Exponential stability for a forecast-assimilation process with unstable dynamics

Data assimilation, a vital process in areas such as numerical weather prediction, integrates observational data into computational models to provide accurate forecasts. In this study, we conceptualize the forecast-assimilation (FA) process as a dynamic-stochastic system driven by time-dependent observational data. The core objective is to investigate the stability of this process with respect to variations in its initial conditions, particularly when the underlying system dynamics, referred to here as the signal, exhibit instability. We provide a rigorous analysis for both linear and nonlinear dynamics to determine conditions under which the FA process remains stable. In the nonlinear case, we identify an exponential semi-group whose stability is used to prove a uniform in time bound on the expected Wasserstein distance between the true FA process and one that is incorrectly initialized. For linear dynamics, we prove that the FA process converges both weakly and in the Wasserstein topology to a "nominal" one. For this, we use a representation of the FA process by means of the classical Kallianpur-Striebel formula. We show that the Wasserstein distance between the FA process correctly initialized and one which is incorrectly initialized converges to 0 exponentially fast provided the wrong initial condition is absolutely continuous with respect to the correct initial condition.