To what extent is your data assimilation scheme designed to find the posterior mean, the posterior mode or something else?

Recently there has been a surge in interest in coupling ensemble-based data assimilation methods with variational methods (commonly referred to as 4DVar). Here we discuss a number of important differences between ensemble-based and variational methods that ought to be considered when attempting to fuse these methods. We note that the Best Linear Unbiased Estimate (BLUE) of the posterior mean over a data assimilation window can only be delivered by data assimilation schemes that utilise the 4-dimensional (4D) forecast covariance of a prior distribution of non-linear forecasts across the data assimilation window. An ensemble Kalman smoother (EnKS) may be viewed as a BLUE approximating data assimilation scheme. In contrast, we use the dual form of 4DVar to show that the most likely non-linear trajectory corresponding to the posterior mode across a data assimilation window can only be delivered by data assimilation schemes that create counterparts of the 4D prior forecast covariance using a tangent linear model. Since 4DVar schemes have the required structural framework to identify posterior modes, in contrast to the EnKS, they may be viewed as mode approximating data assimilation schemes. Hence, when aspects of the EnKS and 4DVar data assimilation schemes are blended together in a hybrid, one would like to be able to understand how such changes would affect the mode- or mean-finding abilities of the data assimilation schemes. This article helps build such understanding using a series of simple examples. We argue that this understanding has important implications to both the interpretation of the hybrid state estimates and to their design.