A continuous three-dimensional model of short-range forecast error covariances

This paper develops a representation for the three-dimensional covariance structure of the observed residuals, or differences, between the short-range forecasts used as trial fields in an operational data assimilation system and the verifying radiosonde data. Assuming homogeneity and isotropy on pressure surfaces, geopotential and wind covariances are fitted to a series expansion employing Bessel functions in the horizontal and vertical normal modes as basis functions. The latter are obtained from linear model equations with an isothermal basic state employing a log-pressure vertical coordinate. Such a formulation allows for a closed analytic form, and therefore a spatially continuous three-dimensional covariance model. The model is used to examine covariances of geopotential as well as transverse and longitudinal wind components. Unlike commonly-used representations employing vertical/horizontal separability, the present model reproduces the observed significant increase of horizontal decorrelation length scales with height. The wind data are used to obtain divergent and solenoidal components and subsequent comparison of stream function and geopotential statistics reveals a high degree of geostrophic balance away from the upper and lower boundaries. It is suggested that the present mathematical framework could be employed in developing analytical non-separable three-dimensional correlation functions for use in operational data assimilation.