Variational Data-Consistent Assimilation

TL;DR

This paper proposes a novel data-consistent 4D-Var data assimilation method that incorporates predictability-aware regularization, improving estimation accuracy and robustness in nonlinear dynamical systems with modest computational overhead.

Contribution

It introduces the DC-WME 4D-Var method, combining data-consistent inversion theory with a weighted mean error map to enhance traditional 4D-Var performance.

Findings

Outperforms standard 4D-Var in error and bias reduction

Maintains robustness under high observation noise

Offers practical scalability with modest computational cost

Abstract

This work introduces a new class of four-dimensional variational data assimilation (4D-Var) methods grounded in data-consistent inversion (DCI) theory. The methods extend classical 4D-Var by incorporating a predictability-aware regularization term. The first method formulated is referred to as Data-Consistent 4D-Var (DC-4DVar), which is then enhanced using a Weighted Mean Error (WME) quantity-of-interest map to construct the DC-WME 4D-Var method. While the DC and DC-WME cost functions both involve a predictability-aware regularization term, the DC-WME function includes a modification to the model-data misfit, thereby improving estimation accuracy, robustness, and theoretical consistency in nonlinear and partially observed dynamical systems. Proofs are provided that establish the existence and uniqueness of the minimizer and analyze how a predictability assumption that is common within the DCI framework helps to promote solution stability. Numerical experiments are presented on benchmark dynamical systems (Lorenz-63 and Lorenz-96) as well as for the shallow water equations (SWE). In the benchmark dynamical systems, the DC-WME 4D-Var formulation is shown to consistently outperform standard 4D-Var in reducing both error and bias while maintaining robustness under high observation noise and short assimilation windows. Despite introducing modest computational overhead, DC-WME 4D-Var delivers improvements in estimation performance and forecast skill, demonstrating its potential practicality and scalability for high-dimensional data assimilation problems.