## 3 Methodology
### Supervised Classification  
Supervised classification is a training based methodology that classifies similar image pixel values into training samples for a determined number of classes. Thus, training samples must be selected based on a homogenous group of image pixels to provide the best separability. Therefore, monitoring the study area and assessing the different land covers is necessary before selecting the training samples. Additionally, applying the appropriate algorithm to identify the homogeneity of the training data to group the pixel values of a dataset is important. Accordingly, the training data selection process and the two different supervised classification algorithms used will be explained further in this section. After the inspection of district 22 in Tehran, five land cover classes were defined: bare land, vegetation, built-up 1, built-up 2, and built-up 3. The three different residential types are shown in Figure 3. Built-up 1 is composed of dense residential areas with mostly two-story old buildings and narrow streets and roads. The buildings are approximately 8 \(\times\) 13 m in size. Built-up 2 is composed of medium density residential areas, including parallel blocks of buildings. The buildings are mostly 12 \(\times\) 18 m in size. These buildings consist of approximately 4-6 floors. Built-up 3 includes individual building with 4-15 floors. Built-up 3 has wider streets than the other areas and plenty of vegetation surrounding the buildings. The buildings are mostly two different shapes, square with a size of 25 \(\times\) 25 m or rectangular with a size of approximately 25 \(\times\) 44 m. For each of these five land cover classes, three training samples were selected. The training samples mainly consist of square shapes of 200 m in length, which are located in different parts of Tehran, as shown in Figure 4.  
The classification process was performed using two algorithms: support vector machine (SVM) and maximum likelihood (ML). Both algorithms are derived from statistical theories. These two methods are commonly used in land cover classification studies [37; 38; 39; 40; 41]; therefore, we intend to evaluate their performance for SAR data imagery. Supervised classifiers were applied to three sets of data: (1) Landsat 8 multi-spectral image; (2) Backscatter values of dual-polarization ALOS-2 and Sentinel-1 data; and (3) the layer stacking of backscatter values and the first three PCTs from ALOS-2 and Sentinel-1 data. Figure 3: Three different residential types (**a**–**c**) extracted from Google Earth images of Tehran city. The maximum likelihood method works based on the assumption that each class is normally distributed. Thus, for each pixel, the probability that it belongs to a specific class is calculated. Then, the pixel is assigned to the class that yields the highest probability as follows [36; 42]:  
\[g_{i}(x)=1 np(\omega_{i})-\
icefrac{{1}}{{2}}1n\Big{|}\sum_{i}\Big{|}-\
icefrac{{1}} {{2}}(x-m_{i})^{T}\!\!\sum_{i}^{-1}(x-m_{i}) \tag{10}\]  
where \(i\) is the number of classes, \(x\) represents the \(n\)-dimensional data (where \(n\) is the number of bands), \(p(\omega_{i})\) denotes the probability that class \(\omega_{i}\) occurs in the image and is assumed to be the same for all classes, \(\left|\Sigma_{i}\right|\) is the determinate of the covariance matrix of the data in class \(\omega_{i}\), \(\Sigma_{i}{}^{-1}\) is its inverse matrix, and \(m_{i}\) represents the mean vector. Support vector machine is a trusted algorithm often used in remote sensing [43; 44]. This algorithm was developed by [PERSON] [45], and its use has increased for land cover classification in recent years. The SVM separates the pixels of an image using optimal hyperplanes that maximize the margin between the classes [46]. The data points closest to a hyperplane are called support vectors. A nonlinear classification can be performed using kernel functions to the support vectors. In this study, the pairwise strategy is used for multi-class classification.