## Materials and Methods
### The k-means/k-means++ Clustering Algorithm
CHC establishes a variance ratio criterion (VRC) from which the optimal number of clusters K is estimated using the following quotient  
\[\mathrm{VRC=\frac{BGSS}{WOSS}\ \frac{p-K}{K-1}} \tag{6}\]  
In (6) K represents the number of clusters, p is the number of pixels in the image, BGSS is the between-group (i.e., inter-cluster) sum of squares and WGSS is the within-group (i.e., intra-cluster) sum of squares. BGSS is made up from the summation of all the possible combinations of the squared distances calculated between two points in different clusters. The computation of WGSS involves the summation of all the possible squared distances in each individual cluster and then adding them up for all the K clusters. Since K is an unknown parameter the VRC was calculated using nine different values from \(K=2\) to \(K=10\) and the one that lead to the maximization of the VRC value was adopted. Maximization requires BGSS (WGSS) to be as greater (smaller) as possible in order to get a better separation of the groups. Since the optimal value of K is image-specific it was found to fluctuate in the 3-7 range for the 2036 input images. Figure 2 shows the way K distributes across the image dataset. There is a peak at \(K=4\) (36% of the cases) followed by \(K=3\) (28% of the cases). In order to use a single value of K the nearest integer from the average over the entire set of input images was established; this value turned out to be \(K=4\). A validation method using bootstrapping was carried out by randomly choosing pixels from the different input images in order to determine a single value of K for the resulting subsample image. This process was replicated several times and the outcome was also \(K=4\) on average. Examples on two different dates whose input images have \(\mathrm{K}=3\) (4 July 2011) and \(\mathrm{K}=7\) (24 April 2013), i.e., the lowest and highest optimal values, respectively, are presented in order to confront their cluster sets with those for \(\mathrm{K}=4\). Individual cluster labeling was carried out over the mean value in an ascending fashion so that cluster 1 has the lowest mean in all sets. Figure 3 shows the input image and clusters 1-3 on 4 July 2011; Figure 4 shows the input image and clusters 1-4 on the same date. Similarly, Figure 5 shows the input image and clusters 1-7 on 24 April 2013 and Figure 6 shows the input image and clusters 1-4 on the same date. Table 1 shows the number of pixels in either set along with the percentage of pixels over the total on these two dates. The only commonality on either date is that the lowest (highest) order cluster includes the coldest (warmest) pixels. The change in the value of \(\mathrm{K}\) yields a reassignment of pixels between the original and the new clusters. A split will occur when the value of \(\mathrm{K}\) increases whereas a consolidation will take place when \(\mathrm{K}\) decreases. On 4 July 2011, the rise in the value of \(\mathrm{K}\) from 3 to 4 leads to a mild decrease in the number of pixels that populated the original cluster 1 that nets in a gain of pixels for the new cluster 2, the original cluster 2 is partly split between the new clusters 2 and 3, and the original cluster 3 is partly split between the new clusters 3 and 4. Initially, cluster 3 was the most populated cluster, including more than a half of the total number of pixels and representing the largest positive standardized anomalies. Following the splitting the most populated of the new clusters is cluster 3, and the largest positive standardized anomalies in cluster 4 account for a fifth of the total number of pixels. On the other hand, cluster 1 remains the less populated cluster after splitting.