Background Error Formulations in Variational Data Assimilation and Implications for Ensemble Prediction in Numerical Weather Prediction

Research project P 13729 Variational Data Assimilation and Ensemble Prediction Martin EHRENDORFER 11.10.1999 State-of-the-art data assimilation systems produce analysed atmospheric states by combining in an optimal way a priori information with observations of the atmosphere. These analyses of the atmospheric flow are subsequently used as initial conditions for the integration of numerical weather prediction models. In variational assimilation systems, the analysis is determined by minimizing a specified objective function J, that describes the distance of the analysis from both the a priori information (also called background information) and the observations (J ideally also accounts for model error). This variational approach is in operational use at major weather forecasting centers, such as, for example, the European Centre for Medium-Range Weather Forecasts (ECMWF). In the research proposed here it is planned to (i) investigate the role of the formulation of the background term in variational data assimilation, and to (ii) study both the impact of the resulting different probability distribution functions (pdfs) and model error descriptions in the context of ensemble prediction. The fundamental issue of question (i) is that in the specification of J knowledge of the background error statistics, denoted as B, is required (in addition to observational and model error statistics). However, flow-dependent statistics B are not readily available if the assimilation problem is formulated variationally. To obtain a. flow- dependent B, approaches based on the correspondence between linear estimation theory and the related variational problems are presently being tested, also in full-scale assimilation systems. Among these approaches is Kalman filter theory, stochastic-dynamic prediction, and various approximated (simplified) forms of the Kalman filter equations. It is proposed here to study these approaches in detail, first, in the context of a completely quasigeostrophic data assimilation system, and, second, in the context of a quasigeostrophic model coupled with the ECMWF variational assimilation system. In both situations, exact and approximated formulations for B can be compared explicitly, and will provide further guidance for full-scale assimilation systems. The fundamental issue of question (ii) relates to the necessity of specifying the initial pdf (of analysis errors) in the context of ensemble prediction. One of the primary purposes of ensemble prediction is the assessment of the time- evolving pdf. The direct and complete assessment of the initial pdf is difficult, but approaches exist for its approximate specification. Specifically, if an assimilated state is produced, the relevant analysis error statistics may, in principle, also be obtained, on the basis of the data assimilation procedure. It is proposed here to study issues in ensemble prediction on the basis of different initial-time pdf descriptions, as they result from the data assimilation experiments of question (i), both from theoretical and practical perspectives (also in collaboration with scientists at the National Center for Atmospheric Research (NCAR) and at ECMWF). These issues in ensemble prediction concern approximations of the initial-time pdf (through, e.g., singular vectors (SVs)), the use of stochastic- dynamic equations, the trade-off between number of SVs and number of integrations, nonlinear corrections to the tangent-linear approximation, as well as issues related to model error. It is planned to study these questions primarily in the context of a quasigeostrophic model, with extensions to a primitive-equation model. In both questions, collaboration with scientists at ECMWF and NCAR is planned, using experience from the preceding project. To carry out the proposed research, funds at a total amount of ATS 2.435.000,(for two graduate students, equipment, material, and travel) are requested for the project duration of 3 years.