## 3 Results  
### Comparison of \(\alpha\) Estimated with the Different LST CDRs  
The optimal values of parameter \(\alpha\) estimated using the three LST datasets are shown in Figure 1. These correspond to the estimate of VHI that leads to the best correlation with the relevant SPEI (scale also shown in Figure 1). Only pixels with statistically significant correlations (\(p<0.1\)) are shown. These are confined to Mediterranean Europe, Southern Africa, North-East Brazil, and a few scattered regions in the Sahel and sub-Saharan Africa. As already pointed out, areas characterized by desert/very arid climates (Sahara, Arabian, and Namib Deserts and the eastern part of the horn of Africa) were previously masked out. Forests in central Africa and parts of Europe are characterized by non-significant correlations between VHI and SPEI (Figure 1) [13]. The patterns of both \(\alpha\) and SPEI time-scale are very similar among the three datasets. Most of the pixels show high values of \(\alpha\), while lower values are confined to relatively small clustered regions (e.g., in east and south Africa and East and Northeast Brazil). Concerning SPEI time-scales, the lower range of 1- and 3-month dominates over most of the Meteosat disk. Nevertheless, longer time-scales are organized in small clusters scattered throughout the region. It is worth noting that these features are present in the maps derived from the three LST datasets. The magnitude and spatial distribution of \(\alpha\) are coherent among these datasets, when either the LST datasets were derived from the same observations with different algorithms (the case of CM-SAF LST CDRs) or from different observations and methodologies (PU and CM-SAF LST CDRs). Figure 1: Parameter \(\alpha\) (**top**) estimated by iterative maximization of Vegetation Health Index (VHI) and Standardized Precipitation-Evapotranspiration Index (SPEI), and respective SPEI time-scale (**bottom**) for which VHI and SPEI have the largest correlation, using Thermal Condition Index (TCI) computed with Princeton University Land Surface Temperature (PU LST) (**left**), Satellite Application Facility on Climate Monitoring (CM-SAF) Physical Mono-Window (PMW) LST (**middle**), and CM-SAF Statistical Mono-Window (SMW) LST (**right**). For a closer look, Figure 2 presents a comparison among values of parameter \(\alpha\) as estimated with the three LST CDRs (PU LST, CM-SAF PMW LST, and CM-SAF SMW LST). The left panels compare the distributions of \(\alpha\) estimated with CM-SAF PMW LST versus PU LST (top), with CM-SAF SMW LST versus PU LST (middle), and CM-SAF PMW versus CM-SAF SMW (bottom). Right panels show histograms of the differences between the same pairs of variables. Results point out that values of \(\alpha\) are well distributed along the diagonal (solid black line) in all three cases. Furthermore, the agreement between \(\alpha\) estimated with CM-SAF datasets is higher, as expected, since these correspond to LST derived from the same observations--both clear-sky only. Values of Root Mean Squared Deviation (RMSD) of \(\alpha\) derived with PU LST and clear-sky PMW and SMW LSTs are of about 0.2. Furthermore, it is worth noticing that the differences between \(\alpha\) estimated with CM-SAF LSTs are smaller, with a RMSD of about 0.1. The average differences of \(\alpha\) estimated with the different LST datasets are negligible, meaning that the use of these datasets is not introducing any overall systematic error in this type of analysis. Figure 3 shows the differences \(\Delta\rho\) between correlations of VHI and SPEI when \(\alpha\) is estimated with the iterative maximisation (\(\rho_{\text{it}}\)) method and when \(\alpha\) is taken as 0.5 (\(\rho_{0.5}\)):  
\[\Delta\rho=\rho_{\text{it}}-\rho_{0.5}, \tag{4}\]  
Since the iterative procedure is designed to obtain the maximum value of correlation, \(\Delta\rho\) is always greater to or equal than 0. It is worth noting that regions where \(\Delta\rho\) is larger match those where values of \(\alpha\) depart more from 0.5.