A comparison of two preconditioner algorithms within the representer-based four-dimensional variational data assimilation system for the Navy coastal ocean model

Four-dimensional variational (4D-Var) data assimilation for operational systems requires the solution of large linear systems that are poorly conditioned in general. In addition to efficient iterative solvers for linear systems, using a good preconditioner is required to guarantee an acceptable solution with a small number of iterations. We consider the assimilation of ocean observations using a weak constraints 4D-Var for the Navy Coastal Ocean Model (NCOM), based on the representer method. Two methods of preconditioning the linear system are implemented, namely the scaling of the linear system by the square root of the observations error variances, and the approximating stabilised representer matrix based on the computation of some representer functions. We evaluate their convergence using criteria such as the norm of the residuals and of the gradient of the cost function, the analysis error evaluated at the observation locations, and finally the convergence of the sequence of analyses. Results from all criteria show that the fastest convergence is achieved with the rescaling preconditioner when the conjugate gradient used to solve the linear system is equipped with a suitable inner product instead of the Euclidean.