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

TLDR

The paper introduces the European Centre for Medium‑Range Weather Forecasts’ 3D‑Var formulation, which uses a spherical‑harmonic expansion analogous to the Bessel‑function expansion of the optimal interpolation scheme and retains a non‑divergent tropical analysis with weak mass‑wind coupling. The authors formulate 3D‑Var via a convolution algebra on the sphere in spectral space, incorporating a non‑separable statistical model and Hough‑mode separation of gravity and Rossby components, and detail the observations, error statistics, and implementation specifics. They demonstrate that all features of the optimal interpolation statistical model can be realized within 3D‑Var, leading to its operational deployment at ECMWF in January 1996.

Abstract

Abstract 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 3D‐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.