## 5 Results
### 2D Results  
An example of SUHI Gaussian fit for the MODIS images of the August 19, 2009, considering the four daily passes of the satellite sensor, is shown in Figure 1, plotting the 39 km \(\times\) 39 km area around the Milan city center. Hours are reported in Central European Summer Time (CEST) date format, where CEST=UTC + 2, _i.e.,_ the local time of the city. For each image, the descriptive parameters for the Gaussian fit of the _SUHI_(\(x\),\(y\)) map are reported in Figure 1. The two spatial extent parameters \(a_{x}\) and \(a_{y}\) define the footprint of the SUHI pattern, with orientation \(\phi\) and center \(x_{0}\), \(y_{0}\): the blue inner ellipse (horizontal cross-section of the Gaussian surface) superimposed on the maps corresponds to A_61%, while the black outer ellipse represents the area A_1K.  
A correlation coefficient (_corr_, reported in Figure 1) was computed between the SUHI values of the image and the Gaussian surface used to fit the SUHI map.  
The four daily passes of the MODIS sensor allow an investigation of the heat island trend during the whole diurnal cycle. The employment of the proposed Gaussian fitting points out quantitatively the magnitude, extent, shape and location of the SUHI pattern and detects the variations during the day. For instance, it is evident the greater footprints and magnitude for the daytime images with respect to the nighttime ones, and the change of orientation. A statistical analysis of MODIS images acquired during different years and fitted using this method is reported in [6].  
Then, the SUHI map retrieved using the Landsat TM data acquired over Milan on August 6, 2009 at 12:00 CEST was processed, allowing to analyze the performance of the fitting procedure for a high resolution image on the same area (pixel 30 m). As in the previous figure, Figure 2 reports again the SUHI map with the Gaussian footprints and the correspondent descriptive parameters.  
Figure 1: Surface urban heat island (SUHI) [K] over Milan (August 19, 2009) using MODIS data (39 \(\times\) 39 pixels), with superimposed the horizontal cross-sections of the Gaussian surface. The descriptive parameters of the Gaussian fitting are also reported. Black pixels are cloudy pixels.  
Figure 2: SUHI [K] over Milan (August 6, 2009) using Landsat TM data (1307 \(\times\) 1307 pixels), with superimposed the horizontal cross-sections of the Gaussian surface. The descriptive parameters of the Gaussian fitting are also reported.  
As expected, the impact of the higher spatial resolution on the fitting representation of the SUHI pattern shows some differences with respect to the lower resolution, such as the growth of the extent parameters and magnitude \(a_{0}\), and therefore of A_61% and A_1K, and a reduction of the correlation coefficient. This is mainly due to the highest temperature values located in the urbanized areas that are preserved inside a pixel of 30 m, while they are smoothed in the averaging process inside a 1 km pixel. Also, clear differences emerge in the detail definition of the SUHI pattern between the two figures, pointing out the different ability to resolve sharp thermal discontinuities in a heterogeneous area. In particular, since the Gaussian fitting produces a smoothed surface, the higher variability of the pixel values for the Landsat TM image leads to a decrease of the correlation coefficient, even if the parameterization preserves the representativeness of the heat island pattern for the Milan area. It is worth to note how the different range of variation of the SUHI values, with higher values for the Landsat TM image, induces an increase of the spatial parameters with respect to the ones of the MODIS image. These aspects are confirmed in the next sub-section.