A combination of Time-Variable Gravity Field Solutions from Multi-Satellite Datasets (1993&ndash;2024) via Least-Squares Collocation

Zhang, Lin; Shen, Yunzhong; Sneeuw, Nico; Saemian, Peyman; Ji, Kunpu; Chen, Qiujie; Wang, Fengwei

doi:10.5194/essd-2026-84

Preprints

https://doi.org/10.5194/essd-2026-84

Preprints

16 Mar 2026

| 16 Mar 2026

Status: this preprint is currently under review for the journal ESSD.

A combination of Time-Variable Gravity Field Solutions from Multi-Satellite Datasets (1993–2024) via Least-Squares Collocation

Lin Zhang, Yunzhong Shen, Nico Sneeuw, Peyman Saemian, Kunpu Ji, Qiujie Chen, and Fengwei Wang

Abstract. Time-variable gravity field solutions from GRACE and GRACE-FO have been successfully applied in hydrological and geophysical studies; however, inter- and intra-mission gaps and limited record length constrain their broader utility. Current approaches involve hydrometeorological-forced machine-learning reconstructions and satellite-tracking-observation combinations; however, the former is constrained by the accuracy and completeness of data inputs, while the latter requires additional filtering due to limited spectral sensitivity, resulting in filtering-dependent solutions. Both approaches neglect covariance information of observation noise and signal, precluding optimal solutions. To address these limitations, this study develops gapless monthly solutions up to degree/order 60 spanning January 1993 to December 2024 using constrained Least-Squares Collocation (LSC), which integrates combination and denoising processes of gravity field solutions without explicit filtering. LSC-based Combined Solutions (LSC-CS) integrates trends, annual and semi-annual variations, and non-seasonal signals from multi-satellite observations (GRACE/-FO, Low Earth Orbit satellites, and Satellite Laser Ranging) without external hydrometeorological inputs, while incorporating covariance matrices of observation errors and combined signals to optimally balance error reduction and signal preservation. Evaluation results indicate that LSC-CS significantly eliminates striping noise and high-degree coefficient noise while effectively preserving low-degree gravity signals (e.g., C20 and C30) and achieving high signal-to-noise ratios. Comparison with three reconstructed products (IGG-SLR-DORIS, RESDCAE, BNML) shows that LSC-CS achieves the lowest sea level budget misclosures, with reductions of 40 %, 2.9 %, and 49 %, respectively. Across 52 major basins, LSC-CS has the smallest water balance errors, with reductions of 4.6 %, 2.6 %, and 1.5 %, respectively. For Antarctic and Greenland ice sheet mass changes, LSC-CS closely match IMBIE estimates, with trend consistency improvements of 46.8 % and 32.7 % over IGG-SLR-DORIS and 48.6 % and 67.4 % over RESDCAE, respectively. The combined monthly gravity field solutions are available at https://zenodo.org/records/18543287 (Zhang et al., 2026).

Received: 30 Jan 2026 – Discussion started: 16 Mar 2026

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.

Download & links

Preprint (PDF, 3551 KB)

Supplement (1052 KB)

Download & links

Lin Zhang, Yunzhong Shen, Nico Sneeuw, Peyman Saemian, Kunpu Ji, Qiujie Chen, and Fengwei Wang

Status: final response (author comments only)

RC1:
'Comment on essd-2026-84', Anonymous Referee #1, 15 Apr 2026
This study presents gap-free monthly gravity field solutions up to degree and order 60 for January 1993-December 2024 using constrained least-squares collocation (LSC), which jointly performs solution combination and denoising without explicit filtering. The proposed product outperforms existing reconstructions based on hydrometeorological forcing and static spatial modes, yielding smaller sea-level budget misclosures, reduced basin-scale water-balance errors, and stronger agreement with IMBIE (Ice Sheet Mass Balance Inter-comparison Exercise) estimates. Overall, the work is meaningful, the methodology is robust, and the results are convincing. I therefore recommend acceptance after some revisions.
Line 75: The time span “1984 to 2022” should be revised to “1984 to 2023.”

Table S2 in the supplementary file presents complex pseudocode and is not visually clear; could the authors replace it with a flowchart to improve readability and better illustrate the workflow?

The labels/captions of Fig. 3(d) and Fig. 3(e) should be checked to confirm whether they are reversed.

Notation and abbreviations should be standardized throughout the manuscript (e.g., TVGF/TVGFS, d/o, SHC/SHCs, NSS/NSSs, capitalization of defined terms) to avoid ambiguity.

A short “Notation” box (or one supplementary table) listing core symbols (, , , , , , etc.) and their dimensions would improve readability..

Section titles could be revised for parallel style and precision (e.g., “Function Model” → “Functional Model”; “Result Evaluations” → “Results and Evaluation”).

Please report the analysis period consistently in all figure captions (e.g., 1994-2020, 2002-2024), as some captions currently list only metrics without the corresponding time span.

A quick language polish is recommended to fix recurrent minor issues, including subject-verb agreement (e.g., results indicates → results indicate) and number/possessive form (e.g., basin’s → basin-scale or basins’). These edits do not affect the methodology but will noticeably improve the manuscript’s professional quality.

One concern is why IGG-SLR-HYBRID was adopted in the combination, whereas IGG-SLR-DORIS was used in the comparison. Since these two models seem to be based on similar methodologies and exhibit comparable performance over 1994–2020, the authors may wish to clarify whether it would be sufficient to include only one of them.

For greater user convenience, the authors are encouraged to reorganize the combined spherical harmonic solutions into the standard monthly .gfc format adopted by the ICGEM website. Moreover, the current 1° gridded data do not include the corresponding uncertainty information or the associated latitude and longitude coordinates. It is therefore recommended that these gridded products be provided in NetCDF format. It would also be highly beneficial if the authors could provide 0.5° gridded data together with their associated uncertainty estimates, thereby offering users a wider range of choices.
Citation: https://doi.org/10.5194/essd-2026-84-RC1
- AC1:
  'Reply on RC1', Yunzhong Shen, 20 Apr 2026
  Thank you very much for your constructive comments. We will carefully revise the manuscript according to your comments. We would like to explain some of the concerns you raised here.
  We have revised the time span to “1984 to 2023”.
  
  We will replace the complex pseudocode in Table S2 of the Supplementary File with a clear flowchart to improve readability and better illustrate the workflow.
  
  Thank you for your careful comment! We have checked Fig. 3(d) and Fig. 3(e) and reversed the labels.
  
  We have standardized the abbreviations throughout the manuscript.
  
  We have added a short “Notation” box in the supplementary material, listing the core symbols and their dimensions to improve readability.
  
  Thank you for your detailed suggestion! We have revised the section titles to ensure a parallel style and improve precision throughout the manuscript.
  
  We have added the analysis period for all figure captions.
  
  Thank you for your helpful suggestions. We have thoroughly proofread the manuscript and corrected the recurring minor language issues, including subject-verb agreement and number/possessive forms, to improve the overall professional quality of the paper.
  
  Thank you for your constructive comments. The difference between IGG-SLR-HYBRID and IGG-SLR-DORIS is that the latter additionally includes Low Earth Orbit (LEO) observations. In our combination framework, LEO information has already been incorporated through the Tongji-LEO2021 model. Therefore, to avoid re-introducing LEO information (and potential redundancy), we only used IGG-SLR-HYBRID in the combination step to isolate the SLR contribution.
  
  For the comparison experiment, we used IGG-SLR-DORIS because both IGG-SLR-DORIS and our combined model contain GRACE, SLR, and LEO information, which ensures a fair comparison. We will clarify this rationale in the revised manuscript.
  Thank you for your valuable comments! For greater user convenience, we will provide the combined spherical harmonic solutions in the standard monthly .gfc format. We will also provide the gridded products in NetCDF format, including latitude and longitude coordinates and corresponding uncertainty information, at both 1° and 0.5° The updated datasets will be uploaded to the Zenodo repository provided in this paper soon.
  
  Citation: https://doi.org/10.5194/essd-2026-84-AC1
RC2:
'Comment on essd-2026-84', Anonymous Referee #2, 29 Apr 2026
The authors generate a long-term TWSA dataset by combining different gravity satellite missions. This work seems unique in comparison to current widely available long-term TWSA dataset, which is generally generated using climatic inputs. It should be valuable for understanding climate change impacts on global terrestrial water storage change from independent satellite observations. I have some minor comments before it can be accepted.
It is interesting to see how it differs from that based on climatic inputs results, for example, Mandal et al., 2025, and/or Li et al., 2021.

Line 24: why the reductions by LSC-CS are not significant, and is this small improvement helpful.

The word evaporation is better to be changed to evapotranspiration.

Since the simulated runoff may differ from the truth. The in situ gauged streamflow data (for example, from GRDC) for some selected river basins can be used for water balance closure, and hopefully it may improve the validation.

A spatial distribution map showing the long-term TWSA trend can be provided for a better understanding of the result, and maybe its differences with other results.

References:
(1) Li et al., 2021, GRL, Long-Term (1979-Present) Total Water Storage Anomalies Over the Global Land Derived by Reconstructing GRACE Data.
(2) Mandal et al., 2025, ESSD, Machine-learning-based reconstruction of long-term global terrestrial water storage anomalies from observed, satellite and land-surface model data
Citation: https://doi.org/10.5194/essd-2026-84-RC2
- AC2: 'Reply on RC2', Yunzhong Shen, 21 May 2026
  
  We sincerely thank the reviewer for the positive assessment and constructive suggestions. The specific revisions corresponding to the comments below will be presented in detail in the revised manuscript.
  1. Thank you. We agree that this difference should be clarified more explicitly. Mandal et al. (2025) is already included in our comparison as the BNML product.
  2. Thank you for this important comment. We would like to clarify that the result in Line 24 refers to the basin-scale water balance evaluation. In this assessment, precipitation, runoff, and evapotranspiration are taken from model-based meteorological/hydrological datasets. Since some of the compared reconstructed products are themselves driven by such meteorological inputs, their water balance residuals may be relatively smaller in this validation framework. Therefore, the improvement of LSC-CS in this metric may appear less pronounced. To make the comparison fairer and reduce the influence of meteorological-input-driven reconstructions, we will further supplement the revised manuscript by incorporating in situ observations for selected river basins where available.
  3. Thank you. We agree and will replace “evaporation” with “evapotranspiration” throughout the manuscript where appropriate.
  4. Thank you for this suggestion. We agree that GRDC streamflow can provide valuable additional validation and will add it in the revised manuscript.
  5. Thank you. We agree that such a map would improve the presentation. We will add a spatial distribution of long-term TWSA trends derived from LSC-CS and include a brief comparison with other representative products in the revised manuscript.
  
  Citation: https://doi.org/10.5194/essd-2026-84-AC2

Lin Zhang, Yunzhong Shen, Nico Sneeuw, Peyman Saemian, Kunpu Ji, Qiujie Chen, and Fengwei Wang

Supplement

https://doi.org/10.5194/essd-2026-84-supplement

Data sets

A-combination-of-Time-Variable-Gravity-Field-Solutions-from-Multi-Satellite-Datasets Lin Zhang et al. https://doi.org/10.5281/zenodo.18543287

Lin Zhang, Yunzhong Shen, Nico Sneeuw, Peyman Saemian, Kunpu Ji, Qiujie Chen, and Fengwei Wang

Viewed

Total article views: 484 (including HTML, PDF, and XML)

HTML	PDF	XML	Total	Supplement	BibTeX	EndNote
309	140	35	484	89	25	30

HTML: 309
PDF: 140
XML: 35
Total: 484
Supplement: 89
BibTeX: 25
EndNote: 30

Views and downloads (calculated since 16 Mar 2026)

Month	HTML	PDF	XML	Total
Mar 2026	107	41	9	157
Apr 2026	125	56	11	192
May 2026	71	41	14	126
Jun 2026	6	2	1	9

Cumulative views and downloads (calculated since 16 Mar 2026)

Month	HTML	PDF	XML	Total
Mar 2026	107	41	9	157
Apr 2026	125	56	11	192
May 2026	71	41	14	126
Jun 2026	6	2	1	9

Viewed (geographical distribution)

Total article views: 493 (including HTML, PDF, and XML) Thereof 493 with geography defined and 0 with unknown origin.

Country	#	Views	%

Latest update: 08 Jun 2026

Download

Preprint (3551 KB)
Metadata XML

Short summary

This study develops monthly gapless, filter-free gravity field solutions from January 1993 to December 2024, which merge multiple satellite datasets without hydrological or climate models, achieving better accuracy than current methods. Assessment of sea level budget closures and water balance at global and basin scales shows major gains. Results well match IMBIE data on ice sheet mass changes in Antarctica and Greenland, showing accuracy for tracking Earth's water and ice systems.


Total:	0
HTML:	0
PDF:	0
XML:	0