# A New Method for Large-Scale Landslide Classification from Satellite Radar
## 2 Materials and Methods  
### Theory: Landslide Detection with SAR  
#### 2.1.1 Synthetic Aperture Radar: Interferometry and Coherence  
Two SAR images acquired by the same satellite system and covering the same area at different times may be combined to form a radar interferogram (e.g., Figure 3a) [31]. To acquire each image, microwave radiation is emitted by the satellite's antenna, back-scattered from the Earth's surface, and recorded again by the satellite as two components: amplitude and phase. An interferogram, such as the one shown in Figure 3, is a map of the difference in phase, \(\Delta\varphi\), between the two images; the repeated bands of colour can be considered as contours of the change in distance between the satellite and the imaged surface. Figure 3 shows an example pre-event interferogram before the 2015 Nepal earthquake, i.e., an interferogram made from two pre-event SAR images. The coherence of a pixel is a measure of the signal to noise ratio of a point. A coherence map can be estimated from two SAR images to assess the spatial coherence in \(\Delta\varphi\) that is shown in the interferogram. SAR coherence, \(\gamma\), is estimated as the correlation between \(\Delta\varphi\) of closely-grouped pixels and can be calculated pixel-by-pixel for a pair of images (denoted \(A\) and \(B\)) from an ensemble of \(n\) pixels using the following equation [32]. \[\gamma=\frac{\frac{1}{n}\left|\sum_{i=1}^{n}A_{i}\cdot\overline{B}_{i}\right|} {\sqrt{\frac{1}{n}(\sum_{i=1}^{n}A_{i}\cdot\overline{A}_{i}\sum_{i=1}^{n}B_{i} \cdot\overline{B}_{i})}} \tag{1}\]  
Figure 2: Timeline showing satellite image acquisition and product release following the 2015 Gorkha earthquake for optical satellite imagery and for SAR. Modified from [9]. Timeline of NASA Advanced Rapid Imaging and Analysis (ARIA) products is taken from [21]. Optical imagery and products are in blue with maps produced by Durham University (DU) and collated maps coordinated with international partners (International Centre for Integrated Mountain Development (ICIMOD), MacDonald, Dettwiler and Associates Ltd, (MDA) and the National Geospatial-intelligence Agency (NGA)) by the British Geological Survey. SAR imagery and products are in green and include the Sentinel-1 (S-1) imagery used in this study and the COSMO SkyMed (CSK) and ALOS-2 imagery of Kathmandu used by the ARIA project. Here, \(i\) is an individual pixel and \(A_{i}\) and \(B_{i}\) are complex representations of phase and amplitude for images \(A\) and \(B\) respectively. The overline indicates the complex conjugate. The factor of \(1/n\) is included here as a scaling factor in order to account for variations between pixels in the size of the ensemble, required for Section 2.1.4. In the widely used 'boxcar' coherence estimation, the ensemble is defined as a square box of pixels centred on the pixel in question. Pixels within the box will sum constructively if they have similar \(\Delta\varphi\) and destructively if not. Examples of areas with low and high estimated coherence are shown in Figure 3b,c respectively. The overall coherence of a pixel is determined by several factors, often illustrated by its decomposition into three components:  
\[\gamma_{total}=\gamma_{temporal}\cdot\gamma_{spatial}\cdot\gamma_{thermal} \tag{2}\]  
where \(\gamma_{\text{temporal}}\) is temporal coherence, \(\gamma_{\text{spatial}}\) is spatial coherence, and \(\gamma_{\text{thermal}}\) is thermal coherence [34]. Decorrelation of any one of these components will lead to decorrelation of the signal as a whole. Thermal coherence is related to noise within the signal, and its decorrelation is generally insignificant [34]. Spatial (or geometric) coherence is dependent on the topography of the target region, the imaging geometry of the satellite, the radar wavelength and the bandwidth of the radar sensor. The temporal component is dependent on the change in scattering properties of a target pixel, which in turn is dependent on modification to the ground surface.