## 5 Discussion and Conclusions  
The spatial constraints provide a valuable complement to the existing conventional and Laplacian constraint methods. The inversion results with spatial constraints are subject to two important parameters (i.e., the radius \(r\) and the threshold \(e\)). The parameter \(e\) determines to what extent the spatial pattern will be smoothed, and the parameter \(r\) affects the spatial scale that will be constrained. Our work suggests that there is no need to precisely estimate the optimal parameters for spatial constraints; instead, a rough range for the empirical parameters is sufficient. As shown in Figure 8, changes in these two factors do not necessarily cause dramatically different inversion results; for instance, the evident signal in the northwest region is similar for different combinations of \(r\) and \(e\). The spatial constraints allow us to employ a more flexible range of parameter selection, but still provide robust inversion results. The densely distributed continuous GNSS stations across the CONUS enable us to study land-water cycle processes with a high spatiotemporal resolution. The constraints are effective in most regions, and the spatiotemporal distribution of the TWS from the GNSS is consistent with those of the GFO and GLDAS. We illustrate the performance of spatial constraints using the six sets of parameters listed in Table 2 and compare the results with conventional and Laplacian constraints. As the spatial distribution of the TWS is not sensitive to changes in the two empirical parameters, the range of parameters for the spatial constraints is not limited to these six sets. The parameters \(r\) and \(e\) fluctuate within a certain range, and the TWS inversion results remain stable, allowing a broader range of \(r\) and \(e\) values. We can apply the spatial constraint inversion method with flexible parameter selection to other regions with a dense distribution of GNSS stations, such as Europe and Japan. The TWS variations inferred from the GNSS vertical displacements using the spatial constraint inversion method still face some challenges and limitations. The GNSS vertical displacement time series contains all the land motion signals, and the unmodelled  
Figure 10: Spatiotemporal distribution of precipitation and daily time series of GNSS-TWS and GLDAS-TWS in the California watershed in January 2019. (**a**) Spatial distribution of total precipitation. (**b**) Daily time series of precipitation. (**c**) Daily time series of GNSS-TWS and GLDAS-TWS. vertical deformation caused by thermal expansion of the monument [61] also affects the isolation of the hydrological loading, which is difficult to accurately separate. Our proposed method only applies flexible spatial constraints, but it cannot reduce the noise in the GNSS measurements. Without doubt, using a preprocessing strategy to denoise the GNSS measurements would improve the inversion not only using the spatial constraints but also using other methods. In addition, using the STDs to evaluate the inversion also has limitations, especially in the inversion with real GNSS data, because the true TWS signal is unknown. In our evaluation, we assume that the signals observed with GRACE/GFO are true; therefore, a small difference in GRACE/GFO would indicate a better result. However, we must recognize that, on the one hand, the TWS variations observed with GRACE/GFO suffer from a low resolution, and on the other hand, they reflect total variations, including the changes in groundwater, which may not be well observed by GNSS measurements. With the increase in global temperatures, droughts and floods are more frequent, severe, and longer lasting [62]. The use of GNSS observations for studying the loading deformation caused by extreme hydrological events, such as storms, hurricanes, and heavy precipitation, will become increasingly important. It is of great significance to conduct research on the inversion of the TWS.