Complex Principal Component Analysis of Antarctic Ice Sheet Mass Balance

: Ice sheet changes of the Antarctic are the result of interactions among the ocean, atmosphere, and ice sheet. Studying the ice sheet mass variations helps us to understand the possible reasons for these changes. We used 164 months of Gravity Recovery and Climate Experiment (GRACE) satellite time-varying solutions to study the principal components (PCs) of the Antarctic ice sheet mass change and their time-frequency variation. This assessment was based on complex principal component analysis (CPCA) and the wavelet amplitude-period spectrum (WAPS) method to study the PCs and their time-frequency information. The CPCA results revealed the PCs that affect the ice sheet balance, and the wavelet analysis exposed the time-frequency variation of the quasi-periodic signal in each component. The results show that the ﬁrst PC, which has a linear term and low-frequency signals with periods greater than ﬁve years, dominates the variation trend of ice sheet in the Antarctic. The ratio of its variance to the total variance shows that the ﬁrst PC explains 83.73% of the mass change in the ice sheet. Similar low-frequency signals are also found in the meridional wind at 700 hPa in the South Paciﬁc and the sea surface temperature anomaly (SSTA) in the equatorial Paciﬁc, with the correlation between the low-frequency periodic signal of SSTA in the equatorial Paciﬁc and the ﬁrst PC of the ice sheet mass change in Antarctica found to be 0.73. The phase signals in the mass change of West Antarctica indicate the upstream propagation of mass loss information over time from the ocean–ice interface to the southward upslope, which mainly reﬂects ocean-driven factors such as enhanced ice–ocean interaction and the intrusion of warm saline water into the cavities under ice shelves associated with ice sheets which sit on retrograde slopes. Meanwhile, the phase signals in the mass change of East Antarctica indicate the downstream propagation of mass increase information from the South Pole toward Dronning Maud Land, which mainly reﬂects atmospheric factors such as precipitation accumulation. the South Paciﬁc region has 7.5-year periodic signals. The correlation coefﬁcient of the low-frequency signal between the meridional wind ﬁeld and SSTA in the equatorial Paciﬁc is 0.77. These results suggest that the changes of the SSTA in the Niño 1+2 region are correlated with the low-frequency signals of the ﬁrst PC.


Introduction
Antarctica shows an important regulatory role in climate changes. The relationship between the ice sheet and climate change over Antarctica has been intensively studied. Studies [1][2][3][4][5][6] indicate there has been an increased warming of the atmosphere over Antarctica during the second half of the 20th century, especially in the western Antarctic Peninsula (WAP) with the rates of 0.4 • C/decade. Turner et al. (2016) [7] and Oliva et al. (2017) [8] analyzed stacked temperature records and noted that the atmospheric temperature has shifted from 0.32 • C/decade to −0.47 • C/decade between 1979-2014. This climate change contributed to the regional mass balance of glaciers. El Niño changes and atmospheric circulation changes in Antarctica have also affected the ice sheet balance [7,9,10].
Some studies suggest that the factors of iceberg calving, basal melt, intrusion of warm saline water on retrograde slopes, and ice discharge have played a dominant role The variation of Earth's gravity field reflects the redistribution of mass inside the Earth. Over a short time (compared with geologic time), the variation of the gravity field can be regarded as mass transfer of the Earth's surface and shallow fluids. GRACE was jointly developed by NASA and The German Research Center for Geoscience (GFZ) and successfully operated for more than 15 years. Its monthly gravity solutions reflect changes of 1-mm geoid fluctuation at a 300-km spatial scale [25,26] and can be used to monitor gravity field variations caused by changes in hydrology and the cryosphere, earthquakes, and glacial isostatic adjustment [27][28][29][30][31].
We used GRACE data Release-06 (RL06) solutions provided by the Center for Space Research (CSR), University of Texas-Austin. The 164 approximately monthly GRACE gravity solutions cover the period from April 2002 to through June 2017 (~14 solutions are missing). Each monthly solution consists of normalized spherical harmonic (SH) coefficients, to degree and order 60. The main improvements in the RL06 products were as follows: (1) a reprocessed GPS constellation consistent with ITRF2014/IGS14; (2) reprocessed RL03 Level-1B products for K-band and star camera instrument data; (3) updated models for ocean tides (FES2014) and Atmosphere and Ocean De-aliasing (AOD1B RL06); and (4) modified parameterization of orbit and instruments. The resulting RL06 time series of monthly GRACE Level-2 spherical harmonics with its underlying processing standards will then serve for the continuation with GRACE-follow-on (GRACE-FO) data with the idea of harmonizing the two time series. This will ensure the consistency between GRACE and GRACE-FO time series for all scientific studies and have improved precision. However, at high degrees and orders, GRACE spherical harmonics are contaminated by noise, including longitudinal stripes, and filtering is still needed. In our study, the smoothness priors method [32,33] was used to remove noise in the spatial domain. Compared with the Gaussian filter, correlated-error filter, and the combined filter (Gaussian with 300-km smoothing and correlated-error), the smoothness priors method has advantages of less Remote Sens. 2021, 13, 480 3 of 19 reduction in signal amplitude at high latitude, preservation of greater detail for shortwavelength components in the result, and less signal distortion at low latitudes.

Glacial Isostatic Adjustment
To compute ice sheet mass trends from GRACE data and interpret them as changes in the water content of hydrologic basins, ocean bottom pressure, or ice sheet mass, it is necessary to remove the effect of the glacial isostatic adjustment (GIA) of the lithosphere and mantle. For the GIA effect in Antarctica, several models have been released since 2012 [34][35][36]. We choose the latest GIA model [37], which consists of normalized spherical harmonic coefficients to degree and order 89, to remove the effect of GIA. The new model used a Bayesian framework with many new GPS time series from 459 sites and 11451 RSL data. Related data are available from the link https://vesl.jpl.nasa.gov/solid-earth/gia/.

Data Processing and Equivalent Water Height
The monthly Release 06 GRACE fields were expressed in the form of spherical harmonic coefficients. We replaced the coefficients of C20 in the GRACE field with satellite laser ranging estimations. The influence of geocenter variations on gravity was also considered [38]. According to Wahr et al. (1998) [19], at an arbitrary point with colatitude θ and longitude λ, the ice mass change can be expressed as where ρ e is the density of the Earth and ρ w is water density. Parameters k n and a are the load Love number and the equatorial radius, respectively; P m n (cos θ) is the nth degree and mth order fully normalized Legendre function. The coefficients c m n and s m n are the normalized SH coefficients.
In order to reduce the leakage of signal originating from the ice sheet to the oceans, an exact averaging kernel ϕ(θ, λ) (1 inside and 0 outside the region) is used based on the grounded portions of the Antarctic drainage data definitions by Zwally [24].
The mass change in vertically integrated water storage averaged over Antarctic region can be re-expressed by a sum of Stokes coefficients as where ϕ c nm and ϕ s nm are the spherical harmonic coefficients of the averaging kernel described above.

Complex Principal Component Analysis
Similar to empirical orthogonal function analysis, principal component analysis (PCA) also decomposes a matrix H composed of n observations at m observation points into the product of the mutually orthogonal spatial function V and the mutually orthogonal time function T; that is,H mn = V mm T mn , where Here, V is a spatial function that does not change with time, and T is a time function that only depends on time changes. v j = (v 1j , v 2j , L, v mj ) T is the jth spatial mode, and t j = (t j1 , t j2 , L, t jn ) is the jth PC. The product of the jth spatial mode and the jth PC can be expressed as Thus, the product of each spatial mode and its corresponding main component reflects the main change in the observation field H captured by the PC under the maximum variance criterion. Since the components are decomposed according to the principle of maximizing variance, they can reflect the changing characteristics of the original observation field H to the greatest extent. The disadvantage of the PCA method is that it can only detect standing waves rather than advancing waves due to the absence of phase information. To overcome this, Horel (1984) [39] introduced phase information into PCA to identify traveling signals and named the method CPCA [33]. CPCA has been widely used in the fields of atmosphere, ocean, and earth science to identify information such as advancing waves of observation fields [40][41][42][43]. The CPCA method combines the observation data and its Hilbert transform into a complex series and then executes the PCA. The CPCA method still performs PCA analysis, which has the same theoretical basis as the PCA method, but the input data contain phase information, so these data can express the wave propagation direction information. This information helps researchers uncover additional information about the observation data field itself.
Before CPCA, a complex observation sequence should first be constructed using a real observation series. For a time varying observation vector u j (t), its Fourier expansion is: here, j represents the location of the observation point, ω is the Fourier frequency, and t is the observation time. In order to describe the propagation characteristics of a time series, it is necessary to construct the imaginary part and convert it into a complex observation sequence. The complex observation sequence can be expressed as: here, we define c j (ω) = a j (ω) + ib j (ω), i = √ −1. Then the Equation (5) can be expanded as: The real part of Equation (6) is the original observation sequence and the imaginary part is the Hilbert transform of the real part, which does not change the amplitude of each component of u j (t). However, the phase of each spectral component is advanced by π/2.
The traditional PCA is the principal component analysis of the real observation vector, whereas the CPCA analysis is the principal component analysis of the constructed complex vector. After the normalization of the complex observation vectors, the average value is subtracted from the complex observation vector of each observation point, and then divided by the standard deviation. The complex correlation matrix of the observation point can be expressed as: here r jk represents the multiple correlation coefficients between the jth and kth observation points. CPCA compresses information using the least complex eigenvector e jn of the correlation matrix (Equation (7)) and the complex principal component p n (t), because the correlation matrix (Equation (7)) is a Hermitian matrix including n real eigenvalues λ i denotes the contribution percentage of the jth principal component.
Observation vector U j (t) can be expressed as the sum of N principal components, where * stands for the complex conjugate, and both complex principal components and complex eigenvectors are orthogonal. The nth complex eigenvector element e jn can be expressed as e jn = U j (t) * p n (t) t = s jn e iθ jn (11) here, e jn indicates the multiple correlation relationship between the jth time sequence and nth principal component, s jn and θ jn are, respectively, correlative order of magnitude and phase, and [· · · ] t signifies the average of times. The time sequence elements of principal components can be expressed as the functional form of amplitude T n and phase Φ n . P n (t) = T n (t)e iΦ n (t) (12)

WAPS
For a time series, we usually need to perform harmonic analysis to determine the trend of changes and the periodic signals contained and then analyze the possible factors affecting its changes. For periodic signals, the harmonic analysis method gives only the frequency information of the signal and not the details of how the signal changes over time. The wavelet amplitude spectral analysis method compensates for this shortcoming by providing frequency and amplitude information of the signal vary with time.
The variation of the ice sheet in Antarctica is the result of interactions of many factors such as the ice sheet, atmosphere, and ocean. Its changes reflect the temporal variations of specific factors represented by different frequency signals as well as long-term nonperiodic changes such as enhanced ice-ocean interaction, ice discharge, and intrusion of warm saline water on retrograde slopes. The PCs can be derived after CPCA analysis and can be expressed by a linear term and many periodic signals. The linear and periodic terms can be easily obtained through harmonic analysis. The linear and periodic terms all contribute to the mass changes of the observation field captured by its PCs. In addition to the linear term, we also need to analyze the time-frequency information of the periodic signals and investigate its possible connections with different climate factors based on the periodic characteristics of the signals. Therefore, after obtaining the evolution information of PCs of mass change in Antarctica, the time-frequency information of quasi-periodic signals required analysis. The WAPS is a useful tool for analyzing the variation in time of periods and the amplitude (energy) of different signals [44]. The advantage of this method is that it is simple to use and we can easily obtain the time-varying amplitude and period, as well as phase information of each quasi-periodic signal. This method has been widely used in time-frequency analysis of sea level changes, glacier mass balance, and other geophysical signal changes. For a more detailed description, see [44][45][46][47].

Results
Each variation field of ice sheet was calculated over the Antarctic using each of the monthly gravity solutions following Equation (3), and then filtered with the smoothness priors method. In this way, we obtained the time sequence of mass changes from April 2002 to June 2017 at each grid point. However, some monthly GRACE gravity solutions were not available due to the data quality. Therefore, the time series of mass change at one point shows discontinuous and sudden transitions. Before applying the CPCA analysis, we need to interpolate the missing data to make the time sequence continuous. Here, we used a spline function to interpolate missing data in the time series. The rate of change of the ice sheet mass based on GRACE data was also estimated. Figure 1 shows the time evolution and trend of the ice sheet mass change after eliminating the effect of GIA on the Antarctic from 2002-2017. It can be seen that the loss of ice in the Antarctica accelerated after 2007. The ice sheet change has the following spatial characteristics: There is a large mass loss area from the Antarctic Peninsula (AP) to Ellsworth (corresponding to the basins B19 to B27). The average annual ice sheet mass loss was 182.9 ± 13.4 Gt in this area, and this amplitude is consistent with estimates of 159 ± 26 Gt in West Antarctica, 33 ± 16 Gt on the Antarctic Peninsula [48], and 179 ± 17.2 Gt in West Antarctica [49]. There is a small mass drop centered in the Ross Land and Victoria area (corresponding to the basins B14 to B15), and Wilkes area (corresponding to the basin B13). The average ice sheet mass loss in these two areas was 12.4 ± 4.6 Gt and 13.4 ± 2.5 Gt per year. The area of mass increase is mainly located in Dronning Maud Land and Enderby Land (corresponding to the basins B4 to B8) areas corresponding to the East Antarctic West Indian Ocean sector. The average ice sheet mass increase in the area was 54.2 ± 5.0 Gt per year. The mass of ice sheets in other areas did not change significantly. The spatial characteristics of the mass change trend are similar to the result of ESA CCI AMB (https://data1.geo.tu-dresden.de/ais_gmb/).
We used the CPCA method to analyze the PCs of ice sheet mass change from 2002 to 2017. Table 1 shows the contributions to the mass change of the first three PCs of ice sheet change during the period of 2002-2017. The eigenvalues corresponding to the first three PCs are 9703.61, 297.32, and 216.33, respectively, and their contributions to the ice sheet balance in the region are 83.73%, 2.57%, and 1.87%, respectively. The results of the PCs analysis show that the main factors influencing the Antarctic ice sheet mass balance change mainly come from the behavior of the first PC. This behavior has absolute dominance over the ice sheet variation in the Antarctic region and explains 83.73% of its changes.  We used the CPCA method to analyze the PCs of ice sheet mass change from 2002 to 2017. Table 1 shows the contributions to the mass change of the first three PCs of ice sheet change during the period of 2002-2017. The eigenvalues corresponding to the first three PCs are 9703.61, 297.32, and 216.33, respectively, and their contributions to the ice sheet balance in the region are 83.73%, 2.57%, and 1.87%, respectively. The results of the PCs analysis show that the main factors influencing the Antarctic ice sheet mass balance change mainly come from the behavior of the first PC. This behavior has absolute dominance over the ice sheet variation in the Antarctic region and explains 83.73% of its changes. corresponding to the second PC of the mass change in Antarctica rapidly drops to 2.57%. The ice sheet mass in the area of basin B21 (Figure 2d) is more sensitive to the behavior corresponding to the second PC. However, change corresponding to the third PC only accounts for 1.87% of the mass change. The ice sheet mass in the area of basins B21 to B23 (Figure 2f) is more sensitive to the behavior corresponding to the third PC. This result shows that the changes in the climatic factors that have recently affected the mass balance of the ice sheets in Antarctica are relatively simple.

The First Three Principal Components
Similar to the PCA analysis method, CPCA also decomposes a matrix composed of n observations at p observation points into the product of the mutually orthogonal spatial function V and the mutually orthogonal time function T. The product of each spatial mode and its corresponding PC reflects the primary change of the observation field H captured by the PC under the maximum variance criterion. Since the components are decomposed according to the principle of maximizing variance, they can reflect the changing characteristics of the original observation field H to the greatest extent. Compared with PCA, the CPCA results not only show the time evolution of the PCs and their corresponding spatial mode characteristics, but also provide phase distribution information of the PCs, thereby enabling CPCA to identify traveling waves. This traveling wave information not only reflects the spatial correlation of mass change, but also indicates the direction from which the factor that affected the mass change originated.
As shown in Figure 2a, the first PC exhibits a rising trend and low-frequency signals, revealing that the main factors affecting the ice sheet changes are linear and low-frequency periodic terms. Figure 2b reflects the spatial characteristics of ice sheet mass changes in basins B20 to 23 of West Antarctica and basins B20 to 23 of East Antarctica. The result shows that the first PC displays a rising trend, and the mass change in basins B20 to 23 of the first spatial mode has a maximum negative value, while the mass variation in basins B5 to 8 has a maximum positive value. Therefore, the product between the first PC and its corresponding spatial mode reflects the mass loss trend characteristics in basins B20 to 23 and the mass increase in the B5 to 8 area. This result is consistent with the spatial charac-

The First Three Principal Components
Similar to the PCA analysis method, CPCA also decomposes a matrix composed of n observations at p observation points into the product of the mutually orthogonal spatial function V and the mutually orthogonal time function T. The product of each spatial mode and its corresponding PC reflects the primary change of the observation field H captured by the PC under the maximum variance criterion. Since the components are decomposed according to the principle of maximizing variance, they can reflect the changing characteristics of the original observation field H to the greatest extent. Compared with PCA, the CPCA results not only show the time evolution of the PCs and their corresponding spatial mode characteristics, but also provide phase distribution information of the PCs, thereby enabling CPCA to identify traveling waves. This traveling wave information not only reflects the spatial correlation of mass change, but also indicates the direction from which the factor that affected the mass change originated.
As shown in Figure 2a, the first PC exhibits a rising trend and low-frequency signals, revealing that the main factors affecting the ice sheet changes are linear and low-frequency periodic terms. Figure 2b reflects the spatial characteristics of ice sheet mass changes in basins B20 to 23 of West Antarctica and basins B20 to 23 of East Antarctica. The result shows that the first PC displays a rising trend, and the mass change in basins B20 to 23 of the first spatial mode has a maximum negative value, while the mass variation in basins B5 to 8 has a maximum positive value. Therefore, the product between the first PC and its corresponding spatial mode reflects the mass loss trend characteristics in basins B20 to 23 and the mass increase in the B5 to 8 area. This result is consistent with the spatial characteristics of mass change in Antarctica, indicating that the first PC captured the spatial pattern of the dominant mass change trend of the ice sheet. In addition to the rising trend, the first PC also exhibits a significant low-frequency periodic signal. Since the amplitude of the quasi-periodic signal is not constant over time, the linear estimation of the ice sheet mass balance in Antarctica contains the contribution of the low-frequency signal to the trend. Compared with the second and third PC, the factor corresponding to the first PC has a larger scale because the area that affects the mass change of Antarctica has the larger scale.
Researchers have pointed out that on the southern Antarctic Peninsula, the ice sheet manifested a slight mass loss trend from 2003 to 2015, while the regional atmospheric climate model result showed that during this period the integral surface mass balance (iSMB, the cumulative sum of each monthly SMB) decreased slightly, suggesting there is no significant increase or decrease in mass in this region at that time [50,51].
Compared with the traditional PCA analysis, the CPCA method introduces phase information, and therefore can provide traveling wave information such as wave propagation direction in the result. From the phase distribution of first PC (Figure 2b), we can see that the factors affecting the mass balance in West Antarctica mainly come from the western AP and the coastal areas of the Amundsen Sea, while the factors affecting the mass balance in East Antarctica mainly come from the inland direction. The arrow lengths in these areas are also the longest, suggesting that the mass changes in the corresponding areas are the most dramatic relative to other areas corresponding the first PC. The phase information direction displays the source of the driving force that affects the mass change, such as enhanced ice-ocean interaction, basal melt, and intrusion of warm saline water on retrograde slopes in West Antarctica, suggesting the propagation progress of ice sheet loss from the ocean-ice interface to the southward upslope in this area. Meanwhile, in East Antarctica, it may reflect the progress of precipitation accumulation from inland to coast. Figure 2d illustrates the spatial mode and phase distribution of the second PC. The phase distribution of the second PC shows that the factors affecting the mass balance are mainly along the latitude line. The phase direction in basins B21-B23 is counterclockwise, while in basin B18 the phase is clockwise. It is somewhat similar to atmospheric circulation at height of 700hPa, which was affected by a disturbance in the Antarctic Peninsula and produced a counterclockwise atmospheric circulation in Basin 21 and Basin 23 region. This data suggests that the impact factors may come from the disturbance of small scale local atmospheric circulation. The phase distribution of the third PC (Figure 2f) shows that the factors affecting the mass balance mainly come from the direction of the Amundsen Sea.
In addition to the 7.5-year periodic signal, the WAPS (Figure 3) shows that the second and third PC also contain the signals with periods < 4 years. The energy of the 7.5-year and 1-year periodic signals are the largest, followed by those of the other periodic signals. Compared with the first PC, the scale factors of the second and third PC are smaller because they affect a smaller area. These results suggest that the second and third PC are related to the anomalies of local small-scale atmospheric and oceanic circulations such the Southern Annular Mode index (Figure 4b,f) and the Amundsen Sea Low (ASL), which affect the ice-ocean interaction, basal melt, intrusion of warm saline water on retrograde slopes, and precipitation accumulation in basins B21 and B22. We also noticed that all of the first three PCs contain the 7.5-year periodic signal, since the CPCA method decomposes the observation matrix according to the maximum variance criterion rather than the period of the signal.  To further determine whether this signal contributed to the linear estimation of mass change, we reanalyzed the mass change using the CPCA method after removing the linear trend from the GRACE data at each point. The low-frequency signals with 5-year and longer periodicity disappeared in the reanalysis results, which indicates that the low-frequency signals with 5-year and longer periodicity do not contribute to the observation field where the trend is zero. This verifies that the low-frequency signals are robust and contributed to the mass change trend in the Antarctic region.
The sea surface temperature anomaly (SSTA) in the equatorial Pacific, the Antarctic Oscillation Index (AO), SAM, ASL Index [60], and the meridian wind speed in the South Pacific region (−80 °S to −40 °S) at a height of 700 hPa were also analyzed using the WAPS method to study the relationships of this low-frequency signal between different datasets. The results of their WAPS are presented in Figures 4 and 5.   Figure 4b shows the WAPS of the SSTA time evolution in the Niño 1+2 region. It can be seen from the figure that the SSTA in the Niño 1+2 region has signals with periods of 2-3 years in addition to the 7.5-year periodic signals. The energy of the 7.5-year periodic signals is comparable to the energy of the 2-3-year periodic signals. The cross-correlation analysis results show that the coefficient between the first PC and the SSTA in the Niño 1+2 region is 0.24. The correlation between the first PC and the SSTA time series after a 48-month moving average is as high as 0.73, which is much larger than the 95% CI of 0.16 (see Table 2). The ice sheet mass changes lag the SSTA by ~9 months, a finding that is similar to the result of Sasgen et al. (2010) [17]. Local climate models such as the AO and SAM, however, do not include this periodic signal, suggesting that the local climate changes have a smaller correlation with the first component of the ice sheet. Meanwhile, the results of the second PC of meridian wind speed ( Figure 5) show that the meridian  [55] studied the ASL, and found that the Amundsen-Bellingshausen Sea (ABS) region exhibits large inter-annual atmospheric circulation variability. This is due, in part, to orographic forcing and in part to its location in the South Pacific, where atmospheric Rossby waves associated with ENSO variability have a year-round influence. The ENSO plays a significant role in determining the depth of the ASL. The most energetic Rossby waves associated with ENSO variability in the Southern Hemisphere occur in spring, and hence the strongest correlations between ENSO variability and the ASL generally occur in this season. In its La Niña phase, in spring, ENSO is associated with a deeper ASL and with warm air advection toward the Antarctic Peninsula and West Antarctica. The ASL is an important circulation feature that influences West Antarctic climate variability. Observations reveal that the ASL has deepened in recent decades with potential impacts on the regional climate through its influence on the meridional wind field. Some studies have suggested that tropical teleconnections have contributed to atmospheric warming in West Antarctica and across the peninsula [6,56], and to sea ice loss in the Bellingshausen Sea [57]. The ASL is probably related to the variability of the Southern Annular Mode (SAM) and ENSO [58,59]. Paolo et al. (2018) [10] noted that studies correlating ENSO tropical forcing with Pacific sector climate indicators, such as the strength of the Amundsen Sea Low, sea-ice extent, and AP temperature, found that correlations with ENSO are significant in some seasons but not in others, with reversals of the sign of the correlation from season to season in some cases. The dominant effect of El Niño on the Amundsen Sea ice shelf mass is the increased basal melting associated with the onshore flow of Circumpolar Deep Water and coastal upwelling as westerly wind stress intensifies.

The Low-Frequency Signals with 5-Year and Longer Periodicity in the First Principal Component
After obtaining the PCs, their time-frequency characteristics can be analyzed once the linear trend has been removed. Figure 3 shows the time-frequency characteristics of the first three PCs. The WAPS of the first PC evolution shows that the PC contains significant periodic signals of 7.5 years and 5 years (Figure 3a). The energy of the 7.5-year periodic signal is the largest, followed by that of the 5-year periodic signal, which occupies the dominant position in the Antarctica mass change of the periodic signals, i.e., apart from linear factors, the low-frequency signals with 5-year and longer periodicity also contribute to the linear estimation of the ice sheet mass balance. The disadvantage of CPCA and WAPS analysis is that they cannot quantitatively distinguish the contributions of linear factors and periodic factors to linear estimation.
To further determine whether this signal contributed to the linear estimation of mass change, we reanalyzed the mass change using the CPCA method after removing the linear trend from the GRACE data at each point. The low-frequency signals with 5-year and longer periodicity disappeared in the reanalysis results, which indicates that the lowfrequency signals with 5-year and longer periodicity do not contribute to the observation field where the trend is zero. This verifies that the low-frequency signals are robust and contributed to the mass change trend in the Antarctic region.
The sea surface temperature anomaly (SSTA) in the equatorial Pacific, the Antarctic Oscillation Index (AO), SAM, ASL Index [60], and the meridian wind speed in the South Pacific region (−80 • S to −40 • S) at a height of 700 hPa were also analyzed using the WAPS method to study the relationships of this low-frequency signal between different datasets. The results of their WAPS are presented in Figures 4 and 5. (g) (h)  Figure 4b shows the WAPS of the SSTA time evolution in the Niño 1+2 region. It can be seen from the figure that the SSTA in the Niño 1+2 region has signals with periods of 2-3 years in addition to the 7.5-year periodic signals. The energy of the 7.5-year periodic signals is comparable to the energy of the 2-3-year periodic signals. The cross-correlation analysis results show that the coefficient between the first PC and the SSTA in the Niño 1+2 region is 0.24. The correlation between the first PC and the SSTA time series after a 48-month moving average is as high as 0.73, which is much larger than the 95% CI of 0.16 (see Table 2). The ice sheet mass changes lag the SSTA by ~9 months, a finding that is similar to the result of Sasgen et al. (2010) [17]. Local climate models such as the AO and SAM, however, do not include this periodic signal, suggesting that the local climate changes have a smaller correlation with the first component of the ice sheet. Meanwhile, the results of the second PC of meridian wind speed ( Figure 5) show that the meridian  Figure 4b shows the WAPS of the SSTA time evolution in the Niño 1+2 region. It can be seen from the figure that the SSTA in the Niño 1+2 region has signals with periods of 2-3 years in addition to the 7.5-year periodic signals. The energy of the 7.5-year periodic signals is comparable to the energy of the 2-3-year periodic signals. The cross-correlation analysis results show that the coefficient between the first PC and the SSTA in the Niño 1+2 region is 0.24. The correlation between the first PC and the SSTA time series after a 48-month moving average is as high as 0.73, which is much larger than the 95% CI of 0.16 (see Table 2). The ice sheet mass changes lag the SSTA by~9 months, a finding that is similar to the result of Sasgen et al. (2010) [17]. Local climate models such as the AO and SAM, however, do not include this periodic signal, suggesting that the local climate changes have a smaller correlation with the first component of the ice sheet. Meanwhile, the results of the second PC of meridian wind speed ( Figure 5) show that the meridian wind in the South Pacific region has 7.5-year periodic signals. The correlation coefficient of the low-frequency signal between the meridional wind field and SSTA in the equatorial Pacific is 0.77. These results suggest that the changes of the SSTA in the Niño 1+2 region are correlated with the low-frequency signals of the first PC.  Turner (2004) [16] concluded that the El Niño-Southern Oscillation might be transmitting its effects from the tropical Pacific Ocean to the Antarctic for the ENSO signals which are found over the Amundsen-Bellingshausen Sea. Turner et al. (2016) [7] analyzed the air temperature changes across the AP and found that abnormal changes in the sea surface temperature (El Niño events) in the Eastern Pacific can cause changes in the strength of Rossby waves that propagate southwards, as well as strengthening of the polar front jets. These researchers showed that during the cooling period (1999-2014) of the AP, the subtropical front that spreads to the AP is reflected by the enhanced polar front jet (60 • S) and cannot affect the interior of the Antarctic based on the 300-hPa zonal wind component. Clem and Fogt (2013) [9] analyzed the relationship between the ENSO and the SAM Index over time, and found that these climate models have spatial dependence with respect to their influence on the AP. There is a clear correlation between ENSO and climate change in the WAP, and there is a clear correlation between SAM and climate change in the northeastern AP.
Paolo et al. (2018) [10] used satellite altimetry data to analyze the relationship between changes in the elevation of ice sheets in the West Antarctica and the local atmospheric circulation driven by El Niño/Southern Oscillation. During strong El Niño periods, the height of the ice sheet accumulation in the Antarctic Pacific sector exceeds the height of the base melting, although net mass loss occurred because the ice lost from the base has higher density than the fresh snow being gained at the surface. The results of this study reflect the impact of El Niño events on the mass loss of the Antarctic ice sheet. When studying changes in the mass balance of the AP, the above studies noted the effects of El Niño changes on atmospheric circulation and temperature changes in the AP and attempted to determine the effect on the mass changes of the ice sheet in the area. They pointed out that studies correlating ENSO tropical forcing with Pacific sector climate indicators, such as the Amundsen Sea Low strength, sea-ice extent, and AP temperature, found that correlations with ENSO are significant for some seasons but not for others, with reversals of the sign of the correlation from season to season in some cases. The dominant effect of El Niño on the Amundsen Sea ice shelf mass is the increased basal melting associated with the onshore flow of Circumpolar Deep Water and coastal upwelling as westerly wind stress intensifies. On interannual timescales, this basal mass loss anomaly, relative to the longer-term mass loss trend, is partially offset by increased snowfall. This precipitation increase is consistent with the northerly wind anomaly during El Niño events, possibly including increased local moisture uptake from the coastal ocean due to a reduction in regional sea-ice concentration.

Conclusions
During the period 2003-2017, the ice sheet in Antarctica exhibited a trend of large mass loss in West Antarctica and mass increase in East Antarctica. The first PC and its corresponding spatial mode from GRACE satellite data successfully captured the spatial pattern of the dominant trend in the mass change of the Antarctic ice sheet. The first PC contains a linear term and low-frequency signals with a period > 5 years, which all contribute to the mass change trends in Antarctica. We also found similar low-frequency signals in the SSTA in the equatorial Pacific Ocean and the second PC of the meridional wind field at 700 hPa over the South Pacific. The correlation results suggest that the low frequency signals with a period > 5 years in the SSTA in the Niño 1+2 region and the first PC of the Antarctic ice sheet mass change have a strong correlation, with the correlation coefficient reaching 0.73, even though the mass changes in Antarctica lag the SSTA in the Niño 1+2 region by~9 months.
The first PC has large-scale factors in both time and space, because it contains the linear terms and low-frequency periodic signals which dominate the spatial pattern of mass change, and hence, its influence range is wide. In contrast, the scale factors of the second and third PC are smaller because they mainly affect the mass changes around the Pine Island and Thwaites regions, and hence, the second and third PC are related to small-scale local atmospheric circulation changes such as the SAM index, ASL, and westerly winds.
The mass change phase signals of West Antarctica indicate the upstream propagation of mass loss information from the ocean-ice interface to the southward upslope over time, which mainly reflects ocean-driven factors such as enhanced ice-ocean interaction and intrusion of warm saline water on retrograde slopes. Meanwhile, the mass change phase signals of East Antarctica reflect the downstream propagation of mass increase information from the South Pole toward Dronning Maud Land and Enderby Land, which mainly reflects atmospheric factors such as precipitation accumulation.