What should an “outer loop” for ensemble data assimilation look like?

Incremental 4D-VAR data assimilation schemes minimize a non-linear log-likelihood cost-function using a Gauss-Newton method in which a series of linear minimization problems are linked together with a non-linear outer-loop. We show that this procedure is an approximation to an outer-loop procedure that naturally arises in ensemble-based state estimation when the ensemble is used to sample the distribution of historical forecasts given that the truth is equal to the best available guess of true state rather than the prior distribution of truth given earlier observations and the model. The approach is illustrated using an extremely simple univariate non-linear model and a somewhat more sophisticated non-linear model of solitons. These simple models are also used to highlight the incompatibility of the outer loop procedure with ensembles that have been designed to sample the prior distribution of truth given earlier observations and the model.