## 2 Theoretical Background
Therefore, the application of GPR has great potential in monitoring road pavements and identifying their local problems [28]. There are several approaches described in the literature concerning the relationship between relative permittivity and the air void content of asphalt mixtures. The most frequently used approach characterises the empirical relationship between these two characteristics. Based on research in Finland, a conclusion has been drawn [33] that there is an exponential relationship between relative permittivity and the air void content (\(AV\)) of asphalt mixtures [34]:  
\[AV=A\cdot\exp(-B\cdot\varepsilon_{AC}) \tag{1}\]  
where \(AV\) is the air void content of the asphalt mixture, \(\varepsilon_{AC}\) is the relative permittivity of the asphalt mixture, and \(A\) and \(B\) are the calibration constants depending on the type of mixture. To determine the values of the parameters in Equation (1), Finnish researchers measured  
Figure 1: Effect of compaction on relative permittivity [33]. the relative permittivity of asphalt samples with various air void contents by applying a percometer probe, and fitted an exponential regression model with a determination coefficient \(R^{2}=0.723\):  
\[AV=272.93\cdot\exp(-1.3012\cdot\varepsilon_{AC}) \tag{2}\]  
After establishing the correlation equation, further measurements were performed but the new data were not in good agreement with the former data. Researchers assumed that this phenomenon is a consequence of the differences in the dielectric characteristics of aggregates, and the inaccuracy of the model could be corrected by shifting the dataset. A \(k\) calibration factor was, thus, introduced, and the final model has the following form:  
\[AV=272.93\cdot\exp(-1.3012\cdot k\cdot\varepsilon_{AC}) \tag{3}\]  
The \(k\) calibration factor was calculated for all in situ drilled core samples, and the values were averaged. This average \(k\) value is valid for asphalt mixtures produced in a certain mix plant using a certain aggregate source. The exponential model shows a good fitting to the drilled asphalt samples. However, in the case of samples with lower relative permittivity values, its predictions become inaccurate. This is why the authors of [12] recently published a new formula with better fitting, even for cases with extreme data values. The proposed Hoegh-Dai (HD) model introduces physical constraints on the possible maximum and minimum values, thereby preventing the model from predicting a negative air void content or a relative permittivity value below 1:  
\[AV=\exp\left(-B\left(D^{\mid\frac{1}{\varepsilon_{AC}-C}-\frac{1}{1-C}\mid}-1 \right)\right) \tag{4}\]  
where \(AV\) is the air void content of the asphalt mixture; \(\varepsilon_{AC}\) is the relative permittivity of the asphalt mixture; and \(B\), \(C\), and \(D\) are the calibration constants determined by minimizing the sum of the square differences between the model and experimental data. If the input relative permittivity value is close to 1 (for air \(\varepsilon_{air}=1.0006\)), then the HD model predicts a 100% air void content, which is physically correct. This way, the model is not sensitive to the unrealistically high air void content of drilled core samples and is suitable for extrapolation below the lowest value of the calibration dataset [12]. With this development, the HD model fits well to drilled core samples with an air void content between 4% and 12% [35] but according to recent experiences, it still overestimates the slope of the relationship function between the air void content and relative permittivity in cases with extreme data values. [PERSON] et al. (2020) [36] proposed that the core sample data predict an inflection point near the extreme values, where the slope of the model function is flattening.