Donald E. Myers

Matrix formulation of co-kriging

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.

Optimization of Sampling Locations for Variogram Calculations

A method is presented and demonstrated for optimizing the selection of sample locations for variogram estimation. It is assumed that the distribution of distance classes is decided a priori and the problem therefore is to closely approximate the preselected distribution, although the dispersion within individual classes can also be considered. All of the locations may be selected or points added to an existing set of sites or to those chosen on regular patterns. In the examples, the sum of squares characterizing the deviation from the desired distribution of couples is reduced by as much as 2 orders of magnitude between random and optimized points. The calculations may be carried out on a microcomputer. Criteria for what constitutes best estimators for variogram are discussed, but a study of variogram estimators is not the object of this paper.

Spatial interpolation: an overview

The interpolation of spatial data has been considered in many different forms. The various forms of kriging are among the best known in the earth sciences although techniques such as inverse distance weighting were and are in use for spatially located data. In the numerical analysis literature various forms of splines and more recently radial basis functions have been developed and used. Because these techniques have been developed in very different contexts the relationship between them has not always been apparent. Various forms of kriging are considered as well as kernel estimators, splines and radial basis functions. By using the dual form of kriging and the positive definiteness property of the variogram connections are shown between splines, kriging and radial basis functions. One of the distinctions between kriging and other interpolators is the incorporation of the support of the samples and explicit estimation of linear functionals such as spatial integrals.

Problems in space-time kriging of geohydrological data

Spatiotemporal variables constitute a large class of geohydrological phenomena. Estimation of these variables requires the extension of geostatistical tools into the space-time domain. Before applying these techniques to space-time data, a number of important problems must be addressed. These problems can be grouped into four general categories: (1) fundamental differences with respect to spatial problems, (2) data characteristics, (3) structural analysis including valid models, and (4) space-time kriging. Adequate consideration of these problems leads to more appropriate estimation techniques for spatiotemporal data.

Space–time analysis using a general product–sum model

A generalization of the product-sum covariance model introduced by De Cesare et al. (Statist. Probab. Lett. 51 (2001) 9) is given in this paper. This generalized model is non-separable and in general is non-integrable, hence, it cannot be obtained from the Cressie-Huang representation. Moreover, the product-sum model does not correspond to the use of a metric in space-time. It is shown that there are simple methods for estimating and modeling the covariance or variogram components of the product-sum model using data from realizations of spatial-temporal random fields.

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.

Interpolation and estimation with spatially located data

Abstract Myers, D.E., 1991. Interpolation and estimation with spatially located data. Chemometrics and Intelligent Laboratory Systems , 11: 209–228. Kriging is a regression method used with irregularly spaced data in 1-, 2- or 3-space for the estimation of values at unsampled locations or for the estimation of the spatial average over a length, area or volume. The estimator is linear in the data and the weights are obtained from a system of linear equations in which the coefficients are the values of variograms or covariance functions quantifying the correlation between data at two sample locations or between a sample location and the location to be estimated. The equations are obtained by minimizing the variance of the error of estimation, the variance being computed from a theoretical model for the correlation function rather than from empirical values as in most regression formulations. Estimation and modeling of this structure function is the most important and potentially the most difficult step in the process. While the method is not implemented in standard statistical packages, public domain software for use on an IBM personal computer or clone is available. The theory is briefly reviewed, practical aspects of the application of the method are discussed and available software and extensions are outlined. The US EPA Dallas Lead Study data is used to illustrate the problems and the method.

To be or not to be... stationary? That is the question

Stationarity in one form or another is an essential characteristic of the random function in the practice of geostatistics. Unfortunately it is a term that is both misunderstood and misused. While this presentation will not lay to rest all ambiguities or disagreements, it provides an overview and attempts to set a standard terminology so that all practitioners may communicate from a common basis. The importance of stationarity is reviewed and examples are given to illustrate the distinctions between the different forms of stationarity.

Pseudo-cross variograms, positive-definiteness, and cokriging

Cokriging allows the use of data on correlated variables to be used to enhance the estimation of a primary variable or more generally to enhance the estimation of all variables. In the first case, known as the undersampled case, it allows data on an auxiliary variable to be used to make up for an insufficient amount of data. Original formulations required that there be sufficiently many locations where data is available for both variables. The pseudo-cross-variogram, introduced by Clark et al. (1989), allows computing a related empirical spatial function in order to model the function, which can then be used in the cokriging equations in lieu of the cross-variogram. A number of questions left unanswered by Clark et al. are resolved, such as the availability of valid models, an appropriate definition of positive-definiteness, and the relationship of the pseudo-cross-variogram to the usual cross-variogram. The latter is important for modeling this function.

Product‐sum covariance for space‐time modeling: an environmental application

In this paper a product-sum covariance for space-time modeling of nitrogen dioxide in the Milan district is introduced. Residuals have been generated for all stations after the removal of daily and seasonal trends, which are readily interpretable, in order to estimate and model the spatial-temporal variogram. The trend component and the residual variogram model have been used to predict the hourly averages of nitrogen dioxide for the first two days of January 1997. GSLIB programs were modified for sample variogram computations, cross-validation and kriging. Copyright