This paper presents the results of an ensemble data assimilation
methodology over the Wasserstein space for high-dimensional nonlinear
dynamical systems, focusing on the chaotic Lorenz-96 model and a
two-layer quasi-geostrophic model of atmospheric circulation. Unlike
Euclidean data assimilation, this approach is equipped with a Riemannian
geometry and formulates data assimilation as a Wasserstein barycenter
between the forecast probability distribution and the normalized
likelihood function. The methodology does not rely on any Gaussian
assumptions and can intrinsically treat systematic model and observation
errors. To cope with the computational cost of the Wasserstein distance,
the paper examines the efficiency of the entropic regularization.
Comparisons with the standard particle and stochastic ensemble Kalman
filters demonstrate that under systematic errors the presented
methodology could extend the forecast skills of nonlinear dynamical
systems.