Inversion of Groundwater Storage Variations Considering Lag Effect in Beijing Plain, from RadarSat-2 with SBAS-InSAR Technology

Abstract

The long-term over-exploitation of groundwater has not only caused the compaction of aquifer thickness and surface deformation but has also further aggravated the loss of groundwater storage (GWS) in Beijing plain. The South-to-North Water Diversion Project (SNWDP) furnishes a new source of water for Beijing. By reviewing related studies, it was found that there are few studies on the realization of GWS estimation based on InSAR technology considering the lag effect. Therefore, in this study, firstly, the long-time series deformation characteristics of Beijing plain were obtained from 46 RadarSat-2 images using small baseline subset interferometric synthetic aperture radar (SBAS-InSAR) technology. Secondly, the seasonal components of surface deformation and hydraulic head change were extracted by means of multichannel singular spectrum analysis (MSSA), verifying the separation accuracy by means of Monto Carlo-SSA (MC-SSA). Finally, for the hydrodynamic delay (aquifer water supply/drainage) of the complex aquifer system, we introduced the time lag cross-correlation (TLCC) approach to correct the hysteresis response of seasonal deformation relative to the variation of the aquifer system head, so as to realize the estimation of aquifer storage properties and GWS loss, even unrecoverable GWS (UGWS). The results showed that the average annual variation of total GWS (TGWS) in Beijing plain was −6.702 × 10⁷ m³, of which the depletion volume of UGWS was −6.168 × 10⁷ m³, accounting for 92.03% of the TGWS. On a temporal scale, the depletion of UGWS lagged behind the total head change, with about one year of lag time. On a spatial scale, in contrast to the north of Beijing plain, the depletion of UGWS in the south only recovered briefly after 2015 and then continued to decline. This further indicated that the process of the decline of middle-deep confined head and long-term GWS loss caused by over-exploitation of groundwater was irreversible. These findings are of great significance to optimize the allocation of groundwater resources, reduce the harm of land subsidence and protect groundwater resources.

1. Introduction

Surface deformation caused by compaction of susceptible aquifer is connected with the decline of groundwater level during the exploitation of groundwater resources [1]. Land subsidence, as a slowly changing and irreversible geological disaster, is becoming increasingly common all over the world [2,3,4], including Mexico, Italy, Iran, Japan, Vietnam and China [5,6,7,8,9,10,11,12,13,14]. Long-term over-extraction of groundwater will lead to the large scale decline of diving level and confined head, decrease in pore water pressure, and the compression of strata under the continuous increase of effective stress. When the balance between the extraction and recharge of aquifer systems cannot be achieved, this may well give rise to the loss of GWS [15,16], form a regional fall funnel in the area with a relatively large amount of exploitation, and even induce surface deformation with a large area and high rate [17,18,19,20].

Land subsidence in Beijing, which has been developing for many years, is principally caused by the excessive exploitation of deep confined water over a long period [21,22]; the area with a cumulative deformation exceeding 100 mm in the Beijing plain was close to as much as 4000 km² from 1955 to 2010 [23]. Groundwater exploitation in Beijing meets two-thirds of the city’s water demand. As is shown in the data, the amount of groundwater exploitation from 2004 to 2013 reached 2.28 × 10⁹ m³ [24]. In order to solve the problem of water shortage in northern China and optimize the allocation of water resources, the SNWDP, the largest water resource allocation project in human history, was officially started at the end of 2002. The middle route, from the Danjiangkou reservoir to Beijing, was officially introduced at the end of 2014. The SNWDP not only brings new water sources to Beijing, but also affects the temporal-spatial response relationship between surface deformation and GWS variation. Therefore, we aim to realize inversion of aquifer storage properties and GWS based on hysteresis correction, by combining time series SBAS-InSAR technology and hydraulic head data, to address this background research gap.

A large number of studies have shown that, compared with optical data, radar data is only minimally affected by the weather, and can be used for flood mapping [25], water area demarcation [26], landslide [27], groundwater [28] and other investigations. Compared with traditional geodetic techniques, the InSAR technique developed in recent years can gradually make it possible to monitor surface deformation related to aquifer system compaction at a low cost and on a large scale [29,30]. In hydrogeology, InSAR data has been extensively used to ascertain aquifer storage properties [17,31,32,33,34,35], predict the change of hydraulic head [36,37] and assess the spatial variability of GWS [38,39,40]. Unlike other traditional interferometric techniques, SBAS-InSAR minimizes the impact of de-correlation and elevation errors by combining multiple sets of small baselines to narrow differences in time and perspective [41,42,43,44]. Therefore, this paper uses time series SBAS-InSAR technology to obtain temporal surface deformation information in Beijing plain.

The compaction of aquifer thickness results in surface deformation, but the degree of compaction is determined by the effective stress in the aquifer. When the effective stress is less than the pre-consolidation pressure (i.e., the maximum effective stress that the aquifer suffered in the past), the compressible soil layer in the aquifer system has elastic (recoverable) compaction; however, it will lead to the rearrangement of particles in the aquitard, as well as inelastic and irreversible compaction. Since the two processes usually occur in the same location at the same time, it is challenging to separate them [1]. A variety of methods, in a great many studies, are used to separate the elastic (seasonal) component and the inelastic (long-term trend) component, such as time series fitting sinusoidal modeling [11,31,45], harmonic sequence decomposition [46], continuous wavelet transform (CWT) [35], principal component analysis (PCA) [17], independent component analysis (ICA) [47] and so on. However, PCA cannot provide obvious spectral resolution for separating appropriate periodic oscillations [35], and CWT is likely to have an edge effect in short time series data [48]. Therefore, in this study, MSSA was used to process InSAR deformation information so that its seasonal oscillation can be obtained.

Due to the hydrodynamic delay between high permeability (aquifer) and low permeability (weak permeable layer) units in an aquifer system, however, the process of surface deformation and compaction of the aquitard is much slower than that of a pumping test [17,49]; moreover, the duration of a pumping test (hours to days) is not enough to completely dehydrate a fine-grained aquifer (weak permeable layer) unit [6]. In addition, by virtue of the low vertical hydraulic conductivity of the weakly permeable layer, the hydraulic head change in the weakly permeable layer may lag behind that of the adjacent aquifer. This can bring about delayed drainage of the aquifer and compaction of compressible soil layers in the aquifer, even after the head changes stabilize, triggering surface deformation. Therefore, in order to make elastic surface deformation and hydraulic head change correspond in timescale, it is necessary to correct the hysteretic response of seasonal deformation. By reviewing relevant studies, it was found that, as for the problem of surface deformation in Beijing plain caused by groundwater overdraft, there are few studies on the characterization of aquifer storage properties and the estimation of UGWS depletion based on considering hysteresis using InSAR remote monitoring technology.

To solve the above problems, based on the temporal sequence characteristics of the surface deformation field of Beijing plain obtained by SBAS-InSAR, this study extracted the seasonal components of surface deformation and hydraulic head change by means of MSSA, verifying the separation accuracy by means of MC-SSA. What is more, the TLCC analysis method was introduced to correct the lag time of seasonal surface deformation, so as to estimate aquifer storage parameters and GWS loss, especially UGWS depletion, under a given hydraulic head change. Finally, we focus on discussing the temporal-spatial response relationship between UGWS depletion and surface deformation in the context of SNWDP. This study, by combining time series InSAR with limited groundwater level monitoring wells, demonstrates the potential to characterize the storage properties of confined aquifers and estimate changes in GWS, which is of great import for the sustainable utilization administration of groundwater resources and the control of subsidence growth in Beijing plain.

The structure of this study is as follows: Section 2 describes the geological structure setting of the study area, the description of the data used, the estimation principles of the hydrogeological parameters and the method of GWS calculation; in Section 3, the process of SBAS-InSAR, together with the methods and principles of MSSA and TLCC, is described; in Section 4, the results are analyzed; Section 5 discusses the results and evaluates the error sources of this study, and considers the prospects for future work; all the work undertaken is summarized in Section 6.

2. Study Area and Datasets

2.1. Study Area

Beijing (Figure 1), is located on the northwestern edge of the North China Plain (NCP) and is situated at the northwestern corner of the north China fault depression of the China Korea quasi-platform. It has been a descending area of platform throughout the Cenozoic era. Beijing, the topography of which is higher in the northwest and lower in the southeast, is composed of an alluvial fan group involving the joint action of five major river systems [50,51,52]. The Beijing plain, with an area of about 6390 km², has a typical continental monsoon climate. The average annual rainfall, temperature and potential evapotranspiration in the region are 580 mm, 10–12 ℃ and 1800 mm, respectively [53]. Quaternary sediments are widely distributed in Beijing plain; from piedmont to plain, the thickness of Quaternary sediments gradually increases, the sediment particles gradually become finer, and the aquifer structure transits from a single layer to a multi-layer (Figure 1 and

Table 1. The division of major aquifers groups in the Beijing plain.

Aquifer Group	Lithology	Thickness (m)	Depth of Bottom (m)
The first aquifer group (Unconfined aquifer)	silt, silty sandy, and sandy clay	15–120	0–50
The second aquifer group	multiple types of gravel, sand, and clay soil	60–95	80–100
The third aquifer group	multiple types of gravel, sand, and clay soil	65–120	100–180
The fourth aquifer group	mainly sand	200–300	180–300

) [19,23,54]. The top area of alluvial and diluvial fan of the main rivers, where sand pebbles are exposed and the conditions of atmospheric precipitation infiltration and river infiltration are good, provides the main area for replenishing groundwater in the plain. Influenced by neotectonic movement, the middle and lower part of the alluvial proluvial fan and alluvial plain area have received huge sediments which are hundreds of meters thick; the thickest of which reach more than 1000 m in the central area of the sedimentary depression [55]. The lithology there has gradually transformed into sand, sand gravel and clayey soil.

The Quaternary pore water in Beijing plain mainly exists in the sand and sand gravel aquifers formed by river alluvial proluvial processes. Quaternary pore water has been exploited in large quantities because of the shortage of Beijing plain surface water resources. As is shown in the data, since 2011, rainfall has increased and the groundwater level in some areas has rebounded, but because the confined water level in the middle and deepest parts is affected by the excessive exploitation of groundwater, the confined water level in most areas is still in decline [56].

2.2. Datasets

RadarSat-2 is a medium resolution radar satellite with a large range image which is equipped with a C-band sensor (wavelength:5.63 cm), and the return period is 24 days. In this study, wide-width data with 30 m nominal resolution was selected, with the descending orbit of right-view imaging and VV polarization mode adopted. The width was 150 km × 150 km, and the time span of 46 images was from 28 January 2012 to 21 October 2016. (The data coverage and vertical baseline of interference relative sets are shown in Figure 2a,b, respectively, and the acquisition parameters of the RadarSat-2 images are shown in

Table 2. Acquisition parameters of RadarSat-2.

SAR Sensor	RadarSat-2	Polarization Mode	VV
Orbit direction	Descending	External DEM	SRTM
Band (wavelength)	C band (5.63 cm)	Spatial resolution of SRTM	90 m
Spatial resolution	30 m	Width range	150 km × 150 km
Return period	24 days	Temporal baseline	<300 d
Average incidence angle	27.10°	Spatial baseline	<300 m
Number of images	46
Temporal period	January 2012–October 2016

). The details of 46 RadarSat-2 SAR images are shown in the Appendix A,

Table A1. Image information table of 46 Radarsat-2 SAR images.

Number	SAR Sensor	Image Acquisition Date	Track	Orbit	Orbit Direction
1	Radarsat-2	20120128	62-26D	21528	Descending
2	Radarsat-2	20120221	63-26D	21871	Descending
3	Radarsat-2	20120409	65-26D	22557	Descending
4	Radarsat-2	20120527	67-26D	23243	Descending
5	Radarsat-2	20120831	71-26D	24615	Descending
6	Radarsat-2	20121018	73-26D	25301	Descending
7	Radarsat-2	20121111	74-26D	25644	Descending
8	Radarsat-2	20121229	76-26D	26330	Descending
9	Radarsat-2	20130122	77-26D	26673	Descending
10	Radarsat-2	20130428	81-26D	28045	Descending
11	Radarsat-2	20130522	82-26D	28388	Descending
12	Radarsat-2	20130615	83-26D	28731	Descending
13	Radarsat-2	20130802	85-26D	29417	Descending
14	Radarsat-2	20130826	86-26D	29760	Descending
15	Radarsat-2	20130919	87-26D	30103	Descending
16	Radarsat-2	20131013	88-26D	30446	Descending
17	Radarsat-2	20131106	89-26D	30789	Descending
18	Radarsat-2	20131130	90-26D	31132	Descending
19	Radarsat-2	20131224	91-26D	31475	Descending
20	Radarsat-2	20140210	93-26D	32161	Descending
21	Radarsat-2	20140306	94-26D	32504	Descending
22	Radarsat-2	20140330	95-26D	32847	Descending
23	Radarsat-2	20140423	96-26D	33190	Descending
24	Radarsat-2	20140610	98-26D	33876	Descending
25	Radarsat-2	20140728	100-26D	34562	Descending
26	Radarsat-2	20140914	102-26D	35248	Descending
27	Radarsat-2	20141101	104-26D	35934	Descending
28	Radarsat-2	20141125	105-26D	36277	Descending
29	Radarsat-2	20141219	106-26D	36620	Descending
30	Radarsat-2	20150205	108-26D	37306	Descending
31	Radarsat-2	20150325	110-26D	37992	Descending
32	Radarsat-2	20150512	112-26D	38678	Descending
33	Radarsat-2	20150629	114-26D	39364	Descending
34	Radarsat-2	20150816	116-26D	40050	Descending
35	Radarsat-2	20150909	117-26D	40393	Descending
36	Radarsat-2	20151003	118-26D	40736	Descending
37	Radarsat-2	20151027	119-26D	41079	Descending
38	Radarsat-2	20151120	120-26D	41422	Descending
39	Radarsat-2	20160107	122-26D	42108	Descending
40	Radarsat-2	20160131	123-26D	42451	Descending
41	Radarsat-2	20160506	127-26D	43823	Descending
42	Radarsat-2	20160530	128-26D	44166	Descending
43	Radarsat-2	20160623	129-26D	44509	Descending
44	Radarsat-2	20160810	131-26D	45195	Descending
45	Radarsat-2	20160903	132-26D	45538	Descending
46	Radarsat-2	20161021	134-26D	46224	Descending

. The external digital elevation model (DEM) data adopts the shuttle radar topography mission (SRTM) data provided by the National Aeronautics and Space Administration (NASA), with a spatial resolution of 90 m. The leveling data set of surface deformation is the second order leveling of the whole plain area carried out by the Geological Department of Beijing.

The results of stratified scale monitoring from 2005 to 2012 show that the main contribution layer of surface deformation in Beijing plain gradually transforms from a shallow layer to a deep one of 100 m, which is in keeping with the change in the main groundwater exploitation layer [57]. Therefore, by combining the coverage of SAR images with the continuity of monitoring time, this study selected 11 groundwater level monitoring points, the hole depth range of which was 82–150 m (the average hole depth was 105.74 m), and the time sampling frequency was daily. Here, we calculated the monthly average groundwater level change as the primary input data of this study. In addition, by establishing a buffer zone with a 100 m radius centered on the groundwater level monitoring point, the average surface deformation of coherent points in the buffer zone was calculated as the surface deformation value of the monitoring point.

3. Methods

In this study, based on time series deformation information obtained by SBAS-InSAR, the seasonal components of deformation and hydraulic head change were decomposed by MSSA. Then, we introduced the TLCC approach to correct the hysteresis response of seasonal deformation relative to the variation of aquifer system head, so as to realize the estimation of aquifer storage properties and GWS loss. The overall methodology of this study is summarized as follows (Figure 3):

3.1. SBAS-InSAR Processing

The InSAR measurement technology acquires the height of the target by calculating the phase difference (interference phase) between two echo signals of the same target [58,59,60,61,62]. Berardino [41] proposed a method to improve the sampling time rate in the separation process of time and Doppler frequency by combining multiple small baseline interferences. The free combination of small baseline pairs can obtain more interference relativities with high coherence, and at the same time, more high coherence points can be selected, so as to weaken the influence of space-time de-coherence and atmospheric phase, and improve the phase unwrapping accuracy [63]. In this paper, we first identify isolated monodic slow filter phase (SFP) pixels in a short time interval by applying the SBAS method, wherein, SFP pixels represent pixels that are almost irrelevant after filtering in a short time interval and are defined by their phase characteristics. Secondly, the stable monodic SFP pixels are estimated by the correlation with the vertical baseline and its phase stability index is determined by the amplitude difference dispersion (the process for selecting stable SFP pixels is shown in the Appendix A.1, Figure A1). Then, by setting the vertical baseline (threshold < 300 m) and Doppler baseline threshold (threshold < 100 hz) to construct the relative interference, multiple small baseline sets were generated, and the deformation phase was calculated according to the phase information contained in each scene of the interference pair according to the following equation:

$$\Phi ={\phi }_{\mathrm{def}}+{\phi }_{\mathrm{topo}}+{\phi }_{\mathrm{orb}}+{\phi }_{atm}+{\phi }_{noise}$$

(1)

where ${\phi }_{\mathrm{topo}}$ is the remnant topographic phase caused by DEM error, ${\phi }_{\mathrm{orb}}$ the phase caused by trajectory inaccuracy, ${\phi }_{atm}$ the atmospheric delay phase, and ${\phi }_{noise}$ the noise caused by thermal noise and fiducial registration error, etc. With other phase differences subtracted, surface deformation information ${\phi }_{\mathrm{def}}$ can be obtained. Moreover, in order to improve the coherence of the differential interferogram, local slope adaptive filtering was used for interference phase filtering and the minimum-cost flow (MCF) method was used for phase unwrapping. Finally, the temporal deformation information of all SFP points could be selected.

3.2. MSSA and MC-SSA

The decomposition of the elastic component in surface deformation is of extreme importance for the inversion of hydrogeological parameters and the management of groundwater resources. MSSA, which is mathematically equivalent to an extended empirical orthogonal function algorithm (EEOFS), has been used as a data analysis method for many years in the field of digital signal processing [64]. The remarkable advantage of MSSA is that it can draw information from brief and noisy time series, and obtain nonlinear trends and periodic oscillations with different amplitudes and frequencies [65,66]. The algorithm is to establish a multichannel trajectory matrix [67] with dimension of $N^{\prime }\times LM$ mainly by embedding the L-dimensional time series into the trajectory matrix {$X_l\left(t\right)$: $l$ = 1, …, $L$; $t$ = 1, …, $N$}, and by setting the sliding lag window $M$. By diagonalizing the lag covariance matrix, and by calculating the $k$ principal components of the eigenvectors and the associated single channel time series, the reconstructed components (RCs), as for a given set of exponents $k$, are obtained by means of the convolution algorithm. Then, according to the RCs, each signal component obtained in the time series of deformation and hydraulic head change can be segregated to gain the seasonal component, nonlinear long-term component and other noise signals with periodic changes [68].

The variables of MSSA must be uniformly sampled in time. Because of the existence of a small number of vacant values in the deformation time series during the study period, the variables used by MSSA, by using the linear interpolation method, are made to be a uniform time series with an equal time interval of 31 days (the change of head is monthly data). Furthermore, MC-SSA was applied to verify whether the disjunct seasonal signals contained false oscillations caused by red noise. In MC-SSA, as for the lth channel of $X$, the maximum likelihood estimation is first used to evaluate the interrelated arguments in the light of the time series $X(t)$, then a collection of mimetic red noise data is produced, with the covariance matrix $C_R$ calculated, and then the covariance matrix of the emulated data is projected onto the basis feature vectors $E_X$ of the primordial data. Finally, the equation is obtained as follows:

$${\Lambda }_R=E_X{}^T{C}_R{E}_X$$

(2)

where ${\Lambda }_R$ measures the similarity between the emulated data set and the primordial data. The confidence interval was set at 95% in this study. If the eigenvalues are outside 95% of the confidence interval, it is confirmed that the seasons of MSSA separation are significantly different from the red noise.

3.3. TLCC Analysis

The TLCC analysis can be used to calculate the correlation coefficient and lag time, separated by MSSA, between the seasonal surface deformation and hydraulic head change. In signal processing, cross-correlation analysis is applied to evaluate the similitude degree of two discrete-time series, as well as a function of the displacement of one sequence relative to the other (sliding dot product or sliding inner product). This kind of analysis is widely applied in a great many fields, such as pattern recognition, single particle analysis, electron tomography, cryptanalysis and neurophysiology [69,70]. In essence, by building the convolution function of two discrete time series, the cross-correlation analysis establishes the cross-correlation matrix containing the expected value of random vectors and the existence of variance, and obtains the cross-cross-covariance function, thereby calculating the cross-correlation coefficient between the two above. Then, by virtue of the maximum of the cross-correlation coefficient (or the minimum if the signal is negatively correlated), the time point at which the signal is best aligned is indicated, and it can be used to determine the delay time between two signals. The time difference ${\tau }_{lag}$ between the time series $b\left(t\right)$ of surface deformation and the time series $h\left(t\right)$ of hydraulic head change can be calculated by the equation below:

$${\tau }_{lag}=\arg \max \left(corr\left(h\left(t\right),b\left(t+\tau \right)\right)\right)$$

(3)

where $corr\left(h\left(t\right),b\left(t+\tau \right)\right)$ is the cross-correlation coefficient which is used to evaluate whether there exists a strong correlation between surface deformation and hydraulic head. In this study, the TLCC analysis method was used to estimate the lag time and correct the lag response of seasonal surface deformation relative to the variation of the aquifer system head.

3.4. Estimation of Aquifer Storage Properties and GWS Loss

3.4.1. Estimation of Aquifer Skeletal Storativity Coefficient

The storage properties of aquifers are an indispensable part of water resources management. The specific storage coefficient $S$ of confined aquifer refers to the amount of the water discharged from the compressible aquifer per unit aquifer volume when the unit head drops [71,72,73,74,75,76,77,78,79,80,81], which can be expressed as:

$$S=\rho g\left(\alpha +n\beta \right)$$

(4)

where $\rho$ is the water density, $g$ the acceleration of gravity, $n$ the porosity, and $\beta$ the compressibility of water. The volumetric aquifer compressibility $\alpha$ expresses the change in relative volume of an aquifer skeleton as pressure changes. Generally, the compressibility of an aquifer system is much greater than that of water. Assuming that water compressibility is omitted, $S_k$ represents the water storage capacity of the skeleton related to the compressibility of the aquifer system, which can be calculated using the relationship between the stress deformation of the aquifer and the change of hydraulic head:

$$S_k=S_{sk}\times b_0=\Delta b/\Delta h$$

(5)

where $S_{sk}$ is the specific skeleton water storage performance, $b_0$ the initial thickness of confined aquifer, $\Delta b$ the variation in the thickness of the confined aquifer, and $\Delta h$ the variation in the hydraulic head. The aquifer storativity coefficient can be divided into inelastic and elastic, the magnitude of which is affected by the change of historical hydraulic head level and surface deformation [46,48]. Two isolated components of the elastic ($S_{ke}$) and inelastic ($S_{kv}$) skeletal storativity coefficient are applied in order to explain this significant difference. The storativity coefficient of the elastic skeleton $S_{ke}$ can be reckoned by the specific value of the seasonal deformation components $\Delta b_s$ to the seasonal hydraulic head component $\Delta h_s$. The equation is as follows:

$$S_{ke}=\Delta b_s/\Delta h_s$$

(6)

Since the loose sand in an aquifer system usually does not undergo inelastic compaction [82], the elastic deformation occurs in an aquifer regardless of whether the hydraulic head is higher than the pre-consolidated hydraulic head [48]. However, because the seasonal deformation only comes up in the aquifer where the hydraulic head is lower than the pre-consolidation hydraulic head, Equation (6) can only be used to estimate the storage capacity of the elastic skeleton in the aquifer system.

3.4.2. Quantization of GWS Depletion

In a confined aquifer, water delivery by pumping derives from water expansion attributed to pore water pressure reduction and compaction of the aquifer system skeleton caused by an effective stress increase [48]. Therefore, in this study, InSAR deformation results were applied to calculate the cumulative deformation volume of the aquifer in the study period, which can be approximately tantamount to the variation of the TGWS in the aquifer system ($\Delta V_{total}$) [83]:

$$\Delta V_{total}=\Delta b_{total}\times S$$

(7)

where $\Delta b_{total}$ is the cumulative surface deformation variables during the study bucket, and $S$ the acreage of the aquifer. After the hydraulic head returns to its initial status, the GWS changes induced by the compaction of elastic soil skeleton begin to recover. Therefore, recoverable GWS (RGWS) in the aquifer system can be expressed as [38]:

$$\Delta V_r=S_{ke}\times \Delta h_{total}\times S$$

(8)

where $\Delta V_r$ is the change amount of RGWS, $S_{ke}$ the elastic skeletal storativity coefficient reckoned by utilizing Equation (6), and $\Delta h_{total}$ is the change amount of total hydraulic head during the study period. When the hydraulic head declines under the pre-consolidation head, the physical rearrangement of particles in the aquifer gives rise to inelastic compaction, which can bring about the depletion of UGWS. The amount of UGWS depletion ($\Delta V_i$) can be estimated by the amount of loss in TGWS minus the amount of change in RGWS:

$$\Delta V_i=\Delta V_{total}-\Delta V_r$$

(9)

In this study, inverse distance weight (IDW) interpolation in ArcGIS is used to conduct spatial interpolation for discrete data in order to explore the spatial distribution of the elastic skeletal storativity coefficient and GWS variation.

4. Results

4.1. Evolution Analysis of Deformation Characteristics

In order to ensure the accuracy of InSAR surface deformation data, leveling data of 80 leveling points (2012–2013) and 14 leveling points (2015–2016) were selected from the second order leveling data network of surface deformation in Beijing plain for verification. By establishing a buffer zone with a radius of 100 m centered on every leveling point, the stable coherent points were screened. Then the deformation value of radar line-of-sight (LOS) direction obtained by SBAS-InSAR processing was converted into a vertical deformation value by dividing by the included angle of LOS direction. Finally, we performed linear regression fitting on the leveling data and SBAS-InSAR vertical deformation results and calculated the goodness of fit R square (R²). The closer the R² value is to 1, the better the fitting degree of the regression line to the observed value (see Appendix A.2. for detailed verification process). The linear regression results of RadarSat-2 and leveling, the results show, were 0.98 and 0.96, respectively (Figure 4a,b), which indicates that the deformation information deduced by RadarSat-2 was of reliable accuracy.

As is shown in the average annual deformation velocity obtained from RadarSat-2 data by virtue of SBAS-InSAR (Figure 5a), the regional surface deformation developed rapidly during the period from January 2012 to October 2016, and the average annual deformation velocity ranged from 0 mm/yr to 139.11 mm/yr, with great differences in spatial distribution. Two major subsidence funnels were formed in the north and the south of Beijing, with a number of deformation centers connected into a whole piece. The largest land subsidence funnel was located in the areas of Chaoyang Jinzhan–Tongzhou (CJT), and the second largest one in the areas of Changping Baxianzhuang–Haidian Shangzhuang (CBHS), where the maximum cumulative deformation reached −626.01 mm, and the area with annual average deformation over 60 mm was 249.77 km². The specific data of cumulative deformation difference in each year are shown in the Appendix A,

Table A2. The differences in cumulative deformation by year.

Year	2012	2013	2014	2015	2016
Annual deformation rate (mm/yr)	0~−127.74	0~−144.44	0~−166.98	0~−127.72	0~−177.06
The maximum cumulative settlement year by year (mm)	−127.74	−270.82	−403.12	−511.81	−626.01
Area with annual deformation velocity greater than 60 mm/yr (km2)	220.48	273.75	300.33	246.92	349.22

. The hydrogeological profile line AA’ crosses the two main subsidence funnels, from CBHS to CJT. The cumulative deformation of profile line AA’ from 2012 to 2016 is shown in Figure 5b, and we found that the increment of cumulative deformation decreased significantly after 2013. The evolution trend of cumulative surface deformation in time series (Figure 5c–g) also shows that, affected by the SNWDP, the annual surface deformation velocity began to slow down significantly after 2014 and reached a minimum of 0–127.42 mm/yr in 2015. The area with annual deformation over 60 mm also reached a minimum of 246.92 km² (53.41 km² less than that in 2014).

4.2. Separation of Seasonal Surface Deformation and Hydraulic Head Variation Components

By comparing the trend of cumulative surface deformation with the groundwater level change of groundwater level monitoring points (Figure 6), it was found that the deformation of most monitoring points was positively correlated with the water level. On the whole, the surface deformation declined to varying degrees, and the downward trend between the deformation and head change at GWL2, 3, 5, 6, 7, 8 and 9 was relatively consistent. Compared with deformation, the hydraulic head of confined aquifer fluctuated more significantly in time series due to the influence of periodic groundwater exploitation and precipitation. Among these points, the head change at GWL2, 3, 5, 6, 7, 8 and 9 showed a downward trend, while GWL1, 4, 10 and 11 showed a stable and even rising trend. It is worth noting that, after the SNWDP formally diverted water to Beijing at the end of 2014, the pumping rates of the groundwater well field decreased, resulting in an increase in the overflow recharge of middle-deep confined aquifers. Therefore, the head of the middle-deep confined aquifers recovered significantly at some groundwater level monitoring points, such as GWL1, 4, 5, 6, 7, 9, 10 and 11. Among them, the maximum head rebound was located at point 6, with a rebound of 5.41 m from December 2014 to March 2016. However, the head change of the middle-deep confined aquifer at GWL2, 3 and 8 still maintained a long-term downward trend, and even decreased again after water head recovery at GWL1, 6, 9 and 10, indicating that the deep groundwater in these areas may still have been over extracted.

In order to further explore the variation relationship and characteristics of seasonal deformation and seasonal hydraulic head, we applied the MSSA approach to decompose the seasonal signals of these two kinds of temporal information, respectively (the delay window was 15), and retained 20 RCs of each group, including low-frequency long-term trend and noise signals. Here, we take the seasonal head of GWL5 and the seasonal deformation of GWL6 (Figure 7a,b), where the red dots and line represent periodic oscillations at this point. By combining characteristics, such as amplitude and wave period length, it can be clearly distinguished from other RCs with different frequencies. According to the variance contribution rate corresponding to the main frequency associated with different RCs (Figure 8a,b), it was found that the variance contribution rate of the seasonal deformation component was about 6.58% while that of the seasonal head component was about 19.97%. In addition, in order to ensure that the separated seasonal signals did not contain stray oscillations caused by atmospheric delay phase, we utilized the MC-SSA to verify the authenticity, through to the different frequency characteristic vector by building a 95% confidence interval (Figure 8c,d), and found that the frequency (f = 0.98) corresponding to the seasonal signals was outside the error bar. This indicates that the seasonal components extracted by MSSA were authentic and dependable.

Therefore, by further comparing the seasonal components of surface deformation and hydraulic head change, obtained by means of MSSA (Figure 9), it was found that the seasonal surface deformation and hydraulic head change of almost all monitoring points had good consistency and fluctuation, with obvious periodic oscillation occurring. By further comparing the amplitude and the position time at the peak/trough of the seasonal deformation and the head change, it was found that there was a lag response of different degrees between the two above, with characteristic spatial differences. Among them, the lag response time of GWL1, 3, 4, 5 and 9 was obviously longer than that of other groundwater level monitoring points, while the peak interval at GWL6, 7, 8, 10 and 11 was significantly smaller than that of other points, which may well be related to the lithology of the aquifer, including the thickness of particulate matter and the thickness of the compressible clay layer. Therefore, it was necessary to obtain the seasonal deformation corresponding to a prescribed head change in time considering the hysteresis influence so that the aquifer storage properties could be further estimated.

4.3. Hysteresis Correction and Estimation of Elastic Skeletal Storativity Coefficient

Surface deformation has the lag different degrees to the change of head (Figure 9). The cross-correlation coefficient and lag time (Figure 10) between the seasonal surface deformation and the change of head through the TLCC were compared and analyzed in the study, and meanwhile, the seasonal deformation through constant displacement was calibrated using the estimated time lag. Moreover, TLCC analysis was used to determine whether there was a strong correlation between surface deformation and the groundwater level change. If the maximum cross-correlation coefficient is greater than 0.5, the seasonal fluctuation near the monitoring well is most likely to be caused by seasonal head change, whereas if the maximum cross-correlation coefficient is less than 0.5, the seasonal deformation or head change may well be disturbed by noise so that the correlation between them is weaker. By comparing the results of the TLCC analysis (Figure 10), it was found that there were spatial differences in the lag response time of different groundwater level monitoring points. The longest lag response time of GWL4 was 244 d, while the shortest lag response time of GWL6 and GWL10 was 0 d. Furthermore, the maximum correlation of all groundwater level monitoring points was greater than 0.5, and the maximum value was 0.96 at GWL7, indicating that there was a strong correlation between seasonal deformation and seasonal hydraulic head change, and the surface deformation there was mainly induced by hydraulic head fluctuation corresponding to the mining depth. The maximum correlation coefficients of GWL1 and 3 were between 0.50 and 0.55, which are smaller than other points, which may be because of noisy interference, unapparent and anomalous seasonal deformation signals, and the deformation presumably caused by deeper hydraulic head fluctuations.

For the purpose of representing the spatial variability of the storativity characteristics of the aquifer system, the elastic skeletal storativity coefficient of 11 monitoring wells were estimated by means of Equation (6) (as shown in

Table 3. The elastic skeletal storativity coefficient calculated according to Equation (6).

Point Number	Elastic Skeletal Storativity (Ske)	Lag Time (Unit: Days)	Thickness of Compressible Layer (Unit: m)
GWL1	0.249 × 10−3	60	150–200
GWL2	0.563 × 10−3	152	150–200
GWL3	0.169 × 10−3	91	100–150
GWL4	0.241 × 10−3	244	100–150
GWL5	0.201 × 10−3	182	100–150
GWL6	0.442 × 10−3	0	150–200
GWL7	0.395 × 10−3	31	200–250
GWL8	0.420 × 10−3	60	150–200
GWL9	0.179 × 10−3	213	100–150
GWL10	0.719 × 10−3	0	50–100
GWL11	0.423 × 10−3	31	100–150

). The calculated results were in the range of the theoretical value domain of the $S_{ke}$ coefficient n × 10⁻⁵–n × 10⁻³ [1,34]. For the sake of exploring the inhomogeneity on the spatial scale of $S_{ke}$, IDW interpolation was applied to obtain the spatial distribution of $S_{ke}$ (Figure 11). The values of $S_{ke}$ ranged from 0.197 × 10⁻³ to 0.719 × 10⁻³, manifesting obvious spatial differences and the aquifer lithology at all points above was composed of 2–3 layers of sand and gravel layers. In general, considering the influence of the geological structure of the Beijing plain, the impact of aquifer permeability in the pluvial fan interaction area became weaker step-by-step, which, to a certain extent, brought about the elastic skeletal storativity coefficients on the space diversity, as well as the uneven subsidence of Beijing.

Among them, the $S_{ke}$ values at GWL2, 6, 8,10 and 11 points were greater than 0.40 × 10⁻³, and the other four points, except GWL6, were located in the less developed area of subsidence, especially GWL10, which did not even show obvious subsidence during the study period. Meanwhile, combined with the structural characteristics of the Beijing plain pore aquifer (Figure 1) and regional subsidence evolution (Figure 5), we found that the reason why $S_{ke}$ at GWL10 was larger than others may be that the thickness of the compressible clay layer at GWL10 was relatively thin (between 50–100 m), the water-rich condition better (between 3000–5000 m³/d) and the pumping rate of groundwater wells small, with the aquifer overflow replenishment fast and hydrodynamic delay small. However, the compressible layer at points GWL2, 6 and 8, which are at the edge of the subsidence funnel, was thicker (between 150–200 m), and there was poorer water-rich condition (between 500–3000 m³/d), so the $S_{ke}$ value was larger than others. In contrast, the $S_{ke}$ values at the other points were less than 0.40 × 10⁻³. GWL1, 3, 4, 5, 7 and 9 points were close to the center of the CJT subsidence funnel. This may be due to the thickness of the compressible clay layer (mostly between 150–200 m), the relatively uniform distribution of lithology and water-rich conditions (all between 1500–3000 m³/d) of the aquifer system, and the long hydrodynamic delay time of the aquifer. In addition, the groundwater pumping rate at these points was large, and the short-term recovery caused by human action cannot change the fact that the middle-deep confined head at these points shows a long-term decline. In the long term, the process of middle-deep confined head declining and land subsidence caused by over-exploitation of groundwater is irreversible.

4.4. Quantization of GWS Depletion

In confined aquifer systems, the variation of TGWS can be approximately equal to the cumulative surface deformation variable. Therefore, by means of Equation (7) in Section 3.4, the loss of TGWS during the study period can be calculated, and the volume of RGWS and the volume of UGWS depletion can be calculated by means of Equations (8) and (9). To characterize the spatial differences of GWS changes, all changes are expressed as equivalent water thickness (EWT), so as to obtain the average distribution of TGWS from January 2012 to October 2016 (Figure 12a), the average volume of RGWS (Figure 12b) and the average volume of UGWS depletion (Figure 12c). During the study period, the loss volume of the TGWS was about −3.276 × 10⁸ m³, and the average annual loss was −6.702×10⁷ m³; the change volume of RGWS was −2.948 × 10⁷ m³, and the average annual change was −0.602 × 10⁷ m³. The depletion volume of UGWS was −3.084 × 10⁸ m³, and the average annual depletion was −6.168 × 10⁷ m³. The depletion of UGWS accounted for 92.03% of the TGWS volume, while RGWS accounted for 7.97%. We compared the annual change GWS in the Beijing Water Resources Bulletin as verification (Figure 13f), which shows that Pearson’s correlation (PC) coefficient was 0.90, indicating that this result was relatively reliable.

5. Discussion

5.1. Temporal Relationship between Deformation and UGWS Loss

To explore the temporal-spatial distribution features of UGWS depletion, the depletion volumes of UGWS from January 2012 to January 2013, from January 2013 to January 2014, from January 2014 to January 2015 and from January 2015 to October 2016, were calculated over the timescale (Figure 13a–e). In light of the comparative analysis of the volume of UGWS depletion year-by-year and the change of average total hydraulic head (Figure 14b), it was found that the overall change of UGWS depletion was not great from 2012 to 2016, and the UGWS depletion lagged behind the change of total hydraulic head with a lag time of about one year. The depletion of UGWS in 2013 (Figure 13b), in terms of the timescale, showed a decreasing trend, but the unrecoverable area showed a trend of expansion compared with that in 2012 (Figure 13a). After 2013, the depletion of UGWS still increased but the depletion rate was significantly slower than before. Even after 2014, the depletion of TGWS and UGWS decreased significantly to varying degrees, which may be the short-term result of the policy of limiting and reducing groundwater overexploitation under the auspices of the SNWDP. However, from 2015 to 2016, the depletion of TGWS and UGWS continued to increase, mainly characterized by an expanding trend in the unrecoverable area of the CBHS subsidence funnel in the north and a deepening trend in the unrecoverable degree of the CJT subsidence funnel in the south.

In addition, by comparing the response relationship between the depletion of UGWS and the change of total hydraulic head in time series (Figure 14b), we also found that the depletion of the corresponding UGWS only decreased in 2014 after the total hydraulic head recovered in 2013. Then the total head recovered to the maximum at the end of 2014, while the depletion of UGWS reached the minimum in 2015. After 2015, the total head decreased again, while the corresponding loss of UGWS increased one year later in 2016. Therefore, we found that under the guidance policy of the SNWDP to reduce over-exploitation of groundwater, the total head in 2016 recovered to a certain extent compared with that in 2015, and it can be predicted that the depletion of UGWS will also decrease afterwards. This also shows that the SNWDP played an important role in alleviating the over-exploitation of groundwater, protecting groundwater resources and controlling regional land subsidence.

5.2. Spatial Relationship between Deformation and UGWS Loss

According to the evolution process of UGWS depletion on the spatial scale (Figure 13a–e), the depletion of UGWS in CBHS was obviously reduced, while the depletion of UGWS in CJT region was still severe because the compressible clay soil layer was thick, the main aquifer particles were fine sand, and the subsidence development was long-term. As the area from CJT is the sub-center of Beijing and densely populated, affected by human activities and dynamic and static loads, the compressible viscous soil layer in the aquifer system in this area has been stressed for a long time, occurs over a wide range, and the deformation is serious. Thus, the depletion of UGWS remains serious. By drawing one profile line AA’ (Figure 13a–e) across the two main subsidence funnels in the north and south, it was found that the depletion of UGWS during 2015–2016 decreased significantly and presented a segmented distribution feature. Therefore, we divided the profile line AA’ into three segments (front, middle and back) according to the degree of rebound (Figure 14a).

The front segment of the profile line (DK < 21 km) was mainly located in the CBHS subsidence funnel in the north Beijing plain. Although the depletion of UGWS in this part increased continuously from 2012 to 2014, the increment decreased year by year, and the depletion of UGWS decreased significantly after 2015, and even continued to decrease after 2016. However, the middle segment of the profile line (21 km < DK < 47 km) mainly passed through the CJT subsidence funnel in the south Beijing plain. Combined with the hydrogeological structure of the profile line (Figure 1c), it was found that the thickness of the compressible clay layer in this part was greater, and the depletion of UGWS decreased significantly in 2015, but then continued to increase in 2016. However, the back segment of the profile line (DK > 47 km) was located in the southeast of the Beijing plain where no significant subsidence has been found. Therefore, the depletion of UGWS in this part fluctuated slightly on the basis of the overall decrease, but the overall change was not obvious.

In summary, it was found that the depletion of UGWS not only fluctuated temporally due to the impact of the SNWDP and groundwater abstraction reduction policy, but also showed differences spatially among different regions. On the whole, the depletion of UGWS decreased after 2015, but the unrecoverable degree of the CJT subsidence funnel in the south still increased compared with the CBHS. Moreover, the CJT is located in the sub-center of Beijing, with a large population density and serried buildings. A large amount of groundwater has been exploited, which further indicates that the soil in this region will continue to be compacted and it is difficult to recover GWS. In consequence, it is urgent to restrict the exploitation of groundwater and control the growth of subsidence.

5.3. Uncertainty Analysis and Future Work

In the process of using SBAS-InSAR monitoring technology to obtain the long-term surface deformation and of utilizing MSSA to obtain the seasonal component of surface deformation and hydraulic head change, there were some uncertainty errors which mainly include the following points:

• As is shown in the study, if the aquifer system has significant horizontal deformation, without considering the horizontal deformation and its contribution to skeleton storage, the estimated elastic skeletal storativity coefficient may well show a deviation which is less than the theoretical value.
• The uncertainty error is connected with the precision of deformation velocity obtained by the SBAS-InSAR technique. This is mainly caused by the atmospheric latency phase, decoloration and registration errors, according to previous research.
• There may be errors caused by human factors with respect to the monitored groundwater level and the real groundwater level.
• As the time series of surface deformation needs to be matched against the head change, the linear interpolation method of supplementing the vacancy value will also produce some errors.
• For coherent points, which are characteristic of significant surface deformation, the long-term trend and the random noise using MSSA may not be able to be completely eliminated from the time series of surface deformation and head change. Although a 95% confidence interval was constructed for verification by means of MC-SSA, there may still be uncertain noise in the seasonal component obtained by separation. Although we have adjusted the lag of the seasonal surface deformation, it is a fact that more frequent InSAR and hydraulic head data sampling (daily or weekly) time series over longer periods are required to obtain more accurate lag times. Other errors may be caused by the fact that the lag time of surface deformation and head change cannot be completely accurately corrected.

6. Conclusions

In order to realize inversion of GWS variation from the perspective of remote sensing, this study, based on the surface deformation field obtained by SBAS-InSAR, combined with head change data, utilized MSSA to separate the seasonal components of the surface deformation and hydraulic head variation. Then the lag response of surface deformation relative to head change was corrected by introducing the TLCC method, so as to estimate the aquifer storage properties and GWS loss under given hydraulic head observations. Finally, we particularly discussed the temporal-spatial response relationship of GWS loss and surface deformation in the context of SNWDP. The main conclusions of this study are as follows:

(1) The average surface deformation velocity of the Beijing plain was up to 139.11 mm/yr, and the maximum cumulative deformation was −626.01 mm from January 2012 to October 2016, forming two main subsidence centers of CJT in the southern plain and CBHS in the northern plain.

(2) After the SNWDP was officially implemented in Beijing at the end of 2014, the head of middle-deep confined aquifers at some points showed a short-term recovery, with a maximum rebound of 5.41 m in two years, and then continued to decline. On this basis, the real seasonal signals of surface deformation and water head were obtained by MSSA, and their fluctuations had spatial differences.

(3) The $S_{ke}$ and the lag time of surface deformation in response to hydraulic head change were spatially different in different places. By introducing TLCC analysis, it was found that the maximum lag time of the two responses was 244 d and the maximum correlation coefficient was 0.96. Meanwhile, the points near the center of CJT subsidence funnel, with a thick compressible clay layer and poor water-richness, had relatively smaller $S_{ke}$ values and the hydrodynamic delay time of aquifer was longer.

(4) This study further quantifies the loss volume and spatial-temporal change of TGWS, RGWS and UGWS from 2012 to 2016. The average annual change of TGWS was −6.702 × 10⁷ m³, and the average annual depletion volume of UGWS was −6.168 × 10⁷ m³, accounting for 92.03% of the TGWS. The depletion of UGWS lagged behind the change of total head with a lag time of about one year. In addition, the depletion of UGWS showed regional differences temporo-spatially. After 2015, the northern CBHS subsidence funnel showed a trend of decreasing UGWS depletion but gradually expansion of the unrecoverable area. However, the UGWS depletion of the southern CJT subsidence funnel only rebounded in the short term after 2015 and then continued to increase.

In general, in the context of the SNWDP, the over-exploitation of groundwater in Beijing plain has been limited and alleviated to some extent. However, the process of over-exploitation of groundwater, leading to the decline of middle-deep confined head and long-term depletion of GWS, is still irreversible. In other words, it is a long and arduous task to prevent and control land subsidence and restore groundwater resources. Nevertheless, it is necessary to wait longer to evaluate the role of the SNWDP in replacing part of the water demand in the capital and slowing down the over exploitation of groundwater.