On the Equivalence of the Cokriging and Kriging Systems

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     

On the pseudo cross-variogram

Normal cross-variograms cannot be estimated from data in the usual way when there are only a few points where both variables have been measured. But the experimental pseudo cross-variogram can be computed even where there are no matching sampling points, and this appears as its principal advantage. The pseudo cross-variogram may be unbounded, though for its existence the intrinsic hypothesis alone is not a sufficient stationarity condition. In addition the differences between the two random processes must be second order stationary. Modeling the function by linear coregionalization reflects the more restrictive stationarity condition: the pseudo cross-variogram can be unbounded only if the unbounded correlation structures are the same in all variograms. As an alternative to using the pseudo cross-variogram a new method is presented that allows estimating the normal cross variogram from data where only one variable has been measured at a point.     

Conditional Indicator Simulation: Application to a Saskatchewan uranium deposit

Mineral deposits frequently exhibit a mixture of rock types in which each type can be identified by a characteristic metal concentration. Such a mixture can be correctly simulated by first reproducing the spatial and geometric configuration of the various rock types in the deposit. Then the grades for each rock type can be jointly simulated and filled in according to their specific coregionalization characteristics. The method of Conditional Indicator Simulation and an uranium—arsenic joint simulation are presented with a detailed, step-by-step application to the Midwest deposit, a high grade uranium deposit in northern Saskatchewan.     

Gaussian Cosimulation: Modelling of the Cross-Covariance

Whenever two or more random fields are assumed to be correlated in reservoir characterization, it is necessary to generate valid cross-covariance models to describe the relationship. The standard methods for constructing covariance matrices for correlated random fields are not very general. In particular, they do not allow one to specify different auto-covariance models for the two fields. It is not possible, for example, for one field to have a Gaussian auto-covariance and the other an exponential auto-covariance, unless the two fields are uncorrelated. The standard approaches also do not allow for nonsymmetric cross-covariance functions. In this report, I present a straightforward method of cosimulation based on the square root of the auto-covariances. The same approach is used for constructing cross-covariance models for the variables. The approach is quite general and does not require symmetry of the cross-covariance. The modelling of the cross-covariance is illustrated with gamma ray and spontaneous potential logs.     

Multivariate Spatial Modeling for Geostatistical Data Using Convolved Covariance Functions

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.     

Understanding airborne radiometric survey signals across part of eastern England

A low‐level airborne radiometric survey provides data on the concentrations of gammaemitting elements including potassium (K), thorium (Th) and uranium (U) in the upper half metre of the soil. Where weathering has not penetrated much beyond this depth, as in the young soils that cover much of England and Wales, the signal is likely to be related to the soil's clay content and its parent material. In these situations radiometric survey could be valuable for mapping soil digitally. We wished to understand how the radiometric signal relates to parent material and soil geochemistry, and to identify the spatial structure, if not the sources, of any unexplained variation. We analysed the joint spatial variation of the airborne gamma signal and high‐resolution soil geochemical survey data across part of eastern England by modelling their coregionalization. We also used reml to assess the joint effects of soil geochemistry and parent material on the radiometric signals of K and Th. The overall correlations of radiometric estimates with soil survey data for K and Th were large, as were the structural correlations for components of variation spatially dependent up to 49 and 16 km for K and Th respectively. This suggests that the radiometric signals for these two elements provide effective estimates of the amounts in the soil and their patterns of distribution. Although class of parent material accounted for a third of of the variance in the radiometric K signal, much of the variation within the classes is explained by geochemistry, suggesting that subtler changes can be detected. A larger proportion of the Th signal was accounted for by parent material. This supports our expectation that radiometric signals for K and Th provide information on parent material in the young landscapes of England and Wales. We are therefore confident that airborne radiometric surveys would be useful for making thematic maps of soil, particularly the soil's texture and closely related properties across England and Wales. Copyright © 2007 Natural Environment Research Council. Published in 2007 by John Wiley & Sons, Ltd.