Ensemble Riemannian Data Assimilation over the Wasserstein Space

In this paper, we present a new ensemble data assimilation paradigm over a Riemannian manifold equipped with the Wasserstein metric. Unlike the Eulerian penalization of error in the Euclidean space, the Wasserstein metric can capture translation and difference between the shapes of square-integrable probability distributions of the background state and observations -- enabling to formally penalize geophysical biases in a state-space with non-Gaussian distribution. The new approach is applied to dissipative and chaotic evolutionary dynamics and its advantages over classic variational and filtering techniques are documented under systematic and random errors.