## 3 Data and Methods
### Estimation of the SUHI  
The SUHI intensity quantifies the heating effects in an urban area, and we computed it as the surface temperature difference between urban (LST\({}_{\text{URB}}\)) and rural (LST\({}_{\text{RUR}}\)) pixels:  
\[\text{SUHI}=\text{LST}_{\text{URB}}-\text{LST}_{\text{RUR}}. \tag{1}\]  
The reference LST\({}_{\text{RUR}}\) represents the mean of LST values associated with the rural pixels in a buffer area, surrounding the urban area, of about 3 km. In such buffer rural areas, water pixels were discarded. The choice of the 3 km buffer is a trade-off that allows, for all the cities, to discard high-altitude rural zones (e.g., Chiang Mai forest in Figure 2) that would lower the LST rural bias. For the computation of Equation (1), to discern built-up and rural pixels at the LST spatial resolution (1 km), urban and rural categories were extracted from the MCD12Q1 product. Figure 3 shows an example of these reference maps for the six cities for 2013, resampled at 1 km pixel size from the native 500 m, used for the computation of the reference LST\({}_{\text{RUR}}\). The thermal characterization of an urban area by means of SUHI maps is also appropriate for comparative analysis among different cities because SUHI intensity is not an absolute temperature, but a temperature difference between rural and urban areas. Therefore, the effects of local weather conditions and other error sources are dampened. To perform SUHI pattern comparisons, a technique based on a two-dimensional least-square Gaussian fit of the heat island was applied. This SUHI fitting technique was proposed in [34] and applied in [10; 13; 27; 35; 36; 37]. The 2D Gaussian surface can be a useful representation of the SUHI pattern when an urban area shows a clear heat island with respect to the rural surroundings, especially if clustered SUHI values are prone to decrease towards the map boundary. The Gaussian fitting of the spatially distributed SUHI(x, y) (\({}^{\circ}\)C) obtained from MODIS data is described as follows:  
\[\text{SUHI(x,y)}=\text{a}_{0}\times\exp\left[-\frac{\left(\left(\text{x}- \text{x}_{0}\right)\cos\phi+\left(\text{y}-\text{y}_{0}\right)\sin\phi\right)^ {2}}{0.5\text{a}_{2}^{2}}-\frac{\left(\left(\text{y}-\text{y}_{0}\right)\cos \phi-\left(\text{x}-\text{x}_{0}\right)\sin\phi\right)^{2}}{0.5\text{a}_{2}^{2 }}\right], \tag{2}\]  
where (x, y) (km) is the pixel location in the map, with the origin (0, 0) in the center of the map. From Equation (2), a nonlinear data-fitting in least-squares sense was performed [35] in order to estimate:  
* the magnitude a0 (\({}^{\circ}\)C) of the SUHI maximum intensity of the study area;
* the spatial extent (a\({}_{\text{x}}\), a\({}_{\text{y}}\)) (km), central location (x\({}_{0}\), y\({}_{0}\)) (km), and orientation (\(\varphi\)) (degree) of the SUHI pattern. A correlation coefficient (r) between the SUHI map from MODIS data and the fitted Gaussian surface was computed to assess the capability of the fitting surface to follow the actual pattern. The two parameters, a\({}_{\text{x}}\) and a\({}_{\text{y}}\) (spatial extent), were also combined to define the overall area of the SUHI. Such an area or footprint is an ellipse obtained by horizontally cross-sectioning the Gaussian surface. The ellipse axes are a\({}_{\text{x}}\) and a\({}_{\text{y}}\), \(\varphi\) is the orientation angle, with the maximum SUHI value located in the footprint center \(\text{x}_{0}\), \(y_{0}\). Figure 3: Urban and rural category maps from MODIS MCD12Q1 product of 2013 for the six cities (1 km pixel size) used for the computation of the reference rural land surface temperature (LST\({}_{\text{RUR}}\)) (rural buffer). The ellipse area defined by the distance from the center, where SUHI decreases to 61% (e\({}^{-1/2}\)) with respect to the maximum value a0, will be defined as Ar61% (km\({}^{2}\)). Similarly, a footprint area where SUHI is greater than a fixed threshold can be defined--for example, greater than 1 \({}^{\circ}\)C (Ar1C) [10; 34].