Non‐Gaussian variational data assimilation with reverse‐lognormal errors

Abstract For the majority of data assimilation (DA) applications, a Gaussian assumption is made to model the behaviour of errors associated with the specific situation. This assumption is generally false in geoscience fields, especially for variables that are positive (semi‐)definite. The three‐dimensional variational (3DVar) data‐assimilation method was traditionally generated through Bayes' theorem with Gaussian multivariate probability density functions; however, over the last 15 years this assumption has been modified to allow for a lognormal distribution, as well as combining the Gaussian and lognormal distributions to form a mixed Gaussian–lognormal distribution. In this article, we adapt this assumption and allow these errors to be reverse‐lognormally distributed. This is done by applying the Bayesian probability framework to derive a reverse‐lognormal 3DVar cost function. With the new reverse‐lognormally distributed 3DVar, we use machine‐learning methods to detect which 3DVar method (Gaussian, lognormal, reverse‐lognormal) is suitable to minimise the errors in that region of the Lorenz 1963 model.