## Abstract

Four-dimensional variational data assimilation (4D-Var) seeks to find an optimal initial field that minimizes a cost function defined as the squared distance between model solutions and observations within an assimilation window. For a perfect linear model, Lorenc showed that the 4D-Var forecast at the end of the window coincides with a Kalman filter analysis if two conditions are fulfilled: (a) addition to the cost function of a term that measures the distance to the background at the beginning of the assimilation window, and (b) use of the Kalman filter background error covariance in this term. The standard 4D-Var requires minimization algorithms along with adjoint models to compute gradient information needed for the minimization. In this study, an alternative method is suggested based on the use of the quasi-inverse model that, for certain applications, may help accelerate the solution of problems close to 4D-Var.

The quasi-inverse approach for the forecast sensitivity problem is introduced, and then a closely related variational assimilation problem using the quasi-inverse model is formulated (i.e., the model is integrated backward but changing the sign of the dissipation terms). It is shown that if the cost function has no background term, and has a complete set of observations (as assumed in many classical 4D-Var papers), the new method solves the 4D-Var-minimization problem efficiently, and is in fact equivalent to the Newton algorithm but without having to compute a Hessian. If the background term is included but computed at the end of the interval, allowing the use of observations that are not complete, the minimization can still be carried out very efficiently. In this case, however, the method is much closer to a 3D-Var formulation in which the analysis is attained through a model integration. For this reason, the method is called “inverse 3D-Var” (I3D-Var).

The I3D-Var method was applied to simple models (viscous Burgers’ equation and Lorenz model), and it was found that when the background term is ignored and complete fields of noisy observations are available at multiple times, the inverse 3D-Var method minimizes the same cost function as 4D-Var but converges much faster. Tests with the Advanced Regional Prediction System (ARPS) indicate that I3D-Var is about twice as fast as the adjoint Newton method and many times faster than the quasi-Newton LBFGS algorithm, which uses the adjoint model. Potential problems (including the growth of random errors during the integration back in time) and possible applications to preconditioning, and to problems such as storm-scale data assimilation and reanalysis are also discussed.

* Current affiliation: Department of Meteorology, University of Maryland, College Park, Maryland.

*Corresponding author address:* Prof. Eugenia Kalnay, Department of Meteorology, University of Maryland 3431, Computer and Space Sciences Building, College Park, MD 20742-2425.

Email: ekalnay@atmos.umd.edu