## 2 Basic Theory  
The temperature assessment of surfaces by using thermal imagery involves the modelling of the radiative properties of the surfaces and of their interactions with the surrounding environment [12, 13]. In this section, a brief summary of the model is presented. Under the formalism adopted here, small letters indicate the spectral quantities, which are functions of the wavelength, while the broadband quantities, integrated or averaged over the spectral range to which the thermal camera is sensitive, are designated by capital letters. The investigated surfaces emit thermal radiation (\(l_{B}\)) depending on their temperature, according to [PERSON]'s law. As real surfaces are not black-bodies, their radiative behavior is controlled by the emissivity (\(\epsilon\)), which is, in general, a function of the wavelength and of the observing angle. The emissivity is generally lower than one; then, surfaces also reflect part of the radiation received by the sky (\(l_{d}\)) and by the surrounding objects (\(l_{sr}\)). The emitted and reflected radiation coming from the investigated surface, together with the radiation directly emitted by the atmosphere layer (\(l_{u}\)), are the main components of the radiation registered by the thermal sensor (\(l_{s}\)). A simplified equation, written in terms of spectral radiances, can model the described situation,  
\[l_{S}=\tau\,\varepsilon\,l_{B}+l_{u}+\tau\,\rho\,F\,l_{d}+\tau\,\rho\,\big{(}1 -F\big{)}l_{sr} \tag{1}\]  
where \(\tau\) is the atmospheric transmittance along the path between the investigated surface and the thermal sensor, \(F\) is the sky view factor representing the portion of the sky that is visible from the surface and \(\rho\)is the spectral reflectance of the surface, which can be computed as \(1-\varepsilon\). As a simplification, the sky view factor is assumed to be equal to unity, which means assuming a negligible contribution from the surrounding objects. Each thermal sensor registers the radiation within a specific portion of the electromagnetic spectrum (\(\Delta\lambda\)); thus, the radiance registered by the sensor is a broadband radiance, which can be defined as,  
\[L_{S}=\frac{\int_{\lambda\lambda}l_{S}\,s\,d\lambda}{\int_{\Delta\lambda}s\,d\lambda} \tag{2}\]  
where \(s\) is the relative spectral response function of the sensor. By substituting Equation 1 in Equation 2 and solving for the \(l_{B}\) term and after some simplifications, the following equation can be written,  
\[\int_{\Delta\lambda}\tau\,l_{B}s\,d\lambda=\frac{\int_{S}s\,d\lambda-\int_{u} \lambda\lambda-\mathrm{P}\int\tau\,l_{d}s\,d\lambda}{\mathrm{E}} \tag{3}\]  
The main theoretical simplification leading to Equation 3 regards the broadband surface emissivity E and the broadband reflectance P [14]. In fact, a grey-body behavior is assumed for all of the surfaces, at least within the \(\Delta\lambda\) spectral range. This assumption is also suggested by a practical consideration, discussed later, about the estimation of the emissivity values for the different materials, which can be found in the images. Furthermore, the surface is assumed to be Lambertian, which means that the dependence of \(\varepsilon\) and \(\rho\) on the observing angle is neglected. To derive the surface temperature from the left side term of Equation 3, a look-up table (LUT) can be built and a second order polynomial relation can be fit (see Figure 1). The temperature range investigated  
Figure 1: Plot of the look-up tables used to model the relations between the surface temperature and the black-body radiance (red line with diamonds, left vertical axis) and between the apparent temperature and the radiance at sensor (blue line with circles, right vertical axis). Vertical axes are in units of W/m\({}^{2}\)/sr. by the LUT is to be chosen according to the thermal image statistics and to the expected temperatures. Within this range, the integral of the radiance is calculated numerically at a step of 0.5 degrees, according to [PERSON]'s law.