Data assimilation: Mathematical and statistical perspectives

Abstract

The bulk of this paper contains a concise mathematical overview of the subject of data assimilation, highlighting three primary ideas: (i) the standard optimization approaches of 3DVAR, 4DVAR and weak constraint 4DVAR are described and their interrelations explained; (ii) statistical analogues of these approaches are then introduced, leading to filtering (generalizing 3DVAR) and a form of smoothing (generalizing 4DVAR and weak constraint 4DVAR) and the optimization methods are shown to be maximum a posteriori estimators for the probability distributions implied by these statistical approaches; and (iii) by taking a general dynamical systems perspective on the subject it is shown that the incorporation of Lagrangian data can be handled by a straightforward extension of the preceding concepts. We argue that the smoothing approach to data assimilation, based on statistical analogues of 4DVAR and weak constraint 4DVAR, provides the optimal solution to the assimilation of space–time distributed data into a model. The optimal solution obtained is a probability distribution on the relevant class of functions (initial conditions or time-dependent solutions). The approach is a useful one in the first instance because it clarifies the notion of what is the optimal solution, thereby providing a benchmark against which existing approaches can be evaluated. In the longer term it also provides the potential for new methods to create ensembles of solutions to the model, incorporating the available data in an optimal fashion. Two examples are given illustrating this approach to data assimilation, both in the context of Lagrangian data, one based on statistical 4DVAR and the other on weak constraint statistical 4DVAR. The former is compared with the ensemble Kalman filter, which is thereby shown to be inaccurate in a variety of scenarios.