The spatial statistics of random geostrophic modes and first-guess errors

It is hypothesized that a data assimilation system with excellent forecast and analysis components and frequent access to good data should have first-guess errors similar to an ensemble of random slow modes with equipartition of energy. The covariance statistics of such a system on a constant f-plane are calculated and shown to have interesting features. Of these, the overall increase with elevation of wind and geopotential errors, the size of temperature variances for given height variances, the increase with elevation of the horizontal velocity scale, and the sharper correlation in the vertical for wind than geopotential, are also found in the empirical studies of first-guess errors over North America by Hollingsworth and Lönnberg. Two major differences are (a) the presence of a maximum wind and geopotential error at the tropopause in the empirical data, and (b) the empirical horizontal spectra for velocity peak at the largest accessible wave length (5000 km) while the theoretical spectra peak at the shortest wave length (1000 km). If it is assumed that basic conditions of the hypothesis are satisfied, the ECMWF forecast and analysis methods must introduce relatively large errors at long wave lengths. A more believable conclusion is that the assumption of “frequent access to good data” is invalidated by data voids outside of the test network.Other results agree with the Hollingsworth-Lönnberg study in that the separable mathematical models used to represent correlations in operational practice misrepresent some important features. The results also suggest that temperature errors in a forecast of high vertical resolution will be underestimated by evaluation of thickness errors between the standard pressure levels.