We summarize the work of [1] to introduce a new class of four-dimensional variational data assimilation (4D-Var) methods grounded in data-consistent inversion (DCI) theory [2-4]. 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 further enhanced using a Weighted Mean Error (WME) quantity-of-interest map to construct the DC-WME 4D-Var method. While theDC 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. These methods are rigorously analyzed as we establish the theoretical 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. REFERENCES [1] Spence, R., Butler, T., Dawson, C., Variational Data-Consistent Assimilation, https://arxiv.org/abs/2511.01759 (in review) [2] Butler, T., Jakeman, J., Wildey, T., Combining push-forward measures and Bayes’ rule to construct consistent solutions to stochastic inverse problems, SIAM Journal on Scientific Computing, 40 (2), A984-A1011, 2018 [3] Pilosov, M. del Castillo-Negrete, C., Yen, T.Y., Butler, T., Dawson, C., Parameter estimation with maximal updated densities, Computer Methods in Applied Mechanics and Engineering, 407, 115906, 2023 [4] del Castillo-Negrete, C., Spence, R., Butler, T., Dawson, C., Sequential maximal updated density parameter estimation for dynamical systems with parameter drift, International Journal for Numerical Methods in Engineering, 126 (3), e7618, 2025