# A Method of Defect Depth Estimation for Simulated Infrared Thermography Data with Deep Learning
## 2 Thermal Consideration and FEM Stimulation
### Temperature and Thermal Contrast Curves  
In transient thermography, previous researchers [6] concluded that the time \(t_{c}\) that corresponds to the maximum temperature contrast \(\Delta T\) has an approximately proportional relationship with the square of the depth of the defect (\(d^{2}\)). Simultaneously, the proportionality coefficient of this relationship \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sign** & **Parameters in Experimental Simulation** & **Real Value** \\ \hline \(\rho\) & Material density & \(1500\frac{k\times k}{m^{3}}\) \\ \(\varepsilon\) & Emissivity & 0.90 \\ \(\mathrm{C_{P}}\) & Constant specific heating & \(1000\frac{l}{k_{S}m}\) \\ L & Specim length & 300 mm \\ W & Specim with & 300 mm \\ H & Specim height & 5 mm \\ \(t_{end}\) & Processing time to finish computations & 9 s \\ \(T_{i}\) & The temperature from initialization & 293.15 k \\ \(T_{amb}\) & The temperature from ambient & 293.15 k \\ \(W_{source}\) & Heat source wastage & 600 Watts \\ \(H_{center}\) & Heat source center & (15 cm, 15 cm) \\ P & Heat pulse density & 100,000 \(\frac{W}{m.m}\) \\ h & The distance between the lamp and specimen & 80 cm \\ S & Time step & 0.0063 s \\ \(t_{pulse}\) & Heating time & 0.0126 s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical properties and parameters of stimulation. rested with the lateral size of the depth: the smaller the defects, the lower the maximum contrast \(\Delta\)T and earlier peak time. As indicated in Figure 4, an interesting case is highlighted. We observed the temperature evaluation of three defects of the same size (18 mm \(\times\) 18 mm) but different depth such as (0.5 mm, 1.0 mm, 1.5 mm). Although the three defects have the same shape, it can be demonstrated that the shallower the defect, the higher the peak temperature value which can be obtained. In this work, five contrast vectors have been extracted above each defect region on the surface from the different points in the defects to reduce the inaccurate influences caused by the small temperature variation. In Figure 5d as indicated in the below, in each defect region, the red cube is a defect point which localized on the corner of the upper left (C1); upper right (C2); center (C3); lower left (C4); lower right (C5). The blue cube is the reference point with respect to each defect point that we extracted. In Figure 5a-c, the data distribution of the thermal contrast curves is illustrated. C1-C5 are thermal contrasts that extracted from each defect point (in Figure 5d) based on the temperature difference between the defect point and reference sound point. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample** & **Row** & **Depth(mm)** & **Shape** & **Defect Size (mm)** \\ \hline \multirow{3}{*}{1} & 1 & 0.5 mm & Quadrangle & Size = 10; 15; 18 \\  & 2 & 0.6 mm & Round & Diameter = 18; 15; 5 \\  & 3 & 0.7 mm & Quadrangle & Size = 5; 10; 18 \\ \hline \multirow{3}{*}{2} & 1 & 0.8 mm & Quadrangle & Size = 5; 10; 15 \\  & 2 & 0.9 mm & Round & Diameter = 18; 15; 10 \\  & 3 & 1.0 mm & Quadrangle & Size = 5; 15; 18 \\ \hline \multirow{3}{*}{3} & 1 & 1.1 mm & Quadrangle & Size = 5; 10; 18 \\  & 2 & 1.2 mm & Round & Diameter = 15; 10; 5 \\  & 3 & 1.3 mm & Quadrangle & Size = 10; 15; 18 \\ \hline \multirow{3}{*}{4} & 1 & 1.4 mm & Round & Size = 5; 15; 18 \\  & 2 & 1.5 mm & Quadrangle & Diameter = 18;10; 5 \\  & 3 & 1.6 mm & Round & Size = 5; 10; 15 \\ \hline \multirow{3}{*}{5} & 1 & 1.7 mm & Round & Diameter = 10; 15; 18 \\  & 2 & 1.8 mm & Quadrangle & Size = 18; 15; 5 \\  & 3 & 1.9 mm & Round & Diameter = 5; 10; 18 \\ \hline \multirow{3}{*}{6} & 1 & 2.0 mm & Round & Diameter = 5; 10; 15 \\  & 2 & 2.1 mm & Quadrangle & Size = 18; 15;10 \\ \cline{1-1}  & 3 & 2.2 mm & Round & Diameter = 5; 15;18 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Defect characteristics of training samples.