Errors in Ensemble Kalman Smoother Estimates of Cloud Microphysical Parameters

Derek J. Posselt University of Michigan, Ann Arbor, Ann Arbor, Michigan

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Daniel Hodyss Naval Research Laboratory, Monterey, California

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Craig H. Bishop Naval Research Laboratory, Monterey, California

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Abstract

If forecast or observation error distributions are non-Gaussian, the true posterior mean and covariance depends on the distribution of observation errors and the observed values. The posterior distribution of analysis errors obtained from ensemble Kalman filters and smoothers is independent of observed values. Hence, the error in ensemble Kalman smoother (EnKS) state estimates is closely linked to the sensitivity of the true posterior to observed values. Here a Markov chain Monte Carlo (MCMC) algorithm is used to document the dependence of the errors in EnKS-based estimates of cloud microphysical parameters on observed values. It is shown that EnKS analysis distributions are grossly inaccurate for nonnegative microphysical parameters when parameter values are close to zero. Furthermore, numerical analysis is presented that shows that, by design, the posterior distributions given by EnKS and even nonlinear extensions of these smoothers approximate the average of all possible posterior analysis distributions associated with all possible observations given the prior. Multiple runs of the MCMC are made to approximate this distribution. This empirically derived average of Bayesian posterior analysis errors is shown to be qualitatively similar to the EnKS posterior. In this way, it is demonstrated that, in the presence of nonlinearity, EnKS algorithms do not estimate the true posterior error distribution given the specific values of the observations. Instead, they produce an error distribution that is consistent with an average of the true posterior variance, weighted by the probability of obtaining each possible observation. This seemingly subtle distinction gives rise to fundamental differences between the approximate EnKS posterior and the true Bayesian posterior distribution.

Corresponding author address: Derek J. Posselt, Department of Atmospheric, Oceanic, and Space Sciences, University of Michigan, 2455 Hayward St., Ann Arbor, MI 48109-2143. E-mail: dposselt@umich.edu

Abstract

If forecast or observation error distributions are non-Gaussian, the true posterior mean and covariance depends on the distribution of observation errors and the observed values. The posterior distribution of analysis errors obtained from ensemble Kalman filters and smoothers is independent of observed values. Hence, the error in ensemble Kalman smoother (EnKS) state estimates is closely linked to the sensitivity of the true posterior to observed values. Here a Markov chain Monte Carlo (MCMC) algorithm is used to document the dependence of the errors in EnKS-based estimates of cloud microphysical parameters on observed values. It is shown that EnKS analysis distributions are grossly inaccurate for nonnegative microphysical parameters when parameter values are close to zero. Furthermore, numerical analysis is presented that shows that, by design, the posterior distributions given by EnKS and even nonlinear extensions of these smoothers approximate the average of all possible posterior analysis distributions associated with all possible observations given the prior. Multiple runs of the MCMC are made to approximate this distribution. This empirically derived average of Bayesian posterior analysis errors is shown to be qualitatively similar to the EnKS posterior. In this way, it is demonstrated that, in the presence of nonlinearity, EnKS algorithms do not estimate the true posterior error distribution given the specific values of the observations. Instead, they produce an error distribution that is consistent with an average of the true posterior variance, weighted by the probability of obtaining each possible observation. This seemingly subtle distinction gives rise to fundamental differences between the approximate EnKS posterior and the true Bayesian posterior distribution.

Corresponding author address: Derek J. Posselt, Department of Atmospheric, Oceanic, and Space Sciences, University of Michigan, 2455 Hayward St., Ann Arbor, MI 48109-2143. E-mail: dposselt@umich.edu
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