Two approaches can be used to solve the variational data assimilation problem. The primal form corresponds to the 3D/4D-Var used now in many operational NWP centres. An alternative approach, called dual or 3D/4D-PSAS, consists in solving the problem in the dual of observation space. Both forms use the same basic operators so that once one method is developed, it should be possible to obtain the other easily provided these operators have a modular form. It has been shown that, with proper conditioning of the minimization problem, the two algorithms should have similar convergence rates and computational performances. In the presence of nonlinearities, the incremental form of 3D/4D-Var extends the equivalence to the so-called 3D/4D-PSAS. The first objective of this paper is to present results obtained with the variational data assimilation of the Meteorological Service of Canada to show the equivalence between the 3D-Var and the PSAS systems. This exercise has forced us to have a close look at the modularity of the operational 3D/4D-Var which then makes it possible to obtain the 3D-PSAS scheme. This paper then focuses on these two quadratic problems that show similar convergence rates. However, the minimization of 3D-PSAS is examined more thoroughly as some parameters are shown to be determining elements in the minimization process. Lastly, preconditioning properties are studied and the Hessians of the two problems are shown to be directly related to one another through their singular vectors, which makes it possible to cycle the Hessian of the PSAS form.

Presently, a preferred minimization for strong-constraint four-dimensional variational (4D-Var) assimilation uses a Lanczos-based conjugate gradient (CG) algorithm. This requires the availability of a square-root of the backgrounderror covariance matrix (B). In the context of weak-constraint 4D-Var, this requirement might be too restrictive for the formulations of the model error term. It might therefore be desirable to avoid a square-root decomposition of the augmented background term. An appealing minimization scheme is the double CG minimization employed, for example, in the grid-point statistical interpolation (GSI) analysis. Realizing the double CG algorithm is a special case of the more general bi-conjugate gradient (BiCG) method for solving non-symmetric problems, the present work introduces a Lanczos-based preconditioning strategy when B, instead of its square-root, is used initially. Implementation of the scheme is done in the context of the GSI analysis system, and preliminary experiments are presented using its 3D-Var version. Comparison of the Lanczos-based CG and the BiCG shows that the algorithms converge at the same rate and to the same solution. Despite the additional computational cost, the importance of the re-orthogonalization step is also shown to be fundamental to any of these CG algorithms. Furthermore, when using the Hessian eigenvectors for preconditioning, the BiCG behaviour is shown to be comparable to that of the Lanczos-CG algorithm. Both schemes construct the same approximation of the Hessian with the same number of eigenvectors, and benefit in the same way from the reduction of the condition number. The efficiency, computational cost, and stability of the three algorithms are discussed. Copyright

Satellite radiance observations combine global coverage with high temporal and spatial resolution, and bring vital information to NWP analyses especially in areas where conventional data are sparse. However, most satellite observations that are actively assimilated have been limited to clear-sky conditions due to difficulties associated with accounting for non-Gaussian error characteristics, nonlinearity, and the development of appropriate observation operators for cloud- and precipitation-affected satellite radiance data. This article provides an overview of the development of the Gridpoint Statistical Interpolation (GSI) configurations to assimilate all-sky data from microwave imagers such as the GPM Microwave Imager (GMI) in the NASA Goddard Earth Observing System (GEOS). Electromagnetic characteristics associated with their wavelengths allow microwave imager data to be highly sensitive to precipitation. Therefore, all-sky data assimilation efforts described in this study are primarily focused on utilizing these data in precipitating regions. To utilize data in cloudy and precipitating regions, state and analysis variables have been added for ice cloud, liquid cloud, rain, and snow. This required enhancing the observation operator to simulate radiances in heavy precipitation, including frozen precipitation. Background error covariances in both the central analysis and EnKF analysis in the GEOS hybrid 4D-EnVar system have been expanded to include hydrometeors. In addition, the bias correction scheme was enhanced to reduce biases associated with thick clouds and precipitation. The results from single observation experiments demonstrate the capability of assimilating all-sky microwave brightness temperature data in GEOS both when the model forecast produces excessive precipitation and too little precipitation. Additional experiments show that hydrometeors and dynamic variables such as winds and pressure are adjusted in physically consistent ways in response to the assimilation.