The level-2 time-variable gravity fields obtained from Gravity Recovery and Climate Experiment (GRACE) and its Follow-On (GRACE-FO) mission are widely used in multi-discipline geo-science studies. However, the post-processing of those gravity fields to obtain a desired signal is rather challenging for users that are not familiar with the level-2 products. In addition, the error assessment/quantification of those derived signals, which is of increasing demand in science application, is still a challenging issue even among the professional GRACE(-FO) users. In this effort, the common post-processing steps and the assessment of complicated error (uncertainty) of GRACE(-FO), are integrated into an open-source, cross-platform and Python-based toolbox called SAGEA (SAtellite Gravity Error Assessment). With diverse options, SAGEA provides flexibility to generate signal along with the full error from level-2 products, so that any non-expert user can easily obtain advanced experience of GRACE(-FO) processing. Please contact Shuhao Liu (liushuhao@hust.edu.cn) and Fan Yang (fany@plan.aau.dk) for more information.
When referencing this work, please cite:
Liu, S., Yang, F., & Forootan, E. (2025). SAGEA: A toolbox for comprehensive error assessment of GRACE and GRACE-FO based mass changes. Computers & Geosciences, 196, 105825. https://doi.org/10.1016/j.cageo.2024.105825
- Auto-collecting GRACE(-FO) level-2 products and related auxiliary files.
- Commony used methodologies and technologies of GRACE(-FO)'s post-processing.
- Types of Error assessment/quantification of GRACE(-FO) based mass change.
- User interface (under construction).
This program homepage is: https://github.com/NCSGgroup/SaGEA.
Use this code to download this project.
git clone https://github.com/NCSGgroup/SaGEA
This project is developed based on Python 3.9 and the dependencies are listed in requirements.txt.
Use these coded to download the dependencies:
python -m pip install -r requirements.txt
To ensure the successful running and validation of programs,
please collect the corresponding auxiliary files and verification files at
https://zenodo.org/records/14087017
and place folders data and validation in the project folder by running this code:
python ./demo/data_collecting/demoCollectAuxiliary.py
Several demo programs are under the direction ./demo/ for users to quickly use and verify.
Users can config the relevant parameters in the corresponding location or an independent JSON file.
Detailed module usage and related scientific explanations will be provided in later chapters.
./demo/data_collecting/demoCollectL2Data.pyprovides an example of collecting GRACE Level 2 products, including auxiliary files such as GAX and low-degrees products. Users can config the collecting parameters from the jason file in./setting/data_collection/CollectL2Data.json../demo/post_processing/demoPostProcessing.pyprovides an example for the post-processing of GRACE data../demo/uncertainty_estimation/demoErrorI.pyprovides an example for the propagation of GRACE error ( variance-covariance matrix) during the post-processing../demo/uncertainty_estimation/demoErrorII.pyprovides an example for the estimation of between-group errors through TCH (Three Corner Hat) technology on GRACE signals../demo/uncertainty_estimation/demoErrorIII.pyprovides an example to gain the post-processing statistical uncertainty of GRACE signals.
Fig. 1:
Data structure of SaGEA Toolbox.
Arrows represent dependency relationships.
SaGEA Toolbox is used for post-processing and error assessment of GRACE level-2 data, with the latter relying on the former, see Fig. 1. Thus post-processing is the core function of SaGEA toolbox.
SaGEA provides comprehensive post-processing methods. For ease to use, we have packaged each method as a function of two data classes:
SHC, representing spherical harmonic coefficients (SHCs);GRID, representing gridded data.
Supporting that users may not be very familiar with the methods and principles we recommend calling the post-processing function through SHC or GRID instead of using them directly, as indicated by the circle in Fig. 1.
Here are listed the attributes and some commonly used methods in SHC() and GRID():
Attributes in SHC()
.value: 2-dimension numpy.ndarray that describes multiple sets of SHCs in shape of (n, (lmax+1)^2), where n
represents the number of sets, lmax represents the maximum degree of SHCs.
The arrangement of SHCs in the second dimension
is like [C(0,0), S(1,1), C(1,0), C(1,1), S(2,2), S(2,1), C(2,0), C(2,1), C(2,2), S(3,3), ... ]
Methods in SHC()
.is_series(): to determine whether the stored SHCs are multiple sets.
.get_lmax(): to get the maximum degree/order of the SHCs.
.get_degree_rms(): to get degree-RMS.
.replace_low_degs(*params): to replace low-degree SHCs with others.
.filter(*params): to apply a spectral filtering on the SHCs.
.convert_type(*params): to convert SHCs from one physical dimension to another.
.geometric(*params): to apply a geometric correction on the SHCs.
.de_background(*params): to deduct a background field.
.add(*params): to add another SHC(), e.g., that of GAD.
.subtract(*params): to subtract another SHC(), e.g., that of GIA.
.expand(*params): to expand the stored SHCs as a linear trend to signals at time epochs (e.g., from GIA trend to
signals).
.synthesis(*params): to harmonic synthesis the stored SHCs into spatial distribution in optional physical
dimensions.
.to_grid(*params): pure harmonic synthesis.
Attributes in GRID()
.value: 3-dimension numpy.ndarray that describes multiple sets of SHCs in shape of (n, nlat, nlon), where n
represents the number of sets, nlat and nlon represents the latitudes and the longitudes of grids.
.lat: 1-dimension numpy.ndarray that describes the geometry latitudes in unit degree.
.lon: 1-dimension numpy.ndarray that describes the geometry longitude in unit degree.
Methods in GRID()
.filter(*params): to apply a spatial filtering on the SHCs (under construction).
.leakage(*params): to apply a leakage reduction.
.seismic(*params): to apply a seismic correction.
.de_aliasing(*params): to fit and deduct long-term aliasing signals.
.integral(*params): to integral in globe or basin and get the results.
.limiter(*params): to set all signals to 1 or 0 according to the threshold.
.to_file(*params) to store as a file (.nc, .hdf5, etc.) locally.
Fan Yang (fany@plan.aau.dk) led the scientific research and conducted rigorous data validation.
Shuhao Liu (liushuhao@hust.edu.cn) contributed to the software architecture design and core functionality implementation.
MIT License
Copyright (c) 2024 NCSG -- Numerical Computation and Satellite Geodesy research group
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
GRACE and GRACE-FO level-2 products can be obtained at open source FTP server ftp://isdcftp.gfz-potsdam.de/. Level-2 products includes GSM, GAA, GAB, GAC, and GAD (The last four products are collectively referred to as GAX.), which are given in fully normalized spherical harmonic coefficients (SHCs) of gravity potential.
-
GSM products represent the estimate of Earth's mean gravity field during the specified timespan derived from GRACE mission measurements.
-
GAA products represent the anomalous contributions of the non-tidal atmosphere to the Earth's mean gravity field during the specified timespan.
-
GAB products represent the anomalous contributions of the non-tidal dynamic ocean to ocean bottom pressure during the specified timespan.
-
GAC products represent the sum of the GAA and GAB coefficients during the specified timespan.
-
GAD products give the SHCs that are zero over the continents, and provide the anomalous simulated ocean bottom pressure that includes non-tidal air and water contributions elsewhere during the specified timespan.
The most used GSM solutions are given by three processing centers, that is, Center for Space Research (CSR), University of Texas at Austin, Jet Propulsion Laboratory (JPL), NASA, and German Research for Geosciences (GFZ), German.
GRACE level-2 GSM solutions lack the three degree-1 coefficients, which are proportional to geocenter motion and can not been ignored for a complete representation of the mass redistribution in the Earth system (Sun et al., 2016). The GRACE-based C20 coefficient is subject to large uncertainties (Chen et al., 2016). Besides, The C30 coefficient is also shown to be poorly observed by GRACE/GRACE-FO when either mission is operating without two fully functional accelerometers (Loomis et al., 2020). Before using the GSM solutions, the degree-1 coefficients needs to be added back, and C20, C30 coefficients with large uncertainties also needs to be replaced with estimates from other techniques, such as satellite laser ranging (SLR).
GRACE level-2 products reflects the distribution of dimensionless geopotential, from which we can obtain the corresponding changes in gravity anomalies, mass changes or other factors through the load theory. As a result, the products are usually get converted to the unit of researcher's interest before further use.
Also, GRACE level-2 products are generally given in the form of SHCs, which can intuitively reflect the signal of different wavelengths (frequency bands). However, it is difficult to directly see in SHCs how the spatial distribution is. Indeed, by performing spherical harmonic synthesis on the SHC, corresponding grid data can be obtained, from which we can easily see the spatial distribution of signals. On the contrary, the corresponding SHC can also be obtained through spherical harmonic analysis of grid data.
Due to the presumed Earth mass conservation and the on-orbit measuring mode, GRACE(-FO) has no ability to obtain the degree-0 and -1 terms of the gravity field (Wu et al., 2012). In addition, another major component, C20, is often not well observed, leading to the common practice of replacing their values with those obtained by Satellite laser ranging (SLR), which has long been relied upon for measuring changes in Earth’s dynamic oblateness (i.e., C20, see Cheng & Ries, 2017). Besides, it was further found by Loomis et al. (2020) that, the C30 coefficient, is also observed by GRACE/GRACE-FO when either mission is operating without two fully functional accelerometers. Therefore, recommendations are also made to replace C30 with that of SLR.
SaGEA toolbox contains the following option to replace the low-degree terms of the gravity field:
- degree-1 terms given by Sun & Ditmar (2016).
- C20 terms given by Cheng & Ries (2017) and Loomis et al. (2020).
- C30 terms given by Loomis et al. (2020).
Due to the existence of high-order noise and correlation error in the GRACE solutions, filtering is a necessary step before apply it on some scientific studies (Wahr et al., 2006). The most usd GRACE filter, isotropic Gaussian filter, was first suggested and applied by Wahr et al. (1998). Other filters based on different principles were raised then like empirical decorrelation filtering (EDF) raised by Swenson & Wahr (2006), DDK filter by Kusche (2007), etc. Among them, EDF was also improved and used by scholars since its initial proposal (Duan et al., 2009). SaGEA toolbox contains the following filtering methods:
- Types of EDFs (Swenson & Wahr, 2006; Chen et al., 2007; Duan et al., 2009).
- Isotropic Gaussian filter (Wahr et al., 1998).
- Non-isotropic Gaussian fielter by Han et al. (2005).
- Fan filter by Zhang et al. (2009).
- DDK filter by Kusche et al. (2007, 2009).
Filters for GRACE can supress the noise at high degrees, but at the same time they could also weaken the signal as well. Spatially speaking, signal where it is strong would leak into some place where it is weaker, for example, the hydrological and atmospheric pressure signals over continents will leak into the oceanic estimates at the coastline ( Wahr et al., 1998). Depending on the signal strength of the study basin, one would consider to reduce the leak-in signal outside the basin (leakage), or to restore the leak-out signal from the basin (bias).
To reduce the leakage, Wahr et al., 1998 for the first time gave an iterative estimation technique to handle it. Here is a brief introduction to the technique to estimate the leakage:
One gives the initial signal a small-scale filter, and harmonic synthesis it into a pre-smoothed spatial distribution. The pre-smoothed spatial signal is set zero outside the interested basin, and then get harmonic analyzed, marked as pre-smoothed SHCs outside. Pre-smoothed SHCs outside then get filtered once again of the same scale with that used in the study, and then get harmonic synthesised it into a new spatial distribution. The new spatial signal inside the interest basin can be seen as the leakage and reduced from the filtered signal. As for the bias, it is usually to gain a scale factor k to calibrate the filtered or smoothed signal. Researchers have raised numbers of method to estimate k, and other methods to restore correct the bias.
SaGEA toolbox contains the following commonly used methods to correct the leakage and bias:
- Iterative (Wahr et al., 1998)
- Multiplicative (Longuevergne et al., 2007).
- Additive (Klees et al., 2007).
- Scaling (Landerer et al., 2012).
- Forward Modeling by (Chen et al., 2015)
- Data-driven (Vishwakarma et al., 2017).
- Buffer zone (Chen et al., 2019)
In the Earth’s gravity anomalies observed by GRACE(-FO), apart from the effects caused by surface mass migration, there are also influences from GIA driven redistribution of solid Earth mass, which cannot be identified and separated directly by GRACE(-FO). GIA’s influence is a long-term phenomenon, with effects extending far beyond the temporal scale of typical GRACE(-FO) observations, which span several decades (Peltier, 2004). Approximately, the impact of GIA can be considered linear, which allows for a simplified correction by predefined models. Thus, to reflect a true mass change, the GIA’s linear approximation has to be subtracted from the observed gravity changes.
SaGEA toolbox provides methods for removing GIA in both spectral (for class SHC) and spatial (for class GRID) domains, including the following commonly used GIA models:
- ICE6G-D (ICE series) by Peltier et al. (2018).
- Geruo2013 by A et al. (2013).
- Caron2018 by Caron et al. (2018).
Due to imperfect tidal models (mainly ocean tides), monthly gravity field solutions from GRACE contain aliasing errors of frequencies much longer than 30 days, such as the S2 (approximately 161 days), P1 (approximately 171 days) and S1 (approximately 322 days) terms (Knudsen, 2003; Han et al., 2007; Seo et al., 2008). In particular, this aliasing error is prominent in high latitudes, and thereby needs to be considered (Chen et al., 2009). The aliasing error still remains as a major error source of the latest gravity product from GRACE(-FO), see Z. Li et al. (2022).
In SAGEA toolbox, either by the Fourier spectrum analysis or by the least square analysis, those aliasing frequency are identified and removed, and S1, S2, P1 are available options.
The geometrical deviation of the actual Earth from the presumed sphere would lead to a bias when converting the geopotential into TWS (J. Li et al., 2017; Ditmar, 2018; Yang et al., 2022). Such bias would become increasingly significant as the latitude increases, or as the topography increases. Correction to this bias is termed as geometrical correction, which indeed consists of an ellipsoid correction and a topography correction.
In SAGEA toolbox, the geometrical correction is implemented following the method of Yang et al. (2022), where one can easily switch it on or off.
It is concluded by Chao & Liau (2019) that, the lowest earthquake magnitude threshold that can be detected by GRACE is the Mw 8.3. This also means, some of largest earthquakes could be directly modelled and removed from GRACE(-FO) monthly gravity fields, to leave clean enough information to reflect expected hydrological or oceanic signals (Tang et al., 2020). This procedure is termed as Seismic correction, which is necessary particularly for investigating the ocean mass change where the seismic clearly biases the estimation. In SAGEA, several seismic events are available for users to choose from, and we use the logarithmic and exponential function model for fitting those seismic events from GRACE(-FO).
Currently, the seismic events available in SaGEA contain Sumatra 2004, Nias 2005, Bengkulu 2007, Maule 2010, Chile 2010, Tohoku-Oki 2011, Sumatra 2012 and Okhotsk 2013. On one hand, SaGEA will continue to update the available seismic events, and on the other hand, users can also edit their own list of seismic events for their need.
In addition to the corrections that are universal and common for all regions, there is some other correction that is specific only for certain study, e.g., to study the global mean ocean mass (GMOM) change. To obtain the complete ocean mass, the non-tidal ocean mass, which have been modelled by ocean general circulation model and removed from the gravity field as prior model, has to be restored. This correction is done by added back another official gravity product, named GAD (the monthly average of ocean bottom pressure anomalies), which we know as GAD recovery.
In the SaGEA toolbox, the GAD model can be easily collected, just like the GRACE level-2 products. And such correction, which one can easily switch on or off, is also integrated into it.
GMAM correction is proposed by Chen et al. (2019) to compensate the offset in Earth mass conservation due to the absence of global mean atmospheric mass (GMAM). Chen et al. (2019) found a constant annual phase lag(for about 10 deg) between GRACE and Altimeter-Argo estimates of GMOM changes. By removing GMAM from the GRACE solutions using atmospheric model (GAA, the monthly average of atmosphere), this annual phase lag is nearly compensated.
In SaGEA toolbox, the GAA model can be also easily collected, and the corresponding correction, which one can easily switch on or off, is integrated into SaGEA toolbox.
Spatial analysis mainly indicates the extraction of basin-scale variability, like TWS (Total Water Storage) changes. It can be done in the spatial or the spectral domain (Swenson and Wahr, 2002), i.e.,
- Geographic-latitude weighted sum of gridded signal in spatial domain.
- Direct product sum of SHCs in spectral domain.
Simply put, it is the use of gridded or spherical harmonic signals, as well as corresponding basin information, to extract target signals. While these two methods are theoretically equivalent, there are differences in practice due to the loss of spatial/spectral conversion.
SaGEA provides both of the two approaches, and users can choose them according to their needs.
The seasonality in time-series of GRACE(-FO) based mass change, i.e., the secular trend, inter-annual periodic signal and seasonal periodic signals, is of particular interest of geoscience studies (Tapley et al., 2019).
SaGEA provides three approaches to gain the seasonality from a time-series (of GRACE and GRACE-FO based mass change), that is,
- Ordinary least squares (OLS).
- Weighted least squares (WLS).
- Fourier analysis.
A majority of publication with GRACE(-FO) prefers to use OLS for extracting seasonality as the error is always presumed as heterogeneous and uniform. Further, WLS makes sense since SaGEA is able to provide the error information, otherwise WLS should be disabled. Another general tool, the Fourier analysis, is also able to obtain the seasonality other than a priori information.
Due to the imperfect background models (Hauk and Pail, 2018; Yang et al., 2021) and on-board instrument error ( Bandikova and Flury, 2014; Flechtner et al., 2016), GRACE(-FO) level-2 monthly gravity product is generated with an intrinsic uncertainty, known as formal error that represents the variance of each SHC. Besides, ITSG solutions provide the full variance covariance matrix of these SHCs (Kvas et al., 2019; Kvas and Mayer-Gürr, 2019). As one has to post-process the level-2 product to derive desired variables (known as level-3 products), the formal error, or the variance-covariance matrix, needs to be propagated to support uncertainty estimation of the level-3 product. Such kind of error is indicated as Error-I and integrated in SaGEA program.
Besides the official GRACE data producers, multiple producers are routinely producing level-2 solutions and contributing them to the International Centre for Global Earth Models (ICGEM). These level-2 products are different from each other, therefore, the discrepancy between these level-2 products shall result in various estimates of desired variables, e.g., TWS estimation. Such a discrepancy is considered as between-group error, which is indicated as Error-II in SaGEA program. As the most popular technique, the TCH (Three-Cornered Hat, Ferreira et al., 2016; Chen et al., 2021) is integrated in SaGEA to consider the Error-II of desired variables.
This category of error might be caused by the non-uniqueness of the post-processing chain and the involvement of corrections with less known uncertaintiesm. As there is no official convention of post-processing chain, each study has adopted its own strategy, resulting in potential large discrepancy (indicated as Error-III in SaGEA) at obtained level-3 products. Error-III is considered as the within-group error since it always relies upon only one set of level-2 product. To quantify it, a large number of ensembles, which reflect various postprocessing chains, are required. As the advanced post-processing module are provided in SaGEA, a comprehensive quantification of Error-III at a diverse and flexible option is possible. Therefore, based on the post-processing module, SaGEA also provides a quantization program for Error-III.
A, G., Wahr, J., Zhong, S., 2013. Computations of the viscoelastic response of a 3-D compressible Earth to surface loading: an application to Glacial Isostatic Adjustment in Antarctica and Canada. Geophysical Journal International 192, 557–572. https://doi.org/10.1093/gji/ggs030
Bandikova, T., Flury, J., 2014. Improvement of the GRACE star camera data based on the revision of the combination method. Advances in Space Research 54, 1818–1827. https://doi.org/10.1016/j.asr.2014.07.004.
Caron, L., Ivins, E. R., Larour, E., Adhikari, S., Nilsson, J., & Blewitt, G., 2018. GIA Model Statistics for GRACE Hydrology, Cryosphere, and Ocean Science. Geophysical Research Letters, 45 (5), 2203–2212. https://doi.org/10.1002/2017gl076644
Chao, B. F., & Liau, J. R., 2019. Gravity Changes Due to Large Earthquakes Detected in GRACE Satellite Data via Empirical Orthogonal Function Analysis. Journal of Geophysical Research: Solid Earth, 124 (3), 3024-3035. https://doi.org/10.1029/2018jb016862
Chen, J.L., Wilson, C.R., Ries, J.C., 2016. Broadband assessment of degree-2 gravitational changes from GRACE and other estimates, 2002-2015. Journal of Geophysical Research: Solid Earth 121, 2112–2128. https://doi.org/10.1002/2015jb012708
Chen, J.L., Wilson, C.R., Tapley, B.D., Grand, S., 2007. GRACE detects coseismic and postseismic deformation from the Sumatra-Andaman earthquake. Geophysical Research Letters 34. https://doi.org/10.1029/2007gl030356.
Chen, J.L., Wilson, C. R., & Seo, K.-W., 2009. S2 tide aliasing in GRACE time-variable gravity solutions. Journal of Geodesy , 83 (7), 679–687. https://doi.org/10.1007/s00190-008-0282-1
Chen, J.L., Tapley, B., Tamisiea, M.E., Save, H., Wilson, C., Bettadpur, S., Seo, K., 2021. Error Assessment of GRACE and GRACE Follow-On Mass Change. Journal of Geophysical Research: Solid Earth 126. https://doi.org/10.1029/2021jb022124.
Cheng, M., & Ries, J. (2017). The unexpected signal in GRACE estimates of c20. Journal of Geodesy , 91 (8), 897-914. https://doi.org/10.1007/s00190-016-0995-5
Ditmar, P., 2018. Conversion of time-varying Stokes coefficients into mass anomalies at the Earth’s surface considering the Earth’s oblateness. Journal of Geodesy , 92 (12), 1401–1412. https://doi.org/10.1007/s00190-018-1128-0
Duan, X. J., Guo, J. Y., Shum, C. K., & Van Der Wal, W., 2009, On the postprocessing removal of correlated errors in GRACE temporal gravity field solutions. Journal of Geodesy, 83(11), 1095–1106. https://doi.org/10.1007/s00190-009-0327-0
Ferreira, V., Montecino, H., Yakubu, C., Heck, B., 2016. Uncertainties of the Gravity Recovery and Climate Experiment time-variable gravity-field solutions based on three-cornered hat method. Journal of applied remote sensing 10, 015015–015015. https://doi.org/10.1117/1.JRS.10.015015.
Flechtner, F., Neumayer, K.H., Dahle, C., Dobslaw, H., Fagiolini, E., Raimondo, J.C., Güntner, A., 2016. What can be expected from the GRACE-FO laser ranging interferometer for earth science applications? Remote sensing and water resources , 263–280. https://doi.org/10.1007/s10712-015-9338-y.
Han, S.-C., Shum, C. K., Jekeli, C., Kuo, C.-Y., Wilson, C., & Seo, K.-W., 2005, Non-isotropic filtering of GRACE temporal gravity for geophysical signal enhancement. Geophysical Journal International, 163(1), 18–25. https://doi.org/10.1111/j.1365-246x.2005.02756.x
Han, S.-C., Ray, R. D., & Luthcke, S. B., 2007. Ocean tidal solutions in Antarctica from GRACE inter-satellite tracking data. Geophysical Research Letters, 34 (21). https://doi.org/10.1029/2007GL031540
Hauk, M., Pail, R., 2018. Treatment of ocean tide aliasing in the context of a next generation gravity field mission. Geophysical Journal International 214, 345–365. https://doi.org/10.1093/gji/ggy145.
Klees, R., Zapreeva, E. A., Winsemius, H. C., & Savenije, H. H. G.， 2007. The bias in GRACE estimates of continental water storage variations. Hydrology and Earth System Sciences, 11(4), 1227–1241. https://doi.org/10.5194/hess-11-1227-2007
Knudsen, P., 2003. Ocean tides in GRACE Monthly Averaged Gravity Fields. Space Science Reviews, 108 (1), 261–270. https://doi.org/10.1023/A:1026215124036
Kusche. J., 2007. Approximate decorrelation and non-isotropic smoothing of time-variable GRACE-type gravity field models. Journal of Geodesy, 81(11), 733–749. https://doi.org/10.1007/s00190-007-0143-3
Kvas, A., Behzadpour, S., Ellmer, M., Klinger, B., Strasser, S., Zehentner, N., Mayer-Gürr, T., 2019. ITSG-Grace2018: Overview and evaluation of a new GRACE-only gravity field time series. Journal of Geophysical Research: Solid Earth 124, 9332–9344. https://doi.org/10.1029/2019JB017415.
Kvas, A., Mayer-Gürr, T., 2019. GRACE gravity field recovery with background model uncertainties. Journal of geodesy 93, 2543–2552. https://doi.org/10.1007/s00190-019-01314-1.
Landerer, F. W., & Swenson, S. C., 2012. Accuracy of scaled GRACE terrestrial water storage estimates. Water Resources Research, 48(4). https://doi.org/10.1029/2011wr011453
Li, J., Chen, J. L., Li, Z., Wang, S., & Hu, X., 2017. Ellipsoidal correction in GRACE Surface Mass Change Estimation. Journal of Geophysical Research: Solid Earth, 122 (11), 9437–9460. https://doi.org/10.1002/2017jb014033
Li, Z., Zhang, Z., & Wang, H., 2022. On tide aliasing in GRACE Time-Variable Gravity Observations. Remote Sensing , 14 (21), 5403. https://doi.org/10.3390/rs14215403
Longuevergne, L., Scanlon, B. R., & Wilson, C. R.. 2010.， GRACE Hydrological estimates for small basins: Evaluating processing approaches on the High Plains Aquifer, USA. Water Resources Research, 46( 11). https://doi.org/10.1029/2009wr008564
Loomis, B. D., Rachlin, K. E., Wiese, D. N., Landerer, F. W., & Luthcke, S. B. (2020). Replacing GRACE/GRACE-FO With Satellite Laser Ranging: Impacts on Antarctic Ice Sheet Mass Change. Geophysical Research Letters, 47 (3). https://doi.org/10.1029/2019gl085488
Peltier, W. (2004). GLOBAL GLACIAL ISOSTASY AND THE SURFACE OF THE ICE-AGE EARTH: The ICE-5G (VM2) Model and GRACE. Annual Review of Earth and Planetary Sciences, 32 (1), 111-149. https://doi.org/10.1146/annurev.earth.32.082503.144359
Seo, K.-W., Wilson, C. R., Chen, J. L., & Waliser, D. E., 2008. GRACE’s spatial aliasing error. Geophysical Journal International, 172 (1), 41–48. https://doi.org/10.1111/j.1365-246X.2007.03611.x
Sun, Y., Riva, R., Ditmar, P., 2016. Optimizing estimates of annual variations and trends in geocenter motion and J2 from a combination of GRACE data and geophysical models. Journal of Geophysical Research: Solid Earth 121, 8352–8370. https://doi.org/10.1002/2016jb013073
Swenson, S., Wahr, J., 2002. Methods for inferring regional surface-mass anomalies from Gravity Recovery and Climate Experiment (GRACE) measurements of time-variable gravity. Journal of Geophysical Research: Planets 107, ETG 3–1–ETG 3–1. https://doi.org/10.1029/2001jb000576.
Swenson, S., Wahr, J., 2006. Post-processing removal of correlated errors in GRACE data, Geophysical Research Letters. https://doi.org/10.1029/2005gl025285.
Tang, L., Li, J., Chen, J. L., Wang, S.-Y., Wang, R., & Hu, X., 2020. Seismic Impact of Large Earthquakes on Estimating Global Mean Ocean Mass Change from GRACE. Remote Sensing , 12 (6), 935. https://doi.org/10.3390/rs12060935
Tapley, B.D., Watkins, M.M., Flechtner, F., Reigber, C., Bettadpur, S., Rodell, M., Sasgen, I., Famiglietti, J.S., Landerer, F.W., Chambers, D.P., et al.,2019. Contributions of GRACE to understanding climate change. Nature Climate Change 9, 358–369. https://doi.org/10.1038/s41558-019-0456-2.
Vishwakarma, B. D., Horwath, M., Devaraju, B., Groh, A., & Sneeuw, N., 2017. A Data-Driven Approach for Repairing the Hydrological Catchment Signal Damage Due to Filtering of GRACE Products. Water Resources Research, 53(11), 9824–9844. https://doi.org/10.1002/2017wr021150
Vishwakarma, B., Devaraju, B., & Sneeuw, N., 2018, What Is the Spatial Resolution of grace Satellite Products for Hydrology?. Remote Sensing, 10(6), 852. https://doi.org/10.3390/rs10060852
Wahr, J., Molenaar, M., Bryan, F., 1998. Time variability of the Earth's gravity field: Hydrological and oceanic effects and their possible detection using GRACE. Journal of Geophysical Research: Space Physics 103, 30205–30229. https://doi.org/10.1029/98jb02844
Wu, X., Ray, J., & van Dam, T., 2012. Geocenter motion and its geodetic and geophysical implications. Journal of Geodynamics, 58 , 44–61. https://doi.org/10.1016/j.jog.2012.01.007
Yang, F., Luo, Z., Zhou, H., & Kusche, J., 2022. On study of the earth topography correction for the GRACE surface mass estimation. Journal of Geodesy , 96 . https://doi.org/10.1007/s00190-022-01683-0
Yang, F., Forootan, E., Wang, C., Kusche, J., Luo, Z., 2021. A New 1-Hourly ERA5-Based Atmosphere De-Aliasing Product for GRACE, GRACEFO, and Future Gravity Missions. Journal of Geophysical Research: Solid Earth 126, e2021JB021926. https://doi.org/10.1029/2021JB021926.
Yang, F., Forootan, E., Liu, S., & Schumacher, M. (2024). A Monte Carlo Propagation of the Full Variance-Covariance of GRACE-Like Level-2 Data With Applications in Hydrological Data Assimilation and Sea-Level Budget Studies. Water Resources Research, 60 (9), e2023WR036764. https://doi.org/10.1029/2023WR036764449.
Yi,S.,& Sneeuw,N. (2022). A novel spatial filter to reduce north south striping noise in GRACE spherical harmonic coefficients. Journal of Geodesy,96(4),23. https://doi.org/10.1007/s00190-022-01614-z.
Zhang, Z.-Z., Chao, B. F., Lu, Y., & Hsu, H.-T., 2009, An effective filtering for GRACE time-variable gravity: Fan filter. Geophysical Research Letters, 36(17). https://doi.org/10.1029/2009gl039459