permeability (q/ks) also has a strong influence on the coupled seepage and deformation. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
18.
Guillaume Larocque Pierre Dutilleul Bernard Pelletier James W. Fyles 《Mathematical Geosciences》2007,39(3):263-288
Coregionalization analysis has been presented as a method of multi-scale analysis for multivariate spatial data. Despite an increasing use of this method in environmental and earth sciences, the uncertainty associated with the estimation of parameters in coregionalization analysis (e.g., sills and functions of sills) is potentially high and has not yet been characterized. This article aims to discuss the theory underlying coregionalization analysis and assess the robustness and limits of the method. A theoretical framework is developed to calculate the ergodic and fluctuation variance-covariance matrices of least-squares estimators of sills in the linear model of coregionalization. To adjust for the positive semidefiniteness constraint on estimated coregionalization matrices, a confidence interval estimation procedure for sills and functions of sills is presented. Thereafter, the relative importance of uncertainty measures (bias and variance) for sills and structural coefficients of correlation and determination is assessed under different scenarios to identify factors controlling their uncertainty. Our results show that the sampling grid density, the choice of the least-squares estimator of sills, the positive semidefiniteness constraint, the presence of scale dependence in the correlations, and the number and range of variogram models, all affect the level of uncertainty, sometimes through multiple interactions. The asymptotic properties of variogram model parameter estimators in a bounded sampling domain impose a theoretical limit to their accuracy and precision. Because of this limit, the uncertainty was found to be high for several scenarios, especially with three variogram models, and was often more dependent on the ratio of variogram range to domain extent than on the sampling grid density. In practice, in the coregionalization analysis of a real dataset, the circular requirement for sill estimates in the calculation of uncertainty measures makes the quantification of uncertainty very problematic, if not impossible. The use of coregionalization analysis must be made with due knowledge of the uncertainty levels and limits of the method. 相似文献
19.
Under the intrinsic coregionalization model if both primary and secondary measurements are available at all sample locations, the conventional geostatistical wisdom is that cokriging provides exactly the same solution as univariate kriging on the primary process alone. However, recent eamples have been given where nonzero secondary cokriging weights have accurred under this spatial dependence structure. This note identifies the conditions under which secondary information is useful under the assumption of intrinsic coregionalization. An illustration is given using a dataset of plutonium and americium concentrations collected from a region of the Nevada Test Site. 相似文献
20.
Since the attenulation of propagating waves through soil/rock is related to the localized material properties as well as the strain developed, the commonly used Rayleigh-type damping model and its variations are not suitable for dynamic finite element analysis of such materials. A linear viscoelastic material model based on the concept of the relaxation spectrum is manipualted in place of the damping model in this paper. The method proposed by Day and Minster11 to transform the convolutional form of the stress–strain relationship to a set of differential operators using the Pade approximant method is generalized to non-scalar waves and implemented for transient finite element analyses. A time-marching scheme is also proposed to incorporate the resultant differential operators into the governing equation of motion. The accuracy related to the Pade approximant method and the time-marching scheme is investigated by critically analysing some scalar wave propagation problems. The proposed technique is further verified using two one-dimensional stress wave propagation problems and a two-dimensional transient propagating wave through an unbounded linear viscoelastic medium. Some encouraging results have been obtained using the proposed technique and guidelines for using this technique are also presented. Comparisons of analytical solutions obtained by Fourier synthesis and numerical results have been provided. 相似文献