# Mining Deformation Life Cycle in the Light of InSAR and Deformation Models
## 1 Introduction
### InSAR Deformations  
The Interferometric SAR (InSAR) technique uses two radar images generated at the same time from different positions or from almost the same position but in two separate moments in order to generate an interferogram of terrain elevation. The deformation of the Earth's surface can be observed by a comparison of two interferograms that comprise the time span of terrain change. The technique isknown as Differential InSAR (DInSAR) and was demonstrated for the first time in 1993 in the domain of terrain deformation [4]. With time, different advanced DInSAR techniques are proposed to tackle the problems of coherence loss and to decrease the atmospheric influence on the radar signal. Most of these techniques are based on the concept of Permanent Scatterer Interferometry (PSI), originally created by [PERSON] et al. [9; 10] like PSInSAR, which was later extended to a new algorithm called SqueeSAR [11]. Different scientific groups developed similar algorithms, like the Small BAseline Subset (SBAS) [12], the improved algorithm for non-urbanised areas StaMPS [13], and others. It is important to mention that most of the PSI approaches assume a linear deformation model for subsidence estimation, which has limited use in longwall mining deformation analyses [14]. [PERSON] et al. [15] pointed out that PSI techniques often experience gaps in the deformation maps due to a misfit between the linear model and nonlinear observed deformations. In the case of mining, subsidence may have a different rate depending on the applied extraction technology. According to [16] (in [17]), in the case of longwall extraction technology, terrain subsidence evolves in three stages: Slow motions in the range of a few millimetres per day in the first six to eight months followed by faster movements in the range of 1 cm per day in the next six to 12 months. After this period, the rate slows to a value of almost zero values. These phases are also known as initial subsidence, main subsidence, and residual subsidence [18]. On the other hand, the shape and progress of the subsidence trough strongly depends on the position, length, and thickness of the layer, the panel width, the dip of the strata of the mining front, the mining extraction rate, the roof control technique, and the geological structure [14; 17]. Despite the expansion of the list of advanced algorithms that may provide more precise results regarding the slow motions at the edges of the subsidence trough, the application of PSI has lower feasibility due to limitations in the capability to measure fast surface deformations [15]. Ambiguity in the unwrapping procedure results in the limitation of the maximum differential deformation rate between two neighbouring permanent scatterers that can be measured by PSI. This value is limited to \(\lambda/4\) over the revisiting time between two consecutive images, where \(\lambda\) is the wavelength of the radar signal. The constellation of the two latest European Space Agency (ESA) radar satellites, Sentinel-1A and 1B, forms a revisit period of six days for the C-band sensors with a wavelength equal to 5.6 cm, which leads to a theoretical maximum deformation rate of 85 cm/year detectable by PSI. The maximum detectable rate of the subsidence trough is dependent also on the special resolution of the radar image [19]. [PERSON] et al. [14] suggested the consecutive DInSAR method with a shorter temporal baseline as the most appropriate for detection of fast terrain motions. A simulated test of the capabilities of the different operating SAR satellites [19] demonstrates that shorter temporal baselines in combination with a higher spatial resolution will determine a more precise subsidence peak. A study that compared the PSI and DInSAR results in the location of the current investigation, namely the Upper Silesian Coal Basin (USCB) in Poland, was presented [20]. This work showed that the SqueeSAR algorithm is not applicable for detection of motions faster than 33 cm/year.