Abstract
While the GRACE (Gravity Recovery and Climate Experiment) satellite mission is of great significance in understanding various branches of Earth sciences, the quality of GRACE monthly products can be unsatisfactory due to strong longitudinal stripepattern errors and other flaws. Based on corrected GRACE Mascon (mass concentration) gridded mass transport time series and updated LDCgam (Least Difference Combination global angular momenta) data, we present a new set of monthly gravity models called LDCmgm90, in the form of Stokes coefficients with order and degree both up to 90. The LDCgam inputs are developed by assimilating degree2 Stokes coefficients from various versions of GRACE (including Mascon products) and SLR (Satellite Laser Ranging) monthly gravity data into combinations of outputs from various global atmospheric, oceanic, and hydrological circulation models, under the constraints of accurately measured Earth orientation parameters in the Least Difference Combination (LDC) scheme. Taking advantages of the relative strengths of the various input solutions, the LDCmgm90 is free of stripes and some other flaws of classical GRACE products.
Measurement(s)  Stokes coefficient • Coefficient 
Technology Type(s)  spherical harmonic expansion • least difference combination • computational modeling technique 
Sample Characteristic  Environment  climate system 
Sample Characteristic  Location  Earth (planet) 
Machineaccessible metadata file describing the reported data: https://doi.org/10.6084/m9.figshare.9924929
Background & Summary
Timedependent gravity from the GRACE (Gravity Recovery and Climate Experiment) twin satellites is of great significance for studies related to changes in land water, ice sheets, sea level rise, ocean circulation, Earthquake dynamics etc.^{1,2,3,4,5,6,7}. GRACE data are routinely provided almost every month (from Apr. 2002 to Jun. 2017, but with 20 months missing) in the form of Stokes coefficients with AOD1B (Atmosphere and Ocean Dealiasing Level 1B) corrections (denoted as GSM) by Center for Space Research (CSR), Deutsches GeoForschungsZentrum (GFZ), Jet Propulsion Laboratory (JPL) and Graz University of Technology (TUG), using a least squares adjustment (LSA) scheme^{8,9,10,11,12,13,14}. There are often limited agreements between GRACEbased results and those obtained by independent observations, mostly attributed to the wellknown strong striped noise patterns caused by the GRACE’s nearpolar orbital inclination and the LSA scheme, which ignores the orthogonality of spherical harmonics and thus leads to correlations of Stokes coefficients^{15,16,17,18,19}. Notable discrepancies can also be found among GRACE products released by different institutes, due to some differences in data processing strategies adopted by them^{8,9,10,11,12,13,19,20,21,22}. Various filtering and destriping methods are proposed to attenuate these stripes, resulting in weaker and distorted signals of interest^{23,24,25,26,27,28}. Moreover, power losses are also found around 3 cycles per year (cpy) and higher in time series of low degree GRACE Stokes coefficients^{21}.
Since 2015, CSR and JPL also provide socalled Mascon solutions using the Mass Concentration blocks (mascons)^{29,30,31,32}, another form of gravity field basis functions. With mascons, some a priori geophysical constraints can be implemented to remove noises from the GRACE observations at the Level2 processing step, which is a much more rigorous approach than the empirical postprocessing filtering and destriping of the LSAbased spherical harmonics. However, the problem of power losses around 3 cpy and higher is not overcome, and notable differences between CSR and JPL mascon solutions still exist (noted by this study).
Mass redistributions will cause changes not only in gravity but also in Earth’s pole coordinates and spin rate, due to conservation of angular momentum^{33,34}. Plenty of studies have explored the links between the timedependent Stokes coefficients and Earth rotational variations, specifically the level of agreement between GRACEbased (C_{21}, S_{21}) series and polar motion, and between SLRbased C_{20} and lengthofday (LOD) variations after contributions unrelated to mass redistributions are excluded^{21,35,36,37,38,39,40,41,42,43,44,45}. Some even made use of these GRACE and/or SLR (Satellite Laser Ranging) coefficient series to improve geophysically based fluid model excitations of polar motion and LOD variations^{21,22,46}. Among these studies, the Least Difference Combination (LDC) of global angular momenta for surficial geophysical fluids of Chen et al.^{21} and Yu et al.^{22} (hereafter termed as LDCgam) seem to have the best performances in both the frequency and time domains, since various versions (CSR, GFZ and JPL) of GRACE and SLR monthly gravity data (RL05) were assimilated into the outputs from various global atmospheric, oceanic, and hydrological circulation models, in the LDC scheme which can extract the best frequency components from various types of data sources provided that one or more proper reference data or models can be established^{21,47}.
To summarize, the currently available GRACE monthly Stokes coefficients are unsatisfactory due to strong longitudinal stripepattern errors and other flaws while assimilating independent related observations may help to improve them. In this study, we used numerical integration to convert Mascon gridded mass to Stokes coefficients and applied necessary corrections as described in the next section. We also prepared for this study an updated LDCgam solution^{48} obtained by similar procedures in Chen et al.^{21} and Yu et al.^{22} but assimilating all RL05 and RL06 GRACE/SLR Stokes coefficients from CSR, GFZ, JPL and TUG, and all RL05 and RL06 Mascon gridded mass fields. Then we put forward the improved monthly gravity model set LDCmgm90, in the form of Stokes coefficients (complete from degree and order 2 to 90) since they are more convenient to use.
Methods
The GRACE monthly data are usually released together with the GRACE AOD1B products, which provide a modelbased dataset (including GAA, GAB, GAC and GAD) that describes the time variations of the gravity potential at satellite altitudes that are caused by nontidal mass variability in the atmosphere and oceans^{49,50,51}. The GAA product describes the monthly nontidal atmospheric mass anomalies simulated by the operational run of the atmosphere model ECMWF (European Centre for MediumRange Weather Forecasts)^{52}, GAB refers to monthly nontidal oceanic mass anomalies simulated by the operational run of the (unconstrained) ocean model OMCT (Ocean Model for Circulation and Tides)^{53} (for RL05) or MPIOM (MaxPlanckInstitute for Meteorology Ocean Model)^{54} (for RL06), GAC is the sum of GAA and GAB, and GAD can be regarded as a revised version of GAC with nontidal atmospheric and oceanic mass anomalies only over ocean areas. GSM is just the gravity residual after GAA and GAB are removed from the GRACE observations (in other words, GSM + GAB + GAA is what GRACE satellites really measure). Consistent with this system, the LDCmgm90 data set also contains five subsets GAA, GAB, GAC, GAD and GSM, all with degree and order up to 90 because higher harmonics are not guaranteed by GRACE’s measurement resolution.
The general procedures to produce the LDCmgm90 are described in Fig. 1, which is explained next.
Step 1: Obtain the LDCmgm degree2 zonal and tesseral potential coefficients
We first obtained elements of the inertia tensor (ΔI_{xz}(t), ΔI_{yz}(t), ΔI_{zz}(t)) through the massredistributionrelated (or massterm) angular momenta LDCgam:
then the corresponding LDCmgm degree2 zonal and tesseral potential coefficients^{31,52}
In Eqs (1) and (2), Ω = 7.292115 10^{−5} rad/s is the mean spin rate of the Earth, k′ = −0.316 is the degree2 load Love number^{33}, M = 5.97236 × 10^{24} kg and a = 6378136.6 m are the mass and mean equatorial radius of the Earth^{55,56}, respectively, ΔT is the change in the trace of the inertia tensor and equals zero in the current case that the global mass is conserved^{33,57}.
The LDCgam provides atmospheric angular momentum (AAM), oceanic angular momentum (OAM) and hydrological angular momentum/cryospheric angular momentum (HAM/CAM), where the HAM/CAM is dominated by but not limited to changes in land water and ice, since all the nonatmospheric and nonoceanic mass redistributions are attributed to it. Therefore, we have the following links (→ means corresponding to):
Massterm AAM → GAA C_{20}, C_{21} and S_{21}.
Massterm OAM → GAB C_{20}, C_{21} and S_{21}.
Massterm HAM/CAM → GSM C_{20}, C_{21} and S_{21}.
Then we can obtain the degree2 GAA, GAB and GSM zonal and tesseral potential coefficients for the LDCmgm90 (please refer to the top half of Fig. 1a). Noting that CSR, GFZ, JPL and TUG all used the same AOD1B products for the given data releases (RL05 or RL06), and the JPL GAA, GAB, GAC and GAD products are the most complete, we thus chose the JPL RL06 GAA, GAB, GAC and GAD products to construct the LDCmgm90.
Step 2: Convert the Mascon gridded mass redistribution to corrected Stokes coefficients
Currently, there are three Mascon solutions CSR Mascon RL05, JPL Mascon RL05 and JPL Mascon RL06, of which the original Mascon gridded data correspond to the GSM products^{29,30,31}. Although the RL05 and RL06 Mascon products are based on different static background geopotential model (which would cause biases among them), we are more interested in the timedependent parts rather than the static ones when using GRACElike products. With these biases removed, a proper combination can extract the best components from these three Mascon solutions since no original single solution is perfect as discussed in Background & Summary.
The Mascon data are represented in the form of equivalent water height Δh(θ, λ, t) on a 0.5 degree longitudelatitude grid but representing the equalarea geodesic grid of size 1 × 1 degree at the equator. The surface density for this thin layer is Δσ(θ, λ, t) = ρ_{w}Δh(θ, λ, t), where ρ_{w} = 1025 kg/m^{3} is the average density of sea water. Then the original Mascon gridded data may be converted to Stokes coefficients by^{58}
where \(k{^{\prime} }_{n}\) is the degreen load Love number (from Table 1 of Wahr et al.^{58}), ρ_{ave} = 5517 kg/m^{3} is the average density of the solid Earth.
The GAA RL05 produced by the ECMWF operational run contains the following two notable jumps^{49,59}:

(1)
Between 20060129 18 h and 20060130 00 h

(2)
Between 20100126 00 h and 20100126 06 h
due to upgrades of the horizontal and vertical resolutions in the ECMWF model, which will lead to opposite jumps in all the corresponding RL05 versions of GSM and Mascon products. Moreover, the RL05 products adopted the nonlinear IERS2010 mean pole correction^{56}, which will cause a longperiod pole tide in C_{21} and S_{21} and should be corrected as suggested by Wahr et al.^{20}. For the two RL05 Mascon products, corrections of the jumps and longperiod pole tide should be applied (see Fig. 1b) while the RL06 data are free of these flaws due to a homogeneous reanalysis of the ECMWF data and the adoption of a linear mean pole model. However, one must keep in mind that whichever RL05 or RL06, GAA and GAB are respectively derived from the ECMWF and OMCT (or MPIOM) operational outputs, which need further refinements as shown in detailed analyses by Chen et al.^{21,47} and Yu et al.^{22}. Thus it would be better to replace them with the LDCcorrected GAA and GAB. Further, GAC = GAA + GAB, and GAD can also be obtained by applying an ocean mask to GAC.
By using Eq. (3) and applying the abovementioned corrections and replacements, we can obtain the corrected Mascon Stokes coefficients as shown in Fig. 1b.
Step 3: Take weighted average of the corrected Mascon Stokes coefficients and obtain the final solutions
The GRACEobserved geopotential V_{obs} may be separated into two parts: the part \({V}_{obs}^{zt}\) including the degree2 zonal and tesseral terms (namely the terms relevant with C_{20}, C_{21} and S_{21}), and the other \({V}_{obs}^{nzt}\) containing all other terms, namely \({V}_{obs}={V}_{obs}^{zt}+{V}_{obs}^{nzt}\). All the CSR, GFZ, JPL and TUG released GRACE data are from the same twin satellites, thus in principle, any overestimate or underestimate of \({V}_{obs}^{zt}\) will cause an opposite effect on \({V}_{obs}^{nzt}\). That is, \({V}_{obs}^{zt}\) and \({V}_{obs}^{nzt}\) must have the same errors for each given version of GRACE data. Based on this reasoning, the weights of the corrected Mascon Stokes coefficients may be estimated as
since C_{20}, C_{21} and S_{21} obtained from LDCgam are the most accurate and may be approximately used as standards to infer errors in other data sets. In Eq. (4), std(x) means standard derivation of x. The corresponding relative weights of the three Mascon solutions can be found in Table 3b.
We can obtain the weighted average of the corrected Mascon Stokes coefficients except for C_{20}, C_{21} and S_{21} as described in the bottom part of Fig. 1b.
Data Records
Availabilities of the data used in this study are summarized in Table 1. While most GRACE and SLR data sets are named after their releasing institutes, the latest GRACE data set computed at TUG is termed ITSGGrace2018 (ITSG for short). Data after Aug. 2016 (7 data points in total) are not provided by all RL06 GRACE products, and are supplemented by the corresponding RL05 ones.
The LDCmgm90 dataset is provided in the netcdf 4.0 format and can be accessed via figshare^{60}, which contains five subsets GAA, GAB, GAC, GAD and GSM, all in the form of Stokes coefficients complete from degree and order 2 to 90.
Technical Validation
The degree2 GSM zonal and tesseral Stokes coefficients from LDCmgm90 and other individual releases are compared in Fig. 2a, while the GSM + GAA + GAB ones are compared in Fig. 2b. One can see the coefficients from LDCmgm are less noisy and free of anomalous signals presented in some other GRACE products, since when combining or assimilating data from different sources, the LDC method can provide a good handle of both the magnitude and phase aspects simultaneously for arbitrary frequency including the lowest frequency component which is usually called the trend of a series^{21,47}. The standard derivations of the original and corrected LDCmgm GSM (C_{20}, C_{21}, S_{21}) with respect with those from other model sets are provided in Tables 2 and 3, respectively. In addition, Figs 1, 3, 4 and Table 4 of Chen et al.^{21} implied that our C_{21} and S_{21} (the corresponding geophysical excitations are denoted as LDCgsc) are the most consistent with the observed polar motion, while Fig. 9 and Table 5 of Yu et al.^{22} suggested our C_{20} agrees the best with the observed lengthofday variations. A further and more independent check of the LDCmgm90 would be to compute the loads from the monthly gravity fields and apply those to GPS time series. However, the complexity of such a check makes it impossible to include in this short data descriptor so that will left for later work.
The mutual differences of geopotential maps for two neighboring months are also compared in Fig. 3. One can see the one corresponding to LDCmgm90 has no stripes, thanks to the Mason solutions used, while those for CSR, GFZ, JPL and TUG (only the map for CSR RL06 is provided here) have strong stripepattern noises, which overwhelm any geophysical signal of interest.
Code availability
The MatLab codes used to generate the LDCmgm90 are available upon request to W. Chen (wchen@sgg.whu.edu.cn).
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Acknowledgements
The editorial board member Dr. Kurt Kjær and two reviewers are highly appreciated for their insightful comments and suggestions, which helped to improve this study. We also thank the CSR, GFZ, JPL and TUG GRACE/SLR teams for making the GRACE/SLR monthly solutions publicly available. The CSR GRACE mascon solutions were downloaded from http://www2.csr.utexas.edu/grace, while the JPL mascon data are available at http://grace.jpl.nasa.gov, supported by the NASA MEaSUREs Program. This study is supported in parts by the National Natural Science Foundation of China (Nos 41874025 and 41474022), and the Fundamental Research Funds for the Central Universities of China (No. 2042016kf0146).
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W. Chen designed the framework of this study and performed part of the numerical computations. J. Luo and N. Yu processed the data and also contributed to numerical computations. J. Ray and J. Li helped to refine the research framework.
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Chen, W., Luo, J., Ray, J. et al. Multipledatabased monthly geopotential model set LDCmgm90. Sci Data 6, 228 (2019). https://doi.org/10.1038/s4159701902397
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