# Depth Retrieval Procedures in Pulsed Thermography: Remarks in Time and Frequency Domain Analyses
### Quantitative Depth Estimation in the Time Domain  
For quantitative depth prediction in the time domain, the most widely used algorithms utilise temporal characteristics of the thermal response received from the tested surface. In particular, these methods are based on the correlation of the defect's depth with a time point at which the hidden feature appears as a temperature difference on the surface of the inspected sample. The idea behind the determination of these time points is based on the fact that, when studying the heat diffusion into the material bulk after the application of a thermal pulse, the time taken for the thermal wave to be reflected back to the surface is inversely proportional to the thermal diffusivity of the material and directly proportional to the square of its depth [6]. The first proposed and the most commonly used method utilises the time point when the maximum thermal contrast between a defective and a preselected reference point occurs, known as Peak Contrast Time [25; 26; 27]. Even though this informative parameter can produce quantitative depth information, one of its major limitations is the fact that the peak contrast appears relatively late, when the 3D heat diffusion around the defect has a great influence on the acquired result. Consequently, different methods have also been proposed investigating the use of earlier temporal points, targeting to the produced accuracy enhancement.  
One of the earliest time points on a temperature-time evolution curve, which is used for defect depth estimations, is the moment when the behaviour of the studied temperature signal above the defect diverges from the behaviour of the reference signal [28]. Usually, as raw thermal data contain noise, the establishment of a threshold level is required to identify this point on the temperature-time curve [29]. Quantitative depth estimations through the aid of this informative parameter have the advantage of early depth prediction, thus possible effects produced from lateral heat diffusion in the defective area are negligible. However, the determination of this value requires a sampling rate high enough to accurately capture the initial time when the temperature in the defective zone diverse from the reference temperature, impeding the providing of accurate results in the case of shallow defects characterisation [30].  
A later time interval but early enough to suppress the negative effects of lateral heat diffusion is the moment when the thermal contrast curve presents its peak slope [31; 32]. This point can be easily found as it corresponds to the peak of the 1 st derivative of the thermal contrast and is commonly referred as Peak Slope Time method. The exact relationship between depth and peak slope time can be determined by solving, for this specific time value, Equation (1) [33]:  
\[t_{peak}=\frac{3.64\mathrm{z}^{2}}{\pi^{2}\alpha} \tag{1}\]  
where \(t_{peak}\) is the time at which the peak in the slope contrast occurs, \(z\) is the defect depth in a sample of thickness \(L\) and \(\alpha\) the thermal diffusivity of the tested material. This parameter is essential for defect depth estimation as experimental and numerical studies have shown that results are unaffected from the size of the defect, while the proportionality coefficient equal to 3.64 was found to be valid regardless the type of the material when \(z/L\leq 0.5\)[33].