On Some Properties of Correlation Functions Used in Optimum Interpolation Schemes

Abstract

Objective analyses using the so-called method of optimum interpolation incorporates statistical information on the variable(s) by means of the covariance or correlation functions. The concern in this contribution is with some properties of the analytic forms of the correlation functions that are used to model the statistical structure. First, some attention is directed to the question of fitting the various analytic forms (containing adjustable constants) to samples of actual correlations. All but one of the candidate forms were indistinguishable on the basis of the residuals of the statistical fitting procedure. Second, the criterion of positive-definiteness of the correlation function is extended to stipulate that the transform (or spectrum) of the function should possess some features of the spectra of actual variables{\textemdash}the most important one being the spectral decay rate at high wavenumber. Again, all but one of the candidate forms (the same one as above) had transforms that were acceptable. Third, the degree of isotropy of the correlation fields is examined, both for scalar variables (geopotential, temperature) and for the wind field. Finally, the imposition of geostrophy requires some special considerations on the form of the correlation function. For all of these properties a variety of suggested analytic forms are compared and conclusions drawn.