## 2 Materials and Methods  
### Preparation of Thermal Barrier Coatings  
To obtain thermal barrier coating samples with varying porosity, atmospheric plasma spraying (APS) equipment was used to deposit TBCs onto carbon steel disc substrates [37; 38]. The diameter of the substrate was 25.4 mm, and the thickness was 3 mm. Yttria-stabilized zirconia (8% Y\({}_{2}\)O\({}_{3}\)-ZrO\({}_{2}\), 8 YSZ) was selected as the material for TBCs, known for its excellent thermal stability and suitability for high-temperature applications. Additionally, 8 YSZ exhibited a reduced susceptibility to phase transformation, particularly the tetragonal-to-monoclinic phase transformation, which could lead to microcracking and degradation of the coating under thermal cycling conditions. The incorporation of yttria (Y\({}_{2}\)O\({}_{3}\)) as a stabilizing agent effectively inhibited this phase transformation, thereby enhancing the overall mechanical durability and longevity of the TBCs. By utilizing 8 YSZ as the ceramic powder, we aimed to ensure the long-term reliability and performance of the TBCs, especially in high-temperature aerospace applications. Moreover, the excellent thermal and mechanical properties of 8 YSZ were conducive to achieving the desired porosity levels and microstructural characteristics essential for our research on nondestructive evaluation and prediction of TBCs' porosity. The ceramic powder had two different particle sizes, as shown in Figure 1, namely 25-55 \(\upmu\)m and 45-106 \(\upmu\)m. These specific particle sizes were selected based on their relevance to the porosity evaluation, as they could influence the formation of pores during the spraying process. TBC porosity samples were controlled by varying the spraying parameters. Specifically, the spraying distance and power were adjusted. The spraying distance refers to the distance between the nozzle of the plasma spraying system and the substrate, and different spraying distances of 50 mm, 70 mm, 90 mm, 110 mm, and 130 mm were chosen. The spraying power levels were at the rates of 33 kW, 38 kW, and 40 kW. These parameter variations effectively regulated the TBCs' porosity. The justification for these choices lay in the need to investigate the impact of different spraying parameters on the resulting porosity levels of the TBCs. By systematically varying the spraying distance and power, we could better understand the relationship between the porosity and the process parameters, providing valuable insights for quality control and evaluation.  
For quantitative characterization of TBC samples with varying porosities, scanning electron microscopy (SEM) was utilized to capture cross-sectional images of the samples. The SEM images facilitated the examination of the microstructure and pore distribution of TBCs. Figure 2 shows an SEM image of the TBC cross-section sprayed by ceramic powder with different particle sizes at the same spraying distance. It is clear that with the increase in powder particle size, the number of pores in the thermal barrier coating increases significantly, and the pore size also increases. This proves that it was valid to obtain samples containing different porosities by adjusting spraying parameters. To accurately assess the porosity, ImageJ module was employed for image analysis and processing, providing a set of image processing tools that enable the measurement of pore area fraction and calculation of the corresponding porosity. By employing the experimental preparation and quantitative characterization methods described above, TBC samples with different porosities were obtained and then utilized for subsequent THz-TDS detection, allowing for precise evaluation and predictive analysis of the porosity of TBCs. In this study, a total of 20 kinds of samples containing different porosities, ranging from 6.94 to 22.09%, were prepared.