LINEARIZING ITERATIVE PROCESSES FOR 4-DIMENSIONAL DATA-ASSIMILATION SCHEMES

The tangent linear model is used in applications including Kalman filt ers and the growth of perturbations. It is also used to define adjoint models in applications such as four-dimensional variational assimilat ion or sensitivity studies. The validity of the tangent linear model f or all of these applications is determined by the period of time and i nitial amplitudes for which perturbation growth remains linear. In thi s 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 al so requires all iterates of the nonlinear process. The tangent lineari zation can also be invalid if too few iterations of the nonlinear proc ess are taken. An approximate linearization which requires only the la st iterate of the nonlinear process was found to be accurate when the correct Linearization has converged. Application of these results to i terative processes occurring in a numerical weather-prediction model r eveals that the approximate linearization can be effective. Specifical ly, 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 iter ative solution of an elliptic equation, the maximum number of iteratio ns allowed by the model was sufficient for both the nonlinear and line arized iterations. Thus the approximate linearization could be used to save CPU time or memory space.