Five types of TVGFS are employed for combination, including (1) ITSG-Grace2018 models up to d/o 60 from GRACE/-FO observations spanning April 2002 to December 2020 (Kvas et al., 2019); (2) Tongji-LEO2021 models up to d/o 40 from six types of LEO satellites covering January 1993 to December 2004 (Chen et al., 2022); (3) IGG-Swarm models up to d/o 40 based on a combination of Swarm A, B and C observations from August 2014 to March 2021 (Lück et al., 2021); (4) IGG-UPWR-SLR models up to d/o 10 from SLR spanning February 1995 to November 2023 (Galdyn et al., 2024); and (5) IGG-SLR-HYBRID models up to d/o 60 from combined SLR and GRACE/-FO observations from January 1993 to December 2020 (Löcher and Kusche, 2021). Among these, ITSG-Grace2018 models offer the covariance matrices, while other products (excluding Tongji-LEO2021) provide the formal errors as initial covariance matrices. In contrast, Tongji-LEO2021 lacks precision information, and an identity matrix is adopted as its initial covariance matrix.

During the GRACE/-FO period, the degree-1 coefficients are restored using Technical Note 13, while the C20 and C30 coefficients are replaced with those of Technical Note 14 (TN14; Loomis et al., 2020). For the pre-GRACE and GRACE-gap periods, the degree-1 coefficients are derived from Löcher et al. (2025), whereas the C20 and C30 coefficients remain unmodified. During the entire period, the GIA effect is removed using the ICE6G-D model (Peltier et al., 2018), and the same mean field from the ITSG-Grace2018 models, spanning January 2004 to December 2009, is removed from all TVGFS.

Three distinctly reconstructed/combined products are utilized for comparisons:

IGG-SLR-DORIS, which only combined satellite-tracking observations from January 1984 to December 2023, including GRACE/-FO, SLR, and DORIS (Löcher et al., 2025);

Notably, IGG-SLR-DORIS is used only for comparison rather than combination because it additionally contains DORIS-related LEO information, which may overlap with the LEO contribution already introduced by Tongji-LEO2021 in our combination framework. Therefore, IGG-SLR-HYBRID is adopted for combination to better isolate the SLR contribution and avoid potential redundancy, while IGG-SLR-DORIS serves as a comparison product with comparable GRACE/-FO, SLR, and LEO-related information.

In this paper, CCM-CS is evaluated across global and regional Terrestrial Water Storage Anomalies (TWSAs), as well as polar ice sheets. Global-scale TWSAs are assessed using the sea level budget (Eq. S4 in the Supplement; Frederikse et al., 2020), wherein TWSAs are isolated by removing contributions from thermosteric expansion (Δhtherm) and mass changes in the Greenland Ice Sheets (GrIS; ΔhGrIS) and Antarctic Ice Sheet (AIS; ΔhAIS) from total Global Mean Sea Level (GMSL; ΔhGMSL). The GMSL is derived from averaged tide-gauge (Frederikse et al., 2020) and satellite altimetry observations (NASA-SSH, 2024), while other components are obtained from Frederikse et al. (2020), which integrates multiple observational datasets within a probabilistic framework. Regional TWS Changes (TWSC) are evaluated through the water balance equation (Eq. S5; Zhang et al., 2023), incorporating three precipitation, four evapotranspiration, and five runoff datasets from land surface models and in-situ sites, with their temporal and spatial resolutions detailed in Table S1 in the Supplement. Furthermore, AIS and GrIS mass changes derived from CCM-CS are validated against IMBIE (Ice Sheet Mass Balance Inter-comparison Exercise, Shepherd et al., 2021) time series of cumulative mass balance, providing reconciled estimates from altimetry, gravimetry, and input-output methods. During the GRACE period, the average of CSR and JPL RL06 mascon is additionally employed to evaluate TWS from CCM-CS. Comprehensive details of all datasets utilized in this study are summarized in Table S1.

For each TVGFS, SHCs of each d/o are fitted with a harmonic model plus NSS (Zhang et al., 2025a), 1yk,ilm=a0,ilm+a1,ilmΔtk+a2,ilmΔtk2+a3,ilmsin2πΔtk+a4,ilmcos2πΔtk+a5,ilmsin4πΔtk+a6,ilmcos4πΔtk+sk,ilm+ek,ilm where yk,ilm and ek,ilm denote the SHCs and corresponding observation errors from the ith model of degree l and order m at month k, with Δtk=tk-t1 representing the temporal interval between the kth month and the reference period. The coefficients a0,ilm, a1,ilm, and a2,ilm indicate constant, linear trend, and acceleration; a3,ilm, a4,ilm and a5,ilm, a6,ilm represent annual and semi-annual oscillations; sk,ilm encompasses the NSS.

By stacking all SHCs of the ith model at kth month into obervation vector yk,i, and denoting the combined observations from all qk models as yk=yk,1Tyk,2T⋯yk,qkTT, where qk accounts for the monthly-varying number of models for combination. Then Eq. (1) is rewritten as, 2yk=Akx+Bksk+ek

where the design and coefficient matrices are defined as Ak=Ak,1Ak,2⋮Ak,qk and Bk=Bk,1Bk,2⋮Bk,qk, with Bk,i and Ak,i (i=1,…,qk) constructed as,

3Bk,i=IMk,iOM-Mk,i4Ak,i=1ΔtkΔtk2sin⁡2πΔtkcos⁡2πΔtksin⁡4πΔtkcos⁡4πΔtkT⊗Bk,i where ⊗ denotes the Kronecker product; IMk,i and OM-Mk,i indicate the identity and zero matrices of size Mk,i×Mk,i and (M-Mk,i)×(M-Mk,i); Mk,i and M denote the number of monthly SHCs from the qkthindividual model and targeted combined model. The dimensional difference (M-Mk,i) arises from the varying quantities of monthly SHCs across different models. x and sk are the combined parameters and NSS, respectively. The combined observation errors and weights are denoted as ek=ek,1Tek,2T⋯ek,qkTT and Pek=blkdiag(Pek,1,Pek,2,⋯,Pek,qk). The covariance matrix Σek=σe2Pek-1, where σe2=blkdiag(σe12IM1,σe22IM2,⋯,σeqk2IMqk) and σei2 represents the time-invariant variance factor for ith model, used to balance magnitude disparities across TVGFS.

Since the NSSs sk vary from month to month, the observation equation after combining various models' SHCs may still be rank-deficient; additional equations or prior information are required to ensure a unique solution. In this paper, we utilize a Gauss-Markov (GM) model to describe the temporal correlations of NSS between r adjacent months (Zhang et al., 2025a), 5sk=∑j=k-rk-1Φk,jsj+wk(k>r) where Φk,j is a state transition matrix from epoch j to k, and wk is the normally distributed process noise with zero mean and covariance matrix Qk. Φk,j and Qk are constructed as follows (Zhang et al., 2025a),

6Φk,j=exp-tk-tjτs⊗IMQk=σs21-∑j=k-rk-1exp-2tk-tjτs⊗DDT where, exp⋅ is the exponential operator; τs and σs2 are the correlation time and variance component. D is a diagonal scale matrix with Dj,j=1/lj, where lj is the degree of jth element of sk (Siemes et al., 2013), accounting for the noise characteristics.

Compared to other temporal smoothing methods (e.g., moving average, spline smoothing, principal component analysis; Schrama et al., 2007; Crowley and Huang, 2020; Zhou et al., 2023), the GM model provides a simple linear expression that can be easily combined with observation equations and enables efficient recursive estimation via KF. By assuming stochastic process noise, the GM model yields time-dependent prediction covariance matrices, which facilitates integration into the CCM. Additionally, the GM model avoids the empirical selection of smoothing parameters required by other methods, such as moving window size, spline fitting order, or number of principal components.

The combined parameters are determined based on the cost function of Tikhonov regularization, 7min:∑k=1nekTΣek-1ek+αxTRx where n is the total months; α and R=DTx^gx^gTTTDT-1 (Siemes et al., 2013; Zhang et al., 2025a) are regularization parameter and matrix, with T denoted as the spherical harmonic analysis matrix and xg defined as the spatial expressions of parameters x^. Once the initial value of the NSS vector in Eq. (2) is given as s^k(0), we can derive the regularized solution to parameter vector x based on Eq. (7), 8x^=∑k=1nAkTΣek-1Ak+αR-1∑k=1nAkTΣek-1yk-Bks^k(0)

The combined NSS vector sk is estimated based on the following cost function, 9min:∑k=1nekTΣek-1ek+sk-s‾kTΣs‾k-1sk-s‾k where s‾k and Σs‾k are predicted NSS and covariance matrix at kth month.

Given Kk=Σs‾kBkTΣek+BkΣs‾kBkT-1 as the gain matrix, s^k is estimated using the Kalman Filtering (KF) process (Ji et al., 2013) with the parameters x^ determined using Eq. (8). The prediction and update steps of the forward KF are shown in Eqs. (10) and (11), 10s‾k=∑j=k-rk-1Φk,js^jΣs‾k=∑j=k-rk-1Φk,jΣs^kΦk,jT+Qk11Kk=Σs‾kBkTΣek+BkΣs‾kBkT-1s^k=s‾k+Kkyk-Akx^-Bks‾kΣs^k=IM-KkBkΣs‾k

Initial s^1 and Σs^1 are empirically determined as specified in Zhang et al. (2024) and (2025a). To mitigate initial value bias, a backward KF process is also implemented (Ji et al., 2013), which is the time-reversed version of the forward process. The final estimates s^k and Σs^k are obtained by averaging the bi-directional KF results. After deriving the combined x^ and s^k, the combined monthly TVGFS is derived through y^k=A‾kx^+s^k, with A¯k=1ΔtkΔtk2sin⁡2πΔtkcos⁡2πΔtksin⁡4πΔtkcos⁡4πΔtkT⊗IM.

Since the estimated equations for x^ and s^k in Eqs. (8) and (11) are mutually nested; the estimation process is iteratively conducted. The iteration process involves updating the variance factors σei2, regularization parameter α, regularization matrix R, and optimal variablesof GM model. Among these, regularization parameter α together with variance factors σek,i2 are iteratively estimated based on the minimum Mean Square Error (MSE; Shen et al., 2012; Ji et al., 2022) and bias-corrected Variance Component Estimation (VCE; Xu et al., 2006). Additionally, the optimal variables of GM model, order r, correlation time τs, variance σs2, are searching by minimizing the cost function over predefined search ranges at grid intervals (Ji et al., 2013), 12L=1n∑k=1nlogdetCk+1n∑k=1nv‾kTCk-1v‾k where v‾k=yk-Akx^-Bks‾k reflects the predicted errors, with their covariance matrix Ck=Σek+AkMx^AkT+BkΣs‾kBkT, Mx^ is the MSE of x^ (Shen et al., 2012; Ji et al., 2022). log⋅ and det⋅ are the natural logarithm and determinant operators.

In the implementation, the model order r is searched over integer values from 1 to 6. For each r, τs and σs2 are jointly searched within a predefined two-dimensional parameter space. Considering the monthly sampling of TVGFS and the need to avoid overly large process noise that may weaken temporal constraints and introduce high-frequency noise, τs is searched from 30 to 360 d with a 30 d interval, while σs2 is searched from 0.001 to 10 cm² with an interval of 0.01 cm².

A concise overview of our study is presented in Fig. 1. The iterative estimation process is detailed in Sect. S2 of the Supplement, with the corresponding flowchart shown in Fig. S5 and core symbols summarized in Table S2. The iteration continues four times until the difference in x^ between consecutive iterations falls below 1 %. Across various iterations, the regularization parameter α and variance factors σei2 of ITSG-Grace2018, Tongji-LEO2021, IGG-Swarm, IGG-SLR-HYBRID, and IGG-UPWR-SLR, for j ranging from 1 to 5, are presented in Table 1, along with the optimal r, τs, and σs2 for the GM model. Since Tongji-LEO2021 exhibits a larger variance than other solutions (except for IGG-UPWR-SLR), its weight in the combination is inherently small, making the specific choice of its initial weight matrix inconsequential, allowing an identity matrix to approximate with negligible impact (Koch, 1999; Teunissen, 2000).

Based on the CCM, we develop a gapless time series of gravity field solutions up to d/o 60 from January 1993 to December 2024 through combining monthly TVGFS from ITSG-Grace2018, Tongji-LEO2021, IGG-Swarm, SLR-UPWR, and SLR-HYBRID models. Figure S1 in the Supplement provides the Root Mean Square (RMS) of spatial TWSAs and SHCs across all months, derived from individual solutions and Combined Solutions based on VCE (VCE-CS), which is the most widely used conventional combination method (Jean et al., 2018; Meyer et al., 2019; Zhong et al., 2021). VCE independently combined monthly TVGFS based on their covariance matrices (Meyer et al., 2019). The results indicate that spatial TWSAs and SHCs at high d/o are seriously polluted by striping noise for both individual TVGFS and VCE-CS, requiring additional filtering.

To maintain the consistency with CCM-CS and given the superior performance of the CCM-based filter over all other common spectral filters (Zhang et al., 2024, 2025a), we denoise the individual solutions using the CCM, setting the model number to one as described in Sect. 3.1. The resulting RMS of spatial TWSAs and SHCs across all months are provided in Figs. 2a–e and 3a–e, with those from CCM-CS presented in Figs. 2g and 3g. Additionally, the average SNRs (Eq. S6) and degree powers (Eq. S7) for SHCs are shown in Figs. 2h and 3h–i. The results indicate that CCM-CS exhibits higher SNRs than other individual solutions, effectively removing the serious noise existing in Tongji-LEO2021 (Fig. 2b) and IGG-Swarm (Fig. 2c) solutions, as well as the residual striping noise over oceans in ITSG-Grace2018 (Fig. 2a) and SLR-HYBRID (Fig. 2e) solutions, which originates from spatial leakage and spectral truncation effects (Siemes et al., 2013). In the spectral domain, CCM-CS also demonstrates remarkable noise reduction in IGG-Swarm (Fig. 3c) and Tongji-LEO2021 (Fig. 3d) solutions, while maintaining excellent consistency with the GRACE/-FO solutions from the ITSG-Grace2018 model in terms of low-degree power spectra (Fig. 3i) that primarily contain TVGF signals. Notably, the low-degree power spectra of SLR-HYBRID are significantly lower than those of CCM-CS. These findings underscore that CCM-CS is more effective than individual solutions in simultaneously suppressing noise and preserving signals.

To further demonstrate the superior noise removal efficiency of CCM-CS compared to VCE-CS, the corresponding results are also provided in Figs. 2 and 3. The results indicate that VCE-CS is significantly affected by striping noise from Tongji-LEO2021 and IGG-Swarm, particularly during the gap period of GRACE, resulting in the spatial distribution (Fig. 2f) and spectral power for SHCs between d/o 30 and 40 (Fig. 3f and h), closely resembling Tongji-LEO2021 (Figs. 2b and 3d) and IGG-Swarm solutions (Figs. 2c and 3c). Meanwhile, the low-degree power spectra of VCE-CS are significantly lower than CCM-CS (Fig. 3i). Notably, CCM-CS estimates the combined parameters over a long-term temporal scale and models NSS by utilizing their temporal correlations, which results in more stable solutions and less influence by the absence of GRACE observations, performing significantly higher SNRs (Fig. 2h) and lower noise in SHCs between d/o 30 and 40 compared to VCE-CS (Fig. 3h).

Additionally, the variations ΔC20 and ΔC30 from CCM-CS and individual solutions are provided in Fig. 4a and b, using the official TN14 product as reference. The results demonstrate that CCM-CS effectively reduces the bias of ΔC20 and ΔC30 from individual solutions. Relative to TN14, the Nash-Sutcliffe Efficiency (NSE; Eq. S8; Nash and Sutcliffe, 1970) for ΔC20 and ΔC30 derived from CCM-CS reach 0.96 and 0.98, outperforming ITSG-Grace2018 (0.34), SLR-UPWR (0.93), and SLR-HYBRID (0.88) for ΔC20, and 0.86, 0.14, and 0.94 for ΔC30, respectively. Since the variations ΔC20 and ΔC30 are critical for accurate polar ice sheet mass estimation, we further present the latitude-weighted mass anomaly series of GrIS and AIS derived from CCM-CS and individual solutions up to d/o 60 in Fig. 5a and b, alongside independent IMBIE products that offer reconciled ice sheet mass estimates from altimetry, gravimetry, and the input-output method. Results reveal that the mass anomalies of GrIS and AIS derived from CCM-CS align more closely with IMBIE relative to individual solutions. Notably, the mass anomalies derived from SLR-HYBRID exhibit significant lower signals during GRACE-FO period for GrIS and after 2005 for AIS, while showing anomalously higher signals during 1993 for GrIS and earlier 1995 for AIS, as highlighted by red dashed rectangles. Meanwhile, ITSG-Grace2018 displays substantial anomalous signals due to poor quality of ΔC20 and ΔC30 coefficients. Based on the good performance of ΔC20 and ΔC30 in CCM-CS, particularly vital for the missing months of TN14, we replace ΔC20 and ΔC30 for months with available TN14 product while maintaining no adjustment for the remaining months.

In this section, CCM-CS is evaluated against three distinct products, including IGG-SLR-DORIS (Löcher et al., 2025), BNML (Mandal et al., 2025), and RESDCAE (Uz et al., 2024), derived from satellite-tracking observations, hydrometeorological models, and a combination of both, spanning from January 1993 to December 2023, January 1993 to December 2024, and January 1994 to December 2020, respectively. IGG-SLR-DORIS is expressed as spherical harmonic coefficients (SHCs) up to d/o 60, while BNML and RESDCAE are presented on 1° spatial grids. Notably, BNML lacks the mass change estimates over Antarctica and Greenland.

The monthly combined TVGFS based on CCM is derived through y^k=A‾kx^+s^k, yielding the monthly covariance matrix Σy^k=A‾kΣx^A‾kT+Σs^k, where Σx^ is computed via covariance propagation based on Eq. (8) and Σs^k is obtained from Eqs. (10) and (11). In the spectral domain, uncertainty of monthly combined solutions is derived from the diagonal elements of covariance matrix Σy^k, while in the spatial domain, uncertainty of monthly TWSA is propagated from Σs^k using the spherical harmonic synthesis matrix. Additionally, the uncertainties of monthly TVGFS from IGG-SLR-DORIS and monthly spatial TWSA from RESDCAE are accessible, with the former propagated from errors of SLR solutions and GRACE/-FO EOFs (Löcher et al., 2025) and the latter computed from standard deviations (Uz et al., 2024). Since the inputs of BNML only include noise-free hydrometeorological datasets, different from those of CCM-CS, its uncertainty is not used for comparison. Across all common months, we compute average uncertainties of TVGFS from CCM-CS and IGG-SLR-DORIS, alongside those of spatial TWSAs from CCM-CS and RESDCAE, through error propagation from monthly uncertainties. The average uncertainties from CCM-CS relative to those from IGG-SLR-DORIS and RESDCAE are provided in Fig. 12a and b. Results demonstrate that CCM-CS exhibits lower uncertainties than IGG-SLR-DORIS solutions, particularly for SHCs at high d/o, while also showing reduced uncertainties relative to RESDCAE over most regions. Average uncertainties across SHCs of all d/o and all spatial TWSA grids are improved by 13.4 % and 14.6 % compared to IGG-SLR-DORIS and RESDCAE solutions, respectively.