An approach to quantifying the efficiency of a Bayesian filter
Data assimilation is the Bayesian conditioning of uncertain model simulations on observations to reduce uncertainty about model states. In practice, it is common to make simplifying assumptions about the prior and posterior state distributions, and to employ approximations of the likelihood function, which can reduce the efficiency of the filter. We propose metrics that quantify how much of the uncertainty in a Bayesian posterior state distribution is due to (i) the observation operator, (ii) observation error, and (iii) approximations of Bayes' Law. Our approach uses discrete Shannon entropy to quantify uncertainty, and we define the utility of an observation (for reducing uncertainty about a model state) as the ratio of the mutual information between the state and observation to the entropy of the state prior. These metrics make it possible to analyze the efficiency of a proposed observation system and data assimilation strategy, and provide a way to examine the propagation of information through the dynamic system model. We demonstrate the procedure on the problem of estimating profile soil moisture from observations at the surface (top 5 cm). The results show that when synthetic observations of 5 cm soil moisture are assimilated into a three-layer model of soil hydrology, the ensemble Kalman filter does not use all of the information available in observations.