Exponential stability for a forecast-assimilation process with unstable dynamics

Area of Science:

• Data assimilation
• Computational mathematics
• Dynamical systems

Background:

• Data assimilation integrates observational data into computational models for accurate forecasting.
• Numerical weather prediction relies heavily on data assimilation processes.
• The stability of these processes is critical, especially when system dynamics are unstable.

Purpose of the Study:

• To conceptualize the forecast-assimilation (FA) process as a dynamic-stochastic system.
• To investigate the stability of the FA process concerning initial condition variations.
• To determine conditions for FA process stability under linear and nonlinear dynamics.

Main Methods:

• Analysis of linear and nonlinear dynamic-stochastic systems.
• Application of an exponential semi-group for nonlinear dynamics analysis.
• Utilizing the Kallianpur-Striebel formula for linear dynamics analysis.

Main Results:

• Identified conditions for FA process stability under linear and nonlinear dynamics.
• Proved a uniform in time bound on the expected Wasserstein distance for nonlinear dynamics.
• Demonstrated weak and Wasserstein topology convergence for linear dynamics.

Conclusions:

• The FA process can remain stable despite unstable dynamics and initial condition errors.
• The Wasserstein distance between correctly and incorrectly initialized FA processes converges exponentially fast under specific conditions.
• This research provides a rigorous framework for understanding FA process stability.