# A Method of Defect Depth Recognition in Active Infrared Thermography Based on GRU Networks
## 2 Theory of AIRT  
In active infrared thermography (AIRT), a controllable thermal flow is applied to the specimen, and the thermal response is able to be measured during a period of time by an infrared camera [20; 21]. The diffusion and transmission of the heat flow are affected when defects exist in the internal structure of the specimen (for example, flat bottom hole defect). This effect will be revealed by the difference in the surface temperature of the specimen before the object reaches thermal equilibrium [22]. Figure 1 illustrates the surface temperature of the specimen under the conditions of existing internal defects and also of without internal defect. The surface temperature distribution of the specimen without defect is showed in Figure 1a. The incident heat flow is partially reflected on the surface of the object; the remaining part continues to diffuse downward; so that a uniform temperature field is formed on the object surface. In Figure 1b, due to the thermal conductivity of the defective region being smaller than that of other regions of the specimen material, the downward transmission of the incident heat flow at the position of defect area is obstructed, and the reflected heat flow is increased; consequently, a relatively hot region is formed on the surface of defect area. On the contrary, it can be seen in Figure 1c that a relatively cold region is formed on the surface of the defective region because of the buried-inside thermal conductivity defect whose thermal conductivity is greater than that of the specimen material.  
The surface temperature distribution of the defect region and sound region are different during different periods of time because the inner thermal flow transfer rates of the regions above would influence the surface temperature distrib  
Figure 1: Surface temperature and heat flow diffusion under different conditions: (**a**) specimen with no defect; (**b**) specimen with thermal insulation defect; (**c**) specimen with thermal conductivity defect.  
transmission in the specimen follows a three-dimensional differential equation of heat conduction, as illustrated below:  
\[\frac{\partial}{\partial x}(\lambda_{x}\frac{\partial T}{\partial x})+\frac{ \partial}{\partial y}(\lambda_{y}\frac{\partial T}{\partial y})+\frac{ \partial}{\partial z}(\lambda_{z}\frac{\partial T}{\partial z})=\rho C\frac{ \partial T}{\partial t} \tag{1}\]  
where \(T\) (K) represents absolute temperature; \(t\) (s) indicates the time variable; \(\rho\) (kg/m\({}^{3}\)) is the density of the specimen; C (J/(kg:K)) is the specific heat capacity; \(\lambda_{x}\), \(\lambda_{y}\), and \(\lambda_{z}\) (W/(m:K)) are the thermal conductivities for the three respective dimensions in Cartesian coordinates. The specimen exchanges heat with the external fluid and swaps heat with the environment by radiation. The boundary condition is described as:  
\[-\lambda\frac{\partial T}{\partial n}=h(T-T_{amb})+\varepsilon\sigma(T^{4}-T^ {4}_{amb}) \tag{2}\]  
where \(n\) represents normal direction outside the surface, \(\lambda\) (W/(m:K)) is the main thermal conductivity, \(T_{amb}\) is the initial environment temperature, \(h\) (W/\(\left(m^{2}\cdot\text{K}\right)\)) is heat transfer coefficient, \(\varepsilon\) represents surface emissivity, and \(\sigma\) is the Stefan-Boltzmann radiation constant. The infrared camera obtains the surface temperature distribution of the specimen, and it is a great advantage to use deep learning to determine the defect characteristics using the temperature data.