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Abstract In many applications of data assimilation, especially when the size of the problem is large, a substantial assumption is made: all variables are well‐described by Gaussian error statistics. This assumption has the advantage of making calculations considerably simpler, but it is often not valid, leading to biases in forecasts or, even worse, unphysical predictions. We propose a simple, but effective, way of replacing this assumption, by making use of transforming functions, while remaining consistent with Bayes' theorem. This method allows the errors to have any value of the skewness and kurtosis, and permits physical bounds for the variables. As such, the error distribution can conform better to the underlying statistics, reducing biases introduced by the Gaussian assumption. We apply this framework to a 3D variational data assimilation method, and find improved performance in a simple atmospheric toy model (Lorenz‐63), compared to an all‐Gaussian technique. 

Abstract We derive a generalization of the Kalman filter that allows for non‐Gaussian background and observation errors. The Gaussian assumption is replaced by considering that the errors come from a mixed distribution of Gaussian, lognormal, and reverse lognormal random variables. We detail the derivation for reverse lognormal errors and extend the results to mixed distributions, where the number of Gaussian, lognormal, and reverse lognormal state variables can change dynamically every analysis time. We test the dynamical mixed Kalman filter robustly on two different systems based on the Lorenz 1963 model, and demonstrate that non‐Gaussian techniques generally improve the analysis skill if the observations are sparse and uncertain, compared with the Gaussian Kalman filter. 

Abstract In this paper we present the derivation of two new forms of the Kalman filter equations; the first is for a pure lognormally distributed random variable, while the second set of Kalman filter equations will be for a combination of Gaussian and lognormally distributed random variables. We show that the appearance is similar to that of the Gaussian-based equations, but that the analysis state is a multivariate median and not the mean. We also show results of the mixed distribution Kalman filter with the Lorenz 1963 model with lognormal errors for the background and observations of the z component, and compare them to analysis results from a traditional Gaussian-based extended Kalman filter and show that under certain circumstances the new approach produces more accurate results.