A review of operational methods of variational and ensemble-variational data assimilation

Variational and ensemble methods have been developed separately by various research and development groups and each brings its own benefits to data assimilation. In the last decade or so, various ways have been developed to combine these methods, especially with the aims of improving the background-error covariance matrices and of improving efficiency. The field has become confusing, even to many specialists, and so there is now a need to summarize the methods in order to show how they work, how they are related, what benefits they bring, why they have been developed, how they perform, and what improvements are pending. This article starts with a reminder of basic variational and ensemble techniques and shows how they can be combined to give the emerging ensemble-variational (EnVar) and hybrid methods. A key part of the article includes details of how localization is commonly represented. There has been a particular push to develop four-dimensional methods that are free of linearized forecast models. This article attempts to provide derivations of the formulations of most popular schemes. These are otherwise scattered throughout the literature or absent. It builds on the nomenclature used to distinguish between methods, and discusses further possible developments to the methods, including the representation of model error.