According to Wahr et al. (1998), surface mass change can be expressed in the form of surface equivalent water height (EWH) as Δσ(θ,λ)=aρe3ρw∑n=0∞2n+11+kn∑m=0nc̃nmcos⁡(mλ)+s̃nmsin⁡(mλ)P̃nm(cos⁡θ), where ρe is average density of the earth, a is the equatorial radius, and ρw is water density. Parameter λ is longitude, θ is colatitude, and P̃nm(cos⁡θ) is the nth-degree and mth-order fully normalized Legendre function. Parameter kn is the load Love number. c̃nm and s̃nm are normalized SH coefficients.

Principal component analysis (PCA) was first formulated in statistics by Pearson (1901). Hotelling (1932) further contributed to PCA development. The utility of PCA has been demonstrated in many scientific fields, and it has several alternate names, such as singular value decomposition (SVD) (Golub and Loan, 1996; Mandel, 1982) and empirical orthogonal function (EOF) analysis (Lagerloef and Bernstein, 1988; Kaihatu et al., 1998; Zhang et al., 2004). Eigenvector analysis and characteristic vector analysis are often used in the physical sciences and other fields.

PCA (Abdi and Williams, 2010; Helena et al., 2000; Wang and Du, 2000) is a multivariate technique that analyzes a data table in which observations are described by several intercorrelated quantitative dependent variables. Its goal is to extract the important information from the table, represent it as a set of new orthogonal variables called principal components, and display patterns of similarity of the observations and variables as points in maps. Mathematically, PCA depends upon the eigendecomposition of positive semidefinite matrices and SVD of rectangular matrices.

Compared with PCA, the CPCA method (Horel, 1984) introduces phase information and it is a useful method for identifying traveling and standing waves (Pfeffer et al., 2010; Kichikawa et al., 2015). CPCA transforms original data and its Hilbert transform into a complex time sequence and conducts principal component analysis by calculating the covariance or complex characteristics vector of the cross-correlation matrix.

For the CPCA, a complex observation sequence should first be constructed, which is different from the PCA. For a time-varying observation vector uj(t), its Fourier expansion is uj(t)=∑ωaj(ω)cos⁡(ωt)+bj(ω)sin⁡(ωt). In this expansion, j stands for the location of the observation point, t is the observation time, and ω is the Fourier frequency. The constructed complex observation vector Uj(t) can be expressed as Uj(t)=∑ωcj(ω)e-iωt. Here, cj(ω)=aj(ω)+ibj(ω),i=-1. According to the definition of cj(ω), Eq. (3) can be expanded as Uj(t)=∑ωaj(ω)cos⁡(ωt)+bj(ω)sin⁡(ωt)+ibj(ω)cos⁡(ωt)-aj(ω)sin⁡(ωt)=uj(t)+ivj(t). The real part of Eq. (4) 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 uj(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 CPCA analysis is the analysis of the constructed complex vector. After 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 r11r12…r1nr21r22…r2n⋮⋯⋯⋮rn1rn2…rnn. Here rjk represents the multiple correlation coefficients between the jth and kth observation points. CPCA compresses information using the least complex eigenvector ejn of the correlation matrix (Eq. 5) and the complex principal component pn(t), because the correlation matrix (5) is a Hermitian matrix including n real eigenvalues λ. λj/∑i=1nλi denotes the contribution percentage of the jth principal component.

Observation vector Uj(t) can be expressed as the sum of N principal components, Uj(t)=∑n=1Nejn∗pn(t), where * stands for the complex conjugate, and both complex principal components and complex eigenvectors are orthogonal. The nth complex eigenvector element ejn can be expressed as ejn=Uj(t)∗pn(t)t=sjneiθjn, where ejn indicates the multiple correlation relationship between the jth time sequence and nth principal component. sjn and θjn are respectively correlative order of magnitude and phase. ⋯t signifies the mean of times. The time sequence elements of principal components can be expressed as the functional form of amplitude Tn and phase Φn. Pn(t)=Tn(t)eiΦn(t)

Mass balance on the QTP is under the influence of climate change, and it exhibits unsteady quasiperiodic change. After obtaining the temporal change series of principal components in the area, the time-varying changes of the periods and amplitude (energy) require analysis. We used the wavelet amplitude-period spectrum (Liu, 1999; Liu and Hsu, 2012; Zhan et al., 2003) to analyze its time–frequency information, and chose the Morlet wavelet (Morlet et al., 2012) as the basic wavelet. The wavelet amplitude-period spectrum reflects the time-varying amplitude and period of each periodic term (or standardized periodic term). This means that, in this spectrum, the location of extreme points corresponds to the instant period of a periodic signal (or quasiperiodic term) at that moment, whereas the extreme point value corresponds to the instantaneous amplitude of a certain period signal at that moment. The wavelet amplitude-period spectrum of a time sequence f(t) is defined as Wψf(a,b)=1aCψ∫-∞∞f(t)ψt-badt,a,b∈R,a≠0, where ψ(t)=e-t22δ2cos⁡(2πω0t),δ,ω0∈R,2πδω0≫1, Cψ=∫-∞∞ψ(t)cos⁡(2πω0t)dt.

Here, the kernel function ψ(t) is the real part of the Morlet wavelet, δ is a constant, and ω0 is the frequency parameter, Cψ is a constant, and a and b are scale factors of period and time, respectively.

In the inland part of the QTP, there are three regions of mass increase: the Qiangtang Plateau (E area), central and eastern parts of the Kunlun Mountains (F area), and Qinghai Plateau (G area). Their respective annual increases were estimated to be 4.5, 5.5, and 3.5 GT and these were much smaller than the 30 GT of Yi and Sun (2014). Many scholars have conducted related research in an attempt to explain the reason behind mass balance change in the region.

Mass balance of the inner Tibetan Plateau (ITP) derived from GRACE data showed a positive rate that was attributable to glacier mass gain, whereas those same glaciers evaluated in other field-based studies showed an overall mass loss. Jacob et al. (2012) deduced glacier mass balance using GRACE data and found a mass increase rate of 7 Gt yr-1 in areas E and F. Yao et al. (2012) observed more than 20 glaciers in the QTP area and concluded that they are shrinking dramatically. Their results indicate that the Himalayas have shown the most extreme glacial shrinkage based on reductions of glacier length and area. The shrinkage is most pronounced in the southeastern QTP, where glacier length decreased at a rate of 48.2 m yr-1 and the area declined at a rate of 0.57 % yr-1 during the 1970s–2000s. The rate of glacial shrinkage decreased from the southeastern QTP to the interior.

Zhang et al. (2013) studied 53 % of the total lake area on the plateau using ICESAT satellite data and found a mass increase rate of 4.95 Gt yr-1. They suggested that the increased mass measured by GRACE was due to increased water mass in lakes. If this rate holds true for all lakes, the total mass variance rate, using the area ratio, is +8.06 Gt yr-1. However, glacier melting into lakes, by itself, should not increase the overall mass and may decrease the mass because a portion of the meltwater would be lost through evaporation or through discharge into rivers that leave the Tibetan Plateau.

Yi and Sun (2014) suggested a relatively large mass rate change in this area, and explained that this change was caused by glacier changes, lake water levels, geologic structural processes, and frozen soil. They stated that, based on model calculation, the change of inland water storage was -3.3 Gt yr-1. The change of negative balance of weakening glacier mass has been confirmed (Bolch et al., 2010, 2012; Yao et al., 2012). According to calculations of Zhang et al. (2013), the increase of lake water is 8.1 Gt yr-1, and the effect of tectonic movement (using simple Bouguer correction) is 0–13 Gt yr-1. The effect of other factors is nearly zero. However, we still lack sufficient observation data of mass balance states in the interior part of the earth in the study region. Thus, the exact Bouguer equilibrium correction requires more data for confirmation.

The effect of soil freezing on mass change in the inland plateau is weak, because the terrain there is flatter than at the plateau edge. The inland area contains numerous lakes and wetlands, which are conducive to fluid convergence. Moreover, when water melts and is lost from frozen soil, soil porosity increases, which captures more water during rainy periods.

Our results indicate that rainfall is the main reason for the mass increase in the study region. There is strong evidence that precipitation over the inland QTP during the past several decades has greatly increased (Yao et al., 2012; Global Precipitation Climatology Project or GPCP, www.esrl.noaa.gov/psd/data/gridded/data.gpcp.html). Through the influence of El Niño, moist air moves westward to the inland plateau through the eastern Himalayas and Qinghai, and this brings rainfall to the inland areas and causes rainfall accumulation in plateau lakes and wetland areas. The inland Plateau, especially the western part of Qiangtang plateau and Kunlun Mountains area, is also influenced by the westerlies and the La Niña phenomenon (Fig. 5a), which further creates the meteorological conditions for rain and snow. Increased temperatures (Qin et al., 2009) accelerate glacial melting in this area. This glacier meltwater enters lakes through runoff. It also explains why on-site observation data of glaciers indicate slight shrinkage, and GRACE observations indicate the reasons for mass increase.

The trend of mass balance change from GRACE data shows that the most negative signal is along the Himalayas and northwestern India. The mass reduction rate of glaciers in the entire Himalayan mountain region is 14 Gt yr-1, and the mass loss of glaciers in the eastern Himalayas was the most dramatic, with the rate of -4.6 Gt yr-1 in the A area and -4.1 Gt yr-1 in the B area. The mass reduction rate in northwestern India (H area) was -13.6 Gt yr-1, whereas Rodell et al. (2009) and Yi and Sun (2014) estimated larger values of -17.7 and -20.2 Gt yr-1, respectively. The reason for this difference is that Rodell et al. (2009) used the data of the earlier RL04 version. Yi and Sun (2014) stated that the RL04 solutions overestimate the glacier melt rate in the Himalayas by as much as 17 %. The difference between our results and those of Yi and Sun (2014) stems from their use of the mascon inverse method in a concise form. Moreover, the filtering method used may attenuate the signal.

Yao et al. (2012) studied glacial change over the past three decades and found that the Himalayas had the most extreme glacial shrinkage based on reductions of both glacier length and area; the shrinkage was greatest in the southeastern QTP (A area), where the length decreased at a rate of 48.2 m yr-1 and the area was reduced at a rate of 0.57 % yr-1. Most negative mass balances occurred along the Himalayas and ranged from -1100 to -760 mm yr-1. This trend of mass change along the Himalayas is consistent with our results. They attributed this change to the weakened Indian monsoon towards the plateau interior.

Thakuri et al. (2014) studied glacier changes on the southern slope of Mt. Everest from 1962 to 2011 (400 km2) using optical satellite imagery. They concluded that the observed glacier shrinkage, upward altitude shift of the snowline, and the negative mass balance (Nuimura et al., 2012) are not only due to warming temperatures, but are also the result of weakened Asian monsoons. Bolch et al. (2011) examined the mass change of glaciers on Mt. Everest using stereo Corona spy imagery (1962 and 1970), aerial images, and high-resolution satellite data (Cartosat-1). They found that glaciers south of Mt. Everest continuously lost mass from 1970 to 2007, but at an increased rate in recent years. Wagnon et al. (2013) arrived at the same conclusion and noted that glacier shrinkage south of Mt. Everest was less than shrinkage of other glaciers in the western and eastern Himalayas and southern and eastern Tibetan Plateau.

Salerno et al. (2015) analyzed the precipitation time series during 1994–2013 reconstructed from seven stations at elevations between 2660 and 5600 m. They found that precipitation has decreased by 47 % during the monsoon period and snowfall has decreased by 10 % in the last 20 years. Salerno et al. (2016) extended this analysis back to the early 1960s and for all regions used, as proxy of the precipitation trend, the surface area variation of glacial lakes. These authors determined that an increase in precipitation occurred until the mid-1990s followed by a decrease until recent years in all Mt. Everest regions.

Studies using different types of data have produced similar results, i.e. negative mass balances and a weakened Indian monsoon along the Himalayas. Our results support these conclusions. The results of CPCA analysis indicate that mass change on the Himalayas and its southern portion are associated with the Indian monsoon climate, and the intensity of this monsoon is weakening. This result is also consistent with the conclusions of Wu (2005). A weakened Indian monsoon brings less humid air to the study region resulting in decreased annual rainfall (Thakuri et al., 2014; Salerno et al., 2015, 2016). The GPCP rainfall data confirms this conclusion. The eastern Himalayas are also affected by El Niño (Fig. 4a) and East Asian monsoons, and no evidence supports the role of westerlies (Fig. 5a) in driving local climate and glacier changes. Glaciers in this area are of a marine type, with masses having large inputs and outputs and strongly affected by changes of the marine climate. The weakened Indian monsoon, strengthening El Niño, and westerlies, combined with the huge topographic landform, exert climatic controls on the distribution of existing glaciers along all Himalayan regions and reduce precipitation there.

Archer and Fowler (2004) indicated that the western Hindu Kush and Karakoram are largely exposed to the arrival of westerly midlatitude perturbations bringing precipitation during winter and early spring, whereas the eastern Himalayas is dominated by summer monsoon precipitation (Syed et al., 2006; Yadav et al., 2012). Their results are similar to those of this study. The results of CPCA indicate that the eastern Himalayas is under the influence of weakened Indian monsoon and El Niño, while the Hindu Kush and Karakoram area is under the influence of a weakened Indian monsoon, westerlies, and La Niña.

Thompson et al. (2000) examined the variability of the South Asian monsoon by analyzing ice core records of the Dasuopu glacier on the QTP. They found evidence of drought conditions and a weak monsoon from 1780 to 1810. Interestingly, according to historical records, at least 600 000 people died in 1972 in just one region of northern India from an epic drought associated with this event. The onset of this event in the Dasuopu cores is concurrent with a very strong El Niño–Southern Oscillation (ENSO) from 1790 to 1793, which was followed by a moderate ENSO event of 1794–1797. These data suggest an association between ENSO and a weakened Asian monsoon.

Arctic amplification may impact midlatitude weather patterns and extremes (Francis and Vavrus, 2012; Screen and Simmonds, 2013), and midlatitude westerlies may increase climate variation and glacier variability in monsoon affected areas of High Asia (Thomas et al., 2014). On large spatial scales, climate change over the QTP may also be connected with hemispheric or global atmospheric circulations including the North Atlantic Oscillation (NAO) and ENSO (Wang et al., 2003). ENSO may influence climate over the southern QTP through linkage with the Indian monsoon (Xu et al., 2010, 2011). The NAO is associated with climate fluctuations over the northern QTP through modulation of the westerlies (Wang et al., 2003; Xu et al., 2010), which is similar to climate change from the westerlies and La Niña in the third principal component.

Yao et al. (2012) studied glacial changes over the last three decades in the QTP and found that glacier recession in the Himalayas was the most dramatic, followed by recession in the inland plateau. Glaciers in the Pamirs had weak balance changes, and some of the glaciers in the plateau of the eastern Pamirs continue to expand. Yao et al. (2012) concluded that the main reason for these changes was the variation of climates with different circulations, which includes effects of the weakened Indian monsoon in the Himalayas, rainfall decreases, effects of the strengthening westerlies in the Pamirs and its eastern portion, and rainfall increases. In the inland plateau, the influences of these two circulations are limited. The two atmospheric circulation patterns, combined with the huge topographic landform, exert climatic controls on the distribution of existing glaciers. The East Asian monsoon only affects glaciers on the eastern margin, such as the Mingya Gongga and those in the eastern Qilian Mountains. The interior of the QTP is dominated by continental climatic conditions, and the sparse glacier distribution and higher ELAs in the continental-climate-dominated interior are consequences of a limited water vapor source from both air masses. They divided glaciers of the Tibetan Plateau into seven regions and categorized them into three climatic transects: (1) southwest–northeast oriented (central Himalayas–Qiangtang Plateau–eastern Qinghai Plateau region), with the weakened Indian monsoon influence northward; (2) southeast–northwest oriented (eastern Himalayas–Qiangtang Plateau–Pamirs region), with the weakened Indian monsoon toward the interior and strengthening westerlies toward the northwest; (3) along the Himalayas, with stronger monsoon influence in the east and weaker monsoon influence in the west.

From the results of the CPCA, the first spatial mode shows that the mass balance of the Himalayas–Pamirs–northwestern India region (transect 3) was the most sensitive to climate change associated with the Indian monsoon, whereas there was little impact of that change on mass balance of the inland plateau. The third spatial mode shows that mass balance of the northwest plateau, including the Kunlun Mountains, is also affected by climate change from the westerlies and La Niña. Another difference between the results of Yao et al. (2012) and those of this study is that climate change from El Niño rather than the weakened Indian monsoon toward the interior affected mass balance along transect 2. We found that the time evolution of the second principal component and El Niño index had a stronger time–frequency correlation.

Yi and Sun (2014) used harmonic analysis of a time series of mass changes in the study region and found a 5-year periodic signal in the Pamirs and Karakorum regions. They analyzed the correlation between mass change, precipitation, El Niño–Southern Oscillation (ENSO) and the Arctic Oscillation (AO), and found that the 5-year undulating signal of mass change is controlled by both the ENSO and AO.

Ke et al. (2017) examined area and thickness changes of glaciers in the Dongkemadi (DKMD) region of the central QTP using Landsat images from 1976 to 2013 and satellite altimetry data from 2003 to 2008. They analyzed relationships between glacier variation and local and macroscale climate factors based on remote sensing and reanalysis data. Their results indicate that glacier change in the DKMD region was dominated by variation of mean annual temperature, and was influenced by the state of the NAO over the past 38 years. The mechanism linking climate variability over the central QTP and state of the NAO is most likely via changes in strength of the westerlies and Siberian High. In addition, ENSO may have been associated with extreme weather (snowstorms) in October 1986 and 2000 which might have led to substantial glacier expansion in the following years. The DKMD is located on the eastern Qiangtang Plateau (the center of transect 2), where area mass balance change is the most sensitive to El Niño in our results.

Yao et al. (2012) considered the effect of the Indian monsoon and westerlies but did not consider El Niño, which was the second major component (16 %) in the study region. Yi and Sun (2014) noted that the 5-year periodic signal in the Pamirs region is related to ENSO, but ignored the effect of La Niña because they did not distinguish the phase information. According to the CPCA, we believe that the mass change in the QTP area is mainly controlled by the Indian monsoon and westerlies but the influence of El Niño and La Niña on the inland areas of the plateau and the Karakorum area is also important. The Indian monsoon mainly affects mass balance change on southern and southwestern QTP, whereas El Niño mainly modifies that change over the eastern Himalayas, Qiangtang Plateau, Pamirs, and eastern Qinghai Plateau area. Mass balance over the western and northwestern QTP is mainly affected by the westerlies and La Niña.