A Practical Method to Estimate Information Content in the Context of 4D-Var Data Assimilation. I: Methodology

Data assimilation obtains improved estimates of the state of a physical system by combining imperfect model results with sparse and noisy observations of reality. Not all observations used in data assimilation are equally valuable. The ability to characterize the usefulness of different data points is important for analyzing the effectiveness of the assimilation system, for data pruning, and for the design of future sensor systems. This paper focuses on the four dimensional variational (4D-Var) data assimilation framework. Metrics from information theory are used to quantify the contribution of observations to decreasing the uncertainty with which the system state is known. We establish an interesting relationship between different information-theoretic metrics and the variational cost function/gradient under Gaussian linear assumptions. Based on this insight we derive an ensemble-based computational procedure to estimate the information content of various observations in the context of 4D-Var. The approach is illustrated on linear and nonlinear test problems. In the companion paper [Singh et al.(2011)] the methodology is applied to a global chemical data assimilation problem.