## 4 Results and Analysis  
### Spatial Distributions of TWS  
We recover the daily TWS variations across the CONUS with a \(1^{\circ}\) grid using three inversion strategies and evaluate their spatial patterns by comparing them to the patterns from the GFO-TWS, GLDAS, and precipitation dataset. We fit the daily TWS time series with Equation (1) using the least squares method at each grid cell and obtain the annual amplitudes. We select the following set of parameters to illustrate the spatial constraint inversion performance: \(r\) = 300 and \(e\) = 60 (SC30060), as they are one of the parameter sets that exhibit good performance in the closed-loop simulation (Table 2). As shown in Figure 7, the spatial distributions of the annual amplitude from various data sources are consistent over many regions. In general, the CC-TWS exhibits the largest amplitude, the SC-TWS displays the smallest amplitude, and the LC-TWS approximately falls between the two in terms of amplitude. The amplitudes of the annual terms in the CC-TWS and LC-TWS exhibit a rougher texture in the spatial domain, resulting in larger amplitudes. Conversely, the SC-TWS shows a smoother spatial distribution of annual amplitudes. The strong signals are mainly distributed in the watersheds of lower Mississippi, the Pacific Northwest, and California. We investigate the spatial distributions of the TWS for \(r\) and \(e\) listed in Table 2 for spatial constraint inversion. As shown in Figure 8, the TWS exhibits consistent spatial distributions across various parameter settings. This result implies that the spatial constraint inversion is less sensitive to the changes in the parameters of r and e, it can provide us a robust inversion with a more flexible range of parameter combinations. Figure 7: Spatial distributions of the GNSS-TWS, GFO-TWS, GLDAS-TWS, and average annual precipitation. Panels (**a**,**b**) represent CC-TWS and LC-TWS, respectively. Panel (**c**) represents TWS from spatial constraints with the parameters \(r\) = 300 km and \(e\) = 60 mm. Panels (**d**–**f**) represent the GFO-TWS, GLDAS-TWS, and average annual precipitation, respectively. The spatial distribution of the annual amplitudes from the LC-TWS and SC-TWS presents some discrepancies in several regions compared with the GFO, GLDAS, and precipitation (Figure 7), e.g., in the Rio Grande watershed region, we observe apparent TWS signals in the inversion using spatial constraints, but those signals are not seen in the LC-TWS and CC-TWS, and neither GFO nor GLDAS show similar signal patterns. We speculate that this discrepancy could be due to the sparse GNSS stations in this region, and therefore the spatial constraints might introduce some unrealistic signals. In addition, we find evident discrepancies in the watersheds of New England and South Atlantic-Gulf, which may be attributed to the local climate characteristics; another reason could be that GNSS is more sensitive to local mass loading, while the GFO is sensitive to large-scale mass changes. We compare the TWS variations with precipitation (Figure 7). Precipitation mainly concentrates in the southeast and northwest of the CONUS; the average annual precipitation total is 1000-2500 mm. The large oscillations of the GLDAS, GFO, and GNSS-TWS in the  
Figure 8: Spatial distributions of TWS when \(r\) and \(e\) take different values for the spatial constraints, the values of \(r\) and \(e\) are listed at the bottom left of each panel. (**a**) represents parameters \(r\) = 150 km and \(e\) = 45 mm, (**b**) represents parameters \(r\) = 200 km and \(e\) = 45 mm, (**c**) represents parameters \(r\) = 250 km and \(e\) = 55 mm, (**d**) represents parameters \(r\) = 300 km and \(e\) = 60 mm, (**e**) represents parameters \(r\) = 350 km and \(e\) = 65 mm, (**f**) represents parameters \(r\) = 400 km and \(e\) = 60 mm, (**g**) represents parameters \(r\) = 450 km and \(e\) = 75 mm, (**h**) represents parameters \(r\) = 500 km and \(e\) = 75 mm. Pacific Northwest watershed region are caused by abundant precipitation.