An approach to quantifying the efficiency of a Bayesian filter

Authors: NEARING, G - University Of Arizona; GUPTA, H - University Of Arizona; Crow, Wade; GONG, W - Beijing Normal University

Submitted to: Water Resources Research
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 3/2/2012
Publication Date: 4/26/2013
Publication URL: http://handle.nal.usda.gov/10113/60026
Citation: Nearing, G.S., Gupta, H.V., Crow, W.T., Gong, W. 2013. An approach to quantifying the efficiency of a Bayesian filter. Water Resources Research. 49(4):2164-2173. DOI: 10.1002/wrcr.20177.

Interpretive Summary: The best possible estimate of critical agricultural variables (e.g., crop yield and soil water availability) are typically based on the merger of independent information acquired from models, remote sensing retrievals and/or ground-based observations. In the geosciences, the merger of information acquired from models with information acquired from observations is generally referred to as "data assimilation." For most agricultural applications, there exists a range of possible data assimilation strategies. Techniques are needed to evaluate various approaches and determine which one integrates information most efficiently (and thus results in, for example, the best-possible estimate of crop yield and/or soil water availability for an agricultural drought monitoring system). This paper describes and applies a novel entropy-based tool for evaluating the performance of a data assimilation system and makes specific recommendations that will eventually improve our ability to accurately estimate agriculturally-relevant variables using remote sensing observations.

Technical Abstract: Data assimilation is defined as the Bayesian conditioning of uncertain model simulations on observations for the purpose of reducing uncertainty about model states. Practical data assimilation applications require that simplifying assumptions be made about the prior and posterior state distributions, and often employ approximations of the likelihood function. We propose a method to quantify the efficiency of these assumptions, and thereby measure the unused information in assimilated observations. Uncertainty is quantified as discrete Shannon entropy, and we define the efficiency of a filter as the difference between entropies of the prior and posterior (normalized by the mutual information between states and observations). Metrics are defined which measure the fraction of entropy in the data assimilation posterior due to: 1) injectivity of the mapping between states and observations, 2) noise in the observations, and 3) inefficiency of the data assimilation algorithm. These metrics sum to account for the entire entropy of the posterior and can be estimated by an observing system simulation experiment. They also provide a method to illustrate the propagation of information through the dynamic system model. This procedure was demonstrated on the problem of estimating profile soil moisture from observations at the surface (top 5 cm). When synthetic observations of 5 cm soil moisture were assimilated into a three-layer model of soil hydrology, it was found that the ensemble Kalman filter did not use all of the available information in observations.