# The challenge of bounded, non-Gaussian, non-linear and multi-scale variables

Current state estimation or data assimilation techniques assume Gaussian uncertainties for both forecasts and observations. However, unbiased observations of bounded variables can be shown to have highly non-Gaussian uncertainties and observation error standard deviations that depend on the value of the unknown true state. In particular, the observation error variance of such observations must tend to zero as the unknown true state tends to zero. Furthermore, observed bounded variables such as wind-speed, clouds, precipitation, fire and ice are highly non-linear functions of mass, momentum, and moisture variables. To simultaneously address both issues, we extend the previously developed Gamma, Inverse-Gamma and Gaussian (GIGG) variation on the Ensemble Kalman Filter (EnKF) to better account for non-linearity. Specifically, we augment the linear regression used by EnKFs to update model state variables from observed variables with a local, low-dimensional, non-linear variational step. In a synthetic Tropical Cyclone wind energy assimilation problem, the approach is shown to profoundly reduce the analysis errors associated with Tropical Cyclone wind-vector fields. The talk also summarizes some other recent improvements to the GIGG filter. In addition, we discuss challenges associated with incorporating features of the non-linear GIGG filter into global variational data assimilation frameworks such as 4DVar.