## 3 Ultrasonic Measurement of Air Void Distribution in Early-Age Concrete
### Determination of Air Void Distribution at Early Stages  
The inverse analysis was conducted on MATLAB code by comparing the theoretical attenuation and experimental attenuation estimated based on Equations (1) and (10) respectively. The fitted attenuation equations for cement paste shown in Section 3.2 were applied as the background attenuation \(\alpha_{a,f}\). The attenuation generated by the aggregate were calculated with Equation (3), depending on the designed aggregate gradation. Based on the description equation for the air void size distribution shown as Equation (4), the attenuation generated by air voids can also be determined with  
Figure 4: The curve fitting of the early-age cement paste attenuation with frequency. (**a**) Curve fitting at Day 1; (**b**) Curve fitting at Day 3; (**c**) Curve fitting at Day 7; (**d**) Curve fitting at Day 14.  
Equation (9). Combining all these three parts, the total theoretical attenuation can be determined and compared with the experimental attenuation determined with the methods mentioned in Section 2.2. Three parameters need to be determined for both the small and large air void size distribution: the volume fraction, average size, and the corresponding standard deviation. The air void size distribution parameters can be obtained by achieving the minimum error between experimental and theoretical attenuation.  
The comparison between the experimental and theoretical attenuation in the frequency domain for Sample Type 1 was demonstrated in Figure 5, where (a), (b), (c) and (d) depict the comparison at Day 1, 3, 7, and 14 respectively. The corresponding inversed air void size distribution and volume fraction were shown in Figure 6, where (a) and (b) represent the change of the air void volume fraction and evolution of air void size distribution with curing ages. Particularly, the air void volume fraction at Day 0 (fresh stage) was measured by the ASTM C138 test, which is in good agreement with the ultrasonic scattering measurement results.  
Similar to the inverse analysis methods used for Sample Type 1, the inverse analysis results were conducted for Sample Types 2 and 3 separately. The changes of air void volume fractions with curing ages are represented in Figures 7a and 8a. Similar to the results shown in Figure 6a, the decreasing trend of the air void volume fraction was observed for all these three samples. Particularly, the air void  
Figure 5: The comparison between theoretical and experimental attenuation during early stages (**a**) The comparison at Day 1; (**b**) The comparison at Day 3; (**c**) The comparison at Day 7; (**d**) The comparison at Day 14.  
Figure 6: Air voids evolution for Sample Type 1. (**a**) The change of air void content with curing age measured with ultrasonic scattering (at early stages Days 1–28) and the ASTM C138 (Day 0); (**b**) Air void size distribution evolution with curing time.  
volume fraction change after Day 14 is much slower compared to the volume change during the first 14 days. The corresponding air void size distribution evolution for Sample Types 2 and 3 are shown in Figures 7b and 8b respectively. The results demonstrate that the volume fraction of both small and large air voids decrease with the curing ages. As mentioned above, the air void volume fraction was observed to decrease with curing age. This phenomenon is in accordance with the results that the air void volume fraction can decrease during hydration age [38], especially on the ITZ area. It is also clear that the decreasing rate of the air void volume fraction declines with the curing age, which is in accordance with the cement hydration speed [39].