## 2 Attenuation-Based Ultrasonic Scattering Measurement Method
### Theoretical Attenuation
#### 2.1.2 Attenuation Generated by Air Voids  
The normal and log-normal distribution are desirable to characterize the size distribution of small air voids and large air voids respectively [15]. The combination of the normal and log-normal distribution function is depicted in Equation (4). The detailed normal and log-normal distributions are shown as Equations (5) and (6) respectively  
\[f=\phi_{21}\times f_{norm}(r_{norm};\mu_{norm},\sigma_{norm})+\phi_{22}\times f _{log-norm}\Big{(}r_{log-norm};\mu_{log-norm},\sigma_{log-norm}\Big{)} \tag{4}\]  
where \(f\) represents the distribution function for the whole air void structure; \(r_{norm}\), \(\mu_{norm}\), \(\sigma_{norm}\) are the radius of the air void, mean value, and standard deviation of the norm size distribution, respectively; \(r_{log-norm}\), \(\mu_{log-norm}\), \(\sigma_{log-norm}\) represent the corresponding parameters for the log-normal distribution. The volume fractions of small air voids and large air voids are \(\phi_{21}\) and \(\phi_{22}\), respectively (\(\phi_{2}=\phi_{12}+\phi_{22}\)).  
\[f_{norm}(r_{norm};\mu_{norm},\sigma_{norm})=\left(1/\sqrt{2\pi\sigma_{norm}^{2} }\right)\exp\Bigl{[}-\left(r_{norm}-\mu_{norm}\right)^{2}/2\sigma_{norm}^{2} \Bigr{]} \tag{5}\]  
\[f_{log-norm}\Big{(}r_{log-norm};\mu_{log-norm},\sigma_{log-norm}\Big{)}=\left( 1/\sqrt{2\pi\sigma_{log-norm}^{2}}\right)\exp[-\left(\ln r_{log-norm}-\mu_{ log-norm}\right)^{2}\Big{/}2\sigma_{log-norm}^{2} \tag{6}\]  
where \(f_{norm}(r_{norm};\mu_{norm},\sigma_{norm})\) and \(f_{log-norm}\Big{(}r_{log-norm};\mu_{log-norm},\sigma_{log-norm}\Big{)}\) represent the normal distribution and log-normal distribution separately, and \(r_{norm}\), \(\mu_{norm}\), \(\sigma_{norm}\) are the radius of the air void, mean value, and standard deviation of the norm size distribution, respectively, while \(r_{log-norm}\), \(\mu_{log-norm}\), \(\sigma_{log-norm}\) represent those parameters for the log-normal distribution.  
Then, the corresponding cumulative function for a certain size air void can be represented as \(F_{2}(a_{i2})\) through the integration of distribution function \(f_{2}(a_{i2})\). Similar to the analysis of aggregate attenuation above, the amount of a certain size air void per unit volume \(n_{i2}\) can be deduced as shown in Equations (7) and (8). Particularly, the air void size distribution function is divided by \(F_{2}(a_{N2})-F_{2}(a_{12})\) for normalization, where \(a_{N2}\) and \(a_{12}\) represent the largest and smallest air void size respectively.  
\[\phi_{i2}=\left[F_{2}(a_{i2})-F_{2}(a_{i2-1})\right]\phi_{2}\bigg{/}\left[F_{ 2}(a_{N2})-F_{2}(a_{12})\right] \tag{7}\]  
\[n_{i2}=\left[F_{2}(a_{i2})-F_{2}(a_{i2-1})\right]\phi_{2}\bigg{/}\left[F_{2}( a_{N2})-F_{2}(a_{12})\right]\frac{4}{3}\pi a_{i2}^{2} \tag{8}\]  
where \(\phi_{2}\) is the volume fractions of the air void (\(\phi_{1}+\phi_{2}=1-\phi\)).  
Introducing Equation (9) into Equation (1), the total theory attenuation can be linked to the air voids' size distribution and aggregate gradation as illustrated in Equation (9).  
\[\alpha_{f,airroid}=\ \frac{1}{2}\sum_{i=1}^{m2}\left[F_{2}(a_{i2})-F_{2}(a_{i2-1} )\right]\phi_{i2}\gamma_{i2,f}^{Sca}\Bigg{/}\left[F_{2}(a_{N2})-F_{2}(a_{12} )\right]\frac{4}{3}\pi a_{i2}^{3} \tag{9}\]