## 2 Methodology  
The principle of the proposed methodology is illustrated by a flowchart, as shown in Figure 1. The methodology is divided into seven steps. From the first step to the fourth step, expertise in infrared thermography methods is applied. In the first step, five experimental campaigns with the microwave thermography method are carried out with five angles of the incident wave (\(\theta_{1}\)): 0\({}^{\circ}\) (normal direction of the microwave antenna), 15\({}^{\circ}\), 30\({}^{\circ}\), 45\({}^{\circ}\), and 60\({}^{\circ}\). It is worth reminding that, in the transmission approach, the infrared camera and the heating source (and the antenna) are not situated on the same side. Details of the experimental set-up are shown in the next part of the present paper. In the second step, the thermograms are obtained from all the tests with the RC wall. Then, one thermogram at an instant is to be selected for each angle of the incident wave. The criteria for selecting the thermogram are based on the thermal contrast being optimal so that the corresponding temperature profile, to be drawn in the third step, gives clear peaks. Multiple choices are possible, but the selected thermogram must be in the heating phase to avoid thermal contrast decrease on the thermogram due to thermal conduction in the detected materials. All the temperature profiles for all \(\theta_{1}\) are drawn on the same axes to facilitate deducingthe values of the shifts of the peaks on the temperature profiles (compared to the case of normal direction) in the fourth step. The shift value for the case of 15\({}^{\circ}\) of the incident wave angle is measured from the peaks of its temperature profile to the peaks of the temperature profile for the case of the normal incident waves. The same way of measurement is used to obtain the shift values for the other cases of incident wave angles. It is worth noticing that the shifts (\(\Delta_{\rm i}\)) are generated by the effect of the refraction of the waves after reaching the steel bars inside the detected RC wall, which was interpreted in the previous study [35]. From step five to step seven, which is the last step, a combination of a physical law and a mathematical analysis method is needed. In step five, a non-linear relation between \(\theta_{\rm i}\) and \(\Delta_{\rm i}\) is created by using the Snell-Descartes law (as presented in Figure 2) and a geometrical relation between the \(\Delta_{\rm i}\), the transmitted waves in the concrete (\(\theta_{\rm i}\)), and the concrete cover thickness (e). Figure 1: Methodology for detecting the concrete cover thickness of one-layer reinforcements in RC wall by microwave thermography method with transmission approach. Figure 2: Refraction of wave by Snell—Descartes’s law. The theory gives the relation between the incident waves and refracted waves as the following equation:  
\[\sqrt{\varepsilon_{1}}\times\sin\theta_{\mathrm{i}}=\sqrt{\varepsilon_{2}}\times \sin\theta_{\mathrm{t}} \tag{1}\]  
where \(\varepsilon_{1}\) is the dielectric constant of medium 1, \(\varepsilon_{2}\) is the dielectric constant of medium 2, \(\theta_{\mathrm{i}}\) is the angle of the incident waves in medium 1, and \(\theta_{\mathrm{t}}\) is the angle of the transmitted waves in medium 2. In our study, medium 1 is the air, and medium 2 is the concrete. The dielectric constant of the air is equal to 1 (\(\varepsilon_{1}=1\)) [31]. The geometrical relation between \(\Delta_{\mathrm{i}}\), \(\theta_{\mathrm{t}}\), and the concrete cover (e) can be illustrated in Figure 3. As shown in Figure 3a, which was discussed in the previous study [35], the concrete in the areas of inference (superposition of the waves) of the refracted waves is hotter than in the other areas. The same study also showed that the steel bars in the detected wall play a role as an obstacle to the transmitted waves, which makes some fractions of the waves totally reflected to the concrete surface.