Journal of Geodesy

, Volume 92, Issue 5, pp 529–544 | Cite as

Deriving time-series three-dimensional displacements of mining areas from a single-geometry InSAR dataset

  • Zefa Yang
  • Zhiwei Li
  • Jianjun Zhu
  • Guangcai Feng
  • Qijie Wang
  • Jun Hu
  • Changcheng Wang
Original Article

Abstract

This paper presents a method for deriving time-series three-dimensional (3-D) displacements of mining areas from a single-geometry interferometric synthetic aperture radar (InSAR) dataset (hereafter referred to as the SGI-based method). This is mainly aimed at overcoming the limitation of the traditional multi-temporal InSAR techniques that require SAR data from at least three significantly different imaging geometries to fully retrieve time-series 3-D displacements of mining areas. The SGI-based method first generates the multi-temporal observations of the mining-induced vertical subsidence from the single-geometry InSAR data, using a previously developed method for retrieving 3-D mining-related displacements from a single InSAR pair (SIP). The weighted least-squares solutions of the time series of vertical subsidence are estimated from these generated multi-temporal observations of vertical subsidence. Finally, the time series of horizontal motions in the east and north directions are estimated using the proportional relationship between the horizontal motion and the subsidence gradient of the mining area, on the basis of the SGI-derived time series of vertical subsidence. Seven ascending ALOS PALSAR images from the Datong mining area of China were used to test the proposed SGI-based method. The results suggest that the SGI-based method is effective. The SGI-based method not only extends the SIP-based method to time-series 3-D displacement retrieval from a single-geometry InSAR dataset, but also limits the uncertainty propagation from InSAR-derived deformation to the estimated 3-D displacements.

Keywords

3-D displacements Mining subsidence InSAR SAR Time series 

1 Introduction

Multi-temporal interferometric synthetic aperture radar (MT-InSAR) techniques, such as persistent scatterer InSAR (PS-InSAR) (Hooper et al. 2004), small-baseline subset InSAR (SBAS-InSAR) (Berardino et al. 2002), and temporarily coherent point InSAR (TCP-InSAR) (Zhang et al. 2012), have been extensively exploited for monitoring the time-series ground surface deformation associated with various geophysical processes and anthropogenic activities (e.g., Alex et al. 2015; Cao et al. 2017; Li et al. 2015; Neri et al. 2009; Perski et al. 2009; Samsonov et al. 2013; Yang et al. 2016, 2017a, b; Zhao et al. 2016). Nevertheless, only a one-dimensional (1-D) time series of surface deformation along the radar line-of-sight (LOS) direction can be obtained from single-geometry InSAR data using the traditional MT-InSAR techniques. Considering that time-series 3-D surface displacements are essential to assess mining-related geo-hazards, the 1-D time series of surface LOS deformation obtained by the conventional MT-InSAR techniques is insufficient when assessing mining-induced damage to infrastructure and understanding the subsidence dynamics of underground mining (Kratzdch 1983; Peng et al. 1992).

A number of approaches have been developed to overcome the limitation of the traditional MT-InSAR techniques in recent years. For example, Samsonov et al. (2013) retrieved the time-series ground surface displacements in the vertical and east directions associated with underground post-mining using ascending and descending ERS/ENVISAT SAR data. In addition, He et al. (2015) mapped the time-series displacements associated with open-pit mining along the vertical and satellite flight directions with ALOS PALSAR images, using the DInSAR and multiple-aperture InSAR (MAI) SBAS techniques. However, significant limitations still exist in these methods. For instance, SAR acquisitions from at least three significantly different imaging geometries are needed in the method of Samsonov et al. (2013) to fully retrieve time-series 3-D displacements. This is unlikely to be met in practice, mainly due to the limited number of SAR sensors. Secondly, the MAI technique adopted in He et al. (2015) is very sensitive to the phase errors caused by interferometric decorrelation, the troposphere, and the ionosphere (Bechor and Zebker 2006; Hu et al. 2012; Jung et al. 2009). Consequently, practical applications of the method demonstrated in Hu et al. (2014) are limited, because of the commonplace interferometric decorrelation in mining areas. Most importantly, only time-series two-dimensional (2-D) displacements along the vertical and east/flight directions, instead of time-series 3-D displacements, are obtained using these two methods. This, to a large extent, hinders the accurate assessment of mining-related damage and the better understanding of the dynamics of mining subsidence.

In 2015, a novel method for estimating 3-D mining-induced displacements from a single InSAR pair (SIP) was proposed by Li et al. (2015b) (hereafter referred to as the SIP-based method). In this method, the proportional relationship between the gradient of the mining-induced vertical subsidence and the horizontal movements is introduced as an extra constraint to stabilize the under-determined system for fully solving 3-D displacements from 1-D LOS deformation. The SIP-based method significantly relaxes the strict requirement of the traditional InSAR-based techniques for retrieving 3-D mining-induced displacements, where synchronous InSAR pairs from at least three significantly different imaging geometries are needed. However, there are also some serious limitations to the SIP-based method. For example, it is only capable of retrieving the 3-D surface displacements in the time period of the single InSAR pair. Moreover, the SIP-based method has a poor resistance to the uncertainties of LOS deformation observations (e.g., noise, unwrapping error, interpolation error), due to the lack of redundant measurements.

In this paper, we propose an approach that estimates the time-series 3-D displacements associated with underground mining from a single-geometry InSAR (SGI) dataset. Hereafter, we refer to this approach as the SGI-based method. The SGI-based method first generates a stack of small-baseline InSAR pairs with the SAR images from a single imaging geometry. The SIP-based method is then applied to retrieve the multi-temporal observations of mining-induced vertical subsidence from the generated small-baseline InSAR pairs. Subsequently, the weighted least-squares (WLS) solutions of the time series of vertical subsidence are estimated based on the SIP-derived multi-temporal observations of vertical subsidence. Having obtained the time series of vertical subsidence, the 2-D horizontal motions in the east and north directions are accordingly estimated using the proportional relationship.

The remainder of this paper is organized as follows. Section 2 discusses the limitations of the SIP-based method. Section 3 describes the rationale and procedures of the proposed SGI-based method for deriving time-series 3-D displacements of mining areas. The experiment undertaken with data from the Datong coal mining area, China, is presented to validate the proposed method in Sect. 4, followed by the discussion in Sect. 5. Finally, our conclusions are drawn in Sect. 6.

2 Limitations of the SIP-based method

2.1 Brief review of the SIP-based method

The InSAR-derived LOS deformation L(ij) at a surface point (ij) is the projection of its 3-D displacements along the vertical W(ij), east E(ij), and north directions N(ij) onto the LOS direction (Hanssen 2001; Hu et al. 2014):
$$\begin{aligned} L\left( {i,j} \right) =\left[ {{\begin{array}{c} {\cos \theta \left( {i,j} \right) } \\ {\sin \theta \left( {i,j} \right) \cdot \cos (\alpha _h -{3\pi }/2)} \\ {-\sin \theta \left( {i,j} \right) \cdot \sin (\alpha _h -{3\pi }/2)} \\ \end{array} }} \right] ^{{\mathrm{T}}}\cdot \left[ {{\begin{array}{l} {W\left( {i,j} \right) } \\ {N\left( {i,j} \right) } \\ {E\left( {i,j} \right) } \\ \end{array} }} \right] ,\nonumber \\ \end{aligned}$$
(1)
where \(\theta \) and \(\alpha _{h}\) represent the incidence angle and heading angle of the SAR sensor, and i and j denote the pixel coordinates of the surface point.
Considering that the mining-induced horizontal motions are approximately proportional to the gradient of the vertical displacement (Peng et al. 1992), the 2-D horizontal movements of the pixel (ij) in the east E(ij) and north N(ij) directions can be approximately expressed by (Li et al. 2015b):
$$\begin{aligned} E\left( {i,j} \right) \;= & {} \frac{b\cdot H\left( {i,j} \right) }{\tan \beta }\cdot \frac{W\left( {i,j+1} \right) -W\left( {i,j} \right) }{R_{\mathrm{E}} } \end{aligned}$$
(2)
$$\begin{aligned} N\left( {i,j} \right) \;= & {} \frac{b\cdot H\left( {i,j} \right) }{\tan \beta }\cdot \frac{W\left( {i+1,j} \right) -W\left( {i,j} \right) }{R_{\mathrm{N}} } \end{aligned}$$
(3)
where b, H, and \(\beta \) denote the horizontal motion constant, the mining depth, and the major influence angle in the mining area, respectively. \(R_{\mathrm{N}} \) and \(R_{\mathrm{E}} \) represent the spatial resolution of the geocoded LOS deformation map in the north and east directions, respectively.
For those pixels located in the last row and column of the mining area (whose size is n rows by m columns), we can safely ignore the contribution of the 2-D horizontal motions to the LOS deformation of these pixels, i.e., \(L(i,j)=cos~[{\theta }(i,j)]~\cdot ~W(i,~j)\) \((i=n\) or \(j=m)\) (Li et al. 2015b). Subsequently, the observation equation between the vertical subsidence and the LOS deformation measurements of the mining area is constructed by substituting Eqs. (2) and (3) into Eq. (1) (see Li et al. 2015b for more details). Rewriting this observation equation in a matrix notation, we obtain:
$$\begin{aligned} \left[ {{\begin{array}{lllll} {{{\varvec{B}}}_1 }&{} {{{{\varvec{B}}}}'_1 }&{} &{} &{} \\ &{} {{{\varvec{B}}}_2 }&{} {{{{\varvec{B}}}}'_2 }&{} &{} \\ &{} &{} \ddots &{} &{} \\ &{} &{} &{} {{{\varvec{B}}}_{n-1} }&{} {{{{\varvec{B}}}}'_{n-1} } \\ &{} &{} &{} &{} {{{{\varvec{B}}}}''} \\ \end{array} }} \right] \cdot \left[ {{\begin{array}{l} {{{\varvec{W}}}_1 } \\ {{{\varvec{W}}}_2 } \\ \vdots \\ {\;{{\varvec{W}}}_n } \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {{{\varvec{L}}}_1 } \\ {{{\varvec{L}}}_2 } \\ \vdots \\ {{{\varvec{L}}}_n } \\ \end{array} }} \right] +\left[ {{\begin{array}{l} {{\varvec{\varepsilon }} _{\mathrm{W}1} } \\ {{\varvec{\varepsilon }}_{\mathrm{W}2} } \\ \vdots \\ {{\varvec{\varepsilon }} _{\mathrm{W}n} } \\ \end{array} }} \right] \nonumber \\ \end{aligned}$$
(4)
where \({{\varvec{W}}}_i =\left[ {W\left( {i,1} \right) ,\;W\left( {i,2} \right) ,\ldots ,\;W\left( {i,m} \right) } \right] ^{\mathrm{T}}\), \(L_i =\left[ {{{\varvec{L}}}\left( {i,1} \right) ,\;{{L}}\left( {i,2} \right) ,\ldots ,\;L\left( {i,m} \right) } \right] ^{\mathrm{T}}\),
$$\begin{aligned} {{{\varvec{\varepsilon }}}} _{\mathrm{W}}i= & {} \left[ {\varepsilon _{\mathrm{W}} \left( {i,1} \right) ,\;\varepsilon _{\mathrm{W}} \left( {i,2} \right) ,\ldots ,\;\varepsilon _{\mathrm{W}} \left( {i,m} \right) } \right] ^{\mathrm{T}} \hbox { (i.e., error vector)},\\ {{\varvec{B}}}_i= & {} \left[ {{\begin{array}{lllll} {A_1 \left( {i,1} \right) }&{} {A_2 \left( {i,1} \right) }&{} &{} &{} \\ &{} {A_1 \left( {i,2} \right) }&{} {A_2 \left( {i,2} \right) }&{} &{} \\ &{} &{} \ddots &{} &{} \\ &{} &{} &{} {A_1 \left( {i,m-1} \right) }&{} {A_2 \left( {i,m-1} \right) } \\ &{} &{} &{} &{} {A_4 \left( {i,m} \right) } \\ \end{array} }} \right] ,\\ {{{\varvec{B}}}}'_i= & {} \mathrm{diag}\left[ {{\begin{array}{lllll} {A_3 \left( {i,1} \right) }&{} {A_3 \left( {i,2} \right) }&{} \cdots &{} {A_3 \left( {i,m-1} \right) }&{} 0 \\ \end{array} }} \right] , \\&\quad (i=1,2,\ldots ,n-1), \\ {{{\varvec{B}}}}''= & {} \mathrm{diag}\left[ {{\begin{array}{lllll} {A_4 \left( {n,1} \right) }&{} {A_4 \left( {n,2} \right) }&{} \cdots &{} {A_4 \left( {n,m} \right) } \\ \end{array} }} \right] , \end{aligned}$$
with
$$\begin{aligned} A_1 \left( {i,j} \right)= & {} \cos \left[ {\theta \left( {i,j} \right) } \right] +b\cdot \frac{H\left( {i,j} \right) \cdot \sin \left[ {\theta \left( {i,j} \right) } \right] }{\tan \beta } \\&\quad \cdot \left[ {\frac{\cos \left( {\alpha _h -{3\pi }/2} \right) }{R_{\mathrm{N}} }+\frac{\sin \left( {\alpha _h -{3\pi }/2} \right) }{R_{\mathrm{E}} }} \right] , \\ A_2 \left( {i,j} \right)= & {} -b\cdot \frac{H\left( {i,j} \right) \cdot \sin \left[ {\theta \left( {i,j} \right) } \right] }{\tan \beta }\cdot \frac{\sin \left( {\alpha _h -{3\pi }/2} \right) }{R_{\mathrm{E}} },\\ A_3 \left( {i,j} \right)= & {} -b\cdot \frac{H(i,j)\cdot \sin \left[ {\theta \left( {i,j} \right) } \right] }{\tan \beta }\cdot \frac{\cos \left( {\alpha _h -{3\pi }/2} \right) }{R_{\mathrm{N}} }, \\&\quad A_{4}(i,j)=\cos [{\theta }(i,j)]. \end{aligned}$$
The back-substitution method is used to solve the vertical subsidence, based on Eq. (4), i.e.,
$$\begin{aligned} \left\{ {{\begin{array}{l} {\hat{{{{\varvec{W}}}}}_n =\left( {\mathbf{{B}''}} \right) ^{-1}\cdot {{\varvec{L}}}_n } \\ {\hat{{{{\varvec{W}}}}}_{n-1} =\left( {\mathbf{B}_{n-1} } \right) ^{-1}\cdot \left( {{{\varvec{L}}}_{n-1} -\mathbf{{B}'}_{n-1} \hat{{{{\varvec{W}}}}}_n } \right) } \\ {\hat{{{{\varvec{W}}}}}_{n-2} =\left( {\mathbf{B}_{n-2} } \right) ^{-1}\cdot \left( {{{\varvec{L}}}_{n-2} -\mathbf{{B}'}_{n-2} \hat{{{{\varvec{W}}}}}_{n-1} } \right) } \\ \vdots \\ {\hat{{{{\varvec{W}}}}}_1 =\left( {\mathbf{B}_1 } \right) ^{-1}\cdot \left( {{{\varvec{L}}}_1 -\mathbf{{B}'}_1 \hat{{{{\varvec{W}}}}}_2 } \right) } \\ \end{array} }} \right. \end{aligned}$$
(5)
Having obtained the estimates of vertical subsidence \(\hat{{\varvec{W}}}_i \) \((i=1,2,\ldots ,n)\), the 2-D horizontal movements of the mining area in the east \({{\varvec{E}}}_i \) and north directions \({{\varvec{N}}}_i \) can be easily estimated on the basis of Eqs. (2) and (3) (Li et al. 2015b), i.e.,
$$\begin{aligned} \left[ {{\begin{array}{l} {{{\varvec{E}}}_1 } \\ {{{\varvec{E}}}_2 } \\ \vdots \\ {{{\varvec{E}}}_n } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{lllll} {{{\varvec{C}}}_1 }&{} &{} &{} &{} \\ &{} {{{\varvec{C}}}_2 }&{} &{} &{} \\ &{} &{} \ddots &{} &{} \\ &{} &{} &{} {{{\varvec{C}}}_{n-1} }&{} \\ &{} &{} &{} &{} 0 \\ \end{array} }} \right] \cdot \left[ {{\begin{array}{l} {{{\varvec{W}}}_1 } \\ {{{\varvec{W}}}_2 } \\ \vdots \\ {\;{{\varvec{W}}}_n } \\ \end{array} }} \right] +\left[ {{\begin{array}{l} {{\varvec{\varepsilon }} _{\mathrm{E}1} } \\ {{\varvec{\varepsilon }} _{\mathrm{E}2} } \\ \vdots \\ {{\varvec{\varepsilon }} _{\mathrm{E}n} } \\ \end{array} }} \right] \end{aligned}$$
(6)
$$\begin{aligned} \left[ {{\begin{array}{l} {{{\varvec{N}}}_1 } \\ {{{\varvec{N}}}_2 } \\ \vdots \\ {{{\varvec{N}}}_n } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{lllll} {-{{{\varvec{C}}}}'_{1} }&{} {{{{\varvec{C}}}}'_{1} }&{} &{} &{} \\ &{} {-{{{\varvec{C}}}}'_{2} }&{} {{{{\varvec{C}}}}'_{2} }&{} &{} \\ &{} &{} \ddots &{} \ddots &{} \\ &{} &{} &{} {-{{{\varvec{C}}}}'_{n-1} }&{} {{{{\varvec{C}}}}'_{n-1} } \\ &{} &{} &{} &{} 0 \\ \end{array} }} \right] \cdot \left[ {{\begin{array}{l} {{{\varvec{W}}}_1 } \\ {{{\varvec{W}}}_2 } \\ \vdots \\ {\;{{\varvec{W}}}_n } \\ \end{array} }} \right] \nonumber \\&\quad +\left[ {{\begin{array}{l} {{\varvec{\varepsilon }}_{{\mathrm{N}}1} } \\ {{\varvec{\varepsilon }}_{{\mathrm{N}}2} } \\ \vdots \\ {{\varvec{\varepsilon }}_{{\mathrm{N}}n} } \\ \end{array} }} \right] \end{aligned}$$
(7)
where \({{\varvec{E}}}_i =\left[ {E\left( {i,1} \right) ,\;E\left( {i,2} \right) ,\ldots ,\;E\left( {i,m} \right) } \right] ^{\mathrm{T}}\), \({{\varvec{N}}}_i =\left[ {N\left( {i,1} \right) ,\;N\left( {i,2} \right) ,\ldots ,\;N\left( {i,m} \right) } \right] ^{\mathrm{T}}\),
$$\begin{aligned} {{\varvec{C}}}_i =\left[ {{\begin{array}{lllll} {-C\left( {i,1} \right) } &{} {C\left( {i,1} \right) } &{} &{} &{} \\ &{} {-C\left( {i,2} \right) } &{} {C\left( {i,2} \right) } &{} &{} \\ &{} &{} \ddots &{} \ddots &{} \\ &{} &{} &{} {-C\left( {i,m-1} \right) } &{} {C\left( {i,m-1} \right) } \\ &{} &{} &{} &{} 0 \\ \end{array} }} \right] , \end{aligned}$$
\({{{\varvec{C}}}}'_i =\mathrm{diag}\left[ {{C}'\left( {i,1} \right) ,\;{C}'\left( {i,2} \right) ,\ldots ,\;{C}'\left( {i,m-1} \right) ,\;0} \right] \), with \(C\left( {i,j} \right) ={b\cdot H\left( {i,j} \right) }/{(\tan \beta \cdot R_{\mathrm{N}} )}\) and \({C}'\left( {i,j} \right) ={b\cdot H\left( {i,j} \right) }/{(\tan \beta \cdot R_{\mathrm{E}} )}\); \({\varvec{\varepsilon }} _{\mathrm{E}}i {=}\big [ \varepsilon _{\mathrm{E}} \left( {i,1} \right) ,\;\varepsilon _{\mathrm{E}} \left( {i,2} \right) ,\ldots ,\varepsilon _{\mathrm{E}} \left( {i,m} \right) \big ]^{\mathrm{T}}\) and \({\varvec{\varepsilon }} _{\mathrm{N}}i =\left[ {\varepsilon _{\mathrm{N}} \left( {i,1} \right) ,\;\varepsilon _{\mathrm{N}} \left( {i,2} \right) ,\ldots ,\;\varepsilon _{\mathrm{N}} \left( {i,m} \right) } \right] ^{\mathrm{T}}\) (\(i=1,2,\ldots ,n-1)\) represent the i-th row error vectors of the estimated horizontal motions along the east and north directions, respectively.

2.2 Large uncertainties in the SIP-estimated 3-D displacements

As mentioned in Sect. 1, the SIP-based method is incapable of retrieving time-series 3-D displacements of mining areas. Moreover, there are no redundant measurements for solving vertical subsidence (see Eq. 4) in the SIP-based method, resulting in the poor resistance to the errors in LOS deformation observations. The theoretical and experimental analyses of the error propagation of LOS deformation observations to SIP-estimated 3-D displacements are described in this section.

2.2.1 Theoretical analysis

To simplify the following theoretical analysis, we rewrite the observation systems of Eqs. (4), (6), and (7) to:
$$\begin{aligned} \mathop {{\varvec{B}}}\limits _{nm\times nm} \cdot \mathop {{\varvec{W}}}\limits _{nm\times 1}= & {} \mathop {{\varvec{L}}}\limits _{nm\times 1} +\mathop {{\varvec{\varepsilon }} _{\mathrm{{{\varvec{W}}}}} }\limits _{nm\times 1} \end{aligned}$$
(8)
$$\begin{aligned} \mathop {{\varvec{E}}}\limits _{nm\times 1}= & {} \mathop {{\varvec{C}}}\limits _{nm\times nm} \cdot \mathop {{\varvec{W}}}\limits _{nm\times 1} +\mathop {{\varvec{\varepsilon }} _{\mathrm{E}} }\limits _{nm\times 1} \end{aligned}$$
(9)
$$\begin{aligned} \mathop {{\varvec{N}}}\limits _{nm\times 1}= & {} \mathop {{{{\varvec{C}}}}'}\limits _{nm\times nm} \cdot \mathop {{\varvec{W}}}\limits _{nm\times 1} +\mathop {{\varvec{\varepsilon }} _{\mathrm{N}} }\limits _{nm\times 1} \end{aligned}$$
(10)
where
$$\begin{aligned}&\mathop {{\varvec{W}}}\limits _{nm\times 1} {=}\left[ {{\begin{array}{llll} {{{\varvec{W}}}_1 }&{} {{{\varvec{W}}}_2 }&{} \cdots &{} {{{\varvec{W}}}_n } \\ \end{array} }} \right] ^{\mathrm{T}}, \\ \mathop {{\varvec{L}}}\limits _{nm\times 1}&=\left[ {{\begin{array}{llll} {{{\varvec{L}}}_1 }&{} {{{\varvec{L}}}_2 }&{} \cdots &{} {{{\varvec{L}}}_n } \\ \end{array} }} \right] ^{\mathrm{T}}, \mathop {{\varvec{E}}}\limits _{nm\times 1} {=}\left[ {{\begin{array}{llll} {{{\varvec{E}}}_1 }&{} {{{\varvec{E}}}_2 }&{} \cdots &{} {{{\varvec{E}}}_n } \\ \end{array} }} \right] ^{\mathrm{T}},\\&\mathop {{\varvec{N}}}\limits _{nm\times 1} {=}\left[ {{\begin{array}{llll} {{{\varvec{N}}}_1 }&{} {{{\varvec{N}}}_2 }&{} \cdots &{} {{{\varvec{N}}}_n } \\ \end{array} }} \right] ^{\mathrm{T}},\\&\mathop {{\varvec{\varepsilon }} _{\mathrm{{{W}}}} }\limits _{nm\times 1} =\left[ {{\begin{array}{llll} {{\varvec{\varepsilon }} _{\mathrm{W}1} }&{} {{\varvec{\varepsilon }}_{\mathrm{{{W}}}2} }&{} \cdots &{} {{\varvec{\varepsilon }} _{\mathrm{{{W}}}n} } \\ \end{array} }} \right] ^{\mathrm{T}},\\&\mathop {{\varvec{\varepsilon }} _{\mathrm{{{E}}}} }\limits _{nm\times 1} =\left[ {{\begin{array}{llll} {{\varvec{\varepsilon }} _{\mathrm{E}1} }&{} {{\varvec{\varepsilon }} _{\mathrm{E}2} }&{} \cdots &{} {{\varvec{\varepsilon }} _{\mathrm{E}n} } \\ \end{array} }} \right] ^{\mathrm{T}}, \\&\mathop {{\varvec{\varepsilon }} _N }\limits _{nm\times 1} =\left[ {{\begin{array}{llll} {{\varvec{\varepsilon }} _{\mathrm{N}1} }&{} {{\varvec{\varepsilon }} _{\mathrm{N}2 }}&{} \cdots &{} {{\varvec{\varepsilon }} _{\mathrm{N}n} } \\ \end{array} }} \right] ^{\mathrm{T}},\\&\mathop {{\varvec{B}}}\limits _{nm\times nm} =\left[ {{\begin{array}{lllll} {{{\varvec{B}}}_1 }&{} {{{{\varvec{B}}}}'_1 }&{} &{} &{} \\ &{} {{{\varvec{B}}}_2 }&{} {{{{\varvec{B}}}}'_2 }&{} &{} \\ &{} &{} \ddots &{} &{} \\ &{} &{} &{} {{{\varvec{B}}}_{n-1} }&{} {{{{\varvec{B}}}}'_{n-1} } \\ &{} &{} &{} &{} {{{{\varvec{B}}}}''} \\ \end{array} }} \right] ,\\&\mathop {{\varvec{C}}}\limits _{nm\times nm} =\left[ {{\begin{array}{lllll} {{{\varvec{C}}}_1 }&{} &{} &{} &{} \\ &{} {{{\varvec{C}}}_2 }&{} &{} &{} \\ &{} &{} \ddots &{} &{} \\ &{} &{} &{} {{{\varvec{C}}}_{n-1} }&{} \\ &{} &{} &{} &{} \mathbf{0} \\ \end{array} }} \right] ,\\&\mathop {{{{\varvec{C}}}}'}\limits _{nm\times nm} =\left[ {{\begin{array}{lllll} {-{{{\varvec{C}}}}'_{1} }&{} {{{{\varvec{C}}}}'_{1} }&{} &{} &{} \\ &{} {-{{{\varvec{C}}}}'_{2} }&{} {{{{\varvec{C}}}}'_{2} }&{} &{} \\ &{} &{} \ddots &{} \ddots &{} \\ &{} &{} &{} {-{{{\varvec{C}}}}'_{n-1} }&{} {{{{\varvec{C}}}}'_{n-1} } \\ &{} &{} &{} &{} \mathbf{0} \\ \end{array} }} \right] . \end{aligned}$$
Assuming that the InSAR-derived observations of LOS deformation \({{\varvec{L}}}\) are independent, we let \({{\varvec{D}}}_{\mathrm{L}} =\mathrm{diag}\left[ {{\begin{array}{llll} {{{\varvec{D}}}_{\mathrm{L}1} }&{} {{{\varvec{D}}}_{\mathrm{L}2} }&{} \cdots &{} {{{\varvec{D}}}_{\mathrm{L}n} } \\ \end{array} }} \right] \) be the covariance of the LOS deformation observations, where \({{\varvec{D}}}_{\mathrm{L}i} =\mathrm{diag}\left[ {\delta _{\mathrm{L}}^2 \left( {i,1} \right) } {\delta _{\mathrm{L}}^2 \left( {i,2} \right) } \cdots \delta _{\mathrm{L}}^2\right. \left. \left( {i,m} \right) \right] \) \((i = 1, 2, \ldots , n)\) and \(\delta _{\mathrm{L}}^2 \left( {i,j} \right) \) represents the variance of the LOS deformation measurement at pixel (ij). Therefore, the covariance of the SIP-estimated vertical subsidence \({{\varvec{D}}}_W \) due to the observation errors of LOS deformation can be estimated based on the error propagation law and Eq. (8):
$$\begin{aligned} {{\varvec{D}}}_W ={{\varvec{B}}}^{-1}\cdot {{\varvec{D}}}_{\mathrm{L}} \cdot \left( {{{\varvec{B}}}^{-1}} \right) ^{\mathrm{T}} \end{aligned}$$
(11)
Subsequently, the covariance of the SIP-derived 2-D horizontal movements in the east \({{\varvec{D}}}_E \) and north directions \({{\varvec{D}}}_E \) can also be estimated based on Eqs. (9) and (10), respectively, i.e.,
$$\begin{aligned} {{\varvec{D}}}_E ={{\varvec{C}}}\cdot {{\varvec{D}}}_W \cdot {{\varvec{C}}}^{\mathrm{T}} \end{aligned}$$
(12)
$$\begin{aligned} {{\varvec{D}}}_N ={{{\varvec{C}}}}'\cdot {{\varvec{D}}}_W \cdot \left( {{{{\varvec{C}}}}'} \right) ^{\mathrm{T}} \end{aligned}$$
(13)

2.2.2 Simulated analysis

A simulated experiment was carried out to intuitively demonstrate the error propagation of InSAR-derived LOS deformation to SIP-derived 3-D displacements. We took the real data from the Qianyingzi coal mining area, as reported in Li et al. (2015b), as an example. To reduce the computational burden, we first resampled the interferometric coherence map used in Li et al. (2015b) with a downsampling rate of two (see Fig. 1e). We then estimated the theoretical standard deviations (STDs) of the LOS deformation observations, namely \(\delta _L \) (see Fig. 1a), on the basis of the down-sampled coherence map (for more details of the method of simulation, please see Tough et al. 1995; Hanssen 2001). Subsequently, we substituted the corresponding parameters of the Qianyingzi mining area, i.e., \({\alpha }_{h}=349.8^{\circ }\), \({\theta }=38.7^{\circ }\), \(R_{\mathrm{N}} \approx 6.5\;\hbox {m}\), \(R_{\mathrm{E}} \approx 6.5\;\hbox {m}\), \({\beta }=59.7^{\circ }\), \(b=0.32\), and \(H\approx 620\;\hbox {m}\) (Li et al. 2015b) into Eqs. (11)–(13) to estimate the covariances of the SIP-estimated 3-D displacements, respectively, based on the estimated theoretical STDs of LOS deformation.

Figures 1b–d depicts the STD maps of the SIP-estimated 3-D displacements in the vertical \({\varvec{\delta }} _W \), east \({\varvec{\delta }} _E \), and north \({\varvec{\delta }} _N \) directions. Figure 1f plots the comparison between the STDs of LOS deformation (red line) and the SIP-estimated 3-D displacements along the vertical (blue line), east (green line), and north (black line) directions along profile \(AA'\) (marked by the black solid line in Fig. 1a). As can be seen from Fig. 1, the errors in the LOS deformation observations are dramatically reduced in the SIP-estimated vertical subsidence, but they are significantly magnified in the SIP-estimated 2-D horizontal motions in the east and north directions.

Furthermore, the mean STD of the LOS deformation observations is about 6 mm, whereas it results in mean STDs of about 0.8, 8.3, and 15.8 mm in the SIP-estimated 3-D displacements along the vertical, east, and north directions, respectively. This again suggests that the errors in the LOS deformation observations are dramatically reduced in the SIP-estimated vertical subsidence, but significantly magnified in the SIP-estimated 2-D horizontal motions, particularly in the north direction. Therefore, it is essential to reduce the error propagation of LOS deformation to the SIP-estimated 3-D displacements, especially to the 2-D horizontal motions.
Fig. 1

a Theoretical STDs \({\delta }_{L}\) of the LOS deformation observations estimated from a real interferometric coherence map (e). bd Estimated STDs of the SIP-derived 3-D displacements in the vertical \({\delta }_{W}\), east \({\delta }_{E}\), and north \({\delta }_{N}\) directions, respectively, due to the theoretical STDs of LOS deformation observations (see a). f Comparisons between the STDs of the LOS deformation and the SIP-estimated 3-D displacements along profile \(AA'\) (the black solid line in a)

3 The SGI-based method for deriving time-series 3-D displacements

The SGI-based method first estimates the time-series vertical subsidence of the mining area using the same rationale as the SBAS-InSAR technique, as described in Berardino et al. (2002). The 2-D horizontal motions of the mining area along the east and north directions are then estimated based on the derived time series of vertical subsidence, using the proportional relationship between the horizontal motion and the gradient of vertical subsidence in the mining area.

3.1 Deriving time-series vertical subsidence of a mining area

We assume that there are \(K+1\) SAR images acquired from a single imaging geometry over the concerned mining area, whose acquired chronological times are \({{\varvec{t}}}=\left[ {t_0 , t_1 , \ldots , t_K } \right] ^{\mathrm{T}}\). M small-baseline InSAR pairs are formed from the \(K+1\) SAR images by setting a given threshold for the spatiotemporal baselines. Processing these M InSAR pairs with the SIP-based method (see Sect. 2.1), respectively, we obtain M multi-temporal observations of vertical subsidence \(\Delta {{\varvec{W}}}=\left[ {{\begin{array}{llll} {\Delta {{\varvec{W}}}_1 }&{} {\Delta {{\varvec{W}}}_2 }&{} \cdots &{} {\Delta {{\varvec{W}}}_M } \\ \end{array} }} \right] \) in the mining area (whose size is n rows and m columns).

We let \({{\varvec{V}}}=\left[ {{{\varvec{V}}}_1 ,\;{{\varvec{V}}}_2 ,\ldots ,{{\varvec{V}}}_K } \right] ^{\mathrm{T}}\) be the vector of the mean subsidence velocity between each two time-adjacent SAR acquisitions, and thus, we can obtain the following system for each pixel (Berardino et al. 2002):
$$\begin{aligned} {{\varvec{F}}}\cdot {{\varvec{V}}}=\Delta {{\varvec{W}}}+{{\varvec{\varepsilon }} }'_W \end{aligned}$$
(14)
where \({{\varvec{F}}}\) is a coefficient matrix depending on the formed small-baseline InSAR pair (see Berardino et al. 2002 for more details), and \({{\varvec{\varepsilon }} }'_W \) represents the error term of this system.
Fig. 2

Geographic location of the Datong mining area (marked by the black dashed line). The solid blue and dashed red rectangles, respectively, denote the footprints of the SAR data used for deriving the time-series 3-D displacements and performing the accuracy evaluation in this study

Due to the small-baseline constraint, the system (Eq. 14) is generally well- or over-determined (Berardino et al. 2002). Accordingly, the WLS solutions of the mean subsidence velocity between each two time-adjacent SAR acquisitions \(\hat{{{{\varvec{V}}}}}\) can be obtained by:
$$\begin{aligned} \hat{{{{\varvec{V}}}}}=\left( {{{\varvec{F}}}^{\mathrm{T}}\cdot {{\varvec{P}}}\cdot {{\varvec{F}}}} \right) ^{-1}\cdot {{\varvec{F}}}^{\mathrm{T}}\cdot {{\varvec{P}}}\cdot \Delta {{\varvec{W}}} \end{aligned}$$
(15)
where \({{\varvec{P}}}\) is the weighting matrix. In this study, the cubic of the coherence corresponding to these selected small-baseline InSAR pairs acts as the weighting matrix, i.e.,
$$\begin{aligned} P=\mathrm{diag}\left[ {{\begin{array}{llll} {{{\varvec{c}}}_1^3 }&{} {{{\varvec{c}}}_2^3 }&{} \cdots &{} {{{\varvec{c}}}_M^3 } \\ \end{array} }} \right] ^{\mathrm{T}} \end{aligned}$$
(16)
where \({{\varvec{c}}}_k \) \((k=1,2,\ldots ,M)\) represents the coherence map of the k-th InSAR pair. We discuss the potential of this weighting scheme in Sect. 5.2. Note that the minimum-norm WLS solutions of the unknown vector \(\hat{{{{\varvec{V}}}}}\) are estimated using the pseudoinverse of the coefficient matrix \({{\varvec{F}}}\) (calculated by singular value decomposition), if this coefficient matrix exhibits rank deficiency (Berardino et al. 2002).
Having obtained the mean subsidence velocity between each two time-adjacent SAR acquisitions \(\hat{{{{\varvec{V}}}}}\), the time series of vertical subsidence \(\hat{{{{\varvec{W}}}}}\left( t \right) =\left[ {{\begin{array}{llll} {\hat{{{{\varvec{W}}}}}\left( {t_1 } \right) }&{} {\hat{{{{\varvec{W}}}}}\left( {t_2 } \right) }&{} \cdots &{} {\hat{{{{\varvec{W}}}}}\left( {t_K } \right) } \\ \end{array} }} \right] \) of the mining area can be estimated by:
$$\begin{aligned} \hat{{{{\varvec{W}}}}}\left( {t_k } \right) =\sum _{l=1}^k {\left( {t_l -t_{l-1} } \right) \cdot \hat{{{{\varvec{V}}}}}_l } ,~~ (k=1,2,\ldots ,K) \end{aligned}$$
(17)
Fig. 3

Temporal and perpendicular baseline distributions of the remaining small-baseline InSAR pairs. The red points denote the acquisition times of the SAR images

3.2 Deriving time-series 2-D horizontal movements of a mining area

Having obtained the time series of mining-induced vertical subsidence \(\hat{{{{\varvec{W}}}}}\left( {{\varvec{t}}} \right) \), the time series of 2-D horizontal movements in the east \(\hat{{{{\varvec{E}}}}}\left( t \right) =\left[ {{\begin{array}{llll} {\hat{{{{\varvec{E}}}}}\left( {t_1 } \right) }&{} {\hat{{{{\varvec{E}}}}}\left( {t_2 } \right) }&{} \cdots &{} {\hat{{{{\varvec{E}}}}}\left( {t_K } \right) } \\ \end{array} }} \right] \) and north \(\hat{{{{\varvec{N}}}}}\left( t \right) =\left[ {{\begin{array}{llll} {\hat{{{{\varvec{N}}}}}\left( {t_1 } \right) }&{} {\hat{{{{\varvec{N}}}}}\left( {t_2 } \right) }&{} \cdots &{} {\hat{{{{\varvec{N}}}}}\left( {t_K } \right) } \\ \end{array} }} \right] \) directions are estimated using the proportional relationship between the horizontal motion and the gradient of vertical subsidence, i.e.,
$$\begin{aligned} \hat{{{{\varvec{E}}}}}\left( {t_k } \right)= & {} {{\varvec{C}}}\cdot \hat{{{{\varvec{W}}}}}\left( {t_k } \right) \end{aligned}$$
(18)
$$\begin{aligned} \hat{{{{\varvec{N}}}}}\left( {t_k } \right)= & {} {{{\varvec{C}}}}'\cdot \hat{{{{\varvec{W}}}}}\left( {t_k } \right) \end{aligned}$$
(19)
where \(k=1,2,\ldots ,K\), and \({{\varvec{C}}}\) and \({{{\varvec{C}}}}'\) are the same as in Eqs. (6) and (7).

It should be pointed out that the multi-temporal observations of 2-D horizontal motions in the east and north directions can also be generated from these small-baseline InSAR pairs using the SIP-based method. In fact, the time series of 2-D horizontal motions can also be estimated using the same rationale as the SBAS-InSAR technique based on their multi-temporal observations. However, in the SGI-based method, we estimate the time series of 2-D horizontal motions using the proportional relationship, in order to reduce the error propagation and the computational time.

4 Experiments and results

4.1 Study area and data processing

4.1.1 Study area

The Datong mining area (outlined by the black dashed line in Fig. 2), which is one of the biggest coal production bases of China, was chosen to test the proposed SGI-based method. The Datong mining area is located in the northwestern part of the city of Datong (see the black circle in Fig. 2), China. The large-scale and long-term underground extraction in the mining area has given rise to a series of environmental problems. For instance, the total area of mining subsidence basins was over \(450~\hbox {km}^{2}\) and more than 25759 surface buildings were damaged up to the end of 2001 (Hou and Zhang 2004). It is therefore critical to monitor the time-series 3-D deformations in this mining area, in order to understand the dynamics of the mining-induced deformation and assess the mining-related damage to infrastructure. To mitigate the potentially adverse effects caused by the interpolation error of the external digital elevation model (DEM) and the atmospheric artifact phase, we chose a study area (whose size is about 1.3 by 1.6 square kilometers) with a relatively flat terrain (see the red point in Fig. 2) in the Datong mining area.

4.1.2 SAR data and available InSAR Pairs

As the L-band PALSAR sensor performs well for monitoring mining-related ground displacements due to its long wavelength (about 23.6 cm) (Alex et al. 2009), we acquired seven ALOS PALSAR ascending SAR acquisitions (frame: 790, path: 454) spanning from July 1, 2007, to May 18, 2008, over the Datong mining area. The footprint of these SAR images is marked by the solid blue rectangle in Fig. 2. Subsequently, 15 InSAR pairs with a perpendicular baseline below 1500 m and a time interval of no more than 138 days were generated. The InSAR pairs with severe decorrelation were then excluded. Finally, 11 small-baseline InSAR pairs were retained. The temporal and perpendicular baselines of these InSAR pairs are plotted in Fig. 3.
Fig. 4

SGI-estimated time series of vertical subsidence with reference to July 1, 2007, which is rewrapped with an interval of 0.1 m, i.e., one color cycle corresponds to 0.1 m subsidence

Fig. 5

SGI-derived time series of the horizontal motion in the east direction with reference to July 1, 2007. Positive and minus values denote the points moving toward the east and west directions, respectively

Fig. 6

SGI-derived time series of the horizontal motion in the north direction with reference to July 1, 2007. Positive and minus values denote the points moving toward the north and south directions, respectively

4.1.3 Data processing

The two-pass D-InSAR procedure was utilized to process the 11 remaining InSAR pairs to generate 11 geocoded LOS deformation maps. In this procedure, a multi-look operation of 1:2 pixels in the range and azimuth directions was carried out, resulting in a resolution of about 6.01 and 6.28 m in the ground range and azimuth directions, respectively. To further suppress the noise in the interferograms, least-squares based filtering (Li et al. 2008) was performed. In addition, the topographic phases of the interferograms were removed using the 1 arc-second Shuttle Radar Topography Mission (SRTM) DEM. The minimum cost flow algorithm (Costantini 1998; Chen and Zebker 2000) was applied to unwrap the differential interferograms with the same unwrapping reference point. A biquadratic polynomial model was applied to mitigate the phase ramps due to possible orbit inaccuracies and atmospheric artifacts (Xu et al. 2014).

4.2 Estimation of time-series 3-D displacements

4.2.1 Generating multi-temporal observations of vertical subsidence

Prior to the generation of the multi-temporal observations of vertical subsidence using the SIP-based method, parameters \({\alpha }_{h}\), \({\theta }\), \(R_{\mathrm{N}} \), \(R_{\mathrm{E}} \), \({\beta }\), b, and H needed to be determined. According to the in situ surveyed geomining condition in the study area, the underground extraction was operated at a mean mining depth H of about 250 m (i.e., \(H\approx 250\;\hbox {m})\) between July 1, 2007, and May 18, 2008. The dip angle of the extracted working panel was about \(4^{\circ }\). Furthermore, the horizontal motion constant b and tangent of major influence angle tan \(\beta \) in the Datong mining area are around 0.3 and 1.8, respectively (i.e., \(b \approx 0.3\) and tan\({\beta }\approx 1.8\)) (Hou and Zhang 2004). The incidence angle and heading angle of the SAR data are \({\theta }=35^{\circ }\) and \(\alpha _h =349.8^{\circ }\), respectively. Furthermore, the spatial spacings of the geocoded LOS deformation maps are \(R_{\mathrm{E}} \approx 6.5\;\hbox {m}\) and \(R_{\mathrm{N}} \approx 6.5\;\hbox {m}\) in the east and north directions. Having acquired the values of these parameters, we applied the SIP-based method to process the 11 geocoded LOS deformation maps to generate the multi-temporal observations of vertical subsidence.

4.2.2 Estimating time-series 3-D displacements

For each pixel in the study area, 11 SIP-derived multi-temporal observations of vertical subsidence were obtained using the seven ascending ALOS PALSAR acquisitions. Furthermore, the six unknown subsidence velocities between time-adjacent SAR acquisitions needed to be estimated at each pixel. Consequently, these six unknown subsidence velocities were easily estimated with Eq. (15), because the constructed system between multi-temporal observations of vertical subsidence and the velocity vector of vertical subsidence between each two time-adjacent SAR images (i.e., Eq. 14) was over-determined. The time series of vertical subsidence in the study area with respect to the reference acquisition of July 1, 2007, was then estimated using Eq. (17). Figure 4 shows the SGI-derived time series of vertical subsidence, which is rewrapped with an interval of 0.1 m corresponding to a color cycle. Having obtained the time series of vertical subsidence, the time series of 2-D horizontal motions in the east and north directions was estimated with the proposed SGI-based method (see Sect. 3.2), and the results are shown in Figs. 5 and 6.

Since the working panel in the study area started extraction at the end of June 2007, and only a small distance was advanced from July 1 to August 16, 2007, the underground mining had only a slight influence on the ground surface, with maximum 3-D displacements of about \(-\,0.08\), 0.02, and 0.03 m in the vertical, east, and north directions (see Figs. 4a, 5a,  6a), due to the bridging function of the overlying rock mass (Kratzdch 1983). With the advancing of the underground mining (see the white arrow in Fig. 4a), the subsidence basin expanded extensively, and its maximum value reached \(-\,0.95\) m in the following 10 months (i.e., by May 16, 2008) (see Fig. 4f). Meanwhile, the maximum absolute horizontal displacement in the east and north directions increased up to 0.11 and 0.34 m, respectively, in this period (see Figs. 5f,  6f).

It is noted that the maximum horizontal motion in the north direction is about three times larger than that in the east direction (see Figs. 5f,  6f). This is most likely due to the following two reasons. Firstly, the accuracy of the estimated horizontal motions in the north direction is less than that in the east direction (see Sect. 2.2), and thus larger errors may be contained in the horizontal motions in the north direction than in the east direction. Secondly, the maximum value of the horizontal motions along the advancing direction of a long-wall working panel is usually smaller than that in its perpendicular direction (He et al. 2015; Peng et al. 1992). In the study area, the surface deformation was induced by mining an along-wall working panel in a roughly east direction (see the white arrow in Fig. 4a). Therefore, it is expected that the maximum horizontal motion in the north direction will be larger than that in the east direction.

4.3 Accuracy evaluation

To fully evaluate the accuracy of the SGI-derived time-series 3-D displacements, we should compare them with corresponding in situ geodetic measurements (e.g., by GPS or leveling). Unfortunately, such geodetic measurements are absent in the study area between August of 2007 and May of 2008. As an alternative, we selected two ascending ALOS PALSAR images from a different orbit (frame: 790, path: 453, whose footprint is marked by the red dashed rectangle in Fig. 2) to roughly evaluate the performance of the proposed SGI-based method. These two PALSAR images were acquired on December 15, 2007, and January 30, 2008. As a result of their small spatiotemporal baseline (i.e., 503 m and 46 days) and insignificant surface change in the winter, the InSAR pair generated by these two SAR acquisitions has very high coherence (with a mean coherence of about 0.85). This, to a large extent, ensures the reliability of the LOS deformation derived from this InSAR pair.

The SIP-based method was subsequently utilized to retrieve 3-D surface displacements from this formed single InSAR pair, and the results are shown in Fig. 7a, d, g (hereafter referred to as SIP-estimated 3-D displacements). We then interpolated the synchronous 3-D displacements between December 15, 2007, and January 30, 2008, with linear scaling from the SGI-estimated time-series 3-D displacements on October 01, 2007, January 01, 2008, and February 16, 2008 (hereafter referred to as SGI-interpolated 3-D displacements). The results are depicted in Fig. 7b, e, h. Figure 7c, f, i represents the differences between the SIP-estimated and the SGI-interpolated 3-D displacements.
Fig. 7

a, d, g Denote the SIP-estimated 3-D displacements between December 15, 2007, and January 30, 2008; b, e, h represent the SGI-interpolated synchronous 3-D displacements; and c, f, i depict the differences between the 3-D displacements derived by the two methods

Figure 7 shows that the SGI-interpolated 3-D displacements roughly agree with the SIP-derived ones in the time period from December 15, 2007, to January 30, 2008. In addition, it also shows that some pixels (e.g., those outside the red dashed circle in Fig. 7a) in the study area are almost unaffected by the underground extraction in this period, and thus, the differences between the SIP-estimated and the SGI-interpolated 3-D displacements at these pixels are nearly equal to zero. If these pixels were included in the dataset for accuracy evaluation, it would unfairly and unreasonably improve the accuracies of the SGI-derived 3-D displacements. Thus, we first masked out pixels with an absolute vertical subsidence of less than 0.02 m (i.e., pixels outside the red dashed circle in Fig. 7a) from the SIP-estimated 3-D displacements. Root-mean-square errors (RMSEs) of 0.023, 0.009, and 0.014 m between the SGI-interpolated and SIP-estimated displacements in the vertical, east, and north directions, respectively, were then obtained. These RMSEs equate to 6.6, 11.4, and 10.5% of the maximum SGI-estimated 3-D displacements in the vertical (0.351 m), east (0.078 m), and north (0.133 m) directions, respectively. This result indicates that the SGI-estimated time-series 3-D displacements are reliable.
Fig. 8

Interferometric coherence (a) and the LOS deformation (b) of the study area generated with the PALSAR images acquired on July 1, 2007, and January 1, 2008

It is noted that some large discrepancies emerge between the SGI-interpolated and SIP-estimated 3-D displacements in this period (e.g., the area marked by the black circle in Fig. 7c), with maximum values of 0.091, 0.082, and 0.165 m. In addition to the inherent model errors of the SGI-based and SIP-based methods, interpolation errors are most likely responsible for these large discrepancies. This is because mining-related deformation is characterized by high nonlinearity (Yang et al. 2017a), whereas the SGI-interpolated 3-D displacements for the accuracy evaluation were linearly interpolated.

5 Discussion

5.1 Comparison of the SIP- and SGI-derived 3-D displacements

Figure 8a, b shows the interferometric coherence and the generated LOS deformation of the InSAR pair formed by the used PALSAR images acquired on July 1, 2007, and January 1, 2008. Due to its large perpendicular baseline (about 875 m) and long temporal separation (184 days) (see Fig. 3), some isolated patches with sufficient coherence for phase unwrapping are surrounded by low-coherence areas (below 0.3 in this study). In this case, the unwrapped phase jumps are most likely to occur in these isolated patches (Costantini 1998; Chen and Zebker 2000). For instance, obvious deformation jumps can be seen in the two isolated patches marked by blue and red ellipses in Fig. 8b. It is believed that these jumps are caused by the unwrapping, and that they are not reliable.

To compare the degree of error propagation at these two isolated patches by the SIP-based and SGI-based methods, we first retrieved the 3-D displacements from the single InSAR pair generated with the PALSAR images acquired on July 1, 2007, and January 1, 2008. The results are shown in Fig. 9a, d, g. The SGI-derived time-series 3-D displacements between July 1, 2007, and January 1, 2008, are also presented in Fig. 9b, e, h for comparison. Fig. 9c, f, i shows the differences between the 3-D displacements derived by the two methods.
Fig. 9

Comparison of the SIP-estimated 3-D displacements in the vertical (a), east (d), and north directions (g) and those estimated by the SGI-based method in the corresponding directions (b, e, h) during the time period of July 1, 2007, to January 1, 2008; c, f, i represent the differences between the 3-D displacements derived by the two methods

Figure 9 shows that significant jumps occur at the two isolated patches and the surrounding pixels in the SIP-estimated 3-D displacements (see the red and blue ellipses), whereas these significant jumps do not emerge in the SGI-derived 3-D displacements. To further check the uncertainty propagation of the SIP-based and SGI-based methods, we selected profile \(BB'\) (marked by the black dashed line in Fig. 9) that crosses these two isolated patches, and calculated the SIP-derived and SGI-derived 3-D displacements during the time period of July 1, 2007, to January 1, 2008 (see Fig. 10). Clearly, the large uncertainty of the LOS deformation at the two isolated patches propagates into the SIP-derived vertical subsidence, resulting in two faulty subsidence troughs (see the blue line marked by the yellow-shaded areas in Fig. 10a). Subsequently, these two faulty subsidence troughs cause large errors in the SIP-estimated horizontal motions in the east and north directions, respectively (see the blue line in Fig. 10b, c). However, these two SIP-estimated faulty subsidence troughs do not occur in the SGI-estimated time series of vertical subsidence (see the red line in Fig. 10a), thereby significantly improving the accuracy of the 2-D horizontal motions compared with the SIP-based method. These results indicate that the SGI-based method performs better in reducing the error propagation of LOS deformation to 3-D displacements than the SIP-based method.
Fig. 10

Comparison of the 3-D displacements in the vertical (a), east (b), and north (c) directions derived by the SIP-based and SGI-based methods along profile BB\('\) (black dashed line in Fig. 9)

5.2 Influence of the weighting scheme on the estimation of time-series 3-D displacements

5.2.1 Theoretical derivation of the weighting scheme

The theoretical variance of the InSAR interferometric phase can be roughly estimated based on its coherence and multi-look number (see Tough et al. (1995) and Hanssen (2001) for more details). Figure 11a plots the theoretical variance \({\delta }_{\varphi }^{2}\) of the interferometric phase with respect to interferometric coherence for the multi-look numbers \(\hbox {ML}=2\) to 8. Generally speaking, the weight of an observation can be estimated by \({\delta }_{\varphi 0}^{2}/{\delta }_{\varphi }^{2}\), where \({\delta }_{\varphi 0}^{2}\) is the variance of the unit weight and \({\delta }_{\varphi }^{2}\) is the variance of the observation. Figure 11b plots the reciprocal of the theoretical variance \(1/{\delta }_{\varphi }^{2}\) with respect to interferometric coherence. Since the theoretical variance is close to zero when the coherence is approaching 1.0, the reciprocal of the theoretical variance then becomes infinite. To clearly present the results, we set the range of the coherence in Fig. 11b to [0, 0.8]. As can be seen, the reciprocal of the theoretical variance increases exponentially with the coherence. Therefore, coherence-based exponential functions may be a good weighting scheme for evaluating the LOS observations.

Figure 11c depicts the weights of nine exponential weighting schemes using the power of coherence. This shows that the weighting schemes of \(P=0.5\) and \(P=c\) are significantly different to the curves of the reciprocal of the theoretical variance with respect to coherence (see Fig. 11b). Furthermore, the weighting schemes using the power of 2 to 8 of the coherence (i.e., \(P=c^{2}\), \(c^{3}\), \(c^{4}\), \(c^{5}\), \(c^{6}\), \(c^{7}\), and \(c^{8})\) are approximately similar to the reciprocals of their theoretical variances. However, it can also be seen in Fig. 11c that the weighting schemes of \(P = c^{5}\), \(P=c^{6}\), \(P=c^{7}\), and \(P=c^{8}\) generate very small weights for the observations with medium coherence (e.g., from 0.3 to 0.6). This would greatly reduce the contribution of such observations (generally considered as reliable) to the time-series estimation of surface deformation. Therefore, the weighting schemes of \(P=c^{2}\), \(P=c^{3}\), and \(P=c^{4}\) would be a good compromise.
Fig. 11

a Theoretical variance of the interferometric phase with respect to coherence; b reciprocal of the theoretical variance with respect to coherence; c weight curves of the nine coherence-based weighting schemes

5.2.2 Influence of the weighting scheme on the estimation of time-series 3-D displacements

As analyzed previously, the weighting schemes of \(P=c^{2}\), \(P=c^{3}\), and \(P=c^{4}\) may be a good compromise compared with the other six schemes. To demonstrate the influence of these three weighting schemes on the SGI-derived 3-D time-series displacements, we replaced the weighting function in Eq. (19) with these three functions (i.e., \(P=c^{2}\), \(c^{3}\), and \(c^{4})\), respectively. Meanwhile, the equal weighting scheme of \(P=0.5\) was also selected to further evaluate the influence of the weighing scheme on the final results. The SGI-based method with these different weighting schemes was then applied to generate the time-series 3-D displacements, respectively.

As presented in Sect. 5.1, obvious deformation jumps occur in the two isolated patches of the LOS deformation map, as generated with the SAR images acquired on July 1, 2007, and January 1, 2008 (see the red and blue ellipses in Fig. 12b). We again selected the time-series 3-D displacements along profile \(BB'\) (black dashed line in Fig. 12b) during this period as an example to demonstrate the degree of error propagation under the different weighting schemes. The results are plotted in Fig. 12.
Fig. 12

SGI-estimated displacements in the vertical (a), east (b), and north (c) directions with the weighting schemes of \(P=0.5\), \(P=c^{2}\), \(P=c^{3}\), and \(P=c^{4}\)

From Fig. 12, we can see that the LOS deformation jumps in the two isolated patches (marked by the two ellipses in Fig. 12b) lead to large uncertainties in the SGI-estimated 3-D displacements (see the blue line) if the weighting scheme of \(P= 0.5\) is used. Compared with the results of the equal weighting (i.e., \(P=0.5\)), the scheme of \(P=c^{2}\) reduces the error propagation from the LOS deformation to the SGI-estimated vertical subsidence (see the magenta curves), e.g., at the locations marked by the light yellow boxes in Fig. 12a. On the other hand, some large gradients of vertical subsidence still occur (e.g., the area in the left light yellow box in Fig. 12a), resulting in large errors in the SGI-estimated 2-D horizontal movements (see the magenta curves in Fig. 12b, c). In addition, it can also be seen in Fig. 12 that the weighting schemes of \(P=c^{3}\) (red curves) and \(P=c^{4}\) (green curves) effectively reduce the error propagation from the LOS deformation to the SGI-estimated vertical subsidence, while no large uncertainties are triggered in their SGI-estimated 2-D horizontal displacements, compared with \(P=c^{2}\).

We now analyze the factors accounting for the different performances of the aforementioned weighting schemes. The seven available InSAR pairs were applied to estimate the mean velocities of the vertical subsidence between each two time-adjacent SAR acquisitions between July 1, 2007, and January 1, 2008 (see Fig. 12). However, due to the large jumps in the LOS deformation generated by the InSAR pair acquired on July 1, 2007, and January 1, 2008, in the two isolated patches, large uncertainties were propagated into the SIP-estimated vertical subsidence. In other words, the observation of vertical subsidence derived from this InSAR pair (hereafter referred to as the vertical subsidence observation of 20070601–20080101) has large errors. When the weighting scheme of \(P= 0.5\) is adopted, the seven multi-temporal observations of vertical subsidence for estimating the mean velocities at an arbitrary point are assigned the same weight (i.e., 0.5). In this case, the vertical subsidence observation of 20070601-20080101 occupies 14.7% (calculated by \(0.5/(7 \times 0.5))\) of all the weights of these seven multi-temporal observations. Since the vertical subsidence observation of 20070601–20080101 at the two isolated patches is overestimated by up to 0.1 m maximum (see Fig. 12a), large uncertainties will inevitably be propagated into the SGI-estimated 3-D displacements under the equal weighting scheme.

When the interferometric coherence-based weighting schemes (i.e., \(P=c^{2}\), \(P=c^{3}\) and \(P=c^{4})\) are adopted, the estimation of the final 3-D displacements can be improved. The mean and the maximum interferometric coherence of the InSAR pair of July 1, 2007, and January 1, 2008, at the two isolated patches are about 0.32 and 0.46, respectively. In contrast, the coherence of the other six InSAR pairs is around 0.83. Therefore, the mean weight of the vertical subsidence observation of 20070601-20080101 at these two isolated patches accounts for around 2.5% (calculated by \(0.32^{2}/(0.32^{2} + 6 \times 0.83^{2}))\) of all the weights of the seven multi-temporal observations. Compared with the equal weighting scheme (i.e., \(P = 0.5\)) of about 14.7%, \(P = c^{2 }\,(2.5\%)\) significantly reduces the error propagation from LOS deformation to vertical subsidence. Nevertheless, the large errors of up to 0.1 m also give rise to large uncertainties in the SGI-estimated 3-D displacements for the weighting scheme of \(P = c^{2}\), especially for the 2-D horizontal displacements (see Fig. 12). If the weighting schemes of \(P = c^{3}\) and \(c^{4}\) are used in the SGI-based method, the vertical subsidence observation of 20070601-20080101 at the two isolated patches occupies only about 0.9 (i.e., \(0.32^{3}/(0.32^{3}+ 6 \times 0.83^{3}))\) and 0.4% (i.e., \(0.32^{4}/(0.32^{4} + 6 \times 0.83 ^{4}))\) of all the weights of the seven multi-temporal observations. Such small weight proportions are capable of mitigating the error propagation from LOS deformation to SGI-estimated 3-D displacements. These results suggest that the weighting schemes of \(P = c^{3}\) and \(P = c^{4}\) are effective in reducing the error propagation.

In fact, the interferometric coherence in most mining areas is generally not very high, due to the decorrelation factors of surface cover variations, large deformation gradients, long spatial baselines, and so forth (Baran et al. 2005; Jiang et al. 2011; Zebker and Villasenor 1992). Generally speaking, large uncertainties occur in the InSAR-derived LOS deformation if the interferometric coherence is less than a certain threshold (i.e., 0.3) (Chen and Zebker 2000). As shown in Fig. 11c, the weighting scheme of \(P = c^{2}\) (green curve) assigns a larger weight to the observations than the weighting scheme of \(P = c^{3}\) (blue curve) when the coherence is below 0.3. This potentially increases the error propagation from LOS deformation to SGI-estimated time-series 3-D displacements. Moreover, the weight difference is small for low and moderate coherence (e.g., from 0.1 to 0.5) in terms of the weighting schemes of \(P = c^{3 }\) and \(P = c^{4}\). Therefore, large errors may also propagate into the estimated time-series 3-D displacements of the mining area where the interferometric coherence is not very high. To make a trade-off, the weighting scheme of \(P = c^{3}\) is a reasonable choice.

6 Conclusions

An approach for deriving time-series 3-D displacements of a mining area with a single-geometry InSAR dataset—the SGI-based method—has been proposed in this paper. Seven ascending ALOS PALSAR images spanning from July 1, 2007, to May 18, 2008, over the Datong mining area of China were selected to test the proposed SGI-based method. The results show that the maximum time-series 3-D displacements in the time period of SAR data acquisition reach 0.95, 0.11, and 0.34 m in the vertical, east, and north directions, respectively. In addition, the comparison between the SGI-derived time-series 3-D displacements and the synchronous ones generated by PALSAR data from a different orbit suggests that the accuracies of the SGI-derived 3-D displacements are about 0.023, 0.009, and 0.014 m in the vertical, east, and north directions, respectively. Such accuracies equate to about 6.6, 11.4, and 10.5% of the maximum SGI-estimated 3-D displacements in the vertical, east, and north directions, respectively.

The SGI-based method allows us to derive mining-related time-series 3-D displacements from a single-geometry InSAR dataset. Compared with the traditional MT-InSAR techniques for retrieving time-series 3-D displacements, in which InSAR data from at least three significantly different imaging geometries SAR are needed, the SGI-based method has less strict requirements for SAR data. Consequently, we believe that it will be a viable alternative technique for the monitoring of time-series 3-D displacements of mining areas. Furthermore, as a result of the redundant multi-temporal observations of LOS deformation and the effective weighting scheme in the SGI-based method, it significantly reduces the error propagation of LOS deformation to time-series 3-D displacements, compared with the previous SIP-based method.

However, we are aware that the proposed SGI-based method was only tested on one real case in this study. Moreover, the accuracy of the SGI-derived time-series 3-D displacements was not evaluated with in-site GPS or leveling measurements, but cross-validated with similar displacements estimated from two PALSAR images of a different orbit. Consequently, it could be said that the potentials and capabilities of the proposed SGI-based method were not convincingly validated and demonstrated. Therefore, in our future research, we will test the SGI-based method in different mining areas, and we will further assess its potential by comparing its results with in situ geodetic measurements.

Notes

Acknowledgements

The work was supported by the National Natural Science Foundation of China (Nos. 41474008, 41474007, and 41404013), the Hunan Provincial Natural Science Foundation of China (No. 13JJ1006), the Primary Research & Development Plan of Hunan Province (No. 2016SK2002), the Major Projects of High Resolution Earth Observation System of China (Civil Part) (No. 03-Y20A11-9001-15/16), the NASG Key Laboratory of Land Environment and Disaster Monitoring (No. LEDM2014B07), and the China Scholarship Council (No. 201506370139). The authors would also like to thank the Japan Aerospace Exploration Agency (JAXA) for providing the PALSAR images of the study area (Nos. 582 and 1390).

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.School of Geosciences and Info-PhysicsCentral South UniversityChangshaChina

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