  Using the recently developed Sinkhorn algorithm for approximating the
Wasserstein distance between probability distributions represented by Monte
Carlo samples, we demonstrate exponential filter stability of two commonly used
nonlinear filtering algorithms, namely, the particle filter and the ensemble
Kalman filter, for deterministic dynamical systems. We also establish
numerically a relation between filter stability and filter convergence by
showing that the Wasserstein distance between filters with two different
initial conditions is proportional to the bias or the RMSE of the filter.
