Study on Annual Signals of Greenland Ice Sheet Mass and Associated Influencing Factors Based on GRACE/GRACE-FO Data

Abstract

As global temperatures rise, the Greenland ice sheet (GrIS) is undergoing accelerating mass loss, with significant implications for sea level rise and climate systems. Using GRACE and GRACE Follow-On (GRACE-FO) RL06 data from April 2002 to May 2023, alongside MARv3.14 regional climate model outputs (ice melting, runoff, rainfall, snowfall, and land surface temperature (LST)), we investigated the drivers of GrIS mass changes. Continuous wavelet transform analysis revealed significant annual signals in all variables except snowfall, with wavelet decomposition showing the largest annual amplitudes for ice melting (58.8 Gt/month) and runoff (44.5 Gt/month), surpassing those of GRACE/GRACE-FO (31.1 Gt/month). Cross-correlation analysis identified ice melting, runoff, rainfall, snowfall, and LST as significantly correlated with GrIS mass changes, with ice melting, runoff, and LST emerging as primary drivers, while snowfall and runoff exerted secondary influences. Temporal lags of 3, 4, 4, 7, and 4 months were observed for ice melting, runoff, rainfall, snowfall, and LST, respectively. These findings highlight the complex interplay of climatic and hydrological processes driving GrIS mass loss.

1. Introduction

As global temperatures continue to rise, the melting of polar ice sheets has become a significant contributor to the rapid rise in global sea levels observed since the beginning of the 21st century [1]. Among these ice sheets, the Greenland ice sheet (GrIS), the largest in the Northern Hemisphere, holds the potential to raise sea levels by approximately 7.4 m if it were to melt completely [2]. According to previous studies, GrIS mass loss is by far the most significant contributor to sea level rise in recent years [3,4]. Consequently, monitoring GrIS mass change is critical for understanding not only global sea level rise but also the global water cycle and the broader dynamics of climate change [5].

Traditional measurement methods encounter difficulty when directly monitoring the mass change of the GrIS and providing accurate quantitative analysis [6]. Advancements in space-based geodetic observation technologies have revolutionized our ability to monitor ice sheet dynamics. Specifically, missions such as the Ice, Cloud, and Land Elevation Satellite (ICESat), Cryosphere Satellite (CryoSat), and the Gravity Recovery and Climate Experiment (GRACE) satellites have provided increasingly accurate measurements of mass changes in polar regions [7,8,9]. Launched in March 2002, the first GRACE satellite mission concluded successfully in October 2017. Its successor, GRACE Follow-On (GRACE-FO), was launched in May 2018 to continue monitoring global gravity field variations and provides high-precision data for monitoring ice sheet mass [10]. Over nearly two decades, the combined GRACE and GRACE-FO time series have generated an invaluable dataset for studying the GrIS and its evolving mass changes [11].

Numerous studies have employed GRACE data to investigate GrIS mass change over different time scales. For example, Barletta et al. [12] found that the mass change of Greenland and Antarctic ice sheets and its influence on sea level are closely related to GIA. After removing the influence of GIA, the mass change rates of Greenland and Antarctic ice sheets are −101 Gt·a⁻¹ and −171 Gt·a⁻¹, respectively. Ran et al. [13] explored the seasonal variations in GrIS mass, revealing that the peak mass anomaly occurs in July, reaching between 80 and 120 Gt. Zou et al. [14] further analyzed the secular, seasonal, and interannual mass changes of the GrIS, finding that the majority of mass loss occurs during the summer months. Some researchers have also focused on long-term trends using different data sources. Forsberg et al. [7] estimated a mass loss of 265 Gt/yr from 2002 to 2015, contributing to a global mean sea level rise of 0.72 mm per year. The summer of 2012 was marked by a record mass loss in the GrIS following a period of accelerated melting. Mankoff et al. [15], using the regional climate model MARv3.12, observed a significant decline in the surface mass balance of the GrIS, from 450 Gt/yr in 1860 to 260 Gt/yr by 2010, while emissions increased substantially during the same period. More recently, Gao et al. [16] analyzed GrIS mass loss from April 2002 to May 2020, incorporating LST, rainfall, and GRACE/GRACE-FO data, and found that anomalous increases in LST and runoff, coupled with decreased rainfall, exacerbated melting in Greenland during the 2010–2017 period.

Much of the current research on GrIS mass change has focused on linear and secular mass change trends, as well as monthly cumulative mass loss [17,18,19,20,21,22]. However, GRACE/GRACE-FO data also reveal important annual signals, which are essential for understanding the temporal dynamics of GrIS mass changes and their relationship with atmospheric and oceanic climate conditions. These changes in ice sheet mass are influenced by climatic factors, including temperature shifts and atmospheric circulation patterns [5]. While several studies have examined the seasonal variation in GrIS mass and its drivers, much of this research is concentrated on the winter and summer seasons, with fewer studies addressing the full annual cycle. Most studies on the drivers of GrIS mass change have focused on quantitative assessments of single drivers (e.g., melting, precipitation) [23]. However, the strength and time lag of the effects of different drivers (e.g., ice melting, runoff, rainfall, snowfall, et al.) of the GrIS mass change have not yet been systematically analyzed. Furthermore, most studies have failed to make full use of time–frequency analysis methods (e.g., wavelet transforms) to analyze driving mechanisms on different time scales. The annual variation characteristics of ice sheet mass and the influence of complex climate dynamics cannot be fully revealed.

To address the above shortcomings, this study is based on GRACE/GRACE-FO satellite data, combining multiple methods such as continuous wavelet transform, wavelet decomposition, and cross-correlation analysis. The relationship between anniversary signals of GrIS mass change and their drivers is systematically investigated. By analyzing the strength and lag effects of LST, ice melting, runoff, rainfall, and snowfall on changes in GrIS mass, uncovering the main drivers influencing changes in GrIS mass, filling the research gap in this area. Through this analysis, we seek to explore the contributions of each of these variables to GrIS mass change, offering a more comprehensive understanding of the complex interactions driving ice sheet dynamics.

The study is organized as follows: firstly, GRACE/GRACE-FO satellite data were used to construct a time series of the GrIS mass change, and this was combined with regional climate models to obtain time series of ice melting, runoff, rainfall, and snowfall. Then, GrIS mass change time series were fitted using linear and quadratic fitting methods, and we compared their performance in capturing the rate of change in mass. In addition, continuous wavelet transform (CWT) was used to analyze the cyclic nature of GrIS mass change and drivers. Afterwards, the annual signals of each variable were extracted by wavelet decomposition and analyzed for comparison. Finally, inter-correlation analyses were used to assess the lags of the drivers to changes in ice sheet mass.

2. Materials and Methods

2.1. Data

2.1.1. Study Area

The Greenland region is located between 59° N and 83° N latitude and 11° W and 74° W longitude. It is bordered to the east by the Greenland Sea, to the west by Baffin Bay, to the north by the Arctic Ocean, and to the southeast by the North Atlantic Ocean. This region represents the largest expanse of continental ice outside Antarctica. Approximately 80% of Greenland lies within the Arctic Circle and is covered in ice. The Greenland region accounts for 10% of the global ice sheet and contributes between 7% and 8% of the world’s freshwater resources [24]. As one of the largest freshwater reservoirs in the world, the continental ice sheet covers an area of 1,833,900 square kilometers, with its northern and southern extremities approximately 2200 km apart. The entire ice sheet is shaped like an inverted bowl and rests upon bedrock. On average, the thickness of the ice sheet measures 1600 m, reaching a maximum thickness of 3367 m [25]. The average elevation of the peripheral region ranges from 1000 to 2000 m, encompassing a total volume of ice and snow amounting to 2.9 million cubic kilometers [26].

2.1.2. GRACE/GRACE-FO RL06 Data

Currently, the Jet Propulsion Laboratory (JPL), the Center for Space Research (CSR), and the Helmholtz Centre Potsdam-German Research Centre for Geosciences (GFZ) provide GRACE/GRACE-FO gravity satellite data [27]. This study primarily utilizes the latest GRACE/GRACE-FO RL06 data released by CSR, with spherical harmonic coefficients truncated to the 60th order. The dataset covers the period from April 2002 to June 2017 for GRACE and from June 2018 to May 2023 for GRACE-FO, totaling 254 months of data. Notably, there are some data gaps, particularly a near one-year gap between GRACE and GRACE-FO.

It is important to note that the reference frame used in the calculation of GRACE and GRACE-FO gravity satellite data is centered on the Earth’s centroid. In this reference frame, the first-order spherical harmonic coefficients (S₁₁, C₁₁, C₁₀), which correspond to surface mass migration and load deformations, are all zero. Therefore, the first-order coefficients (S₁₁, C₁₁, C₁₀) proposed by Swenson et al. [28] are applied to correct for the Earth’s centroid variation during data preprocessing. Due to the influence of satellite orbit, ocean, atmospheric models, and other factors [29], the C₂₀ term calculated from satellite data is often inaccurate. Thus, the C₂₀ term derived from satellite laser ranging (SLR) is used to replace the C₂₀ term in the GRACE/GRACE-FO spherical harmonic coefficients [30].

Furthermore, due to errors from satellite orbit, ocean and atmosphere models, and spherical harmonic terms in the global gravity field, significant north–south stripe errors occur when using time-varying gravity field data to invert surface mass. To mitigate these errors, a combined approach using CHENP4M6 deconvolution filtering and Gaussian filtering with a 250 km radius was employed [31,32]. GRACE/GRACE-FO data are also sensitive to glacial isostatic adjustment (GIA) signals, which must be subtracted when studying surface mass changes. Accordingly, a GIA model was applied to correct for the post-ice rebound effect in the time-varying GRACE/GRACE-FO signals [33,34]. To ensure a complete dataset for analyzing ice sheet mass changes, singular spectrum analysis (SSA) was used to interpolate missing data [35], resulting in 254 months of continuous data.

2.1.3. Surface Mass Balance (SMB) Data

Mass balance methods are widely used to study changes in ice sheet mass. These methods determine mass changes by calculating both surface and basal mass inputs (e.g., rainfall, snowfall) and outputs (e.g., ice melting, runoff, evaporation, ice discharge). Snowfall, rainfall, ice melting, and runoff are referred to as the ice sheet surface mass balance (SMB). The SMB data used in this study are derived from the regional climate model (Modèle Atmosphérique Régional, MARv3.14), developed by the Climatology Laboratory at the University of Liège’s Faculty of Geography [36]. MARv3.14 data, which can be accessed at http://ftp.climato.be/fettweis/MARv3.14/Greenland/ (accessed on 1 January 2025), provide monthly 1 × 1 km SMB data for ice sheet and tundra regions. The model has a spatial resolution of 1 km in Greenland, and the time span used in this study ranges from 2002 to 2023. The SMB data from the MAR model primarily include ice melting, runoff, rainfall, snowfall, evaporation, and surface water. Evaporation and surface water exhibit less variability compared to other variables and are therefore excluded from this analysis [37].

2.1.4. Land Surface Temperature (LST) Data

To explore the relationship between GrIS mass change and LST variations, this study uses temperature data from the Global Historical Climatology Network version 2 (GHCN) and the Climate Anomaly Monitoring System (CAMS), published by the National Oceanic and Atmospheric Administration (NOAA) [38]. This dataset differs from several existing surface air temperature datasets in two key aspects: (1) it utilizes a combination of two extensive individual datasets of station observations sourced from the Global Historical Climatology Network version 2 and the Climate Anomaly Monitoring System (GHCN + CAMS). This allows for regular updates in near real-time, supported by a substantial number of stations; and (2) it employs unique interpolation methods, including the anomaly interpolation approach that incorporates spatially and temporally varying temperature lapse rates derived from observation-based reanalysis for topographic adjustment. The LST data, which can be accessed at https://psl.noaa.gov/data/gridded/data.ghcncams.html (accessed on 1 January 2025), have a spatial resolution of 0.5° × 0.5° and cover the period from April 2002 to May 2023.

2.2. Methodology of Analysis

Gaussian filtering is a filtering technique that utilizes a spatial smoothing kernel function. It is primarily employed to mitigate the higher-order errors associated with spherical harmonic coefficients. Define $W$ as a function that depends only on the angle of $\alpha$ between two points $(\theta ,\lambda )$ and $({\theta }^{\prime },{\lambda }^{\prime })$, as follows $W\left(\theta ,\lambda ,\theta \prime ,\lambda \prime \right)=W\left(\alpha \right)$, among $\cos \alpha =sin\theta sin{\theta }^{\prime }+cos\theta cos\theta \prime \mathrm{c}\mathrm{o}\mathrm{s}(\lambda -{\lambda }^{\prime })$. The formula is:

$$\overline{∆\rho }\left(\theta ,\lambda \right)={\displaystyle \frac{2a{\rho }_{ave}\pi }{3}}\sum _{l=0}^{\infty }\sum _{m=0}^l{\displaystyle \frac{2l+1}{1+k_l}}{W}_l\widetilde{P_{lm}}\left(sin\theta \right)[∆C_{lm}\cos \left(m\lambda \right)+∆S_{lm}\mathrm{s}\mathrm{i}\mathrm{n}(m\lambda )$$

(1)

Among them:

$$W_l={\int }_0^{\pi }W\left(\alpha \right)P_l\left(cos\alpha \right)sin\alpha d\alpha$$

(2)

And $P_l=\widetilde{P_{lm=0}}/\sqrt{2l+1}$ is a Legendre polynomial.

The aforementioned smoothing method was initially proposed by Jekeli [39] to address the issue of poor precision in the high-order terms of GRACE spherical harmonic coefficients, with the aim of enhancing estimates of changes in the Earth’s gravitational field. Wahr et al. [40] were the first to explore the potential application of this method for processing GRACE spherical harmonic coefficient data, yielding favorable results. Jekeli [39] introduced a normalized Gaussian mean kernel function, which ensures that the global integral of $W$ is equal to 1, and it is expressed as follows:

$$W\left(\alpha \right)={\displaystyle \frac{b}{2\pi }}{\displaystyle \frac{exp[-b(1-\mathit{cos}\alpha \left)\right]}{1-e^{-2b}}}$$

(3)

Among them:

$$b={\displaystyle \frac{\mathit{ln}2}{1-\mathit{cos}(r/\alpha )}}$$

(4)

where $r$ represents the weight at $\alpha =0$, which corresponds to the distance from the Earth’s surface when reduced by half; this is referred to as the Gaussian filter radius and is employed recursive relations to compute coefficient $W_l$:

$$\left\{\begin{array}{c}{W}_0={\displaystyle \frac{1}{2\pi }}\hfill \\ W_1={\displaystyle \frac{1}{2\pi }}[{\displaystyle \frac{1+e^{-2b}}{1-e^{-2b}}}-{\displaystyle \frac{1}{b}}]\hfill \\ W_{l+1}=-{\displaystyle \frac{2l+1}{b}}{W}_l+W_{l-1}\hfill \end{array}\right.$$

(5)

As demonstrated in Equations (3) and (4), the Gaussian smoothing weight coefficient is influenced by both the filtering radius and its order. Consequently, when selecting the filter radius, it is essential to not only minimize banding errors but also to retain as much of the genuine signals from the higher-order terms as possible.

The spatial correlation error in the spherical harmonic coefficients of GRACE significantly impacts the inversion results, particularly contributing to the north–south banding error. To address this issue of spatial correlation in GRACE errors, Swenson and Wahr proposed a method to mitigate the north–south banding error by reducing the correlation between spherical harmonic potential coefficients [41], employing a polynomial decorrelation filtering technique. The fundamental principle of this decorrelation filtering method is as follows: first, a specific number $m$ is determined; subsequently, odd-order and even-order coefficients are fitted with polynomials in succession. The resulting fitting values are treated as errors and are then subtracted from the original spherical harmonic coefficients. The specific derivation process is outlined as follows: when fitting $\mathsf{\Delta }{C}_{\mathrm{lm}}$ of a fixed order, if the sliding window is set to 5, the requirements for $\mathsf{\Delta }{C}_{\mathrm{l}-4,\mathrm{m}}$, $\mathsf{\Delta }{C}_{\mathrm{l}-2,\mathrm{m}}$, $\mathrm{\Delta }{C}_{\mathrm{l},\mathrm{m}}$, and $\mathrm{\Delta }{C}_{\mathrm{l}+4,\mathrm{m}}$ are determined. Subsequently, a cubic polynomial fitting is applied.

$$\mathsf{\Delta }{C}_{\mathrm{m}}\left(l\right)=a_0+a_{1}l+a_2{l}^2+a_3{l}^3$$

(6)

Since the fitting value is considered an error, the spherical harmonic coefficient after decorrelation filtering is denoted as $\mathsf{\Delta }{\widehat{C}}_{\mathrm{lm}}=\mathsf{\Delta }{C}_{\mathrm{lm}}-\mathsf{\Delta }{C}_{\mathrm{m}}\left(l\right)$. We apply the same procedure for $\mathsf{\Delta }{S}_{\mathrm{lm}}$, which will yield $\mathsf{\Delta }{\widehat{S}}_{\mathrm{lm}}=\mathsf{\Delta }{S}_{\mathrm{lm}}-\mathsf{\Delta }{S}_{\mathrm{m}}\left(l\right)$.

Chambers [42] and Chen et al. [43] have enhanced the previously described sliding decorrelation filtering method. In this paper, we propose a novel non-sliding decorrelation filtering method (PnMm). The fundamental concept of this approach is as follows: the low-order term coefficients (specifically, the first m order coefficients) remain unchanged, while a polynomial fit is conducted on all odd-order terms (and even-order terms) that are equal to or exceed the degree of the $m$-th order coefficients. Subsequently, these fitted values are subtracted from the original spherical harmonic coefficients, with these fitting values being treated as errors.

Although Gaussian filtering is effective in suppressing high-frequency errors, and decorrelation filtering can eliminate correlation errors, each method has its inherent limitations when applied independently. To achieve a more effective reduction of banding and correlation errors, the strengths of both filtering techniques can be fully utilized by integrating Gaussian filtering with decorrelation filtering. In this combined approach, decorrelation filtering is initially employed to remove coefficient correlations, followed by Gaussian smoothing to mitigate high-frequency noise.

CWT can be used to infer the periodic characteristics of the signal by observing the distribution of the wavelet coefficients, so this paper uses CWT to study the periodic change in ice sheet mass. The CWT of the time series $X_m$ ($m$ = 1, 2,..., $N$) is defined as [44,45].

$$W_m^x\left(S\right)=\sqrt{\delta t/s}{\sum }_{m^{\prime }=1}^N{x}_{m^{\prime }}{\psi }_{{\omega }_0}\left[\left(m^{\prime }-m\right){\displaystyle \frac{\delta t}{s}}\right]$$

(7)

where $W_m^x\left(S\right)$ wavelet coefficients, $\delta t$ and ${\psi }_{{\omega }_0}$ are time scales and the wavelet generating function, $s$ is the wavelet scales, and $m^{\prime }$ and $\sqrt{{\displaystyle \frac{\delta t}{s}}}$ are inverse time and normalization factors, respectively. The main feature of the CWT is its ability to provide both time and frequency information about the signal. The basic idea is to select a suitable wavelet function based on the signal characteristics of the time series; the wavelet is then used as a bandpass filter, and the characteristic period of the filter is linear with the wavelet scale. It is worth noting that the Morlet wavelet is used in this paper. It is expressed as follows:

$${\psi }_{{\omega }_0}\left(\eta \right)=K{e}^{i{\omega }_0\eta }{e}^{-{\displaystyle \frac{1}{2}}{\eta }^2}$$

(8)

where $K$ ($K={\pi }^{-1/4}$) denotes the normalization parameter, and $\eta (\eta =s\cdot t)$ and ${\omega }_0$ denote time and frequency, respectively. When ${\omega }_0=6$, it provides a good balance between the local resolution of time and frequency.

To further analyze the annual variability in GrIS mass, this study employs the wavelet decomposition method to extract the annual signals of various variables. Wavelet decomposition allows a signal to be separated into time series with different frequency components by stretching and transforming the wavelet function. This method decomposes the original signal (S) into low-frequency approximations (A) and high-frequency details (D). The approximation sequence captures the trend characteristics of the original signal, while the detailed sequence represents its finer variations. After the first-level decomposition, the signal is expressed as S = A1 + D1. The second-level decomposition further breaks down the approximate component A1 from the first-level decomposition into A2 and D2, resulting in S = A2 + D2 + D1, where A1 = A2 + D2. This process continues for N levels, with the original signal decomposed as S = AN + DN + ... + D2 + D1, where D1, D2,..., DN represent the detail components obtained at each level of decomposition, and AN represents the final approximation at level N [46]. For this study, the time series are decomposed into scales corresponding to the following frequencies: D1 (2–4 months), D2 (4–8 months), D3 (8–16 months), and D4 (16–32 months). The approximation A4 represents the long-term trend component. To effectively capture the annual signals, we apply a four-level decomposition, with the corresponding frequencies summarized in

Table 1. Frequencies corresponding to the scales of the Meyer wavelet.

Level	n	Period (Months)	Range (Months)
D1	1	2	2–4
D2	2	4	4–8
D3	3	8	8–16
D4	4	16	16–32
A4	5	32	>32

[47].

3. Results

3.1. GRACE Data Processing Results

When using GRACE/GRACE-FO data to derive changes in Earth’s surface mass, these changes are typically represented as equivalent water height [10]. Taking the February 2004 CSR RL06 data as an example, this paper describes the processing of spherical harmonic coefficients from the GRACE/GRACE-FO gravity field using Gaussian filtering, decorrelation filtering, and combined filtering methods. Figure 1a shows the inversion results of global surface mass changes using the raw, unfiltered GRACE/GRACE-FO data. Figure 1b–d display the inversion results using spherical harmonic coefficients with Gaussian filter radii of 200 km, 250 km, and 300 km, respectively.

The results in Figure 1 indicate that as the Gaussian filter radius increases, the suppression of north–south stripe errors and high-frequency noise becomes more pronounced. At a filter radius of 250 km, the geophysical signal is clear, but some residual stripe errors and high-frequency noise remain in mid-latitude regions. A 300 km, the Gaussian filter radius effectively eliminates stripe errors and high-frequency noise, resulting in a clearer geophysical signal, although the signal strength at high latitudes (Greenland and Antarctica) weakens as the filter radius increases. In conclusion, Gaussian filtering is highly effective in suppressing north–south stripe errors (mainly high-frequency noise), and its suppression effect strengthens with increasing filter radius. However, this comes at the cost of slightly weakening the signal itself. Therefore, a 250 km Gaussian filter radius offers a balance, effectively suppressing stripe errors while retaining a valid signal.

Figure 2 presents the global surface mass changes derived from the GRACE/GRACE-FO raw data and processed using the decorrelation filtering method. The results show a significant reduction in north–south stripe errors compared to the unfiltered raw data. However, due to higher-order term errors in the spherical harmonic coefficients, some residual north–south stripe errors persist near the equator. Combining the results from Figure 1 and Figure 2 reveals that the north–south stripes generated by different filtering methods exhibit varying error distributions and densities, which contribute to different signal leakage patterns. Specifically, signal leakage in high-latitude regions (e.g., Antarctica and Greenland) is notably weaker in Figure 2a,b compared to Figure 2c, whereas north–south stripe errors near the equator are more pronounced in Figure 2a,b than in Figure 2c. These findings indicate that decorrelation filtering alone is insufficient to fully suppress stripe errors.

The combined Gaussian and decorrelation filtering results demonstrate that high-frequency errors and systematic correlation errors in the spherical harmonic coefficients of the GRACE/GRACE-FO gravity field are challenging to suppress simultaneously with a single filtering method. However, when spatial smoothing and decorrelation filtering are used in combination, a more effective suppression is achieved. To this end, several filter combination modes were tested by integrating Gaussian filtering with a 250 km radius and different decorrelation filtering methods, including Duan, CHENP4M6, Chambers2012, and Swenson. The results, shown in Figure 3, reveal that Figure 3a contains only minor north–south stripe errors, while signal leakage is more pronounced in Figure 3c,d. In contrast, Figure 3b exhibits no obvious stripe errors or signal leakage. Consequently, for subsequent GrIS mass change analyses, a combination model of 250 km Gaussian filtering and CHENP4M6 decorrelation filtering was selected.

3.2. Trend Analysis of Ice Sheet Mass Changes

Based on the research of both domestic and international scholars on GRACE/GRACE-FO, the time series of GrIS mass change can be obtained by applying the combination filtering process and selecting the appropriate boundary for the study area, followed by GIA correction. Assuming no errors, the time series derived from GRACE/GRACE-FO ice sheet mass changes can be classified into secular, seasonal, and interannual signals according to their spectral characteristics. Secular signals primarily reflect long-term mass trends or periodic fluctuations lasting more than 10 years. Seasonal signals include annual and semi-annual cycles, which are driven by factors such as the Earth’s orbit. Interannual signals fall between seasonal and secular time scales and primarily represent fluctuations in ice sheet mass from year to year.

To further analyze trends in the GrIS mass time series, linear and quadratic curve fitting methods based on least squares estimation were applied to the GrIS mass change data from 2002 to 2023, as shown in Figure 4. The analysis reveals both positive and negative mass changes, where positive values indicate an increase in GrIS mass, and negative values indicate a decrease. Overall, the GrIS has been in mass deficit throughout the study period, with the rate of loss following a pattern of slow to fast changes, and then slow again, before accelerating once more. The linear rate of mass change for the GrIS is −131.4 ± 2 Gt/yr, while the quadratic fit yields a rate of −152.7 ± 8 Gt/yr, where ±2 and ±8 represent fitting error estimates. Notably, GrIS mass loss sharply increased in 2012, stabilized in the two years following, and continued to decrease thereafter. However, results may vary due to differences in data sources, time periods, and data processing methods.

The effectiveness of the linear and quadratic curve fits can be assessed based on the coefficients obtained. The coefficient of determination for the quadratic fit is 0.987, while the linear fit yields 0.885, indicating that the quadratic fit provides a better representation of the data.

To further investigate the drivers of GrIS mass change, Figure 5 displays the time series of ice melting, runoff, rainfall, snowfall, LST, and ice sheet mass change after removing secular trends. This adjustment allows for a clearer reflection of short-term fluctuations and anomalous changes in ice sheet mass. The figure reveals that ice melting and runoff exhibit similar trends. Ice melting and GRACE/GRACE-FO show opposite trends, while runoff and GRACE/GRACE-FO share similar trends. Snowfall and rainfall have less influence on ice sheet mass changes compared to ice melting, runoff, and GRACE/GRACE-FO. Specifically, GrIS mass changes are primarily driven by ice melting and runoff, with snowfall and rainfall exerting secondary effects.

Additionally, LST trends in the opposite direction to ice sheet mass, with an increase in LST corresponding to a decrease in ice sheet mass. This suggests that rising temperatures contribute to accelerated melting of the ice sheet. Notably, the GrIS mass exhibits an accelerated decline between 2010 and 2012, with the most significant decrease occurring in 2012.

3.3. Periodic Signal Analysis of GrIS Mass Change and Its Drivers

To further explore the relationship between GrIS mass change and both SMB and LST from 2002 to 2023, the CWT method was applied to identify cyclical signals in each variable. The results are presented in Figure 6, which displays the CWT energy spectra for each time series. In the figure, a thick black solid line or thicker indicates regions where the red noise test passed the 95% confidence level. The area below the thin black solid line represents the wavelet influence cones, where edge effects may significantly impact the data [44].

Figure 6 presents the results of the continuous wavelet transform (CWT) analysis for GrIS mass, ice melting, rainfall, runoff, snowfall, and LST time series. The wavelet power spectra for LST and GrIS mass changes show a high degree of similarity, with the signal peaks predominantly occurring in the 10- to 14-month period range. A distinct annual signal is observed across the entire time domain, and the corresponding spectral energy passes the 95% confidence test, confirming significant annual signals in both LST and GrIS mass changes. These findings align with those of Bian et al. [31], who also identified strong annual signals in LST and GrIS mass variations using the CWT method.

Additionally, ice melting, rainfall, and runoff show distinct annual signals throughout the observation period, while snowfall exhibits less pronounced annual variability over several years. GrIS mass changes, ice melting, runoff, rainfall, and snowfall also display periodic signals in the 4- to 6-month range; however, these signals have weaker energy spectra and are less significant. In summary, the results highlight that the annual signal is the predominant feature across the entire time domain.

3.4. Comparison of Annual Signals for GrIS Mass Change and Control Variables

According to Section 3.3, the annual signals of each variable dominate the entire time domain; to investigate the correlation of GrIS mass changes with the annual signals of various climate variables, wavelet decomposition method is used to decompose the original time series of each variable. As can be seen from Table 1, the D3 band contains annual signals, and Figure 7 shows the time series of each variable annual signal in the D3 band.

Figure 7 illustrates substantial annual amplitude variations in GRACE/GRACE-FO, ice melting, and runoff compared to smaller variations in rainfall and snowfall. As shown in

Table 2. The mean annual amplitudes between GRACE/GRACE-FO-derived ice sheet masses change, ice melting, rainfall, runoff, and snowfall.

Variables	GRACE/GRACE-FO	Ice Melting	Runoff	Rainfall	Snowfall
Annual mean amplitude (Gt/month)	31.1	58.8	44.5	4.1	7.4

, the annual amplitudes of ice melting (58.8 Gt/month) and runoff (44.5 Gt/month) are significantly larger, being 1.9 and 1.4 times that of GRACE/GRACE-FO (31.1 Gt/month), respectively. Additionally, ice melting and runoff exhibit similar annual phase changes, and their amplitudes surpass those of rainfall (4.1 Gt/month) and snowfall (7.4 Gt/month). These findings highlight that runoff and ice melting are the primary drivers of GrIS mass changes, with snowfall and rainfall having secondary effects.

Further analysis of Figure 7a,b reveals that the annual phase of LST (land surface temperature) is opposite to that of GRACE/GRACE-FO, indicating a clear link between rising temperatures and increased ice mass loss. Notably, during the 2022–2023 period, Greenland experienced sustained temperature increases, culminating in a peak in May 2023, when GrIS mass loss also reached its maximum. These observations underscore the critical role of temperature in accelerating GrIS mass loss.

Correlation analysis was used to further explore the correlation between the annual signals of the variables and GrIS mass changes. For the time series $x_1\left(t\right)$ and $x_2\left(t\right)$, the cross-correlation function is provided by [48]:

$$\rho \left(T\right)={\displaystyle \frac{{\sigma }_{12}\left(t\right)}{\sqrt{{\sigma }_{11}{\sigma }_{22}}}}$$

(9)

where $\rho \left(T\right)$ is between −1 and 1, the absolute magnitude responds to the degree of correlation between two time series, $T$ is the delay of the time series $x_1\left(t\right)$ relative to the time series $x_2\left(t\right)$, ${\sigma }_{12}\left(t\right)$ is the mutual covariance of time series $x_1\left(t\right)$ and $x_2\left(t\right)$, and ${\sigma }_{11}$ and ${\sigma }_{22}$ are the autocovariances of time series $x_1\left(t\right)$ and $x_2\left(t\right),$ respectively. By studying the $\rho \left(T\right)$ distribution of the time series, it is possible to analyze the correlation of the time series in the time domain. Generally, the correlation coefficient ${\rho }_{max}$ with the largest absolute value from $\rho \left(T\right)$ is used to represent the degree of correlation between the two sets of time series; ${\rho }_{max}$ with the corresponding delay T then reflects the delay in time of the two timeseries [49].

Figure 8a–e illustrate the normalized cross-correlation between GrIS mass changes and ice melting, rainfall, runoff, snowfall, and LST in the D3 frequency band.

Table 3. Correlation of GRACE/GRACE-FO-derived ice masses change with ice melting, rainfall, runoff, snowfall, and LST.

	Variables	Ice Melting	Rainfall	Runoff	Snowfall	LST
Lags
−10		−0.841	−0.788	−0.851	0.023	−0.888
−9		−0.861	−0.685	−0.851	0.372	−0.788
−8		−0.668	−0.355	−0.612	0.627	−0.459
−7		−0.292	0.065	−0.211	0.712	−0.005
−6		0.160	0.459	0.244	0.594	0.450
−5		0.568	0.724	0.632	0.302	0.782
−4		0.825	0.790	0.852	−0.084	0.899
−3		0.863	0.644	0.844	−0.456	0.771
−2		0.672	0.329	0.610	−0.705	0.435
−1		0.299	−0.070	0.213	−0.711	−0.018
0		−0.155	−0.446	−0.242	−0.591	−0.462
1		−0.565	−0.688	−0.627	−0.248	−0.768
2		−0.821	−0.746	−0.842	0.160	−0.866
3		−0.854	−0.608	−0.828	0.513	−0.732
4		−0.654	−0.312	−0.589	0.711	−0.406
5		−0.275	0.062	−0.191	0.707	0.024
6		0.177	0.417	0.256	0.511	0.443
7		0.576	0.659	0.629	0.189	0.739
8		0.810	0.725	0.825	−0.165	0.835
9		0.819	0.597	0.793	−0.457	0.709
10		0.603	0.310	0.544	−0.620	0.396

summarizes the corresponding lags and correlation coefficients obtained from the cross-correlation analysis. The results indicate that ice melting, rainfall, runoff, snowfall, and LST are significantly correlated with GrIS mass changes, with maximum correlation coefficients of 0.863, 0.790, 0.852, 0.712, and 0.899, respectively. These peak correlations correspond to lag times of 3, 4, 4, 7, and 4 months for ice melting, rainfall, runoff, snowfall, and LST, respectively. Furthermore, the analysis confirms that ice melting, runoff, and LST exert the strongest influence on GrIS mass changes, while snowfall and rainfall play secondary roles.

According to the results of cross-correlation analysis, the influence of driving factors on the lag in mass change of the ice sheet is further examined. Melting typically coincides with rising temperatures; during warmer seasons, the surface of the ice sheet begins to melt. Water then penetrates through the surface or flows along cracks within the ice sheet, and it takes time for this process to manifest significant changes in mass. Some of this meltwater may percolate or refreeze within the ice sheet or underground, contributing to a hysteretic effect on mass change. Runoff refers to the outflow of meltwater from the ice sheet into oceanic or groundwater systems. The flow of meltwater from the surface can be delayed; particularly during periods when meltwater accumulates, there exists a lag between water reaching the edge of the ice sheet and runoff that ultimately enters oceanic bodies. Due to complex drainage systems within the ice sheet, complete outflow can take months. Portions of this meltwater may even remain stored in groundwater systems for extended periods, further exacerbating delays in mass changes associated with melting processes. The impact of rainfall on ice sheet mass is primarily manifested through an acceleration in snowmelt and a direct increase in water volume. While rainfall hastens melting processes, its effects on ice sheet mass are not immediately observable but become apparent over time. Rainfall rapidly melts the snow layer atop the ice sheet, while groundwater infiltration and outflow along the edges further contribute to this dynamic. Collectively, these processes result in a lag in changes to ice sheet mass. In contrast, snowfall typically requires an extended period before its effects are reflected in alterations to ice sheet mass. As snow accumulates, it forms multiple layers that influence the surface characteristics of the ice sheet. The accumulation of snow undergoes seasonal deposition and compaction processes that ultimately translate into changes in ice sheet mass. Land surface temperature directly influences both melting of the ice sheet and accumulation of snow. As temperatures rise, melting begins at the surface of the ice sheet; however, melted ice and snow can temporarily refreeze, forming either an additional layer of ice or a water body. This phenomenon may take several months before resulting in significant changes to overall ice sheet mass.

4. Discussion

Previous studies on GrIS mass changes have predominantly focused on short-term time series and seasonal trends, with most investigations emphasizing winter and summer variations. In contrast, periodic and annual signals have received comparatively less attention. For instance, Shamshiri et al. [50] employed continuous wavelet transform and cross-wavelet transform to investigate the common frequencies and relative phases between the GrIS mass change time series and the IST time series. Their analysis revealed a strong shared frequency between these variables. Furthermore, by fitting the GrIS mass time series, it was determined that a quadratic fit provided superior results, which aligns with the conclusions drawn in this study.

Bevis et al. [9] indicate that the negative phase of the North Atlantic Oscillation (NAO) is linked to an increase in surface melting of ice sheets and a corresponding acceleration in mass loss. Additionally, this phase is associated with heightened flow rates and thinning of the ice within the sheet, which are extending further inland. Their results indicated that the topographic characteristics of southwest Greenland make it more sensitive to atmospheric forcing, predicting this region will become a major contributor to sea level rise within 20 years. Ramillien et al. [51] conducted an analysis that revealed significant interannual fluctuations in ice sheet mass change throughout the study period, which are closely associated with climatic factors. It was emphasized that subsequent studies should incorporate additional climate, temperature, and precipitation data to achieve a more comprehensive understanding of the various influences on changes in ice sheet mass.

This study utilized the latest GRACE/GRACE-FO RL06 data (2002–2023) provided by CSR, complemented by the MARv3.14 regional climate model and GHCN + CAMS global land surface temperature (LST) data, to investigate GrIS mass changes and their periodic variations. Firstly, an in-depth analysis of the time series of GrIS mass changes was carried out. The results of the study demonstrate that the GrIS shows significant mass loss trend over the past decades; however, the rate of change exhibits phase characteristics. In particular, in 2012, the GrIS experienced a rapid melting event. Subsequently, the rate of loss of ice mass slowed between 2013 and 2018. However, in 2019, the Greenland ice sheet once again experienced a rapid melting phenomenon. The above results show that GrIS mass changes exhibit significant volatility across time, which was consistent with the results of Sasgen et al. [52].

To further explore the annual signal of mass change in the GrIS and its driving factors, periodic signals in GrIS mass change, ice melting, runoff, rainfall, snowfall, and LST were analyzed using continuous wavelet transform (CWT), and annual signals were extracted via wavelet decomposition. The results show that ice melting and runoff are the main factors affecting the mass change of the GrIS. For example, a sharp increase in ice melting and runoff in 2012 led to a severe loss of mass of the GrIS that year, and a decrease in precipitation and an increase in ice melting and runoff in 2019 combined to cause a severe loss of mass of the GrIS that year. In addition, in 2013, ice melting and reduced runoff allowed the GrIS to experience a slight mass accumulation during the year. LST contributes to the mass loss of the GrIS to some extent: the main manifestation is that the increase in temperature will lead to an increase in runoff and ice melting, which will affect the mass change of the GrIS. Ruan [37] analyzed the relationship between the mass change of the GrIS and the variation in MAR model variables in summer and winter: it was found that rainfall and snowfall are not the main factors affecting the mass loss of the GrIS; the amount of summer melt is the main factor affecting the mass loss of the GrIS. The conclusions are basically the same as those obtained in this study, but this study combined temperature data; thus, the driving factors affecting the mass change of the GrIS are analyzed more comprehensively. Furthermore, the refreezing of melted ice and snow significantly impacts the mass change of the GrIS. During the summer months, the influence of refreezing on surface moisture is particularly pronounced. The GrIS experiences melting of ice and snow, resulting in the formation of surface water (meltwater). If this meltwater is not drained or absorbed into the ice sheet promptly, it will refreeze during fall and winter as temperatures decline. The layer formed by this refreezing process affects the rate of mass change within the ice sheet. The melting process absorbs energy from solar radiation, while refreezing releases heat. Although refreezing temporarily mitigates meltwater loss, it also alters the thermodynamic properties of the ice sheet. In the short term, refreezing may slow down mass loss; however, over a longer period, this newly formed layer could accelerate changes in mass within the ice sheet. If surface water undergoes repeated cycles of melting and refreezing throughout an extended seasonal cycle, it may lead to continuous adjustments in the structure of the ice sheet—ultimately exacerbating its overall mass change.

Moreover, the analysis demonstrated that various factors affect GrIS mass changes differently and that lag effects are present, with ice melting, runoff, rainfall, snowfall, and LST influencing ice mass change at varying temporal offsets. By analyzing the lag of each driving factor relative to the mass change of the GrIS, we can better help people to improve the understanding of the mass change of the GrIS. Based on previous studies, the annual signal and driving factors of mass change of the GrIS are studied in detail from the perspective of climate and temperature factors. Although some preliminary conclusions were drawn in this study, the limitations of the data, as noted by Florent Cambier et al. [53], suggest that further investigation is warranted regarding the primary patterns of ice sheet mass change and their relationship with climate indices (e.g., North Atlantic Oscillation (NAO), Greenland Blockage Index (GBI), Atlantic Multi-Year Intergenerational Oscillation (AMO)) as well as meteorological parameters (e.g., temperature, precipitation, and albedo) in future research. Such efforts could further elucidate the complex interactions between climatic, hydrological, and geophysical factors affecting GrIS mass dynamics.

5. Conclusions

As global climate change intensifies, the loss of mass of the GrIS has become one of the main drivers of global sea level rise. Assessing and analyzing the impact of climatic factors on changes in the mass of the GrIS has become a central issue in current research. In this context, the present study, based on GRACE/GRACE-FO satellite data and characterization of mass changes in the monitoring of the GrIS, focuses on analyzing the drivers of changes in the mass of the GrIS and its annual changes. The key findings of this study are as follows: firstly, the time series data for land surface temperature (LST), ice melting, runoff, rainfall, snowfall, and GrIS mass changes exhibit significant cyclical variations. Further analysis utilizing continuous wavelet transform (CWT) revealed that GrIS mass changes, ice melting, runoff, rainfall, and LST all display prominent annual signals; however, the annual signal for snowfall is relatively weak and negligible. To further investigate the annual variations in these variables, wavelet decomposition was employed to extract their respective annual signals. The results indicate that the annual amplitudes of ice melting and runoff demonstrate substantial variability that significantly exceeds those of rainfall and snowfall. Moreover, LST exhibited an annual phase that is inversely related to GrIS mass changes, suggesting a strong correlation between rising temperatures and increased ice mass loss. Finally, cross-correlation analysis indicated that ice melting, runoff, rainfall, snowfall, and LST all exert significant influences on GrIS mass changes with lag times of 3, 4, 4, 7, and 4 months, respectively. Among these factors, ice melting, runoff, and LST were identified as primary drivers of mass loss while snowfall and rainfall had secondary effects.

While this study highlights the interplay of climatic and hydrological processes influencing GrIS mass loss, the dominant role of warmer temperatures, ice melting, and runoff in mass loss was emphasized. These findings provide important insights for further understanding of GrIS mass dynamics. However, reliance on regional climate models and LST data still has certain limitations. Future research should incorporate ice discharge data, observed precipitation, and ice core records from meteorological stations, such as those of the Danish Meteorological Institute (DMI), to refine our understanding of GrIS mass dynamics and improve predictions of its contribution to global sea level rise.