This series of three papers describes a cross‐validation method to estimate the spatial covariance structure of intrinsic or nonintrinsic random functions from point or spatially averaged data that may be corrupted by noise. Any number of relevant parameters, including nugget effect, can be estimated. The theory, described in this paper, is based on a maximum likelihood approach which treats the cross‐validation errors as Gaussian. Various a posteriori statistical tests are used to verify this hypothesis and to show that in many cases, correlation between these errors is weak. The log likelihood criterion is optimized through a combination of conjugate gradient algorithms. An adjoint state theory is used to efficiently calculate the gradient of the estimation criterion, optimize the step size downgradient, and compute a lower bound for the covariance matrix of the estimation errors. Issues related to the identifiability, stability, and uniqueness of the resulting adjoint state maximum likelihood cross‐validation (ASMLCV) method are discussed. This paper also describes the manner in which ASMLCV allows one to use model structure identification criteria to select the best covariance model among a given set of alternatives. Practical aspects of ASMLCV and its application to synthetic data are presented in paper 2 (Samper and Neuman, this issue (a)). Applications to real hydrogeological data (transmissivities and groundwater levels) have been presented elsewhere, while hydrochemical and isotopic data are analyzed by ASMLCV in paper 3 (Samper and Neuman, this issue (b)).

Paper 2 of this three‐part series uses synthetic data to investigate the properties of the adjoint state maximum likelihood cross‐validation (ASMLCV) method presented in paper 1 (Samper and Neuman, this issue (a)). More than 40 synthetic experiments are performed to compare various conjugate gradient algorithms; investigate the manner in which computer time varies with ASMLCV parameters; study the effect of sample size and choice of kriging points on ASMLCV estimates ; evaluate the ability of various model structure identification criteria to help select the most appropriate semivariogram model among given alternatives; study the conditions required for parameter identifiability, uniqueness, and stability; quantify the statistics of cross‐validation errors; test hypotheses concerning the distribution and autocorrelation of these errors; and illustrate the computation of approximate quality indicators for ASMLCV parameter estimates.

Paper 3 of this three‐part series presents applications of our adjoint state maximum likelihood cross‐validation (ASMLCV) method to real data from aquifers. The Madrid basin in Spain serves as the source of information about 11 hydrochemical variables (pH, electrical conductivity, silica content, and the concentration of major ions) and two isotopes (oxygen 18 and carbon 14). Due to a lack of sufficient vertical resolution, our analysis is restricted to the horizontal plane. With the exception of oxygen 18 and silica, the variables appear to be free of a horizontal drift. No discernible directional effects are seen. All variables exhibit a large nugget effect which is indicative of background noise. We conclude that more detailed and careful sampling in three dimensions is required if groundwater quality information is to become less prone to such noise and thereby more useful in the context of quantitative hydrogeological analyses. Despite the existing noise, we are able to confirm geostatistically some (though not all) of the hypotheses advanced by others about hydrochemical evolution and isotope changes in the basin. The ability of ASMLCV to filter out spatial variations from part of the measurement noise is illustrated on carbon 14 data. The same data are also used to investigate the utility of model structure identification criteria in selecting the best among a set of alternative semivariogram models.

No abstract

No abstract