## 4 Approach
### DRL-Based Formulation for the VarDA System  
In this study, we reformulate the traditional VarDA system into an interactive environment compatible with DRL, allowing an agent to learn a policy for adaptively scaling the spatio-temporal variances of the **B**-matrix. This customized environment, referred to as the spatio-temporal variance-weighting environment for the VarDA system (abbreviated as VarDA W-ST), is illustrated in Figure 1, where the DA process is marked by circled steps 1-7, and the agent-environment interaction is indicated by steps (1)-(3). The system first estimates the static background error covariance matrix, **B\({}_{\text{NMC}}\)**, using the NMC method [29], which analyzes 500 forecast difference pairs following a 50-day spin-up period (1). Starting from the initial condition, the Lorenz-96 model is integrated forward for 6 h to obtain the background state at time \(t\), denoted as \(\mathbf{x}_{t}^{b}\) (2), which serves as the prior for the assimilation step. The first forecast is initialized with the predefined state from Equation (15), while subsequent forecasts use the analysis \(\mathbf{x}_{t}^{t}\) from the previous cycle as the initial condition. Observations \(\mathbf{y}_{t}^{o}\) are assimilated every 6 h by incorporating the innovation \(\mathbf{d}_{t}=\mathbf{y}^{o}-\mathcal{H}(\mathbf{x}_{t}^{b})\) into the VarDA system to correct the background state (3). To enable spatially adaptive variance scaling and improve numerical stability, **B\({}_{\text{NMC}}\)** is first normalized globally to yield **B\({}_{\text{norm}}\)** (4), which is then partitioned into **B\({}_{\text{norm}}^{(ck)}\)** for each spatial chunk \(ck=1,2,\ldots,CK\), where \(CK\) is the total number of chunks. Here, **B\({}_{\text{norm}}^{(ck)}\)** represents the portion of the globally normalized **B\({}_{\text{norm}}\)** corresponding to the \(ck\)-th spatial chunk. Such normalization is widely adopted in the VarDA system to eliminate spatially inhomogeneous variances, improve the conditioning of **B**, and ensure consistent variance tuning across variables and scales, thereby facilitating chunk-wise modulation strategies [19]. In step (1), the current analysis state \(s_{t}=\mathbf{x}_{t}^{a}\) is passed to the agent, which outputs a chunk-wise scaling vector \(\mathbf{W}=[w_{1},\ldots,w_{CK}]\) (step (2)). This action is then applied to **B\({}_{\text{norm}}^{(ck)}\)** to produce the rescaled matrix **B\({}_{\mathbf{W}}\)**, where **B\({}_{\mathbf{W}}^{(ck)}=w_{ck}\cdot\mathbf{B}_{\text{norm}}^{(ck)}\)** for each chunk \(ck=1,2,\ldots,CK\), and all chunk-wise scalings are performed simultaneously to form the complete **B\({}_{\mathbf{W}}\)** (5), step (3)). Initially, the scaling vector is randomly sampled. In later cycles, it is adaptively generated  
Figure 1: Schematic of the VarDA W-ST environment, which formalizes the adaptive variance rescaling process in a DRL-compatible framework. Circled numbers (1)-7 represent the sequential steps of the VarDA process, while blue arrows (steps (1)–(3)) indicate the agent–environment interaction loop. This process corresponds to the logic outlined in Algorithm 1.  
by the policy network trained using DRL-AST (see Section 4.4). The analysis state \(\mathbf{x}_{t}^{a}\) is then computed by the VarDA system ( 6 ) based on the background state, innovation, and the rescaled covariance matrix \(\mathbf{B}_{\mathbf{W}}\). This updated analysis \(\mathbf{x}_{t}^{a}\) is used as the initial condition for the next forecast ( 7 ), thereby continuing the sequential forecast-assimilation process. Building on this structure, we formalize the process of spatio-temporal variance rescaling--based on the static \(\mathbf{B}\)-matrix estimated via the NMC method--as an MDP defined by the following components:  
* **State \(s_{t}\)** represents the analysis state \(\mathbf{x}_{t}^{a}\) produced by the VarDA system at time step \(t\).