Nonlinearity in Data Assimilation Applications: A Practical Method for Analysis

A new method to quantify the nonlinearity of data assimilation problems is proposed. The method includes the effects of system errors, measurement errors, observational network, and sampling interval. It is based on computation of the first neglected term in a ‘‘Taylor’’ series expansion of the errors introduced by an extended Kalman filter, and can be computed at very little cost when one is already applying a second-order (or higher order) Kalman filter or an ensemble Kalman filter. The nonlinearity measure proposed here can be used to classify the ‘‘hardness’’ of the problem and predict the failure of data assimilation algorithms. In this manner it facilitates the comparison of data assimilation algorithms and applications. The method is applied to the well-known Lorenz model. A comparison is made between several data assimilation algorithms that are suitable for nonlinear problems. The results indicate significant differences in performance for more nonlinear problems. For low values of V, a measure of nonlinearity, the differences are negligible.