Second-order approximation to the 3DVAR cost function: application to analysis/forecast

The analysis/forecast cycling scheme used in operational weather prediction centers involves repeated application of a 3DVAR assimilation phase followed by a forecast phase. In this paper we first derive a closed-form expression for the complete Hessian of the 3DVAR objective function. This facilitates the use of Newton-like algorithms in the 3DVAR assimilation phase. It is also shown that this algorithm includes the standard first-order method as a special case. Notwithstanding the method used in this optimization phase, if the iterative optimization is terminated prematurely before convergence (generally for the sake of economy), the resulting analysis incurs an error that will corrupt the initial condition of the model and thereby affect the forecast. Using Lorenz’s maximum simplification form of the barotropic vorticity equation, we examine the assimilation error and its dependency on the following factors: observational and background error covariances and the optimization strategy.