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 Eulerian penalization of error in the Euclidean space, the Wasserstein metric can capture translation and shape difference between square integrable probability distributions of the background state and observations, enabling to formally penalize geophysical biases in a non-Gaussian state-space. The new approach is applied to dissipative and chaotic evolutionary dynamics with a wide range of applications in Earth system models. Its advantages over classic variational and particle filter techniques are documented under systematic errors and non-Gaussian state-spaces.