Non-Gaussian based Maximum Likelihood Ensemble Smoothers

Recently the Maximum Likelihood Ensemble Filter (MLEF) has been extended to allow for lognormal as well as reverse lognormal errors that is based upon the non-Gaussian extension of the Kalman Filter. Therefore, the next logical step is to extend this to the Maximum Likelihood Ensemble Smoother (MLES). It has been shown recently that it is possible to express MLES as a 4D matrix equation that then uses the Hessian preconditioner of MLEF, as such it is possible to extend this formulation to the lognormal and reverse lognormal distributions. We shall show the derivations for these new ensemble smoothers and demonstrate their improvement in analysis error over the Gaussian fits all formulation with some toy problems.