## 4 Results and Analysis
### Temporal Variation Features of TWS  
We assess the temporary variations in the TWS for all the watersheds (Figure 9). The watershed mean TWS variations are computed using a cosine latitude weighting scheme. We focus on the monthly mean changes; therefore, we average the daily time series into a monthly time series. The TWS variations in the CONUS vary significantly over time in different regions. Figure 9 shows the monthly TWS time series from January 2018 to August 2022 in the 18 watersheds and the CONUS. As shown in Figure 9, the time series variations in the CC-TWS, LC-TWS, and SC-TWS are generally consistent, indicating that the impact of different constraints strategies on the TWS time series is minor over regional scales. The temporal variations in the GNSS-TWS, GFO-TWS, and GLDAS-TWS are consistent, particularly in the regions with large oscillation amplitudes, such as the watersheds of lower Mississippi, the Pacific Northwest, and California. However, it is inconsistent in the Rio Grande watershed due to the fact that the GNSS stations are sparsely distributed.  
Figure 9: Time series of monthly GNSS-TWS, GFO-TWS, and GLDAS-TWS in the 18 watersheds and the CONUS. (**a**) represents the CONUS. (**b**) represents watersheds 1-9, identified by HUC-2. (**c**) represents watersheds 10-18, identified by HUC-2.  
We compare the annual amplitudes of the GNSS-TWS with the GFO-TWS and GLDAS-TWS to quantitatively evaluate the performance of the GNSS inversion strategies. The TWS time series mainly show seasonal variations; therefore, we fit the TWS time series using Equation (1) and obtain the amplitudes of the annual terms in the TWS time series for each watershed. As shown in Table 3, the CC-TWS, LC-TWS, and SC-TWS exhibit amplitudes that are comparable to those of the GFO-TWS and GLDAS-TWS in most of the watersheds. However, we find that the amplitudes of the annual term in the SC-TWS is nearly four times larger than the GFO-TWS in the Rio Grande watershed, probably due to the sparse distribution of GNSS stations. The amplitudes of the annual terms in the CC-TWS, LC-TWS, and SC-TWS are about 0.3 times larger than those in the GFO-TWS in the Great Basin watershed; this large discrepancy may be due to tectonic activity and climate types, and the GNSS vertical displacements may be contaminated by non-hydroological factors. We calculate the STDs of the time series between the GNSS-TWS and GFO-TWS in order to quantify the consistency of the TWS inverted from various constraints strategies and the GFO. As shown in Table 4, the STDs of the SC-TWS across 13 watersheds are smaller than those of the LC-TWS. The STDs of the SC-TWS and LC-TWS are at the same level, and the deviations are within 5 mm in the remaining five watersheds and the CONUS.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**HUC-2**} & \multicolumn{3}{c}{**STD (mm)**} \\ \cline{2-4}  & **CC** & **LC** & **SC30060** \\ \hline
1 & 69 & 70 & 65 \\
2 & 55 & 57 & 55 \\
3 & 51 & 53 & 55 \\
4 & 46 & 48 & 47 \\
5 & 40 & 38 & 39 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The STD between time series of GNSS-TWS and GFO-TWS. CC and LC represent conventional and Laplacian constraints, respectively. SC30060 represents the parameters \(r=300\) km and \(e=60\) mm for spatial constraints.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**HUC-2**} & \multicolumn{3}{c}{**Amplitude (mm)**} \\ \cline{2-5}  & **CC** & **LC** & **SC30060** & **GFO** & **GLDAS** \\ \hline  
CONUS & 22.28 \(\pm\) 5.78 & 21.55 \(\pm\) 6.09 & 21.13 \(\pm\) 6.10 & 60.65 \(\pm\) 4.55 & 51.61 \(\pm\) 3.03 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The amplitudes of the annual terms in GNSS-TWS, GFO-TWS, and GLDAS-TWS time series. CC and LC represent conventional and Laplacian constraints, respectively. SC30060 represents the parameters \(r=300\) km and \(e=60\) mm for spatial constraints.