## 4 Linear Law  
Based on the procedure described in Sections 2 and 3, more than \(10^{7}\) linear conversion models have been calibrated by varying the number \(m\) and, thus, the combinations of data considered in the calibration process. It is worth noting that the coefficient of variation related both to the concrete strength and the rebound index, varies from each combination to another even if they are acquired on the same structure.  
The main objective is to study the variation of the coefficients of the linear empirical law. Thus, in Figure 1 the terms \(b_{i}\) and \(a_{i}\) are plotted for several values of the number \(m\) of data utilized for the calibration. Each point of the plots corresponds to a different combination of data and, thus, to a different linear model. It is worth mentioning that the values of the coefficients vary significantly. However, a clear relationship can be observed between such coefficients, which have been evaluated by means of a linear regression analysis. The results of such analyses have been reported in each plot with the related coefficient of determination. The values are similar and do not depend on the number \(m\) of considered data. Moreover, considering other values of \(m\)--whose results have not been plotted for the sake of brevity--it is possible to recognize the following relationships between the coefficients:  
\[b_{i}=-0.0237a_{i}+0.5920 \tag{11}\]  
With a coefficient of determination equal to 0.9977, the excellent accuracy of the linear relationship in fitting the numerical values is shown.  
In Figure 2 the relationship between the coefficients of the linear models has been plotted, also considering the values of the coefficient of variation of the concrete strength and of the rebound index evaluated for each combination of data composed by \(m=15\) measurements. The plots show that for increasing values of the coefficient of variation of the concrete strength, the values of \(b_{1}\) tend to increase while the values of \(a_{1}\) decrease, but the relationship between such coefficients remains almost the same. Comparable behavior is observed for increasing values of the coefficient of variation of the rebound index, but the latter coefficient is characterized by a very low variation. The described trends are also detected for other values of \(m\) but are not plotted for the sake of brevity.  
Figure 3 shows the variation of the RMSE with respect to the coefficient of variation of the concrete strength and of the rebound index for two different values of \(m\). It is observed that even for a higher given value of \(m\), the range of variability of RMSE decreased, and it would not be concluded for the examined case that a direct relationship exists between the RMSE and the considered coefficients of variation.  
Figure 1: Relationship between the coefficients of the calibrated linear empirical laws \(b_{l}\) vs. \(a_{l}\) for different value of \(m\) considered data: (**a**) \(m=11\); (**b**) \(m=15\); (**c**) \(m=18\); (**d**) \(m=22\).  
Figure 2: Calibration of the empirical model based on the data acquired in \(m=15\) points of the investigated structure: (**a**) Plot of the coefficients of the calibrated linear empirical laws and of the coefficient of variation of the concrete strength; (**b**) Plot of the coefficients of the calibrated linear empirical laws and of the coefficient of variation of the rebound index.