Three-Dimensional Covariance Functions for NOGAPS Data

Abstract

Height innovation data for a 2-month period from NOGAPS was analyzed to obtain height prediction and observation error covariances. Different methods of weighting the data in least squares approximations of the spatial covariance data were investigated using the second-order autoregressive (SOAR) correlation function, both with and without an additive constant (varying with pressure level). Based on the properties of the derived covariance matrices and the SOAR parameters, the SOAR without an additive constant was used for the horizontal approximations. The vertical correlations were fit using a combination of SOAR plus an additive constant and a transformation of the logP coordinate to another coordinate to achieve a best fit. The resulting three-dimensional approximation is partially separable, being the product of the horizontal covariance function (which depends on height) and the vertical correlation function. Figures demonstrating various aspects of the process and the results are given.