Data assimilation strategies for state‐dependent observation error variances

Abstract

The Ensemble Kalman Filter (EnKF) and 4D‐Var Data Assimilation (DA) approaches require that a fixed observation error variance be specified for each observation. To highlight the need to consider the state dependence of observation error variances, we prove that the error variance of unbiased observations of bounded variables tends to zero as the unknown true value of the variable approaches the bound. How then, should state‐dependent observation error variances be specified for the EnKF and 4D‐Var? In an idealized system, three distinct strategies for choosing the observation error variance R are considered: (a) choose R to be the ensemble mean of the observation error variances associated with each member of an ensemble forecast, (b) choose R to be the observation error variance that would occur if the truth was equal to the average of the ensemble mean and the observed value, or (c) (impractically) choose R to be the true observation error variance associated with the (unknown) true state. It is shown that choice (c) is the worst choice while (a) is the best choice. It is then shown that the Kalman gain of the EnKF is the best linear unbiased estimator of the state only when its R is the mean of all the observation error variances implied by the prior distribution of truth. This is a general result that supports the idealized experiment's findings. Because EnKF and 4D‐Var Gaussian assumptions are grossly inaccurate for near zero semi‐positive‐definite variables with state‐dependent R, the article also compares the performances of two variations of the EnKF (the ln‐EnKF and the GIGG‐EnKF) that correct aspects of these inaccuracies including the tendency of R to diminish as the truth approaches zero. It was found that the GIGG‐EnKF out‐performs the ln‐EnKF by a significant margin.