## 5 Experimental Results  
### Baselines  
Our study aims to improve the consistency between the initial conditions and the numerical model dynamics by adaptively rescaling the static **B**-matrix within the VarDA framework, ultimately facilitating enhanced forecast performance without incurring additional computational costs. To achieve this, we compare our method against two representative baselines:  
1. Empirical variance rescaling by a constant factor (named CON for short): this method is the most widely used in operational VarDA systems, recognized for its high assimilation performance and straightforward implementation. Despite its lack of a rigorous theoretical foundation, extensive validation in operational environments has established it a practical and reliable benchmark. 2. An observation-dependent spatio-temporal update vector named CUTE reported by the authors of [31] (named CUTE-ST for short): CUTE-ST has been shown to outperform traditional adaptive techniques in both theoretical and practical evaluations. By leveraging background-observation error covariance within the existing VarDA cycle, it realizes spatio-temporal adaptivity in the **B**-matrix without introducing additional computation, rendering it an efficient and suitable baseline for comparison. While other advanced methods exist, they were excluded from our comparative analysis due to specific limitations. The LSTM-based approach proposed by [PERSON] et al. [39], although an innovative application of DL in DA as described in Section 2, is not directly applicable to the diagnosis of the **B**-matrix. Additionally, hybrid DA methods can enhance performance by integrating ensemble-based flow-dependent covariances with static background error covariances. However, their primary drawback is the substantial computational cost associated with a larger ensemble size. Given that our study focuses on achieving adaptive variances adjustment in the **B**-matrix without increasing computational burdens, these two alternative methods were not included as control baselines. The basic setup and introduction of the two baselines are presented below. \begin{table}
\begin{tabular}{l c l} \hline \hline
**Parameter** & **Value** & **Description** \\ \hline Scaling factor bounds \((l,u)\) & \((1\times 10^{-4},3.6)\) & Range for the action space in DRL-AST \\ Learning rate & \(7\times 10^{-4}\) & Adam optimizer learning rate \\ Batch size & 128 & Number of samples per training batch \\ PPO epochs & 4 & Number of PPO update steps per iteration \\ Total training steps & \(10^{7}\) & Total environment steps for training \\ Discount factor \((\gamma)\) & 0.998 & Future reward discount factor \\ GAE parameter \((\lambda)\) & 0.95 & Smoothing factor in GAE \\ PPO clip parameter \((\xi)\) & 0.2 & Clipping threshold in PPO \\ Entropy coefficient & 0.01 & Entropy regularization for exploration \\ Value loss coefficient & 0.5 & Weight of value function loss \\ Max gradient norm & 0.5 & Gradient clipping threshold \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Hyperparameters utilized in DRL-AST training. CON: this baseline meticulously selects a specified constant factor within the range of 0.05 to 3.16 through an extensive series of empirical experiments. It is imperative to underscore that in instances where the observational error attains a conspicuous magnitude indicative of its unreliability, the initial value should be set to 0. However, given that the weight is contingent upon the inverse of the variance, the utilization of 0 as the denominator is precluded. Consequently, our study employs a suitably diminutive value of 0.05. The upper limit is derived from the paradigm presented by the authors of [60], wherein the forgetting factor (\(\rho\), following the original notation therein) spans the interval of 0.1 to 1, thereby resulting in a variance inflation factor (\(\alpha=\rho^{-1/2}\)) that ranges from 1 to 3.16. To balance computational cost and resolution, the incremental step size for value traversal was set to 0.05. 2.