Markov Models for Cross-Covariances

Markov models based on various data screening hypotheses are often used because they reduce the statistical inference burden. In the case of co-located cokriging, the commonly used Markov model results in the cross-covariance being proportional to the primary covariance. Such model is inappropriate in the presence of a smoothly varying secondary variable defined on a much larger volume support than the primary variable. For such cases, an alternative Markov screening hypothesis is proposed that results in a more continuous cross-covariance proportional to the secondary covariance model. A parallel development of both Markov models is presented. A companion paper provides a comparative application to a real data set.     

Ordinary Cokriging Revisited

This paper sets up the relations between simple cokriging and ordinary cokriging with one or several unbiasedness constraints. Differences between cokriging variants are related to differences between models adopted for the means of primary and secondary variables. Because it is not necessary for the secondary data weights to sum to zero, ordinary cokriging with a single unbiasedness constraint gives a larger weight to the secondary information while reducing the occurrence of negative weights. Also the weights provided by such cokriging systems written in terms of covariances or correlograms are not related linearly, hence the estimates are different. The prediction performances of cokriging estimators are assessed using an environmental dataset that includes concentrations of five heavy metals at 359 locations. Analysis of reestimation scores at 100 test locations shows that kriging and cokriging perform equally when the primary and secondary variables are sampled at the same locations. When the secondary information is available at the estimated location, one gains little by retaining other distant secondary data in the estimation.     

Ordinary Cokriging Revisited

This paper sets up the relations between simple cokriging and ordinary cokriging with one or several unbiasedness constraints. Differences between cokriging variants are related to differences between models adopted for the means of primary and secondary variables. Because it is not necessary for the secondary data weights to sum to zero, ordinary cokriging with a single unbiasedness constraint gives a larger weight to the secondary information while reducing the occurrence of negative weights. Also the weights provided by such cokriging systems written in terms of covariances or correlograms are not related linearly, hence the estimates are different. The prediction performances of cokriging estimators are assessed using an environmental dataset that includes concentrations of five heavy metals at 359 locations. Analysis of reestimation scores at 100 test locations shows that kriging and cokriging perform equally when the primary and secondary variables are sampled at the same locations. When the secondary information is available at the estimated location, one gains little by retaining other distant secondary data in the estimation.     

Modeling Spatial Variability with One and Multidimensional Continuous-Lag Markov Chains

The continuous-lag Markov chain provides a conceptually simple, mathematically compact, and theoretically powerful model of spatial variability for categorical variables. Markov chains have a long-standing record of applicability to one-dimensional (1-D) geologic data, but 2- and 3-D applications are rare. Theoretically, a multidimensional Markov chain may assume that 1-D Markov chains characterize spatial variability in all directions. Given that a 1-D continuous Markov chain can be described concisely by a transition rate matrix, this paper develops 3-D continuous-lag Markov chain models by interpolating transition rate matrices established for three principal directions, say strike, dip, and vertical. The transition rate matrix for each principal direction can be developed directly from data or indirectly by conceptual approaches. Application of Sylvester's theorem facilitates establishment of the transition rate matrix, as well as calculation of transition probabilities. The resulting 3-D continuous-lag Markov chain models then can be applied to geo-statistical estimation and simulation techniques, such as indicator cokriging, disjunctive kriging, sequential indicator simulation, and simulated annealing.     

Collocated Cokriging Based on Merged Secondary Attributes

There exist many secondary data that must be considered in in reservoir characterization for resource assessment and performance forecasting. These include multiple seismic attributes, geological trends and structural controls. It is essential that all secondary data be accounted for with the precision warranted by that data type. Cokriging is the standard technique in geostatistics to account for multiple data types. The most common variant of cokriging in petroleum geostatistics is collocated cokriging. Implementations of collocated cokriging are often limited to a single secondary variable. Practitioners often choose the most correlated or most relevant secondary variable. Improved models would be constructed if multiple variables were accounted for simultaneously. This paper presents a novel approach to (1) merge all secondary data into a single super secondary variable, then (2) implement collocated cokriging with the single variable. The preprocessing step is straightforward and no major changes are required in the standard implementation of collocated cokriging. The theoretical validity of this approach is proven, that is, the results are proven to be identical to a “full” approach using all multiple secondary variables simultaneously.     

Comparison of approaches to spatial estimation in a bivariate context

The problem of estimating a regionalized variable in the presence of other secondary variables is encountered in spatial investigations. Given a context in which the secondary variable is known everywhere (or can be estimated with great precision), different estimation methods are compared: regression, regression with residual simple kriging, kriging, simple kriging with a mean obtained by regression, kriging with an external drift, and cokriging. The study focuses on 19 pairs of regionalized variables from five different datasets representing different domains (geochemical, environmental, geotechnical). The methods are compared by cross-validation using the mean absolute error as criterion. For correlations between the principal and secondary variable under 0.4, similar results are obtained using kriging and cokriging, and these methods are superior slightly to the other approaches in terms of minimizing estimation error. For correlations greater than 0.4, cokriging generally performs better than other methods, with a reduction in mean absolute errors that can reach 46% when there is a high degree of correlation between the variables. Kriging with an external drift or kriging the residuals of a regression (SKR) are almost as precise as cokriging.     

A comparative study of hydraulic conductivity estimations using geostatistics

Three approaches for estimating the hydraulic conductivity (K) of the Trifa aquifer, Morocco were investigated: (1) kriging of the K values obtained from pumping tests, (2) cokriging of the pumping test data with electrical resistivity data as a secondary variable, and (3) cokriging of the pumping test data with the slope of the water table. Gauss-transformed values of the variables are used because they provide more robust variograms and transformed values of the primary and secondary variables show correlations higher than the raw values, which is beneficial in cokriging. In cokriging with electrical resistivity, two zones are considered since the geological deposits are different from the north to the south of the aquifer, which is reflected in different correlations between the variables. Comparison of the three approaches is based mainly on the estimation errors, and to a lesser degree on the cross-validations of the corresponding variogram models and general considerations, like the measurements’ reliability and aquifer make-up. The best-estimated K is given by cokriging with the slope of the water table and is therefore preferred for further use in groundwater flow modeling. Thus, electrical resistivity or the slope of the water table can both be used as secondary variables to estimate K, especially in heterogeneous aquifers with lateral variations in lithology, as is the case of the Trifa aquifer.     

Universal cokriging under intrinsic coregionalization

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.     

Conditioning Geostatistical Operations to Nonlinear Volume Averages

The all-important process of data integration calls for algorithms that can handle secondary data often defined as nonlinear averages of the primary (hard) data over specific areas or volumes. It is suggested to approximate these nonlinear averages by linear averages of a nonlinear transform of the primary variable. Kriging of such nonlinear transforms, followed by the inverse transform, allows exact reproduction of all original data, both of point support and nonlinear volume averages. In a simulation mode, the previous cokriging provides the mean and variance of a conditional distribution from which to draw a simulated value, which is then backtransformed into a simulated value of the primary variable. The nonlinear averaged data values are then reproduced exactly. The direct sequential simulation algorithm adopted does not call for using any Gaussian distribution.     

Cokriging of aquifer transmissivities from field measurements of transmissivity and specific capacity

This paper presents a new application of the cokriging technique for constructing maps of aquifer transmissivity from field measurements of transmissivity and specific capacity. The technique is illustrated using data from Yolo Basin, California. Cokriging is well-suited for estimating undersampled variables. To improve the accuracy of the estimation, cokriging considers the spatial auto-correlation of the variable to be estimated and the spatial cross-correlation between the variable to be estimated and other, better-sampled variables. Consequently, in regions that lack data of the variable to be estimated, accurate estimation can still be made on the basis of auto- and cross-correlation. In addition, estimation variances can be obtained with a little additional computation effort.     

Data Configurations and the Cokriging System: Simplification by Screen Effects

Large cokriging systems arise in many situations and are difficult to handle in practice. Simplifications such as simple kriging, strictly collocated and multicollocated cokriging are often used and models under which such simplifications are, in fact, equivalent to cokriging have recently received attention. In this paper, a two-dimensional second-order stationary random process with known mean is considered and the redundancy of certain components of the data at certain locations vis-à-vis the solution to the simple cokriging system is examined. Conditions for the simple cokriging weights of these components at these locations are set to zero. The conditions generalise the notion of the autokrigeability coefficient and can, in principle, be applied to any data configuration. In specific sampling situations such as the isotopic and certain heterotropic configurations, models under which simple kriging, strictly collocated, multicollocated and dislocated cokriging are equivalent to simple cokriging are readily identified and results already available in the literature are obtained. These are readily identified and the results are already available in the literature. The advantage of the approach presented here is that it can be applied to any data configuration for analysis of permissible simplifications in simple cokriging.     

Comparison of kriging techniques in a space-time context

Space-time processes constitute a particular class, requiring suitable tools in order to predict values in time and space, such as a space-time variogram or covariance function. The space-time co-variance function is defined and linked to the Linear Model of Coregionalization under second-order space-time stationarity. Simple and ordinary space-time kriging systems are compared to simple and ordinary cokriging and their differences for unbiasedness conditions are underlined. The ordinary space-time kriging estimation then is applied to simulated data. Prediction variances and prediction errors are compared with those for ordinary kriging and cokriging under different unbiasedness conditions using a cross-validation. The results show that space-time kriging tend to produce lower prediction variances and prediction errors that kriging and cokriging.     

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     

Deriving Constraints on Small-Scale Variograms due to Variograms of Large-Scale Data

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.     

A comparison between cokriging and ordinary kriging: Case study with a polymetallic deposit

Cokriging is applied to the estimation of mineral resources in a polymetallic deposit. Several major steps, which should be taken in using cokriging, are highlighted as necessary practical considerations. The case study is related to an ultramafic copper-nickel deposit. Six elements, Cu, Ni, Au, Ag, Pt, and Pd, occurring in the deposit, are partitioned into three subgroups and the elements within each group are simultaneously estimated on the basis of over 4000 drill assays. A comparison was made between ordinary kriging and cokriging methods through cross-validation. The results show that cokriging has significantly improved the estimates of resources by reducing the overall estimation error by over 15% and the variance of error by over 20%.     

Which Models for Collocated Cokriging?

When a target variable is sparsely sampled, compared to a densely sampled auxiliary variable, cokriging requires simplifications. In its strict sense, collocated cokriging makes use of the auxiliary variable only at the current point where the target variable is to be estimated; in the multicollocated form, it also makes use of the auxiliary variable at all points where the target variable is available. This paper looks for the models that support these collocated cokrigings, i.e., the models in which the simplification resulting from the collocated forms does not result in any loss of information. In these models, the cross-structure between the two variables is shown to be proportional to the structure of the auxiliary variable, not to the structure of the target variable as is often assumed (except, of course, when all structures are proportional). The target variable depends on the auxiliary variable and on a spatially uncorrelated residual. Collocated cokriging simplifies to the simple method, which consists in kriging this residual. The strictly collocated cokriging corresponds to the particular case where the residual has a pure nugget structure, but it is then reduced to the single regression at the target point. Except for this trivial case, there are no models in which strictly collocated cokriging is exactly a cokriging.     

Joint simulation of multiple variables with a Markov-type coregionalization model

Many applications are multivariate in character and call for stochastic images of the joint spatial variability of multiple variables conditioned by a prior model of covariances and cross- covariances. This paper presents an algorithm to perform cosimulation of such spatially intercorrelated variables. This new algorithm builds on a Markov-type hypothesis whereby collocated information screens further away data of the same type, allowing cosimulation without the burden of a full cokriging. The proposed algorithm is checked against a synthetic multi-Gaussian reference dataset, then against a multi-Gaussian cosimulation approach using full cokriging. The results indicate that the proposed algorithm perform as well as the full cokriging approach in reproducing the univariate and bivariate statistics of the reference set, yet at less cpu cost.     

On Some Simplifications of Cokriging Neighborhood

Choosing the cokriging neighborhood is often difficult. A poor choice, ignoring influent data, can result in a loss of information as well as in artifacts in simulations based on cokriging. Then it is convenient to use if possible, or to refer to models that lead to simplified cokriging neighborhood. We essentially consider the case of two stationary variables, a target variable and an auxiliary one. By examining possible simplifications, we set up a list of models (essentially models with residuals) that, in general or under specific configurations, lead to simplifications of cokriging neighborhood. Collocated, dislocated, and other types of neighborhood are identified, that are optimal in some models and configurations. Possible extensions to cokriging with unknown means, and to more variables, are included.     

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.