Non-Linear Global Ice and Water Storage Changes from a Combination of Satellite Laser Ranging and GRACE Data

Highlights

What are the main findings?

• A combined SLR and GRACE gravity model spanning 1995–2024 reveals significant non-linear mass changes, identifying specific trend reversal dates for global ice and water reservoirs, such as the 2004 peak in Svalbard and the 2021 trend reversal in the Antarctic Peninsula.
• The analysis demonstrates that linear trend models fail to stabilize even with 30 years of data, whereas models incorporating acceleration parameters achieve stabilization for most polar regions after 15–20 years.

What are the implications of the main findings?

• This study proves that extending the satellite gravimetry record back to 1995 using SLR allows for the accurate detection of climate-driven hydrological events, e.g., the 1997/1998 El Niño, prior to the GRACE mission launch.
• Incorporating acceleration terms into long-term gravity models provides a more reliable metric for monitoring climate change impacts than linear trends alone, particularly for detecting the onset of rapid ice mass depletion or recovery.

Abstract

Determining long-term changes in global ice and water storage from satellite gravimetry remains challenging due to the limited temporal coverage of high-resolution missions. Here, we combine Satellite Laser Ranging (SLR) and Gravity Recovery and Climate Experiment (GRACE) data to reconstruct large-scale, non-linear mass variations from 1995 to 2024, extending gravity-based observations into the pre-GRACE era while preserving spatial detail through backward extrapolation. The combined model reveals widespread and statistically significant accelerations in global water and ice mass changes and enables the identification of key turning points in their temporal evolution. Results indicate that in Svalbard, a non-linear transition in ice mass balance occurred in late 2004, followed by a pronounced acceleration of mass loss due to climate warming. Glaciers in the Gulf of Alaska exhibit persistent mass loss with a marked intensification after 2012, while in the Antarctic Peninsula, ice mass loss substantially slowed and a potential trend reversal emerged around 2021. The reconstructed mass anomalies show strong consistency with independent satellite altimetry and climate indicators, including a clear response to the 1997/1998 El Niño event prior to the GRACE mission. These findings demonstrate that integrating SLR with GRACE enables robust detection of non-linear, climate-driven mass redistribution on a global scale and provides a physically consistent extension of satellite gravimetry records beyond the GRACE era.

1. Introduction

Long-term observations of the Earth’s water cycle and the mass balance of glaciers and ice caps using satellite gravimetry constitute one of the significant achievements in contemporary geodesy, making a substantial contribution to various scientific fields, including hydrology [1,2], climatology [3,4], glaciology [5], and oceanography [6].

Since 2002, the Gravity Recovery and Climate Experiment (GRACE) [7] mission has played a crucial role, elevating the level of precision in observing these changes to a previously unattainable level. Through global measurements, the GRACE mission has also provided a foundation for understanding the climate system and its interactions with other factors. For more than 15 years, the mission has generated a series of hydrological and cryospheric models, whose significance was confirmed by the launch of its successor, GRACE Follow-On [8], after a one-year break in data collection. As of now, excluding this break, the global community benefited from 24 years of high-resolution models of Earth’s gravity field describing changes in land hydrology, oceans, and ice resources in Antarctica, Greenland, and other regions. Although the GRACE mission has greatly enhanced our understanding of gravitational field changes since 2002, the question remains: how can we learn about what happened before that?

Prior to the GRACE mission, there were alternative methods for determining gravity field models. Laser ranging measurements to passive spherical satellites [9] were used for monitoring Earth’s rotation and defining global reference systems [10]. When placing such satellites in orbit, the possibility of determining changes in the gravity field was also considered. Initially, from the 1980s onward, only the Earth’s oblateness [11] could be determined. Nevertheless, over the years and with the growing constellation of passive satellites, it became possible to increase the resolution of the models [12,13,14]. Although not as high as with GRACE data, it was possible to estimate large-scale gravity changes.

Even with the advancements of the GRACE mission, there remains significant interest in long-term Satellite Laser Ranging (SLR) gravity field models. This is aided by the fact that the part of low-degree zonal spherical harmonics (C₂₀ and C₃₀) [15,16,17,18] is still being replaced in GRACE data, and SLR data can provide long-term continuous models. Realistically, with the deployment of satellites such as LAGEOS-2 (1992) and Stella (1993) [9], the constellation was sufficiently large to obtain more than just the Earth’s oblateness. However, the number of laser ranging stations and, consequently, the number of observations did not allow for reliable estimates in the early years.

The majority of the mass transport in the Earth system is included in low-degree gravity field parameters, which can be well determined from SLR [12,13,14]. Even though SLR data alone cannot provide a high resolution, it can serve as a foundational basis incorporating large-scale variations, with additional information sourced from other determinations. However, there are few high-resolution temporal models of the gravity field for the period before 2002. One out of such models is the IGG-SLR-HYBRID, where spherical harmonics above degree-6 are replaced by empirical orthogonal functions derived from the GRACE period [19] and rescaled by SLR data. Recently, combined GRACE-SLR models have emerged with supplementary data through deep learning [20]. Scientists have also attempted to determine models before the GRACE mission using low Earth orbiting (LEO) satellites [21]. Models directly based on the SLR technique expanded to degree and order 10 only allow for the determination of large-scale mass changes due to their limited spatial resolutions [22]. Other estimations rely on direct methods addressing specific regions such as polar areas, countries, or river basins [23].

Given a relatively long time series of Earth mass changes, trends can be determined by fitting linear slope functions. Additionally, the identification of significant changes in global freshwater availability can be explored [24]. It turned out that most of the changes in freshwater resources could be explained as the consequences of natural, anthropogenic, or climate-related transformations. Local variations in river basins were also examined based on monitoring data from the GRACE mission [25]. Since the beginning of the GRACE mission, a great variety of research has focused on the Greenland and Antarctic regions [26,27,28,29,30], where changes were also determined using trend fitting. However, during that period, the time series was insufficiently long for the calculated accelerations to attain statistical significance. It took almost a decade before attempts were made to identify accelerating areas of non-linear behavior [30,31,32,33], with limitations due to the length of the observation period and the variability of the ice sheet cover [3]. These accelerations were also assessed in the context of sea level rise investigations [34]. In the case of continental changes, the results were inconclusive [35] or exhibited discrepancies relative to land surface models, but in test areas such as the Caspian Sea, the gravity-based results were consistent with altimetric data [36,37,38,39].

By combining long-term SLR with GRACE-derived spatial information, this study addresses the limited spatial detail of pre-GRACE gravimetry and delineates global regions with significant accelerations or decelerations in water storage change. For this purpose, we use SLR data as a basis for large-scale changes, and GRACE data fitted using a quadratic function as a complement to improve the spatial resolutions of the models. The validation results show high compliance with external data such as satellite altimetry and climate parameters, including Sea Surface Temperature Anomalies (SSTA). They also demonstrate consistency with a hydrological model, as well as with IMBIE estimates for Greenland and Global Mean Ocean Mass (GMOM).

2. Satellite Gravimetry Ice and Water Mass Estimates

Obtaining a long-term time series dataset reaching back to the 1990s with high spatial resolution is impossible using a single technique due to inherent limitations. Combining SLR and GRACE data directly is feasible; however, the resolution of these datasets may vary meaningfully, resulting in incomplete consistency at epoch boundaries. A suitable approach may involve combining these datasets, drawing on the strengths of each technique. The stable determination of low-degree spherical harmonics and the availability of data before 2002 are characteristic of SLR data. Through this, we can discern long-term, large-scale changes. On the other hand, GRACE mission data excel in capturing small, even short-term changes attributable to higher-degree spherical harmonics.

By leveraging the benefits of both techniques, we opt to merge these datasets, extending GRACE data while anchoring it with SLR data. The representation of interannual variability remains challenging; nevertheless, the long-term evolution and secular trends of the mass redistribution signal are primarily controlled by the low-degree and order spherical harmonic components. For this purpose, we utilize GRACE data provided by the Combination Service for Time-variable Gravity Fields (COST-G) [40] available from April 2002 to December 2023, and long-term time-variable Earth’s gravity fields derived using a combination of SLR data available from January 1995 to December 2023, corresponding to 3-month solution SLR S from [22]. We determine the Earth’s gravity field coefficients from SLR observations based on two high-orbiting LAGEOS satellites and up to seven low-orbiting satellites: Starlette, Stella, AJISAI, LARES, Larets, BLITS, and Beacon-C [9]. The gravity field has been estimated in the celestial mechanics approach [41], which utilizes the orbit perturbations and variational equations of motion. We derive the SLR-only gravity field solutions with the expansion up to a degree and order 10 using the development version of the Bernese GNSS Software Version 5.3 [42]. The spherical harmonic coefficients of the gravity field are derived with simultaneous estimation of Earth rotation parameters, satellite orbits, and station coordinates. For LAGEOS-1/2, we generated the 10-day orbital arcs and 1-day arcs for other satellites. The pseudo-stochastic pulses in the along-track direction are set up once-per-revolution for low-orbiting satellites [43] to compensate for the inconsistencies in the atmospheric density models. The once-per-revolution empirical orbit parameters in along-track are estimated for all satellites in order to absorb mismodeled albedo and solar radiation pressure, or the not explicitly modelled Yarkovsky and Yarkovsky–Schach effects [44,45]. Different weights are assigned to SLR observations for various satellites: 8 mm for LAGEOS-1/2, 15 mm for LARES, 20 mm for Starlette and Stella, 25 mm for AJISAI, 30 mm for Larets and BLITS, and 50 mm for the non-spherical Beacon-C. For details related to SLR data processing, see [14,22].

The Earth’s gravitational potential can be represented as a series of spherical harmonic coefficients:

$$V\left(r,\varphi ,\lambda \right)={\displaystyle \frac{G{M}_E}{R}}{{\displaystyle \sum }}_{n=0}^{\infty }{\left({\displaystyle \frac{R}{r}}\right)}^{n+1}{{\displaystyle \sum }}_{m=0}^n{P}_{nm}\left(\sin \varphi \right)\left(C_{nm}\cos m\lambda +S_{nm}\sin m\lambda \right),$$

(1)

where $r, \varphi , \lambda$ are the spherical coordinates in the reference frame, $G{M}_E$ is the product of the gravitational constant and the Earth’s mass, $R$ is the semi-major axis of the Earth, $C_{nm}$ and $S_{nm}$ are the Stokes coefficients of spherical harmonics of degree n and order m, and $P_{nm}$ are the fully normalized Legendre polynomials. The final model was decorrelated and smoothed using a decorrelation (DDK3) filter [46,47].

The most commonly used representation for gravity field comparison is the Equivalent Water Height [48] (EWH), which represents the temporal (e.g., monthly) variations within the Earth’s water cycle. By determining this parameter for each longitude and latitude, we can study its spatial distribution. Comparisons are then made in areas characterized by intense variations in the EWH parameter. From the individual spherical harmonic values, the EWH quantity is calculated as:

$$EW{H}_{\left(\varphi ,\lambda \right)}={\displaystyle \frac{M_E}{4\pi R^2{\rho }_w}}{{\displaystyle \sum }}_{n=2}^{\infty }{\displaystyle \frac{2n+1}{1+k_n}}{{\displaystyle \sum }}_{m=0}^n{P}_{nm}\left(\sin \varphi \right)\left(∆C_{nm}\cos m\lambda +∆S_{nm}\sin m\lambda \right),$$

(2)

where $k_n$ are the elastic load Love numbers of degree $n$, $M_E$ is the Earth’s mass, ${\rho }_w$ is the density of water, and $∆C_{nm}$ and $∆S_{nm}$ are temporal variations in the Stokes coefficients. The ∆ indicates that the signal of the static gravity field has been subtracted. In this study, we used the static coefficients of GOCO06s [49] for this purpose.

To compare the available models from different sources and techniques, the spherical harmonic coefficients are derived using consistent models following the International Earth Rotation Service and Reference Systems (IERS) Conventions 2010 [50] with Atmosphere and Ocean Dealiasing (AOD) Models RL06 [51]. We divide both datasets into two periods:

(1)

From the beginning of 1995 to the start of the GRACE mission (2002.4);

(2)

From the start of the GRACE mission (2002.4) to the end of the analyzed data (2023.12).

We split the spherical harmonics into three groups: from degrees 2 to 4, from 5 to 10, and from 10 to 60. In

Table 1. The description of the proposed SLR+GRACE model, divided by time and periods for different spherical harmonics (SH).

SH	Time
	January 1995–April 2002	April 2002–December 2023
2–4	SLR-only	SLR-only
	raw estimates
5–10	SLR-only with fitted annual and semiannual signals and the second-degree polynomial with annual step pulses * for January 1995–April 2002	GRACE from COST-G
	fitted annual and semiannual signals and the second-degree polynomial with annual step pulses for the period January 1995–December 2023
11–60	GRACE from COST-G	GRACE from COST-G
	backward extrapolation of the full deterministic model from the period April 2002–December 2023	fitted annual and semiannual signals and the second-degree polynomial with annual step pulses

* annual step pulses are deterministic parameters (step functions) estimated once per year to capture anomalies not represented by the quadratic + seasonal terms.

, we provide a detailed description of the model. Examples of modeling in each group are shown in Figure 1. Following the strategy applied in the gravity field models used in this study, the solutions were computed without estimating degree-1 coefficients, which represent the geocenter motion. SLR cannot recover the secular component of geocenter translation because, during the realization of the International Terrestrial Reference Frame (ITRF), any long-term geocenter drift is absorbed into the station velocity field [52]. As a result, degree-1 information from SLR is reliable only on short timescales, and its reconstruction was intentionally excluded from the present analysis, which focuses on the variability of higher-degree gravity components.

The combination scheme presented in Table 1 leverages the complementary strengths of both techniques. The long-wavelength gravity field (SH 2–4) is anchored entirely in SLR observations, reflecting the fact that SLR is currently the most accurate technique for determining low-degree spherical harmonics not limited to C₂₀ and C₃₀, thereby ensuring consistency across the full time series. For intermediate wavelengths (SH 5–10), we integrate fitted SLR solutions for the early period, seamlessly merging them with GRACE data from 2002 onwards. Crucially, this approach allows for the reproduction of annual and semiannual signals as observed in GRACE data and their backward propagation onto the pre-GRACE period, while the long-term trend for each spherical harmonic is derived directly from the SLR solution. The high-resolution component (SH 11–60) relies on the backward propagation of the deterministic signal structures (including the dominant seasonal spatial modes) identified during the GRACE mission operational phase. The high-resolution component (SH 11–60) relies on the backward propagation of deterministic signal structures (including the dominant seasonal spatial modes) identified during the GRACE operational period. This approach implicitly assumes that these seasonal spatial patterns remain representative when extrapolated back in time. Consequently, the pre-2002 high-degree reconstruction should be interpreted as a mode-based extrapolation that primarily preserves the large-scale geometry of the GRACE-era seasonal signal, while it may not fully capture changes in the spatial pattern itself, shifts in relative amplitudes between subregions, or pronounced interannual departures from the seasonal cycle. Although the annual step pulses provide additional temporal flexibility, localized high-frequency events (e.g., flash floods in small sub-basins) and small-scale interannual variability occurring before 2002 may be attenuated or spatially smeared—especially if they do not project onto the retained modes and/or do not imprint on the low-degree field (approximately ≤ 10 degree and order of SH) constrained by SLR.

The annual step pulses, as listed in Table 1, are deterministically estimated parameters corresponding to discrete temporal shifts. The intention is to allow the model to account for structural changes or anomalies that occur on a yearly basis (e.g., due to large-scale climatic events or rapid changes in ice mass loss), particularly those that are visible in the spherical harmonic coefficients. A yearly spacing was selected to avoid over-parameterization and to preserve parameter identifiability. Using denser pulses (sub-annual) would increase the degrees of freedom and could become difficult to separate from other short-timescale adjustments often used in geodetic satellite orbit determination, thereby absorbing modeling residuals rather than true interannual variability.

Mathematically, we define the design matrix A to include, in addition to deterministic components like trend and seasonal terms, a set of step functions

$$H_k\left(t\right)=\left\{\begin{array}{ll}1,\hfill & \mathrm{if} t\ge t_k\hfill \\ 0,\hfill & \mathrm{otherwise} \hfill \end{array}\right.$$

(3)

$$\begin{gathered} y\left(t\right)={\beta }_0+{\beta }_1{t}_s+{\beta }_2\cos \left(2\pi t_s\right)+{\beta }_3\sin \left(2\pi t_s\right)+{\beta }_4\cos \left(4\pi t_s\right)+{\beta }_5\sin \left(4\pi t_s\right)+{\beta }_6{t}_s^2 \\ +{{\displaystyle \sum }}_{k=1}^N{\gamma }_k{H}_k\left(t\right)+\epsilon \left(t\right) \end{gathered}$$

(4)

where

• $t$ is the time of observation (epoch), expressed in decimal years;
• $t_s=t-t_0$ is the time centered around the reference epoch;
• $t_0$ is the reference epoch;
• ${\beta }_0,{\beta }_1$ are the intercept and linear trend;
• ${\beta }_2,{\beta }_3$ are seasonal amplitudes;
• ${\beta }_4,{\beta }_5$ are the semiannual signal amplitudes;
• ${\beta }_6$ is the coefficient of the quadratic (second-order polynomial) term;
• ${\gamma }_k$ are the magnitudes of the estimated annual step pulses;
• $\epsilon \left(t\right)$ is the residual;
• $N$ is the number of years in the estimation interval.

This formulation allows the model to fit annual deviations in a flexible manner, thereby improving the fit in regions where the mass change signal is affected by unmodeled variability or local phenomena.

3. Validation

To assess the reliability of the proposed SLR+GRACE reconstruction particularly in the pre-GRACE period, we conduct a multi-step validation using independent observations and complementary datasets. Because the reconstruction aims to extend GRACE-like information back in time, the validation focuses on whether the derived mass anomalies reproduce physically consistent variability and agree with external references across different temporal scales. We first examine the relationship between water-storage anomalies and the El Niño–Southern Oscillation (ENSO) using SSTA. We then compare the reconstructed signals with alternative gravity-field products and with altimetry-based water-level variations over selected reservoirs, ensuring a consistent spatial sampling. Finally, we assess the robustness of estimated trends and accelerations by analyzing their sensitivity to the length of the available record.

3.1. Validation with Sea Surface Temperature Anomaly

The ENSO, occurring every 2 to 7 years, demonstrates a close relationship between oceans and atmospheres [53,54]. Its initial phase is typically detectable using the SSTA indicator Niño 3.4 (5°N–5°S) (170–120°W), which indicates anomalies relative to the long-term average temperature during the period from 1991 to 2020. Even without a full understanding of the causes of this phenomenon, its effects are undeniably evident. El Niño induces droughts in North [55] and South America [56], while La Niña intensifies trade winds, strong convection, and rains over Indonesia [57].

Given the significant dependence of mass anomalies on precipitation, evapotranspiration, local temperature, and prevailing conditions, a tight correlation of this data with indicators signaling ENSO is possible. This has been described in research primarily linking them with data obtained from the GRACE mission [58,59] but also using in situ river level and precipitation data [60].

The duration of EWH recordings from GRACE and GRACE Follow-On [8] is considerably shorter compared to the typically over 40-year timespan of ENSO records. Therefore, to enhance the reliability of the assessment, it is necessary to have a longer dataset that predates the GRACE period. This study extends the time series of GRACE and GRACE Follow-On data by over 7 years using SLR data. Utilizing data from 1995 allows for the determination of mass changes for the Amazon basin and other regions, as well as validation of relationships during one of the most intense El Niño periods in recorded history, which occurred around the 1997/1998 transition [61]. To accomplish this, we illustrate SSTA in Figure 2, with values marked above and below 0.5 degrees, which represent potential El Niño and La Niña periods. To qualify as El Niño or La Niña, the index must persist for at least 5 months above/below this temperature threshold. Figure 2 also depicts the estimation of mass anomalies for the Amazon region after removing the trend, annual, and semi-annual signals. We applied the 5-month moving average filter to keep consistency with other results [62]. We can identify events related to the ENSO phenomena by retaining only the residuals, which are typically delayed relative to SSTA. As demonstrated by other studies, this delay classically ranges from 4 to 6 months, during which correlations are the highest. The calculated correlations for the proposed model relative to SSTA for the pre-GRACE era are −0.72, −0.70, and −0.69 for delays of 7, 6, and 5 months, respectively. For the GRACE era, the correlations are −0.56, −0.54, and −0.49 for delays of 7, 6, and 5 months, respectively. The results are comparable to other studies [20], although correlations may be slightly lower and more delayed due to the SLR solution being smoothed. Overall, Niño3.4 SSTA provides an independent and physically meaningful benchmark for evaluating the reconstructed interannual water-storage variability over the Amazon basin. The consistently negative correlations with an expected lag of ~5–7 months indicate that the proposed reconstruction captures both the sign and timing of the ENSO-related hydrological response. In particular, the strong 1997/1998 El Niño episode is clearly visible in Figure 2 and is followed by a pronounced anomaly in the Amazon mass series after the inferred delay. The fact that this event remains detectable in the SLR+GRACE reconstruction suggests that its imprint is largely expressed at spatial scales captured by the low-degree field (roughly up to degree/order 10) that is robustly constrained by SLR.

3.2. Validation with Different Models and Altimetry Data

The comparison between altimetry data and satellite gravimetry data occurs once the altimetry data have been corrected for temperature and salinity effects on water. This is because temperature and salinity influence the water density, which affects the sea surface height (SSH) but not its mass, which is obtained from gravimetric measurements. The difference between these measurements is called the steric contribution, which can be corrected using in situ measurements of these parameters. For this analysis, aimed at verifying the determined model by achieving high correlation consistency, we do not consider the steric parameter. We utilize areas classified as significantly accelerating in Figure A1 for the proposed SLR+GRACE model. These areas are usually associated with changes in surface water. In addition to the proposed SLR+GRACE model, we employ two alternative gravity field solutions derived from SLR. The IGG-SLR-HYBRID model provides monthly gravity field solutions by combining SLR-derived spherical harmonic coefficients with empirical orthogonal functions obtained from GRACE data [19], covering the period from late 1992 to 2021. Furthermore, we use the IGG-SLR-DORIS model, which extends the IGG-SLR-HYBRID solution by incorporating DORIS observations, thereby prolonging the time span back to 1984 and forward to the end of 2023. All the models considered in this analysis have been filtered using the DDK3 filter [46]. For this data, we do not use any leakage corrections. As altimetry data, we utilize the Global Water Measurement portal (https://blueice.gsfc.nasa.gov/gwm (accessed on 12 December 2025)), from which we extract lake height variations. This platform consolidates measurements from multiple altimetry missions that have monitored water surface heights over the years. To ensure a consistent spatial reference between the datasets, all gravimetric time series used in the comparison (Figure 3 and

Table 2. Correlations between the proposed SLR+GRACE model, IGG-SLR-HYBRID, IGG-SLR-DORIS, COST-G and altimetry data for individual water reservoirs, as well as for the pre-GRACE period (1995–2002), GRACE period (2002–2021), and the entire period (1995–2021). The insignificant correlations (p-value > 0.05; red).

	1995–2021			2002–2021				1995–2002
	SLR+GRACE (This Study)	IGG-SLR-HYBRID	IGG-SLR-DORIS	SLR+GRACE (This Study)	IGG-SLR-HYBRID	IGG-SLR-DORIS	COST-G	SLR+GRACE (This Study)	IGG-SLR-HYBRID	IGG-SLR-DORIS
Caspian Sea	0.93	0.91	0.92	0.97	0.93	0.92	0.99	0.51	0.64	0.78
Lake Michigan	0.82	0.50	0.59	0.83	0.68	0.77	0.89	0.64	0.38	0.43
Lake Hulun	0.71	−0.18	0.04	0.53	0.53	0.49	0.53	0.32	0.14	−0.09
Lake Victoria	0.63	0.49	0.53	0.90	0.79	0.79	0.92	0.44	−0.05	0.15
Lake Therthar	0.64	0.50	0.44	0.58	0.34	0.39	0.72	0.76	0.52	0.33

) were extracted from the specific grid cell of the SLR+GRACE, IGG-SLR-HYBRID, and IGG-SLR-DORIS solutions corresponding to the locations of the altimetry measurements. Consequently, the correlations reported in Table 2 refer to point-based values (single grid cells) rather than basin-averaged signals.

We observe a high level of consistency for the proposed SLR+GRACE model, particularly during the pre-GRACE period (1995–2002), even for relatively small water bodies, as reflected by the correlation coefficients listed in Table 2. For this period, the correlations amount to 0.51, 0.64, 0.32, 0.44, and 0.76 for the Caspian Sea, Lake Michigan, Lake Hulun, Lake Victoria, and Lake Therthar, respectively. In comparison, the IGG-SLR-HYBRID model yields correlation coefficients of 0.64, 0.38, 0.14, −0.05, and 0.52, while the IGG-SLR-DORIS solution shows values of 0.78, 0.43, −0.09, 0.15, and 0.33 for the same reservoirs.

Over the entire period common to all datasets (1995–2021), the Caspian Sea exhibits consistently high correlations across all gravity field solutions, reaching 0.93 for the proposed SLR+GRACE model, 0.91 for IGG-SLR-HYBRID, and 0.92 for IGG-SLR-DORIS. This agreement indicates that long-term mass-driven water-level variations in large reservoirs are robustly captured by all models. In contrast, for smaller reservoirs characterized by pronounced water-level changes prior to the GRACE era, notable discrepancies emerge between the solutions. For Lake Hulun, the proposed SLR+GRACE model achieves a correlation of 0.71 over the full period, whereas the IGG-SLR-HYBRID and IGG-SLR-DORIS models yield substantially lower values of −0.18 and 0.04, respectively.

Similar behavior is observed for Lake Therthar, where the proposed model maintains a moderate correlation of 0.64 for the full period, compared to 0.50 for IGG-SLR-HYBRID and 0.44 for IGG-SLR-DORIS. Visual inspection of the corresponding time series (Figure 3) indicates that the proposed SLR+GRACE model is able to reproduce the long-term water-level variations. The markedly different correlations for some smaller water bodies (e.g., Lake Hulun and Lake Therthar) likely reflect differences in the underlying parameterization of the high-degree signal. The IGG-SLR-HYBRID and IGG-SLR-DORIS products represent monthly variability primarily through a set of GRACE-derived EOF patterns whose scaling factors are adjusted to SLR/DORIS tracking data. Because SLR/DORIS observations can robustly constrain only a limited number of EOFs, the recovered signal may become incomplete in regions where a substantial fraction of the variability is distributed across higher-order EOFs or where the dominant spatial modes deviate from the GRACE-era climatology, particularly outside the GRACE time frame. In contrast, the proposed SLR+GRACE reconstruction is designed to preserve the long-wavelength and multi-decadal evolution that is most strongly expressed in low degrees, which helps to reproduce the non-seasonal component seen by altimetry more consistently; however, strictly local short-term anomalies remain expectedly smoothed.

Although the correlations are high, minor amplitude discrepancies in the pre-GRACE era (visible in Figure 3) may result from the extrapolation of GRACE climatology. Non-seasonal, local hydrological anomalies specific to the pre-2002 period are naturally smoothed out in the reconstruction if they are not significant enough to impact the lower harmonic degrees derived from SLR.

Additionally, we performed a direct comparison of SLR+GRACE-derived EWH anomalies with the Land and Surface Discharge Model (LSDM) [63] in the Amazon basin (Figure 4). This comparison, shown in the top and bottom panels, reveals consistent seasonal and interannual variability between the two datasets over the full analysis period, including the pre-GRACE era.

Quantitatively, for the entire analysis period (1995–2023), the comparison yields a root mean square (RMS) difference of 74 mm and a correlation coefficient of 0.87, indicating strong overall agreement. Focusing on the pre-GRACE period (1995–2002), the RMS is 97 mm with a correlation of 0.87, which demonstrates a notable level of agreement even before the availability of GRACE satellite data. After the GRACE mission launch (2002–2023), the RMS improves significantly to 64 mm, and the correlation rises to 0.91. The lower panel in particular highlights interannual variability that aligns well with known ENSO events (Figure 4), further supporting the physical basis of the mass signals observed.

To verify the reliability of the reconstructed mass changes, particularly in the pre-GRACE era (1995–2002), we compared the proposed combined SLR+GRACE solution with independent estimates from the Ice Sheet Mass Balance Inter-comparison Exercise (IMBIE) for Greenland [29], as well as with two established SLR-based models: IGG-SLR-DORIS [64] and IGG-SLR-HYBRID [19]. We applied identical processing standards to the analyzed model: decorrelation using a DDK3 filter, removal of GIA utilizing the ICE-6G_D model [65], and correction of signal leakage using the Forward Modeling technique [66]. Furthermore, since the IGG-SLR-DORIS model includes degree-1 coefficients while the proposed SLR+GRACE solution is defined without degree-1, these coefficients were set to zero in the IGG-SLR-DORIS model to ensure full consistency between the datasets.

As illustrated in Figure 5 (top panel), there is a high level of agreement among the proposed solution, the IGG-SLR-HYBRID model, the IGG-SLR-DORIS model, and the IMBIE ensemble estimate over the entire study period. Crucially, during the pre-GRACE phase (1995–2002), all datasets consistently capture the long-term mass trends and interannual variability. This mutual consistency between independent methodologies confirms the robustness of the reconstructed ice mass balance signals in the early satellite gravimetry era.

The bottom panel of Figure 5 presents the comparison of GMOM anomalies [67]. The long-term trends and seasonal signals show remarkable consistency among the proposed solution, the IGG-SLR-DORIS model, and the IGG-SLR-HYBRID model. The strong correlation and amplitude concordance among these independent approaches confirm that the regional non-linear features and accelerations identified in this study are reliably captured and represent a robust surface mass redistribution consistent with established gravity field models.

3.3. Methodological Validation: Sensitivity to Record Length

Our time series, due to their nearly 30-year length, enable proper estimation of parameters such as trend and acceleration [3] and define years with the trend reversal. Figure 6 shows mass anomaly, trends, and acceleration in mass change for the Svalbard region as a function of record length. The trend is not always a reliable indicator, as even with over 25 years of data, its stabilization can be difficult to achieve when estimating a trend alone without an acceleration component. On the other hand, acceleration is much more reliable, showing good stabilization after 15–20 years of observation when estimated together with the linear coefficients, which corresponds to the trend. Therefore, for a proper description of the long-term time series of global gravity field changes, the model with linear and acceleration coefficients is more suitable than the linear trend alone because it reaches its stabilization after 15–20 years, depending on the region. Additional plots showing results as a function of record length are included in Figure A2.

4. Water Storage Accelerations

Water storage changes are often non-linear, because they are driven by time-varying hydroclimatic forcing on interannual to decadal scales and, in some cases, by anthropogenic regulation (e.g., reservoir impoundment and operational changes). To capture such curvature, accelerations, and possible trend reversals rather than assuming a purely linear evolution, we computed model the EWH [48] time series using a deterministic function that includes a second-degree polynomial term in addition to annual and semiannual components from 1995 to 2024. We extracted only those accelerations whose absolute values exceeded 2σ (1.6 mm/year²), representing deviations from the mean that fall outside the 95% confidence interval, indicating statistically significant values. Subsequently, timestamps were determined at which maxima and minima of the function occur. These extrema were identified using a quadratic model of the form, where the extremum occurs at $t_{ext}=-{\displaystyle \frac{{\beta }_1}{2{\beta }_6}}$, the point at which the first derivative is zero. This analytical approach enables the detection of the point at which the rate of change vanishes and thus reveals when a trend reversal occurs. These values are visualized in Figure 7 and shown in

Table 3. The trend, acceleration, and the year of the global extremum of the function for regions from Figure 7. The trend and acceleration were calculated based on the fitted annual and semiannual signals.

	Trend [cm/yr]	Acceleration [cm/yr2]	Extremum [yr.mth]
Lake Michigan	0.60 ± 0.07	0.118 ± 0.007	December 2006
Lake Therthar	−1.34 ± 0.05	0.077 ± 0.005	February 2018
Caspian Sea	−3.05 ± 0.08	−0.139 ± 0.004	April 2000
Lake Hulun	0.03 ± 0.04	0.093 ± 0.003	April 2009
Lake Ramos Mexia	−0.86 ± 0.04	0.081 ± 0.004	September 2014
Porto Primavera Dam	0.96 ± 0.07	−0.128 ± 0.007	January 2013
Lake Hansali	0.66 ± 0.05	−0.123 ± 0.002	May 2012
Lake Victoria	1.38 ± 0.06	0.093 ± 0.006	January 2002

. In this section, we focus solely on areas outside the polar zones.

We delimitate regions that exhibit significant acceleration and are associated with water reservoirs. Most of these areas have either reached their minimum (Lake Michigan, Lake Therthar, Lake Hulun, Lake Victoria, and Lake Ramos Mexia) or maximum function (Caspian Sea, Porto Primavera Dam, and Lake Hansali). The highest trend is observed in results related to the Caspian Sea (−3.05 cm/year). This region also demonstrates the highest acceleration value (0.14 cm/year²) and reached its maximum in 2000. The positive trends are observable for Lake Michigan, Lake Victoria, and the location of the Porto Primavera Dam, where both trends (0.60, 1.38, and 0.96 cm/year, respectively) and accelerations (0.12, 0.09, and −0.13 cm/year², respectively) are positive (Table 3). The timing of these extrema provides further insight into the dynamics of mass change. Specifically, the date of a minimum or maximum indicates whether the process is intensifying or decelerating. For example, in the Caspian Sea, a positive trend is observed up to the year 2000, after which a depletion begins; conversely, Lake Michigan shows a reversal from a negative to a positive trend around 2007. These extrema, therefore, mark points of trend reversal and are particularly informative for interpreting long-term hydrological behavior. For improved readability and interpretation, the extrema have also been separated by type (maxima and minima), and these visualizations are provided as Appendix A (Figure A6 and Figure A7).

4.1. Antarctica Region—Accelerating Ice Mass Depletion

In both the Antarctic and Greenland regions, it is possible to identify areas with different trends and accelerations after removing the GIA effect [68]. In Figure 8, selected areas exhibit high accelerations but differ in trends. In the eastern Antarctic, for Area of Interest (AOI4), mass is being accumulated (with a trend of 1.92 cm/year), whereas for AOI3 it is decreasing (with a trend of −1.12 cm/year), with the onset of acceleration occurring in similar years (2006–2012). Conversely, examining western Antarctica, significant mass loss is observed (

Table 4. The trend, acceleration, and the year of the global extremum of the function for regions from Figure 8. The trend and acceleration were calculated based on the fitted annual and semiannual signals.

	Trend [cm/yr]	Acceleration [cm/yr2]	Extremum [yr.mth]
AOI1	−3.35 ± 0.06	0.149 ± 0.003	August 2020
AOI2	−11.34 ± 0.08	−0.156 ± 0.006	February 1973
AOI3	−1.12 ± 0.07	−0.145 ± 0.004	July 2005
AOI4	1.92 ± 0.04	0.077 ± 0.004	January 1997
AOI5	−1.32 ± 0.03	0.062 ± 0.003	December 2019

). For the most remote part of the Antarctic Peninsula (AOI1), it appears that the ice mass loss completely decelerated and the trend reversed around 2021. In the remaining identified area in western Antarctica (AOI2), continuous loss of mass is observed, with the maximum of the function occurring before the period from which the data are determined, and with a trend of −11.34 cm/year. During the same periods, in terms of function behavior, we achieved consistency with an external analysis conducted in 2014 [31] and also in 2009 [69] during the GRACE era and before [70]. To better illustrate the spatial distribution of extrema and support the interpretation of trend reversals and accelerations across Antarctica, separated maps of maxima and minima are provided in the Appendix A (Figure A8).

4.2. Arctic Region—Accelerating Ice Mass Depletion

For the Greenland region, we utilized areas that have a similar size and ice formation [71] and were recognized as significant in terms of ice mass acceleration, as shown in Figure 9 and

Table 5. The trend, acceleration, and the year of the global extremum of the function for regions from Figure 9. The trend and acceleration were calculated based on the fitted annual and semiannual signals.

	Trend [cm/yr]	Acceleration [cm/yr2]	Extremum [yr.mth]
British Columbia + Alaska	−5.54 ± 0.05	−0.091 ± 0.004	January 1979
Greenland NO	−2.79 ± 0.08	0.164 ± 0.006	December 2001
Greenland NW	−6.65 ± 0.09	−0.172 ± 0.007	February 1990
Greenland SE	−13.3 ± 0.06	0.052 ± 0.008	September 2138
Svalbard	−2.07 ± 0.10	−0.223 ± 0.005	October 2004

. We also considered the Svalbard and North American regions, which were also highlighted in the analysis as being significant. In the Svalbard region, the fitted function attains its maximum in the mid-2000s (October 2004). From this point onward, the time series exhibits a clear transition to an accelerating ice mass loss, with the acceleration becoming pronounced from that epoch and persisting to the present day, consistent with independent observational evidence of climate-driven warming [72,73]. A similar phenomenon is observed in the Gulf of Alaska Glaciers. It is possible to observe a substantial ice mass loss from the beginning of the acquired observations, with summer seasons characterized by greater mass outflow than accumulation in winter seasons [74]. The extremum of the fitted function is located in 1979, i.e., outside the observational time span, and therefore represents a purely mathematical extrapolation of the quadratic model rather than an empirically observed feature. Additionally, the acceleration intensifies after 2012 and persists until the present day.

Most areas identified in Greenland exhibit long-term mass loss. In the north (NO) and north-west (NW) regions, this loss significantly accelerated around 2004. The southern area (SE) is characterized by continuous loss without periods of greater accumulation. The Canadian Arctic Archipelago glaciers, which also experience significant accelerations. To provide a clearer view of the timing and spatial variability of extrema in the Greenland, Svalbard, and North American regions, we include separate maps of maxima and minima in the Appendix A (Figure A9).

5. Discussion and Summary

In the above study, we delineate continental regions where significant accelerations in mass change have occurred over the past 30 years. We utilize long-term SLR data as a basis and complement it with GRACE data. This allows us to use a long time series of data and avoid limitations in detecting accelerations due to climate variability. Our time series, due to their nearly 30-year length, enable proper estimation of parameters such as trend and acceleration [3] and define years with the trend reversal. The consistency between our reconstructed Greenland and GMOM changes and independent external datasets (IMBIE and IGG-SLR-based solutions; Figure 5) further supports the physical robustness of the proposed SLR+GRACE reconstruction, including in the pre-GRACE period.

Significant accelerations in continental areas are typically associated with water reservoirs, whose inflows are heavily exploited (e.g., the Caspian Sea, the Nile, Lake Victoria), or locations on rivers where artificial reservoirs in the form of dams have been established (e.g., Porto Primavera Dam). This analysis also reveals accelerations in polar regions, largely attributable to climate change. For Svalbard, the inferred turning point around ~2004 can plausibly be interpreted in the context of the strong sensitivity of its low-elevation hypsometry to relatively modest changes in summer melt and rain/firn processes [70,71], together with evidence for a mid-2000s shift in summer atmospheric circulation and associated surface mass-balance conditions over the archipelago [73,75]. This interpretation is also consistent with the influence of Atlantic Water heat advection in the West Spitsbergen Current on local climate and sea-ice conditions around Svalbard [76] and with recent observations linking regional ocean/sea-ice variability to the spatial pattern and recent spread of Svalbard glacier mass loss [77]. For the Antarctic Peninsula (AOI1), long-term mass changes are widely linked to ice-dynamic adjustments following reductions in buttressing (e.g., ice-shelf collapse and associated glacier speed-up), which can generate non-linear trajectories and delayed responses [78,79,80]. The apparent slowdown/reversal inferred around ~2021 may additionally reflect the superposition of dynamic discharge with short-lived surface-mass-balance anomalies, since snowfall-driven SMB variability can be large on interannual time scales; recent RACMO-based results indicate an Antarctic-wide SMB increase in the early 2020s, particularly in 2022 [81]. Consistently, GRACE/GRACE-FO analyses reported a pronounced Antarctic Peninsula mass gain in 2021–2022 primarily attributed to anomalously enhanced precipitation [82].

In most areas, we observe considerable acceleration, uncompensated by ice accumulation regions. This is similarly noted in Greenland and its surrounding regions, such as Svalbard or the Gulf of Alaska Glaciers. Long-term time series also exhibit a high correlation with independent data on SSTA. In the residuals of mass changes in the Amazon region, it is possible to detect every unnaturally dry and wet season, which is linked to El Niño and La Niña events. It is noteworthy to emphasize the high dependency between the proposed model and ENSO events before the GRACE era (El Niño around 1997/1998), where the model relies on SLR data. The calculated correlations for the proposed model relative to SSTA for the pre-GRACE era are −0.72, −0.70, and −0.69 for delays of 7, 6, and 5 months, respectively. For the GRACE era, the correlations are −0.56, −0.54, and −0.49 for delays of 7, 6, and 5 months, respectively. Finally, the proposed combined SLR+GRACE model is much better replicated by satellite altimetry data, as evidenced by high correlations prior to the GRACE mission.

We observe a high consistency, particularly for the pre-GRACE period, even for smaller water bodies. The correlation between altimetry and gravity models is 0.51, 0.64, 0.32, 0.44, and 0.76 for the Caspian Sea, Lake Michigan, Lake Hulun, Lake Victoria, and Lake Therthar (Iraq), respectively. It is worth noting that the pre-GRACE spatial details at scales primarily carried by SH degrees > 10 are conditional on the assumption that the GRACE-era seasonal spatial modes remain representative when extrapolated backward in time. Consequently, localized high-frequency hydrological events and small-scale interannual variability that do not project onto the SLR-constrained low degrees (approximately 2–10) may be attenuated or spatially smeared in the pre-2002 fields. Accordingly, for the pre-GRACE period we primarily emphasize consistency in long-term trends, accelerations, and multi-decadal evolution, rather than localized short-term anomalies. However, since the acceleration parameters are primarily driven by long-wavelength mass redistribution and multi-decadal temporal curvature, they remain robust for the large-scale regions analyzed in this study. The long-term models can, thus, delimitate areas with statistically significant accelerations and indicate the year of the trend reversal. We also identify potential minima and maxima of the function, where deceleration/growth attenuation or slowdown/decay cessation may occur.