## 2 Attenuation-Based Ultrasonic Scattering Measurement Method
### Experimental Setup and Experimental Attenuation Measurement  
The basic setup for ultrasonic scattering measurement is demonstrated in Figure 1. The two Olympus 5077 transducers (Olympus Co., Center Valley, PA, USA) are located on the opposite sides of the concrete specimen (Prepared by the authors in Benedict Lab, Houghton, MI, USA) as the pulser and receiver separately. The couplant B2 from Olympus (Center Valley, PA, USA) was also used to mitigate the coupling effect between the specimen surface and ultrasonic transducers. The ultrasonic wave centered at 500 kHz frequency is applied in this study, which is appropriate for the detection of the air voids and has been used in the concrete study by different researchers [29; 30]. Ultrasonic waves generated by the excitation voltage can transmit through the specimen and keep being reflected at the transducer-specimen interface. The wave signal captured by the receiver transducer is recorded by an oscilloscope (Olympus Co., Center Valley, PA, USA) and displayed on a computer (Dell Co., Round Rock, TX, USA). Particularly, the first two transmission waves \(S_{1}(t)\) and \(S_{2}(t)\), as shown in Figure 1, are used for the experimental attenuation calculation.  
The first and second transmission waves referred to as \(S_{1}\) and \(S_{2}\) were first converted from the time domain to the frequency domain by Fast Fourier Transform on MATLAB (R2015a, MathWorks Co., Natick, MA, USA), as shown in Figure 2. It is clear that the amplitude of \(S_{2}\) is significantly reduced along the whole frequency range compared to the first transmission wave \(S_{1}\). The experimental attenuation is estimated from Equation (10) proposed by Sears [31]. The irrelevant attenuation caused by the wave diffraction was eliminated by using the diffraction correlations as shown in Equation (10).  
\[\alpha(f)=\frac{1}{2d}\left[\ln\biggl{(}\frac{s_{1}(f)}{s_{2}(f)}\biggr{)}- \ln\biggl{(}\frac{D(s;d)}{D(s;3d)}\biggr{)}\right] \tag{10}\]  
where \(\alpha(f)\) is the experimental attenuation of the specimen, which is expressed with wave frequency \(f\); \(d\) is the specimen thickness; and \(s_{1}(f)\) and \(s_{2}(f)\) are the first and second transmission waves respectively. \(D(s;d)\) and \(D(s;3d)\) are proposed by [PERSON] [32], which are the simplified closed-form expressions for the diffraction correction.  
Figure 1: Demonstration of the ultrasonic scattering measurement of air void size distribution in concrete samples.  
Before the investigation of ultrasonic attenuation on the concrete specimen, the attenuation measurement from the ultrasonic test setup was validated with a thin polymethylmethacrylate sample. The sample was placed between the \"pulser\" and \"receiver\" transducers, and the couplant was also used to reduce the coupling effect. The first and second transmission waves were recorded and converted to the frequency spectrum with Fourier transformation for attenuation calculation. The attenuation was calculated with Equation (10) and the measurement results are shown in Figure 3. The relationship between ultrasonic attenuation and wave frequency is demonstrated by the fitting curve of \(y=1.725\times 10^{-5}\times x+10.958\), where \(x\) and \(y\) represent the wave frequency (Hz) and wave attenuation (Np/m), respectively. The transmission speed can also be measured based on the arrival time of the first transmission signal and the sample thickness. The measured transmission speed is 2717 m/s; the corresponding attenuation per wavelength can then be calculated as 0.0469 NP. The results are close to the measured values (2782 m/s and 0.045 NP) in reference [10], and the published values (2750 m/s and 0.041 NP) in reference [33], which demonstrate the accuracy of attenuation measurement from this system.