## Materials and Methods
### The k-means/k-means++ Clustering Algorithm
On 24 April 2013, the reduction in the value of \(\mathrm{K}\) from 7 to 4 leads to the transfer of some of the pixels in the original cluster 2 to the new cluster 1, the rest of the pixels in the original cluster 2, the original cluster 3 and a small portion of the original cluster 4 are amalgamated into the new cluster 2, most of the contributions to the new cluster 3 comes from the amalgamation of the original clusters 4 and 5, and the original clusters 6 and 7 are amalgamated into the new cluster 4 with contributions from the original cluster 5 as well. After consolidation cluster 1 remains the less populated cluster and the new cluster 3 is the more populated cluster with half the total number of pixels. The new cluster 4, which includes the largest positive anomalies, represent more than a third of the total number of pixels. Figure 2: Distribution of the optimal number of clusters across the image dataset. Figure 4: Input image (**a**) and clusters 1–4 (**b–e**) on 4 July 2011. The color scale is the same for all the panels. Figure 3: Input image (**a**) and clusters 1–3 (**b–d**) on 4 July 2011. The color scale is the same for all the panels. Figure 5: Input image (**a**) and clusters 1–7 (**b–h**) on 24 April 2013. The color scale is the same for all the panels. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Cluster \#**} & \multicolumn{4}{c}{**Date**} \\ \cline{2-5}  & \multicolumn{2}{c}{**4 July 2011**} & \multicolumn{2}{c}{**24 April 2013**} \\ \hline
**1** & 465,021 (9.22\%) & 311,922 (6.18\%) & 86,072 (1.71\%) & 182,495 (3.62\%) \\
**2** & 1,813,436 (35.94\%) & 748,304 (14.83\%) & 228,091 (4.52\%) & 622,275 (12.33\%) \\
**3** & 2,767,078 (54.84\%) & 2,984,436 (59.15\%) & 405,573 (8.04\%) & 2,523,610 (50.02\%) \\
**4** & & 1,000,873 (19.84\%) & 1,092,484 (21.65\%) & 1,272,155 (34.03\%) \\
**5** & & & 1,549,494 (30.71\%) & \\
**6** & & & 1,175,199 (23.29\%) & \\
**7** & & & 508,622 (10.08\%) & \\ Total & 5,045,535 (100.00\%) & 5,045,535 (100.00\%) & 5,045,535 (100.00\%) & 50,455,535 (100.00\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Number of pixels included in each cluster for two selected dates that have the optimal number of clusters not coinciding with the fixed value of 4 used in the paper. Similar figures for the number of clusters set to 4 are also shown. The representativeness of each individual cluster within its group is presented in round brackets and it is estimated through the percentage over the total number of pixels in each image. Figure 6: Input image (**a**) and clusters 1–4 (**b–e**) on 24 April 2013. The color scale is the same for all the panels. The numbers in Table 1 were used to estimate the information entropy [63] as follows  
\[S=-\sum_{i}p_{i}\ln\left(p_{i}\right) \tag{7}\]  
In (7) \(p_{i}\) stands for the \"probability\" of each cluster--estimated as the ratio between the number of pixels in the cluster and the total number of pixels in the input image. Given that the lower the value of the entropy the higher the information provided relation (7) can be used as an absolute value for comparisons between cluster sets regardless of the value of \(K\). With the aid of Table 1 the calculation of the information entropy on 4 July 2011 gave 0.92 and 1.09 for \(K=3\) and \(K=4\), respectively. In other words, a fewer number of clusters as estimated by the VRC gave more information. On the other hand, the value of the entropy on 24 April 2013 was 1.68 and 1.09 for \(K=7\) and \(K=4\), respectively. This particular case stresses that an optimal number of clusters does not necessarily imply the best distribution of information across them.