## 5 Discussion and Conclusions
The improvement in GNSS data processing and the enrichment of the GNSS inversion methods will better serve meteorological and hydrological management departments in the future. We propose a new inversion method with flexible spatial constraints for daily TWS inversion using the GNSS vertical displacements from January 2018 to August 2022 across the CONUS in this study. Our closed-loop simulation demonstrates that the spatial constraint inversion has an advantage in recovering the annual oscillation amplitude of the TWS. Additionally, the TWS time series from spatial constraints are closer to the input signals. We use conventional, Laplacian, and spatial constraints to investigate the spatiotemporal distributions of the TWS across the CONUS and compare them with the GFO-TWS and GLDAS-TWS. The results show that the spatial distributions of the annual amplitude from different constraint strategies are consistent with those of the GFO and GLDAS over many regions, but still present some discrepancies in the regions with sparse GNSS stations. The temporal variations in GNSS-TWS from different constraint strategies show good consistency with GFO and GLDAS, and the STDs between SC-TWS, LC-TWS and GFO are at the same level. The spatial constraint inversion proposed in this study can be an alternative tool for TWS inversion. Conceptualization, D.M. and T.X. ; methodology, D.M. ; software, P.Y. ; validation, P.Y. and D.M. ; formal analysis, P.Y. and D.M. ; investigation, P.Y. ; resources, D.M. ; writing--original draft preparation, P.Y. ; writing--review and editing, D.M. and P.Y. ; visualization, P.Y. ; supervision, T.X. ; project administration, D.M. and T.X. ; funding acquisition, D.M. and T.X. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (Grant No. 42192534 and 41904081). **Data Availability Statement**: GNSS vertical displacements time series are downloaded from the Nevada Geodetic Laboratory, available at [[http://geodesy.unr.edu/gps_timeseries/tenv/IGS14/](http://geodesy.unr.edu/gps_timeseries/tenv/IGS14/)]([http://geodesy.unr.edu/gps_timeseries/tenv/IGS14/](http://geodesy.unr.edu/gps_timeseries/tenv/IGS14/)) (accessed on 30 December 2020). Gravity Recovery and Climate Experiment Follow-On mason data are downloaded from the Center for Space Research website, available at [[https://www2.csr.utexas.edu/grace/RL06_macons.html](https://www2.csr.utexas.edu/grace/RL06_macons.html)]([https://www2.csr.utexas.edu/grace/RL06_macons.html](https://www2.csr.utexas.edu/grace/RL06_macons.html)) (accessed on 27 October 2022). Global Land Data Assistance System Noah datasets are available from Goddard Earth Sciences Data and Information Services Center: [[https://disc.gsfc.nasa.gov/datasets?keywords=GLDAS](https://disc.gsfc.nasa.gov/datasets?keywords=GLDAS)]([https://disc.gsfc.nasa.gov/datasets?keywords=GLDAS](https://disc.gsfc.nasa.gov/datasets?keywords=GLDAS)) (accessed on 30 December 2022). Meteorological precipitation products are available from NOAA's Physical Sciences Laboratory: [[https://pssl.noaa.gov/](https://pssl.noaa.gov/)]([https://pssl.noaa.gov/](https://pssl.noaa.gov/)) (accessed on 20 December 2022).