Positive definiteness is not enough

Geostatisticians know that the mathematical functions chosen to represent spatial covariances and variograms must have the appropriate type of positive definiteness, but they may not realize that there are restrictions on the types of covariances and variograms that are compatible with particular distributions. This paper gives some examples showing that (1) the spherical model is not compatible with the multivariate lognormal distribution if the coefficient of variation is 2.0 or more (even in 1-D), and (2) the Gaussian covariance and several other models are not compatible with indicator random functions. As these examples concern quite different types of random functions, it is clear that there is a general problem of compatibility between spatial covariance models (or variograms) and a specified multivariate distribution. The problem arises with all distributions except the multivariate normal, and not just the two cited here. The need for a general theorem giving the necessary and sufficient conditions for a covariance or a variogram to be compatible with a particular distribution is stressed.     

Computationally efficient restricted maximum likelihood estimation of generalized covariance functions

Computational aspects of the estimation of generalized covariance functions by the method of restricted maximum likelihood (REML) are considered in detail. In general, REML estimation is computationally intensive, but significant computational savings are available in important special cases. The approach taken here restricts attention to data whose spatial configuration is a regular lattice, but makes no restrictions on the number of parameters involved in the generalized covariance nor (with the exception of one result) on the nature of the generalized covariance function's dependence on those parameters. Thus, this approach complements the recent work of L. G. Barendregt (1987), who considered computational aspects of REML estimation in the context of arbitrary spatial data configurations, but restricted attention to generalized covariances which are linear functions of only two parameters.     

Coregionalization by Linear Combination of Nonorthogonal Components

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.     

Geostatistical Modelling of Cyclic and Rhythmic Facies Architectures

A Pluri-Gaussian method is developed for facies variables in three dimensions to model vertical cyclicity related to facies ordering and rhythmicity. Cyclicity is generally characterised by shallowing-upward or deepening-upward sequences and rhythmicity by the repetition of facies at constant intervals along sequences. Both of these aspects are commonly observed in shallow-marine carbonate successions, especially in the vertical direction. A grid-free spectral simulation approach is developed, with a separable covariance allowing a dampened hole-effect to capture rhythmicity in the vertical direction and a different covariance in the lateral plane along strata, as in space-time models. In addition, facies ordering is created by using a spatial shift between two latent Gaussian functions in the Pluri-Gaussian approach. Rapid conditioning to data is performed via Gibbs sampling and kriging using the screening properties of separable covariances. The resulting facies transiograms can show complex patterns of cyclicity and rhythmicity. Finally, a three dimensional case study of shallow-marine carbonate deposits at outcrop shows the applicability of the modelling method.     

Some asymptotic properties of kriging when the covariance function is misspecified

The impact of using an incorrect covariance function on kriging predictors is investigated. Results of Stein (1988) show that the impact on the kriging predictor from not using the correct covariance function is asymptotically negligible as the number of observations increases if the covariance function used is compatible with the actual covariance function on the region of interestR. The definition and some properties of compatibility of covariance functions are given. The compatibility of generalized covariances also is defined. Compatibility supports the intuitively sensible concept that usually only the behavior near the origin of the covariance function is critical for purposes of kriging. However, the commonly used spherical covariance function is an exception: observations at a distance near the range of a spherical covariance function can have a nonnegligible effect on kriging predictors for three-dimensional processes. Finally, a comparison is made with the perturbation approach of Diamond and Armstrong (1984) and some observations of Warnes (1986) are clarified.     

Maximum Likelihood Estimation of Spatial Covariance Parameters

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.     

Regionalized favorability theory for information synthesis in mineral exploration

This paper presents a regionalized method for the estimation of a favorability function through generalization of all relevant variables (explanatory and target) into random functions. The new method allows the use of cross-covariance functions in addition to ordinary covariances for extracting spatial joint information, which is virtually overlooked in the conventional analyses. The optimal weights for a favorability equation are derived from solving a generalized eigen-system established by the maximization of covariances between a favorability function and the principal components of a set of pre-selected target variables. Various correlation coefficients may be computed to assist in interpretation of the favorability estimates. Both favorability functions and correlation coefficients may be estimated for a point or a block. The regionalized favorability theory can be compared to cokriging in that both use the sample-sample covariances to account for the sample-sample relations and the point-sample covariances to account for the point-sample configurations. The new technique is demonstrated on a test case study, which involves the integration of geochemical, airborne-geophysical, and structural data sets for the target selection of hydrothermal gold-silver deposits.     

Polynomial expansions in likelihoods for spatial data: A case study

This paper illustrates the computational benefits of polynomial representations for quantities in the likelihood function for the spatial linear model based on the power covariance scheme. These benefits include a comprehensive study of likelihoods and maximum likelihood estimators for data. For simplicity, we focus on a relatively simple covariance scheme and data observed at equal intervals along a transect; we briefly indicate how generalizations to more complicated covariance functions and higher dimensions will operate.     

Limiting stochastic operations for stationary spatial processes

A natural extrapolation of stochastic operations (continuity and differentiation) already described in time domain (one-dimensional case) is established for spatial processes (two- or three-dimensional case). If stationarity decision is assumed, the continuity and differentiability (in the mean square sense) of a spatial process depends on the continuity and differentiability of the correlation function at the origin. Spatial processes described by stationary random functions are not continuous (in the mean square sense) when the covariance function presents a nugget effect, and they are not differentiable when the same covariance function is described by a spherical or an exponential covariance (models which are often used in geostatistics).     

Calculation of the Inverse of the Covariance

In reservoir characterization, the covariance is often used to describe the spatial correlation and variation in rock properties or the uncertainty in rock properties. The inverse of the covariance, on the other hand, is seldom discussed in geostatistics. In this paper, I show that the inverse is required for simulation and estimation of Gaussian random fields, and that it can be identified with the differential operator in regularized inverse theory. Unfortunately, because the covariance matrix for parameters in reservoir models can be extremely large, calculation of the inverse can be a problem. In this paper, I discuss four methods of calculating the inverse of the covariance, two of which are analytical, and two of which are purely numerical. By taking advantage of the assumed stationarity of the covariance, none of the methods require inversion of the full covariance matrix.     

Response of Glacier and Lake Covariations to Climate Change in Mapam Yumco Basin on Tibetan Plateau during 1974–2003

The study of spatial and temporal covariances of glaciers and lakes would help us to understand the impact of climate change within a basin in Tibet. This study focuses on glacier and lake variations in the Mapam Yumco(玛旁雍错)Basin (covering 7 786.44 km2)by Integrationg series of spatial data from topographic maps and digital satellite images at four different times 1974, 1990, 1999,and 2003. The results indicate that (1) decreased lakes, retreated glaciers, enlarged lakes and advanced glaciers co-exist in the basin during the last 30 years; (2) glacier recession was accelerated in recent years due to the warmer climate; (3) lake areas in the basin are both reduced and enlarged by an accelerated speed with more water supplies from speeding melt glaciers or frozen ground in the last three decades.     

The Characterization Problem for Isotropic Covariance Functions

Isotropic covariance functions are successfully used to model spatial continuity in a multitude of scientific disciplines. Nevertheless, a satisfactory characterization of the class of permissible isotropic covariance models has been missing. The intention of this note is to review, complete, and extend the existing literature on the problem. As it turns out, a famous conjecture of Schoenberg (1938) holds true: any measurable, isotropic covariance function on ^d (d 2) admits a decomposition as the sum of a pure nugget effect and a continuous covariance function. Moreover, any measurable, isotropic covariance function defined on a ball in ^d can be extended to an isotropic covariance function defined on the entire space ^d .     

Nonparametric estimation of generalized covariances by modeling on-line data

Structural analysis of data displaying trends may be performed with the help of generalized increments, the variance of these increments being a function of a generalized covariance. Generalized covariances are estimated primarily by parametric methods (i. e., methods searching for the best coefficients of a predetermined function), but also may be computed by one known nonparametric alternative. In this paper, a new nonparametric method is proposed. It is founded on the following principles: (1) least-squares residues are generalized increments; and (2) the generalized covariance is not a unique function, but a family of functions (the system is indeterminate). The method is presented in a general context of a k order trend in R^d, although the full solution is given only fork = I in Rⁱ. In Rⁱ, higher order trends may be developed easily with the equations included in this paper. For higher dimensions in space, the problem is more complex, but a research approach is proposed. The method is tested on soil pH data and compared to a parametric and nonparametric method.     

Nonseparable Space-Time Covariance Models: Some Parametric Families

By extending the product and product–sum space-time covariance models, new families are generated as integrated products and product–sums. These include nonintegrable space-time covariance models not obtainable by the Cressie–Huang representation. It is shown how to fit the spatial and temporal components of the models as well as the probability density function. The methods are illustrated by a case study.     

Transition probability-based indicator geostatistics

Traditionally, spatial continuity models for indicator variables are developed by empirical curvefitting to the sample indicator (cross-) variogram. However, geologic data may be too sparse to permit a purely empirical approach, particularly in application to the subsurface. Techniques for model synthesis that integrate hard data and conceptual models therefore are needed. Interpretability is crucial. Compared with the indicator (cross-) variogram or indicator (cross-) covariance, the transition probability is more interpretable. Information on proportion, mean length, and juxtapositioning directly relates to the transition probability: asymmetry can be considered. Furthermore, the transition probability elucidates order relation conditions and readily formulates the indicator (co)kriging equations.     

Dynamic stochastic estimation of physical variables

A fundamental problem facing the physical sciences today is analysis of natural variations and mapping of spatiotemporal processes. Detailed maps describing the space/time distribution of groundwater contaminants, atmospheric pollutant deposition processes, rainfall intensity variables, external intermittency functions, etc. are tools whose importance in practical applications cannot be overestimated. Such maps are valuable inputs for numerous applications including, for example, solute transport, storm modeling, turbulent-nonturbulent flow characterization, weather prediction, and human exposure to hazardous substances. The approach considered here uses the spatiotemporal random field theory to study natural space/time variations and derive dynamic stochastic estimates of physical variables. The random field model is constructed in a space/time continuum that explicitly involves both spatial and temporal aspects and provides a rigorous representation of spatiotemporal variabilities and uncertainties. This has considerable advantages as regards analytical investigations of natural processes. The model is used to study natural space/time variations of springwater calcium ion data from the Dyle River catchment area, Belgium. This dataset is characterized by a spatially nonhomogeneous and temporally nonstationary variability that is quantified by random field parameters, such as orders of space/time continuity and random field increments. A rich class of covariance models is determined from the properties of the random field increments. The analysis leads to maps of continuity orders and covariances reflecting space/time calcium ion correlations and trends. Calcium ion estimates and the associated statistical errors are calculated at unmeasured locations/instants over the Dyle region using a space/time kriging algorithm. In practice, the interpretation of the results of the dynamic stochastic analysis should take into consideration the scale effects.     

克立格估计的分析解释与协方差函数的代数确定

首先引入利用旋转面作为基函数的函数逼近概念, 在此基础上经过复杂的矩阵推导证明泛克立格法可表示为传统的带权最小二乘多项式拟合与以旋转面作为基函数的函数逼近, 并在一定条件下(随机场高度连续无块金效应) 论证了协方差(即旋转面) 的参数可通过数学分析的方法确定, 给出了以高斯函数为例确定协方差函数的两个准则.      

On the velocity covariance for steady flows in heterogeneous porous formations and its application to contaminants transport

We consider groundwater steady flow in a heterogeneous porous formation of random and stationary log-conductivity Y = ln K, characterized by the mean 〈Y〉, and the two point correlation function C _Y which in turn has finite, and different horizontal and vertical integral scales I and I _v , respectively. The fluid velocity V, driven by a given head drop applied at the boundary, has constant mean value U ≡ (U, 0, 0). Approximate explicit analytical expressions for transverse velocity covariances are derived. The adopted methodology follows the approach developed by Dagan and Cvetkovic (Spatial moments of kinetically sorbing plume in a heterogeneous aquifers, Water Resour. Res. 29 (1993) 4053) to obtain a similar result for the longitudinal velocity covariance. Indeed, the approximate covariances of transverse velocities are determined by requiring that they have the exact first order variances as well as zero integral scale (G. Dagan, Flow and Transport in Porous Formations (Springer, 1989)) , and provide the exact asymptotic limits of the displacement covariance of the fluid particles obtained by Russo (On the velocity covariance and transport modeling in heterogeneous anisotropic porous formations 1. Saturated flow, Water Resour. Res., 31 (1995) 129). Comparisons with numerical results show that the proposed expressions compare quite well in the early time regime, and for Ut/I >100. Since most of the applications, like assessing the effective mobility of contaminants or quantifying the potential hazards of nuclear repositories, require predictions over higher times the proposed approximate expressions provide acceptable results. The main advantage related to such expressions is that they allow obtaining closed analytical forms of spatial moments pertaining to kinetically sorbing contaminant plumes avoiding the very heavy computational effort which is generally demanded. For illustration purposes, we consider the movement of one contaminant species, and show how our approximate spatial moments compare with the numerical simulations.     

Revisiting the accurate calculation of block-sample covariances using Gauss quadrature

Block-sample covariances may be calculated by discretizing a block into regularly spaced grid points, computing punctual covariance between each grid point and the sample, then averaging. Gauss quadrature is a better, more accurate method for calculating block-sample covariance as has been demonstrated in the past by other authors (the history of which is reviewed herein). This prior research is expanded upon to provide considerably more detail on Gauss quadrature for approximating the areal or volumetric integral for block-sample covariance. A 4 × 4 Gauss point rule is shown to be optimal for this procedure. Moreover, pseudo-computer algorithms are presented to show how to implement Gauss quadrature in existing computer programs which perform block kriging.     

Multivariable spatial prediction

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.