# Mining Deformation Life Cycle in the Light of InSAR and Deformation Models
## Materials and Methods
### Sentinel-1 Data and DInSAR Methodology
#### 2.1.5 Geospatial and Statistical Analyses  
The vertical subsidence retrieved from the DInSAR processing was stacked monthly (for time aggregation details, see Section 2.3.1) and was smoothed based on the convolution procedure from the Scientific Computing Tools for Python (SciPy) module [44]. Each pixel of the image (subsidence raster) \(I_{i}\) was multiplied by a weights kernel \(W_{j}\) to produce the output pixel value \(C_{i}\):  
\[C_{i}=\ \sum_{j}I_{i+k-j}W_{j}, \tag{3}\]  
where \(k\) is a coordinate of kernel \(W\) origin. In this case, we used the 5 \(\times\) 5 normalised box blur kernel. The following process produces a new raster, where each pixel value is calculated as an average value from the pixel value and its neighbouring pixels from the original image. The nearest border pixels were  
Figure 5: An example of trend removal for the descending orbit, displacements between 2 nd and 8 th of January 2017: Original data (**left**), trend function (**middle**), and smoothed data (**right**). The example is the local (image) coordinate system; deformation values in [m]. Please note different deformation scales in the original and smoothed images. calculated using an extended neighbourhood in places where the kernel required values outside the image boundaries. At the next step, rasters were reclassified to the binary image (0 for no subsidence and 1 for subsidence), and a polygonisation algorithm was run to build the polygons. This Geospatial Data Abstraction software Library (GDAL)-based algorithm [45] creates vector polygons for all connected regions of pixels sharing a common pixel value in the raster. Further processing was based only on the following polygon statistics: (1) Area affected by significant subsidence, and (2) location and value for the maximum monthly subsidence. The described procedure was also applied to the vertical displacement grids created based on subsidence modelling (Section 2.2), and the comparison between the observation and forecast results is presented in Section 4. The extent of the deformation zone for the two sources was spatially observed, and the overlapped (O) area (\(A_{O}\)) for each month was compared with the original area (\(A_{D}\)), which was delineated from DInSAR (D) and from the modelled (M) data (\(A_{M}\)):  
\[A_{O/D}=\frac{A_{O}}{A_{D}}*100\%\text{ and }A_{O/D}=\frac{A_{O}}{A_{D}}*100\%. \tag{4}\]  
The results represent the deviation between the subsidence extent derived from the two types of data. We used the value of the cumulative maximum subsidence \(d_{w}\) (the subsidence corresponds to the vertical component \(d_{U}\) of \(d_{LOS}\); Equation (2)) within the affected area for each month \(d_{w}(month)\) to estimate the maximum subsidence velocity \(v_{w}\) in millimetres per day:  
\[v_{w}=\frac{d_{w}(month)}{dt}, \tag{5}\]  
where \(dt\) is the number of days for the current month based on the time span of the interferograms included in the stack, which may vary if some images are missing and will result in 12-day interferograms. To indicate the possible non-linear motion in the motion, the acceleration is also calculated as follows:  
\[a_{w}=\frac{dv_{w}}{dt}, \tag{6}\]  
where \(dv_{w}\) is the change in of the vertical velocity between every two successive months. The results from our study were validated with geodetic in-situ data (Section 2.3.2). A quantitative assessment of the satellite radar measurements and modelling compared to the levelling data as ground truth was conducted using a statistical approach for the extracted values from DInSAR and the model grids for the locations of the levelling benchmarks.