The ECMWF implementation of three-dimensional variational assimilation (3D-Var). I: Formulation

In the first of this set of three papers, the formulation of the European Centre for Medium-Range Weather Forecasts (ECMWF) implementation of 3D-Var is described. In the second, the specification of the structure function is presented, and the last is devoted to the results of the extensive numerical experimentation programme which was conducted. The 3D-Var formulation uses a spherical-harmonic expansion, much as the ECMWF optimal interpolation (OI) scheme used an expansion of Bessel functions. This formulation is introduced using a convolution algebra over the sphere expressed directly in spectral space. It is shown that all features of the OI statistical model can be implemented within SD-Var. Furthermore, a non-separable statistical model is described. In the present formulation, geostrophy is accounted for through a Hough-modes separation of the gravity and Rossby components of the analysis increments. As in OI, the tropical analysis remains essentially non-divergent and with a weak mass-wind coupling. The observations used, as well as their specified statistics of errors, are presented, together with some implementation details. In the light of the results, 3D-Var was implemented operationally at the end of January 1996.