## 1 Introduction  
Numerical weather prediction (NWP) serves as the cornerstone of global weather forecasting, with its accuracy highly dependent on the quality of initial conditions [1, 2]. The nonlinear dynamics of atmospheric systems mean that minor initial errors can exponentially amplify, resulting in substantial deviations in short-term and extreme weather event predictions [3, 4]. This sensitivity to initial conditions is a hallmark of chaotic systems that are often characterized by Lyapunov exponents, which quantify the rate at which infinitesimally close trajectories in phase space diverge [5]. A positive maximum Lyapunov exponent, the inverse of Lyapunov time, indicates exponentially growing sensitivity to perturbations, underscoring the necessity of precise initial conditions in enhancing forecast reliability. Additionally, the persistence of coherent structures in chaotic multi-attractor systems, such as the atmospheric-oceanic system, relates to the Kolmogorov-Arnold-Moser(KAM) theorem, which describes the stability of invariant tori in weakly perturbed Hamiltonian systems [6]. While the atmosphere is not a strictly Hamiltonian system, the presence of quasi-stationary regimes suggests that KAM theory may offer insights into large-scale flow predictability and the persistence of dominant weather patterns. Error accumulation becomes particularly critical in medium-range to long-range forecasts due to factors such as errors in initial conditions derived from sparse observations, limitations in physical process parameterization, inadequate representation of cross-scale interactions, constraints in vertical discretization, and inherent imperfections in theoretical modeling [7; 8]. These challenges underscore the need for advanced assimilation strategies that account for the dynamical constraints imposed by chaotic system behavior. Data assimilation (DA) is a critical technique for integrating short-range predictions with observational data to improve forecasting skills. As NWP models advance toward higher resolutions, the demand for high-quality initial conditions becomes increasingly pressing, further highlighting the crucial role of data assimilation in improving forecast performance. Consequently, optimizing data assimilation systems to minimize errors due to uncertainties in initial conditions and enhance the predictive accuracy of NWP remains a fundamental scientific challenge in the field [9]. DA seeks to construct the most accurate depiction of atmospheric or oceanic states by integrating all available observations and model information. In this context, the estimation of initial conditions from sparse and noisy observations in NWP can be regarded as an ill-posed inverse problem, and DA provides a framework to mitigate the difficulties arisen from the non-uniqueness, instability, and ill conditioning of such a problem [10; 11; 12; 13]. Among existing DA techniques, variational data assimilation (VarDA) stands out for its ability to directly incorporate unconventional observations, such as satellite and radar data, while maintaining physical and dynamical consistency with NWP models through nonlinear constraints. These attributes render VarDA indispensable for atmospheric state estimations in NWP, with three-dimensional variational assimilation (3D-Var) being particularly favored in operational rapid-update systems for its computational efficiency [14; 15; 16]. This method provides a correction of the model forecast by minimizing a cost function, integrating the background state, observations, and their Gaussian uncertainty statistics [17]. Operationally, the background state is derived from a short-term numerical forecast; thus, forecast errors are somehow approximate to background errors (the background state minus the 'true' state). However, the 'true' state, being empirically inaccessible, is instead represented by its statistical correlations, notably the background error covariance matrix (**B**-matrix), which inherently describes the probability distribution function of forecast errors.