Application of the quasi-inverse method to data assimilation

Four-dimensional variational data assimilation (4D-Var) seeks to find an op timal initial field that minimizes a cost function defined as the squared d istance between model solutions and observations within an assimilation win dow. 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 condit ions are fulfilled: (a) addition to the cost function of a term that measur es the distance to the background at the beginning of the assimilation wind ow, and (b) use of the Kalman filter background error covariance in this te rm. The standard 4D-Var requires minimization algorithms along with adjoint models to compute gradient information needed for the minimization. In thi s study, an alternative method is suggested based on the use of the quasi-i nverse model that, for certain applications, may help accelerate the soluti on of problems close to 4D-Var. The quasi-inverse approach for the forecast sensitivity problem is introduc ed, and then a closely related variational assimilation problem using the q uasi-inverse model is formulated (i.e., the model is integrated back ward b ut changing the sign of the dissipation terms). It is shown that if the cos t 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-V ar-minimization problem efficiently, and is in fact equivalent to the Newto n 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 o bservations that are not complete, the minimization can still be carried ou t 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 integrat ion. For this reason, the method is called "inverse 3D-Var" (I3D-Var). The I3D-Var method was applied to simple models (viscous Purgers' equation and Lorcnz model), and it was Found that when the background term is ignore d 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 me thod and many times faster than the quasi-Newton LBFGS algorithm, which use s the adjoint model. Potential problems (including the growth of random err ors during the integration back in time) and possible applications to preco nditioning, and to problems such as storm-scale data assimilation and reana lysis are also discussed.