Computation of observation sensitivity and observation impact in incremental variational data assimilation

We discuss the computation of observation sensitivities and observation impact for incremental variational data assimilation (VDA), accounting for the inner and outer loops. To fully account for the outer loops, a second-order adjoint of the data assimilation system is required, which makes it impractical for an operational data assimilation system. However, some approximations can be made that allow useful results to be obtained with multiple outer loop iterations, in particular, for observation impact studies. Two algorithms are presented to compute the adjoint of the inner loop minimization, and their merits are discussed. Validation results are given for both of these algorithms. We show that one algorithm, based on the adjoint of an approximation of the inverse of the Hessian of the cost function, can also be used to investigate some convergence aspects of the incremental VDA inner loop. Because it is computationally inexpensive, the proposed algorithm could be used to monitor an operational system routinely. We give some numerical results illustrating the impact of observations in successive outer loop iterations.