## 5 Results and Discussion
### Gap-Filling of the TWSA Between GRACE and GRACE-FO  
The GAIN model is then applied to fill the gap between GRACE and GRACE-FO. The GAIN model consists of two primary components: a generator and a discriminator, both implemented as fully connected neural networks. The generator and discriminator each have three layers, with each layer consisting of 50 neurons. The training process employs key hyperparameters: a batch size of 28, a hint rate of 0.9, an alpha value of 100 for weighting the mean squared error term in the generator's loss function, and a total of 1000 training iterations. The model is trained adversarially, whereby, in each iteration, a mini-batch of data is sampled with missing values replaced by random noise, and the generator produces gap-filling values that are combined with observed data to form a \"completed\" dataset. The discriminator then attempts to distinguish between observed and gap-filling values, guided by the hint vector, while the generator and discriminator are alternately optimized using the Adam optimizer. Figure 6 illustrates the performance of three machine learning algorithms--GAIN, LSTM, and KNN--in filling missing values within the TWSA data, specifically addressing missing rates of 10%, 20%, and 30%. Figure 5: The mean TWSA from March 2022 to November 2022 in the PRB. In this analysis, artificial missingness is deliberately introduced into the FPI datasets to simulate real-world scenarios where data gaps often occur. The results of Figure 6a-c indicate that GAIN consistently outperforms both LSTM and KNN across all missingness rates, with a particularly marked advantage observed at higher missingness levels. This superior performance underscores GAIN's capability to effectively capture the underlying data patterns, thereby enhancing its reliability in maintaining data integrity during financial analyses. Overall, the findings suggest that GAIN represents a promising approach for filling missing values in TWSA datasets, offering significant advantages over traditional methods, especially in scenarios with substantial data gaps. This reinforces the importance of selecting advanced gap-filling techniques for robust data management and analysis in hydrological studies. Table 2 shows the performance of each algorithm in filling missing values. The analysis of RMSE improvements shows that the GAIN method consistently outperforms both LSTM and KNN across different missing data rates. At a 10% missing rate, GAIN achieves a 57.14% improvement over LSTM and a 73.27% improvement over KNN. For a 20% missing  
Figure 6: Different missing rate true and predicted value scatter plot, (**a**) 10% missing rate; (**b**) 20% missing rate, (**c**) 30% missing rate. rate, GAIN demonstrates a 46.83% improvement over LSTM and a 65.25% improvement over KNN. At a 30% missing rate, GAIN outperforms LSTM by 48.84% and KNN by 66.29%. These results underscore GAIN's effectiveness in reducing RMSE, offering a more reliable approach for handling missing data compared to LSTM and KNN. The effectiveness of GAIN imputation is compared against baseline algorithms, including LSTM and KNN, using both quantitative metrics and qualitative assessments. The findings demonstrate the superior capability of GAIN in filling missing TWSA data across the Pearl River Basin, particularly in terms of preserving data patterns and providing robust imputation across different missing data scenarios. The TWSA time series for the PRB is modeled and gap-filling using GAIN to fill the 33-month GRACE/FO data gap between 2002 and 2023. The original TWSA data, the GAIN-filling values, and the precipitation measurements are as shown in Figure 7.  
Figure 7 shows a consistent and smooth transition between the original TWSA data and the GAIN gap-filling values, suggesting that the imputation effectively fills in the missing data without introducing abrupt or unrealistic changes. The gap-filling data points are well-aligned with the surrounding trend, maintaining continuity and coherence in the TWSA pattern.