The GIGG-EnKF: Ensemble Kalman Filtering for highly skewed non-negative uncertainty distributions.

Craig H Bishop*
Naval Research Laboratory

Near-zero non-negative variables such as aerosol, water vapor, cloud, and precipitation have uncertainty distributions that are skewed and better approximated by gamma and inverse-gamma probability distribution functions (pdfs) than Gaussian pdfs. Current Ensemble Kalman Filters (EnKFs) yield suboptimal state estimates for these variables. Here, we introduce a variation on the EnKF that accurately solves Bayes’ theorem in univariate cases where the prior forecasts and error prone observations given truth come in (gamma, inverse-gamma) or (inverse-gamma, gamma) or (Gaussian, Gaussian) distribution pairs. Its multivariate extension is similar to an EnKF. We refer to it as the GIGG-EnKF or GIGG where GIGG stands for Gamma, Inverse-Gamma and Gaussian. The GIGG-EnKF enables highly skewed uncertainty distributions to be accurately handled without the need for observation bias inducing lognormal or Gaussian anamorphosis non-linear transformations. In the case that all observations are treated as Gaussian, the GIGG-EnKF gives identical results to the original EnKF. A multi-grid-point and multi-variable idealized system was used to compare and contrast the data assimilation performance of the GIGG with that of both the perturbed observation and deterministic forms of the EnKF. This test system features variable types whose uncertainty distributions approximate Gaussian, gamma and inverse-gamma distributions. The normalized analysis error variance of the GIGG ensemble mean was found to be significantly smaller than that of the EnKFs. The higher moments of the analyzed ensemble distributions were tested by subjecting the ensemble members to non-linear “forecast” mappings. The normalized mean square error of the mean of the corresponding GIGG forecast ensemble was found to be less than a 3rd of that obtained from either form of the original EnKF. It is also shown how a subsequent “ensemble of 4DVARs” type procedure allows the GIGG filter to produce remarkably accurate solutions to Bayes’ theorem even when the posterior distribution is multi-modal.

*Preference: Oral