## 3 Conversion Models and Calibration Procedures  
The reliability of the assessed concrete strength depends on the assumption of a relationship between the acquired rebound number and the actual concrete strength. The available codes [12] suggest performing a calibration procedure of the conversion model that correlates the RH measurement with the strength of concrete cores extracted at the same position of the RH tests. Many empirical formulations have been proposed which make use of several models and a high number of coefficients for the same model. The main empirical models are the power law [9]:  
\[f_{c}=a_{r}\ R^{b_{r}} \tag{2}\]  
the linear law [22, 23, 24]:  
\[f_{c}=b_{l}\ R+a_{l} \tag{3}\]  
the exponential law [9]:  
\[f_{c}=a_{e}\ e^{b_{e}R} \tag{4}\]and the polynomial law [25; 26; 27]:  
\[f_{c}=a_{p}R^{2}+b_{p}R+c_{p} \tag{5}\]  
Several authors proposed favorable values for the coefficients of Equations (2)-(5) based on the results of experimental studies, which were performed in many cases in laboratory conditions or in existing buildings. However, the proposed values arguably vary in wide ranges and a conversion model suitable for any kind of application has not yet been successfully established. Indeed, to improve the accuracy of the adopted model, a calibration procedure should be performed by minimizing the estimation error of the concrete strength on a set of in-situ strength data acquired by means of compression tests on drilled cores \(f_{c,\,s}\). In the present paper, the laws of Equations (3)-(5) are investigated. In detail, the coefficients of the linear empirical law are obtained by minimizing the following mean square error:  
\[\epsilon_{l}=\frac{\sum_{i=1}^{m}[Ln(f_{c,s_{i}})-(b_{l}Ln(R_{i})+a_{l})]^{2} }{m} \tag{6}\]  
where \(f_{c,s_{i}}\) is the in-situ strength evaluated on the basis of the strength of the \(i\)-th core, \(R_{i}\) is the rebound index registered at the same position of the \(i\)-th core and \(m\) is the number of drilled cores considered in the calibration procedure. The exponential law may be written using logarithmic notation as follows:  
\[Ln\ (f_{c})=Ln(a_{e})+b_{e}\ R_{i} \tag{7}\]  
where the coefficients are calibrated based on the minimization procedure of the following mean square error:  
\[\epsilon_{e}=\frac{\sum_{i=1}^{m}[Ln(f_{c,s_{i}})-(Ln(a_{e})+b_{e}R_{i})]^{2} }{m} \tag{8}\]  
In the case of the polynomial law the mean square error can be written as:  
\[\epsilon_{p}=\frac{\sum_{i=1}^{m}[Ln(f_{c,s_{i}})-\big{(}a_{p}R_{i}^{2}+b_{p} R_{i}+c_{p}\big{)}]^{2}}{m} \tag{9}\]  
The influencing factors on the rebound index make it quite difficult to define a conversion model which can be applied to all testing conditions, as has been reported by several authors [9; 15]. The present work aims to highlight that, even if the same experimental conditions are ensured, the effects of some influencing factors are minimized and variability in the coefficients of conversion models still remains, which can be predicted by means of ad hoc empirical laws that have been proposed for different analytical formulations of the conversion model. In the experimental campaign examined here, a significant number of cores have been extracted with the final aim of acquiring a deep knowledge of the concrete compressive strength. However, in the present paper, in order to consider the general practical conditions, which are usually characterized by a limited number of drilled cores due to logistic and economic reasons, the available \(N\) data have not been utilized to calibrate a single empirical conversion model. The data have been rearranged in order to build several datasets of possible results of the experimental investigations, which have been utilized to calibrate a huge number of conversion models. In detail, the data acquired during the experimental tests have been utilized to calibrate the coefficients of three different analytical formulations, i.e., the linear law, the exponential law and the polynomial law.