## 4 Result Analysis and Discussion
### Determination of the Shifts' Values (\(\Delta_{\mathrm{i}}\)) from the Thermograms  
As mentioned in the methodology section of this paper, the selection of the thermograms is very important in the procedure for determining the \(\Delta_{\mathrm{i}}\). Five thermograms, selected from the detection of the RC wall, with five different angles of the incident waves (or the direction of the microwave antenna), are shown in Figure 6. All the selected thermograms were not obtained at the same instant of heating, although the same heating duration was used for the tests: 172 s when \(\theta_{\mathrm{i}}=0^{\circ}\), 243 s when \(\theta_{\mathrm{i}}=15^{\circ}\), 288 s when \(\theta_{\mathrm{i}}=30^{\circ}\), 290 s when \(\theta_{\mathrm{i}}=45^{\circ}\), and 281 s when \(\theta_{\mathrm{i}}=60^{\circ}\). Figure 5: Angles of refracted waves (\(\theta_{\mathrm{i}}\)) in the function of the incident waves (\(\theta_{\mathrm{i}}\)) and the dielectric constant of the concrete (\(\varepsilon_{2}\)). On all the thermograms, we can see the heated areas due to the inference of the refracted waves between the steel bars inside the detected concrete wall. In order to obtain the temperature profiles for determining the shifts' values, a horizontal segment (the same length on all the thermograms) is placed at the same position (in the vertical and horizontal direction) on all the thermograms. The obtained temperature profiles for all the angles of the incident waves (\(\theta=\theta_{\rm i}\)) are presented in Figure 7. The horizontal axis corresponds to the dimension (the length) of the detected RC wall. Figure 6: Thermograms from the detection of RC wall by transmission approach at different instants for different angles of incident waves (\(\theta_{\rm i}\)): (**a**) Thermogram at 172 s when \(\theta_{\rm i}=0^{\circ}\); (**b**) Thermogram at 243 s when \(\theta_{\rm i}=15^{\circ}\); (**c**) Thermogram at 288 s when \(\theta_{\rm i}=30^{\circ}\); (**d**) Thermogram at 290 s when \(\theta_{\rm i}=45^{\circ}\); (**e**) Thermogram at 281 s when \(\theta_{\rm i}=60^{\circ}\). In order to deduce the shifts' values, the bottom peaks (which correspond to the real positions of the steel bars in case the normal direction of the microwave antenna is used) of the temperature profiles have to be defined. However, the forms of the temperature profiles do not facilitate the determination of the peaks' positions. For this reason, convenient fitting curves were needed for all the temperature profiles. In our study, moving the average with a period of 5 was used. Other fit functions (such as linear, exponential, logarithmic, polynomial, and power) have also been tried, but the moving average gives the best-fit curves. Other periods (3, 4, and 6) have also been tried, but period 5 gives the best-fit curves. It is necessary to note that the main objective of using the moving average fit curves is to facilitate the determination of the peaks on the temperature profile curves (after eliminating the multiple pulses) but to avoid a shift between the obtained fit curves and the original temperature profile curves. The peaks were hard to determine with period 3, while period 4 gave a better appearance of some peaks. Period 6 gave good peaks too (compared to period 5), but small shifts (compared to the initial temperature curves) started to appear, and the higher the period was used, the bigger shifts appeared. The fitted curves of the temperature profiles, after the application of the moving average, for all the angles of the incident waves are shown in Figure 8.  
It is worth noting that 100 cm in the horizontal axis corresponds well to the 1 m width of the detected wall. The minor gridlines of 1 cm lead to a good observation of the shifts (from the peaks for different angles of the incident waves compared to those of the normal incident wave). On the new graphs of the temperature profiles, the peaks clearly appear.