Bridging classical data assimilation and optimal transport: the 3D-Var case

Abstract

Abstract. Because optimal transport (OT) acts as displacement interpolation in physical space rather than as interpolation in value space, it can avoid double-penalty errors generated by mislocations of geophysical fields. As such, it provides a very attractive metric for non-negative, sharp field comparison – the Wasserstein distance – which could further be used in data assimilation (DA) for the geosciences. However, the algorithmic and numerical implementations of such a distance are not straightforward. Moreover, its theoretical formulation within typical DA problems faces conceptual challenges, resulting in scarce contributions on the topic in the literature. We formulate the problem in a way that offers a unified view with respect to both classical DA and OT. The resulting OTDA framework accounts for both the classical source of prior errors, background and observation, and a Wasserstein barycentre in between states which are pre-images of the background state and observation vector. We show that the hybrid OTDA analysis can be decomposed as a simpler OTDA problem involving a single Wasserstein distance, followed by a Wasserstein barycentre problem that ignores the prior errors and can be seen as a McCann interpolant. We also propose a less enlightening but straightforward solution to the full OTDA problem, which includes the derivation of its analysis error covariance matrix. Thanks to these theoretical developments, we are able to extend the classical 3D-Var/BLUE (best linear unbiased estimator) paradigm at the core of most classical DA schemes. The resulting formalism is very flexible and can account for sparse, noisy observations and non-Gaussian error statistics. It is illustrated by simple one- and two-dimensional examples that show the richness of the new types of analysis offered by this unification.