Some Remarks on Spatial Correlation Function Models

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Abstract
The method of optimal interpolation, which is widely used in meteorological data assimilation, relies very much on good approximations of spatial correlation functions. Therefore, many models for such functions have been developed. These models should fulfill certain mathematical constraints; particularly, they should be positive-definite functions. For the classes of homogeneous and isotropic processes, the positivity property and its consequences are reviewed. A special class of correlation models based on so-called spatial autoregressive processes is critically examined. It is shown that models of this type are not positive definite on the meteorological relevant spaces. Some other models taken from the literature are shown to lack this property also. Three strategies to obtain models that have the appropriate mathematical properties are outlined. Abstract The method of optimal interpolation, which is widely used in meteorological data assimilation, relies very much on good approximations of spatial correlation functions. Therefore, many models for such functions have been developed. These models should fulfill certain mathematical constraints; particularly, they should be positive-definite functions. For the classes of homogeneous and isotropic processes, the positivity property and its consequences are reviewed. A special class of correlation models based on so-called spatial autoregressive processes is critically examined. It is shown that models of this type are not positive definite on the meteorological relevant spaces. Some other models taken from the literature are shown to lack this property also. Three strategies to obtain models that have the appropriate mathematical properties are outlined.

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