#### 4.2. InSAR and Leveling Comparison

Leveling data are available in May 2007, March 2008, September 2008 and March 2009 from 13 benchmarks along a 20 km section of the Jing-Hu High Speed railway in this area. InSAR results are compared with the leveling survey results. Comparison between InSAR and leveling is not straightforward due to different reference systems. Leveling reference point for Tianjin is the Liqizhuang bedrock point located in Tianjin institution of surveying and mapping. Liqizhuang bedrock point is a deep stratum benchmark, with its base 1088 m underground in Cambrian formations [

41]. The InSAR results are referenced to the mean value of phases first. Then, the areas with the greatest uplift signals are chosen as the InSAR reference area. Comparison between leveling and InSAR displacements can be made through a double difference in time and space [

42]. The spatial reference for InSAR and leveling comparison is leveling point BM2010, which exhibited the least subsidence among all leveling points. The temporal reference for InSAR and leveling comparison is set at the first leveling campaign in May 2007. Firstly, LOS InSAR displacements are linearly interpolated at the leveling times. Secondly, Interpolated LOS InSAR results are double differenced using the spatial and temporal reference. Finally, LOS Envisat and ALOS results are decomposed into east-west and vertical components by ignoring north-south horizontal movement for a polar orbit SAR satellite [

43]. To be consistent with InSAR results, the leveling results are re-referenced to BM2010 at May 2007 as well. It should be noted that first Envisat acquisition is later than first leveling campaign, so InSAR and leveling results are temporally re-referenced to March 2008 when necessary.

The RMS differences are 9.7 mm between ASAR LOS and leveling, 8.8 mm between PALSAR and leveling, and 8.3 mm between vertical component and leveling (

Figure 4a–c). Slightly improvement is seen after decomposition of horizontal movement from vertical (

Figure 4c,f). The reason lies in the fact that the accuracy of Leveling data, interpolated ALOS and Envisat displacement is limited. On the one hand, the accuracy of interpolated ALOS and Envisat displacement is limited by seasonal movement. For example, ALOS displacement on 1 March 2009 is interpolated from displacements of 22 January and 9 March 2009, while Envisat displacement on 1 March 2009 is interpolated from displacements of 27 February and 3 April 2009. The water level began to drop in late March, so the interpolated Envisat displacement is greater than ALOS in March 2009. It seems that the decomposition of horizontal and vertical movement is not highly effective in the presence of seasonal displacement. On the other hand, the leveling dates available are only accurate to months, and the first day of the month is assumed as the leveling date for a campaign. However, it still can be seen that interpolated ALOS and Envisat displacements generally follow the displacement trend of benchmarks, although with less displacement (

Figure 4d,e). It is likely that the vertical displacement is dominant to horizontal displacement along the leveling line, and the projection from vertical to LOS results in displacement reduction for both ascending and descending tracks.

#### 4.3. Model of West Subsidence

The western subsidence is modeled using Mogi source solution in a semi-analytical approach [

44,

45].

where

${u}_{z}\left(x,y\right)$ is the vertical displacement at

$\left(x,y\right)$,

$\Delta {V}_{i}$ is the volume change at reservoir element

$i$,

${g}_{z}\left(r,d\right)$ is the vertical green/influence function,

$\nu $. is the Poisson’s ratio,

${r}_{i}$ is horizontal distance between surface point

$\left(x,y\right)$ and reservoir element

$({x}_{i},{y}_{i})$,

$d$ is reservoir depth. The reservoir is divided into equal squares.

A semi analytical inversion is performed for Mogi source array to determine the best fit distribution of source volume change for LOS annual displacements (annual rates times one year). Modeled surface displacements are the ensemble contributions from each source. Poisson’s ratio of 0.25 is set for this study. Reservoir depth is fixed at about 200 meters as ground water is pumped from a depth of 100–300 m in Tianjin [

7]. The linear least-squares inversion procedure has been adopted to estimate the source volume change.

It is well known that InSAR observations may contain global bias due to uncertainties in reference level or other uniform signals (e.g., sediment compaction). Accordingly, a constant offset is allowed in model inversion to accommodate the global bias.

In the process of inversion, regularization on the source strength is often needed to ensure a reliable inversion with a faithful representation of the source. Directly modeled reservoir volume changes (

Figure 5a) seem noisier than the regularized source (

Figure 5b). This can be caused by displacement uncertainty due to noise and other error sources. Besides, the semi analytical approach allows approximation of reservoir change and efficient computations, but it does not assure the physical process that must involve a continuous contracting/inflating volume. A Laplacian regularization based process was applied to smooth the source (

Figure 5b).

where

$G$ is the Green’s function,

$L$ is the Laplacian smooth operator,

$F$ is the smoothing factor (weight), and their product

$LF$ is the smoothing matrix used in model inversion.

$S$ is the source volume change,

$O$ is the offset term for global bias, and

$D$ is the observed displacement. In order to balance the roughness of source volume change and model fit to displacement, the L curve method (

Figure 6) is adopted to find the best smoothing factor [

46].

The modeled source volume change is about −7694 to +1678 m

^{3} for each element (300 m square) per year (

Figure 7), equivalent to water volume change of −85,000 to +18,000 m

^{3}/km

^{2}/year, further equivalent to water storage change of −85 to +18 mm·year

^{−1} in height. Unfortunately, to the best of our knowledge,

in situ measurements of fluid volume from production wells is unpublished and is not available for comparison with our modeling results. Ground water storage (GWS) change from GRACE satellite measurement is about −17 to −22 mm·year

^{−1} in North China Plain [

47,

48,

49], equivalent to −17,000 to −22,000 m

^{3}/km

^{2}/year. The GRACE measurement covers Beijing, Tianjin, Hebei and Shanxi of ~370,000 km

^{2}, while our model is in an area of 860 km

^{2} in Tianjin. Simulated recoverable groundwater storage depletion is −30,000 m

^{3}/km

^{2}/year (equivalent to 30 mm·year

^{−1} in height) from 1970 to 2008 [

50], equivalent to −2700 m

^{3} for each element (source) per year. Contour line of −2700 m

^{3} is superimposed on source volume change (

Figure 7d). Line −2700 m

^{3} coincides with sharp reduction of both the water extraction volume and subsidence observed. Modeled source volume change show local water storage increase up to 1678 m

^{3}. InSAR observations show subsidence of −168.5 to −2.9 mm in model area (

Figure 7a). The volume increase seems to contradict with the fact that only subsidence is observed in this area. From the perspective of semi-analytical modeling, any surface deformation is the ensemble contribution of all model sources via their influence functions. Therefore, for a surface point, the superposition of a remote contracting source with large volume decrease, and a nearby inflating source with small volume increase, may still generate a surface subsidence. From the perspective of physical process, aquifer can benefit from precipitation infiltration, irrigation return, or other vertical or horizontal recharge. For instance, rainfall in summer recharge local ground water aquifers in North China Plain by 12–29 mm seen from GRACE and 25 mm seen from ground boreholes [

48].

The horizontal displacement can be modeled using horizontal Green’s function and the best-fit source volume change.

where

${g}_{r}\left(r,d\right)$ is the horizontal green/influence function. Modeled horizontal displacements can be substantial, with maximum annual horizontal displacements reach up to 74 mm (

Figure 8). Maximum horizontal displacements are located along a semi closed oval around the modeled source center. The modeled maximum horizontal displacement is 36% of the maximum vertical displacement. Theoretical ratio of maximum horizontal to maximum vertical displacement is 38% for a single Mogi source [

51]. A field based aquifer test in Nevada shows that the horizontal displacements reach 8 mm and vertical displacements reach 12 mm within the first 22 days of pumping before reaching the steady state pumping [

52]. Thus, the horizontal displacement induced by aquifer abstraction can be significant. Horizontal displacements are believed to be a cause of ground fissures [

53]. Ground fissures associated with subsidence has caused severe infrastructure damage in Taiyuan Basin of China [

54]. Hence, it is worth measuring the horizontal displacements due to ground water depletion, including Tianjin.