Treating Nonlinearities in Data-Space Variational Assimilation

Abstract

One goal of four-dimensional variational (4D-Var) state estimation is to utilize the longest time window that maximizes the observational constraints to improve predictive skill; unfortunately, nonlinearities are present in geophysical flows and limit the time in which the linear approximation is valid. For weakly nonlinear flows, updating the background trajectory, relinearizing, and repeating the minimization is a way to lengthen the time window. This so called “outer-loop” requires special consideration when minimizing the solution in data-space. This discussion provides a review of the relevant theory and presents two data-space cost functions: the standard cost-function that becomes unconstrained during additional outer-loops and a modified function that preserves the original constraint. Experiments with the Lorenz (J Atmos Sci 20:130–141, 1963) model show that unconstrained outer-loops perform similarly to sequentially applied 3D-Var assimilations by overfitting the observations and producing state estimates with poor predictive skill. Evaluating the posterior error covariances, the analysis error, and minimum cost function illustrate how overfitting degrades the solution. This is an important lesson for assimilation schemes: minimizing the model data residuals without proper constraint does not provide the optimal solution. By properly constraining the data-space outer-loop, adjoint-based methods will be well constrained over time windows that are longer than those required by linearity.