Linearizing iterative processes for four‐dimensional data‐assimilation schemes

The tangent linear model is used in applications including Kalman filters and the growth of perturbations. It is also used to define adjoint models in applications such as four‐dimensional variational assimilation or sensitivity studies. The validity of the tangent linear model for all of these applications is determined by the period of time and initial amplitudes for which perturbation growth remains linear. In this work we examine the validity of various linearizations of a class of iterations known as fixed‐point iterations. The tangent linearization is found to converge more slowly than the nonlinear iteration, and also requires all iterates of the nonlinear process. The tangent linearization can also be invalid if too few iterations of the nonlinear process are taken. An approximate linearization which requires only the last iterate of the nonlinear process was found to be accurate when the correct linearization has converged. Application of these results to iterative processes occurring in a numerical weather‐prediction model reveals that the approximate linearization can be effective. Specifically, the linearization of the iteration for the mid‐trajectory position for semi‐Lagrangian advection was found to converge more slowly than the nonlinear iteration, for our choice of initial guess. For the iterative solution of an elliptic equation, the maximum number of iterations allowed by the model was sufficient for both the nonlinear and linearized iterations. Thus the approximate linearization could be used to save CPU time or memory space.