The GFZ-RL07 AO model is the official de-aliasing product for all existing satellite gravity missions. It consists of an atmospheric component based on operational and reanalysis (ERA5; ) datasets of the ECMWF, and an oceanic component derived from unconstrained simulations using the MPIOM (Max Planck Institute for Meteorology Ocean Model; ) ocean model, which is consistently forced by the corresponding atmospheric fields of the ECMWF. Unlike CRA-40, ERA5 is based on the Integrated Forecasting System (IFS) Cycle 41r2 with 4D-Var data assimilation, which provides hourly output at 31 km horizontal resolution and includes 137 vertical levels up to 0.01 hPa. The data assimilation system utilizes observations from more than 200 satellite instruments and conventional sources, including selected data from FY-3B/C . MPIOM uses a 1° tri-polar Arakawa-C grid with 40 vertical levels and newly includes cavities underneath the Antarctic ice-shelf and SAL (self-attraction and loading; ) feedback. GFZ RL07 AO model provides non-tidal atmospheric and oceanic components with 3 h temporal resolution and spherical harmonic expansion up to degree/order 180 alongside selected tidal constituents slower than 6 h. GFZ-RL07 is accessible via https://isdc.gfz-potsdam.de/esmdata/aod1b/ (last access: 29 March 2024) and is used here for comparison with CRA-LICOM.

Temporal gravity field (Level-2) using GRACE Level-1b products can be used to assess the accuracy of CRA-LICOM for current satellite gravity missions. GRACE Level-1b products, including along-track range(-rate), accelerometer, star camera attitude, and reduced dynamic orbit data, are available at https://podaac.jpl.nasa.gov/dataset/GRACE_L1B_GRAV_JPL_RL03 (last access: 20 December 2024). Additionally, the latest version (RL06) of GRACE Level-2 temporal gravity fields (in terms of spherical harmonic coefficient) from CSR (Center for Space Research from the University of Texas at Austin, Texas, USA), JPL (Jet Propulsion Laboratory, USA), and GFZ are used for further validation, accessible via https://icgem.gfz-potsdam.de/home (last access: 5 January 2025).

Altimeter is widely used to monitor sea level change that consists of the steric and non-steric compartments, where the latter is mainly caused by mass change. As Argo is capable of measuring the steric sea level, the altimeter combined with Argo can reflect the non-steric sea level, theoretically equivalent to GRACE-derived ocean mass change plus the currents, that is, the AO model . Therefore, Altimeter and Argo are often used to validate GRACE as well as its underlying AO model . In this study, an ensemble of three Argo products is adopted, that is, BOA , EN4 , and SIO , covering the upper ocean above 2000 m. Altimeter data are collected from AVISO at a resolution of 0.25° × 0.25° and 10 d intervals, with further calibration of GIA (Glacier Isostatic Adjustment, see ) as suggested.

Ocean Bottom Pressure (OBP) data from the Deep Ocean Assessment and Reporting of Tsunamis (DART) system are utilized for validation. We download the quality-controlled and de-tided OBP data with a 15 s resolution from DART for the period 2002–2018. After processing the OBP into hourly mean data, we select timestamps every six hours starting from 00:00, such as 00:00, 06:00 (UTC 00:00), and so on. Finally, we obtained OBP datasets that included 68 locations between the years 2002 and 2023, mainly distributed over the Pacific and Atlantic Oceans. In addition, the global temperature and salinity obtained from Argo are used to further validate the simulated oceanic conditions, which are critical for accurate OBP simulations. Such monthly Argo data span 2005–2020 with a horizontal resolution of 1° × 1° and a vertical resolution of 27 layers (depths of up to 2000 m), which are available at https://apdrc.soest.hawaii.edu/projects/Argo/data/gridded/On_standard_levels/index-1.html (last access: 14 December 2024).

The additional data set includes GLDAS (Global Land Data Assimilation System, https://ldas.gsfc.nasa.gov/gldas/, last access: 1 February 2025), which is based on advanced land surface modeling and data assimilation techniques to merge satellite- and ground-based observations into the model. GLDAS provides high-quality global land surface fields to support the investigation of change in TWS . In this study, we extract TWS (3 h and 1° × 1°) from GLDAS to approximate the component of the global hydrology (H) signal to be compared with the AO component, as indicated by CRA-LICOM.

The input surface pressure fields represent a mixture of tidal and non-tidal constituents, which should be isolated as a first step. The logic behind the isolation is the fact that the model can often better predict tides. As limited by Nyquist sampling law, the expected tidal signals extracted from CRA-40 (6 hourly) must be slower than the semi-diurnal tide, which includes P1 (14.9589314° h⁻¹), S1 (15.0000000° h⁻¹), K1 (15.0410686° h⁻¹), N2 (28.4397295° h⁻¹), M2 (28.9841042° h⁻¹), L2 (29.5284789 deg/h) and T2 (29.9589333° h⁻¹). Then, a point-wise tidal pressure can be obtained from a summation of all these frequency-dependent tides ζs (assuming a tide s with frequency ωs, amplitude ξs and phase δs) following 1ζ(θ,λ,t)=∑sξs(θ,λ)cos⁡[ωst+χs-δs(θ,λ)]=∑s[As(θ,λ)cos⁡(ωst)+Bs(θ,λ)sin⁡(ωst)], where (θ,λ) denotes the spherical coordinate (colatitude, longitude) of the point and t denotes an arbitrary time epoch. In particular, χs is the Warburg phase correction documented in ; the time reference where δs=0 is selected as 1 January 2007 00:00:00. And in Eq. () the amplitude and phase can be translated into the coefficients As(θ,λ) and Bs(θ,λ) to enable the ordinary least squares (OLS) solution. In this study, the OLS is configured by terms of trending C(θ,λ), tidal ζ(θ,λ) and non-tidal D(θ,λ) signals: 2P(θ,λ,t)=∑s[As(θ,λ)cos⁡(ωst)+Bs(θ,λ)sin⁡(ωst)]+C(θ,λ)t+D(θ,λ), where parameters (As,Bs,C,D) are fitted from the “observations”, i.e., the time series of surface pressure P(θ,λ,t). Subsequently, each tide ζs (its amplitude and phase) can be recovered from its coefficients (As,Bs) via Eq. (). Be aware that a high-pass filter (with a time window of 3 days) is applied to P beforehand, to damp non-tidal signals first and stabilize the tidal estimations. Here, the tide constituents are fitted from the years 2007–2014, consistent with the period used in AOD1B RL06 .

To accurately reflect the non-tidal atmospheric gravity field change, one has to exploit layered observations to account for contributions from both surface and upper air anomalies. To this end, two types of air mass integration are required: (1) a surface integration that considers the air mass as a thin layer, that is, by neglecting the vertical structure of air; (2) a 3-D vertical integration of all mass columns to obtain the upper air contribution. Regardless of either type of integration, the first step is to obtain the “inner integral”, i.e., In, which is often degree dependent (the degree of spherical harmonic expansion); see . For surface integration, In is treated as 3Ins(r,θ,λ)=raen+2ΔP0(r,θ,λ)g(r,θ,λ)=a(θ,λ)+ζ(θ,λ)+h(θ,λ)aen+2⋅ΔP0(r,θ,λ)g(r,θ,λ), where (r,θ,λ) is the spherical coordinate (radial distance, colatitude, and longitude) of the evaluated point. In the case of a realistic Earth, the radial distance r consists of the ellipsoidal radius a, the geoid undulation h , and the topography ζ (e.g., ). In addition, in Eq. () ae denotes the Earth's mean radius, e.g., ∼ 6 378 136.6 km; g(r,θ,λ) is the gravity acceleration of the evaluated point. ΔP0(r,θ,λ) indicates the pressure difference between two layers, but because only the surface layer is adopted here so that ΔP0(r,θ,λ) equals the surface pressure.

By contrast, the 3-D vertical integration is able to consider the vertical structure of the atmosphere by taking advantage of multi-level atmospheric input fields. Here, CRA-40, in terms of pressure levels, is used that yields 4Inv(r,θ,λ)=∑k=0kmax⁡(a(θ,λ)+ζ(θ,λ)+h(θ,λ)+zk(θ,λ)ae)n+2⋅ΔPk(θ,λ,rk)gk(θ,λ,rk), where the inner integral is discretized into k-layers, and for CRA-40, it has maximal kmax⁡=47 layers. ΔPk(θ,λ,rk) indicates pressure difference between adjacent layers kth and (k-1)th. Comparing Eq. () to Eq. (), the zk emerges as the geometric distance from the evaluated point to the surface, which must be solved from an accumulation of geopotential differences between adjacent pressure layers. To this end, the multi-level humidity and temperature fields are required to obtain the geopotential height difference; see for more details.

It is worth mentioning that because CRA-40 is given in terms of pressure level, integration with Eq. () might face risks of “outliers”: there could be some cases in which the pressure of a layer is greater than the surface pressure. In other words, the specific isobaric surface goes through the interior of the Earth, which is obviously unreasonable in physics. It is relevant to calibrate this outlier. Otherwise, the modeling quality would be significantly degraded. To this end, we propose a calibration method, which yields 5Vk(θ,λ)=Vk-1(θ,λ),∀ΔPk(θ,λ,rk)<0, where Vk indicates an arbitrary variable in the kth layer, which could be either pressure, temperature, or humidity. In this manner, outliers can be identified and fixed with a reasonable approximation (the neighboring value at the next pressure level aloft). One can see our preliminary work for more details on the calibration approach for atmospheric pressure-level reanalysis, while this study will focus on the evaluation and validation of the AO model.

An accurate modeling of the non-tidal atmospheric gravity field requires us to account for both the direct gravitational effect and the indirect Earth's deformation effect. Therefore, a combination of the hypothetical thin-layered air (with two further corrections) and the upper air is necessary. The implementation of the combination follows the method proposed by , i.e., 6In=(Ins-Intide+InIB)+11+kn(Inv-Ins), where the kn indicates the degree-dependent load Love number, and the quantity in the first bracket indicates the non-tidal surface air that accounts for both the direct and indirect effects. By contrast, the other quantity in the second bracket indicates the upper air that accounts for only the direct effect, which makes sense since it cannot lead to Earth's deformation. In addition, corrections are made to the surface integral that includes: (i) tide removal as indicated by Intide, which can be modeled by Eq. (), and (ii) Inverted Barometer (IB) correction as indicated by InIB, which is introduced to only include the static contribution to OBP into the ATM coefficients; please see for more details.

We used the low-resolution global configuration of LICOM3.0 to obtain 3 hourly, 360 × 218 tri-polar (equivalent to 1° on average) OBP data spanning 2002–2024, as depicted in Fig. b. The ocean model adopts primitive equations, comprising the full form of Navier-Stokes equations, continuity equations, conservation equations for temperature and salinity, and the equation of state for seawater. These equations are discretized on a tri-polar grid and 30 vertical layers. The model workflow consists of three main blocks:

Initialize: This block manages the reading of grids, initial states, and parameters. The initial conditions use global climatology for temperature and salinity from PHC3.0 (Polar Science Center Hydrographic Climatology; )

More information on LICOM3.0 can be found in the Appendix . In this paper, the model was spun up for five cycles from 2002–2023. Then, at the end of the 5th cycle, an integration from 1 January 2002, to existing months in 2024 was conducted and analyzed. LICOM reached equilibrium within six spin-up cycles (see Fig.  in Appendix ), suggesting that no artificial drift remains in the system. The OBP in LICOM3.0 is the sum of atmospheric pressure and the vertical integration of seawater density between dynamic sea level and ocean bottom, computed as 7Pb=Pa+H0ρ(1)g+∑k=130ρ(k)Δz(k)g, where Pb is the OBP, Pa is atmospheric pressure, H0 is dynamic sea level, ρ and ρ(1) are seawater density and its value at surface level, g (= 9.80665 m s⁻²) is the gravity acceleration, k is vertical layer number, and Δz is vertical layer thickness. Note that although Eq. () is applied in 3D at time t and position (θ,λ), the dimension notation is omitted for readability.

For the same reason as atmospheric gravity field modeling, ocean tides must be estimated and removed from the oceanic contribution to OBP. Be aware that LICOM3.0 does not directly simulate the lunisolar gravitational tides in the oceans. Hence, oceanic tidal fluctuations are solely induced by periodically varying atmospheric forcing. Furthermore, because atmospheric pressure is set to zero in the model's momentum equations, the resulting oceanic tides are mainly driven by tidal variations in other atmospheric forcings, such as solar radiation (e.g., ) and wind (e.g., ), rather than barometric pressure loading. Therefore, the amplitudes and fluctuations of the simulated oceanic tides are likely much weaker than in reality. For example, the global mean amplitude of S2 simulated by LICOM3.0 in this study is approximately 101 Pa, compared to ∼103 Pa in the tidal models of .

In addition to the seven tidal constituents mentioned in Sect. , S2 (30.0000000° h⁻¹), R2 (30.0410667° h⁻¹) and SK₃ (45.0410760° h⁻¹) are also removed using the T_TIDE package , due to a higher sampling rate of OBP, i.e., 3 h. In this study, tide fluctuations are calculated annually for 2002–2023. Furthermore, be aware that the effect of atmospheric loading must be removed beforehand. Hence, the overall formula to obtain non-tidal oceanic contribution to OBP follows 8Pbnt=(Pb-Pa)-tide(Pb-Pa), where Pbnt is the non-tidal oceanic contribution to OBP, and the second term indicates the removal of tides.

Because of the Boussinesq approximation in the momentum equations of LICOM, mass conservation is no longer preserved, as density changes within this volume-conservative model. To ensure mass conservation, a global mass correction has to be implemented following the method proposed by . This correction involves subtracting the mean OBP across the entire ocean domain at each time step, which yields 9Pbdyn=Pbnt-1/Aoceans∫PbntdA, where Pbdyn is the non-tidal dynamic OBP, which will be used in Eq. () as well to obtain the inner integral InO of the ocean in Eq. (). Be aware that the ultimate temporal resolution of Pbdyn is down-sampled from native 3 to 6 h to be consistent with that of the atmospheric forcing field.

In this section, a straightforward comparison (against the official GFZ-RL07 product) is made at the product level itself. To assess the temporal performance of CRA-LICOM, temporal correlations and biases were evaluated in both the spectral and spatial domains.

First, in the spectral domain, the Stokes coefficients of CRA-LICOM and GFZ-RL07 products were analyzed for all degrees up to 60, for example. Figure  presents the mean temporal correlation coefficients of the Stokes coefficients per degree. At lower degrees (n<15), CRA-LICOM exhibits correlations exceeding 0.8, demonstrating fairly good agreement with GFZ-RL07 for gravity signals on a medium to large spatial scale. The consistency of low degrees is important since it is known that the AO model, as well as the GRACE gravity field, has its major energy at those degrees. As the degree increases, the correlation gradually decreases but remains statistically significant at the confidence level of 99 %. The peak correlation occurs at degree 5 for coefficient C (0.89) and degree 6 for coefficient S (0.90). Then, the standard deviation (SD) ratios of the two products were also analyzed to evaluate the variability biases; see the dashed curves in Fig. . CRA-LICOM generally slightly underestimates the variability coefficients compared to GFZ-RL07, with the SD ratios stabilizing over 90 % throughout the spectrum. Despite a decline at lower degrees, the ratio is constantly increasing from degrees 30 to 100, eventually reaching around 100 % at degree 100 (not shown), indicating CRA-LICOM's capability to reproduce high-frequency signals effectively.

Furthermore, the Stokes coefficients of ATM, representing atmospheric effects, and OCN, representing the dynamic oceanic contribution to OBP, are also shown in Fig. . Compared to OCN, the correlations between CRA-LICOM and GFZ-RL07 for ATM are consistently higher across all degrees, with smaller variation biases. As GLO reflects the combined effects of both ATM and OCN, its correlation and SD deviations lie between those of ATM and OCN. The correlation of ATM gradually decreases with increasing degree. For the first 20°, ATM maintains a high correlation, with coefficients exceeding 0.9; by around degree 50, the correlation drops to approximately 0.6. For OCN, the correlation of OCN peaks at degree 5 or 6, and since the peak the correlations gradually decline, reaching around 0.6 near degree 30. Furthermore, CRA-LICOM shows greater variability in ATM compared to GFZ-RL07. The SD ratio of ATM-C ranges from 105 % to 124 %, and that of ATM-S ranges from 106 % to 128 %. In contrast, CRA-LICOM exhibits weaker variability in OCN. For degrees below 30, the SD ratio remains around 80 %, gradually increasing thereafter but never exceeding 100 %. These findings suggest that atmospheric gravity variability in CRA-LICOM is closely aligned with those in GFZ-RL07, while discrepancies in oceanic gravity variations persist, likely reflecting uncertainties between models.

Then, evaluations are made in the spatial domain by projecting two products onto pressure fields on a regular grid 1° × 1°. To be consistent with the 1-month resolution of the present satellite gravity mission, the time-series pressure fields are decomposed into a frequency variability <60 d and another frequency variability >60 d. Figure a illustrates the temporal correlation coefficients for that <60 d, from which we see: (1) nearly all are statistically significant at a 99 % confidence level; (2) the correlations decrease with decreasing signal levels; (3) the correlations are substantially higher overland (0.85 on average) than over the ocean (0.63 on average). The high correlation over land indicates an overall consistency of atmospheric forcing fields employed, despite a few exceptions, such as Central Africa and the Northern region of South America, where the correlation coefficient degrades to around 0.7. We attribute this degradation as a consequence of the remaining S2 atmospheric tide in CRA-LICOM since the spatial pattern resembles that of S2 at a high similarity, see . In contrast, S2 has been removed from GFZ-RL07. For oceanic regions, mid- to high-latitudes have a correlation coefficient above 0.7, while it is lower at mid- to low-latitude oceans, particularly the Atlantic Ocean, and marginal seas exhibit weaker correlations down to 0.3, indicating a larger discrepancy between the two ocean models in these areas.

Furthermore, Fig. c illustrates the root mean square error (RMSE) of global temporal variability at a time window of <60 d. The global mean RMSE is found to be 1.79 hPa, while most pronounced biases are observed in the continental shelves and marginal seas of the Arctic Ocean, offshore China, and Hudson Bay, with RMSE peaks of up to 15 hPa. These discrepancies may result from differences in how the models represent topography, parameterization schemes (particularly wind stress), and the bottom friction law. The elevated RMSE in the Ross Sea can be attributed to LICOM's limited ability to simulate OBP in this region. Furthermore, notable biases are evident in the Southern Ocean, where the RMSE averages around 4 hPa, smaller than the SD value, approximately 8 hPa. These findings suggest that CRA-LICOM effectively captures consistent temporal variability amplitudes across most regions, except the marginal seas near continental shelves. Figure b and d demonstrate the correlations and biases for periods >60 d, revealing stronger correlations and smaller biases globally, while this has little impact on satellite gravity because its spectrum is slower than the aliasing frequency. The average correlation coefficients are 0.89 for the land and 0.71 for the ocean regions, with a global mean RMSE of as low as 1.29 hPa. This confirms the improved agreement between CRA-LICOM and GFZ-RL07 for longer periods. However, model uncertainties are more pronounced at higher frequencies, which poses a challenge for OBP simulations in the context of de-aliasing satellite gravity observations.

The observation of satellite gravity on orbit along the track, for example, the intersatellite K-band range rate (KBRR), is extremely sensitive to the geophysical process over regions where satellites fly . Therefore, KBRR, especially its residuals after removing essential background geophysical signals, including AO, can be an effective indicator of the quality of the AO model . In particular, since the postfit KBRR residuals, obtained after least-square adjustment of the temporal gravity field, is more sensitive than the prefit KBRR to the mis-modeling error, we select the postfit KBRR residuals as the metric in this study. Here, we use data from Sect.  to calculate the postfit KBRR residuals for an initial diagnosis of two AO products, i.e., GFZ-RL07 and CRA-LICOM. All data processing to obtain the final KBRR residuals is manipulated by our open source Python software (namely PyHawk, https://github.com/NCSGgroup/PyHawk.git, last access: 1 February 2025, see also ), which indeed has achieved a complete data processing chain from Level-1b raw data to Level-2 temporal gravity fields.

Figure  illustrates the spatial map of the KBRR residuals in terms of gridded root mean square (RMS), where an arbitrary month is selected as an example. By comparing Fig. a to b, both scenarios, as expected, have demonstrated the successful removal of the major temporal gravity signals, so that the overall residuals are displayed as white noise. However, for example, in Fig. a, there are still some places where the residuals are obviously larger than the average, suggesting a greater uncertainty of the recovered signals in these places. In this sense, we may find in Fig. b that the uncertainty of CRA-LICOM is slightly stronger than that of GFZ-RL07. For a statistical study, we derive their differences in Fig. c. Since AO is utilized as a prior model to be removed from KBRR observations and the only difference between Fig. a and b is the AO model, one can always expect a smaller KBRR residual if the prior model better reproduces the reality. In Fig. c, the global mean of the differences is found to be -12.81 nm s⁻¹, which, as a negative value, suggests that CRA-LICOM is slightly more noisy. However, since the state-of-the-art KBRR is insensitive to noise less than ∼ 100 nm s⁻¹, the slight degradation of CRA-LICOM relative to GFZ-RL07 cannot be captured. Likewise, the global RMS of Fig. c is only 39.85 nm s⁻¹, which is also less than ∼ 100 nm s⁻¹. Furthermore, we exclude meaningless values by setting a threshold of 100 nm s⁻¹ in Fig. c; as a result, for the remaining data, the proportion (relative to the entire map) of positive and negative grids is 0.28 % and 2.38 %, respectively. The small proportion indicates that a majority of their differences are insensible, while GFZ-RL07 slightly outperforms CRA-LICOM due to a higher proportion of negatives. Also, be aware that GFZ-RL07 is better temporally resolved (3 h), which should also be responsible for its superiority.

As one step further than in the previous section, we recover Earth's temporal gravity fields up to a degree/order of 60 for five years. The period from 2005–2010 was selected as an example to take advantage of GRACE's stable performance. Then, a series of standard post-processing steps of the gravity fields obtained were performed, which include (but are not limited to): (1) conversion of the gravity field to the equivalent height of water (EWH) on a gridded map of 0.5° × 0.5°, (2) replacement of low degree Stokes coefficients , (3) spatial filtering, DDK3, to damp the noise , (4) removal of glacial isostatic adjustment , etc. Subsequently, a linear regression is performed to extract the climatology and residuals, to indicate the signal and noise level of the obtained gravity field. All the aforementioned post-processing and signal/error analyses are conducted using our open source Python software (called SaGEA, https://github.com/NCSGgroup/SaGEA, last access: 1 February 2025), which follows a standard workflow to handle the Level-2 gravity fields, see for more details.

Figure  presents a detailed comparison between the latest official CSR gravity field product, which adopts the GFZ-RL07 AO model, and our derived gravity field solution (hereafter referred to as CRA-LICOM for consistency), which utilizes the CRA-LICOM AO model. The first two rows of the figure demonstrate that both solutions achieve comparable signal magnitudes. Specifically, the secular trend exhibits a high spatial correlation coefficient of 0.94, while the annual amplitude shows an even stronger correlation of 0.96, reinforcing that the CRA-LICOM constitutes a viable alternative AO model for current satellite gravity missions. However, a notable distinction arises in the noise characteristics of the two solutions. As evident in Fig. , our gravity field product exhibits a systematically higher noise level compared to the CSR solution, although this elevated noise remains within the documented uncertainty range of GRACE-derived gravity solutions . Furthermore, the spatial distribution of errors in Fig. e–f reveals a striking resemblance to the patterns observed in Fig. , particularly in regions such as the Ross Sea, Indonesian archipelago, Arctic coastal zones, Black Sea, and Baltic Sea. This spatial coherence strongly suggests that (1) the uncertainty of the AO model is a dominant contributor to the overall uncertainty in the derived gravity field and (2) the current AO model exhibits reduced reliability in these specific regions due to a likely incomplete representation of ocean dynamics or atmospheric coupling. A comprehensive discussion of these limitations, including potential avenues for model refinement, will be provided in Sect. . In addition to the AO model, we acknowledge that our processing skill for GRACE gravity field recovery, although robust, does not yet match the optimization of CSR in noise suppression. And this can also be responsible for the higher noise in our product.