Asymptotic behavior of the forecast-assimilation process with unstable dynamics

TL;DR

This paper provides a rigorous mathematical explanation for how observation assimilation stabilizes unstable dynamics in systems like weather prediction, even with partial observations of unstable modes.

Contribution

It introduces a rigorous theoretical framework demonstrating stabilization effects of observation assimilation without requiring full observation of all system modes.

Findings

Assimilation stabilizes unstable dynamics.

Partial observations of unstable modes are sufficient for stabilization.

Theoretical results align with numerical evidence.

Abstract

Extensive numerical evidence shows that the assimilation of observations has a stabilizing effect on unstable dynamics, in numerical weather prediction and elsewhere. In this paper, we apply mathematically rigorous methods to showing why this is so. Our stabilization results do not assume a full set of observations and we provide examples where it suffices to observe the model's unstable degrees of freedom.