Gaussian quadrature 4D‐Var

Abstract

AbstractA new incremental four‐dimensional variational (4D‐Var) data assimilation algorithm is introduced. The algorithm does not require the computationally expensive integrations with the nonlinear model in the outer loops. Nonlinearity is accounted for by modifying the linearization trajectory of the observation operator based on integrations with the tangent linear (TL) model. This allows us to update the linearization trajectory of the observation operator in the inner loops at negligible computational cost. As a result the distinction between inner and outer loops is no longer necessary.The key idea on which the proposed 4D‐Var method is based is that by using Gaussian quadrature it is possible to get an exact correspondence between the nonlinear time evolution of perturbations and the time evolution in the TL model. It is shown that J‐point Gaussian quadrature can be used to derive the exact adjoint‐based observation impact equations and furthermore that it is straightforward to account for the effect of multiple outer loops in these equations if the proposed 4D‐Var method is used. The method is illustrated using a three‐level quasi‐geostrophic model and the Lorenz (1996) model.