# Matched-Filter Thermography
## 4 Matched Filtering in Thermography
### Thermal-Wave Radar (TWR)
#### 4.1.1 Theory and Modeling
The spectral energy of the radiometric response from the steel sample surface can be divided into two parts: the energy due to excitation generated diffusive waves at the sample surface  
\[-\kappa\frac{\partial\theta(z,\omega)}{\partial z}\bigg{|}_{z\ =\ 0}\ =\ \Im\{f(t)\}\ =\ F(\omega),\]3 Simulated (**a**) spectral responses and (**b**) Thermal-wave radar (TWR) subtractive surface temperature evolution over blind holes of different thicknesses; (**c**) simulated TWR outputs for 100 \(\upmu\)m thick and semi-infinite steel samples; (**d**) variation of TWR peak delay time with opaque sample thickness in normal and subtraction modes; (**e**) effect of TWR excitation waveform parameters on detectability of defects; (**f**) variation of TWR peak delay time with absorber depth in turbid medium [21]. (surface energy absorption) and the coherently accumulated energy due to back-interface interacted thermal-waves. Surface energy absorption increases the sample surface temperature and therefore produces a radiometric signal whose spectral energy distribution exhibits the damped behavior expected under the envelope of diffusion (a low pass filtering action). On the other hand, the energy due to back-interface interacted thermal-waves in finitely thick opaque solids is delayed (phase-shifted) with respect to the surface absorption energy part. In a semi-infinite sample, the radiometric signal energy (Figure 3a, curve 7) is only due to surface energy absorption and no delayed back-interface interacted thermal-wave contribution is expected. As a result, a relatively uniform ESD is observed, characterized by the diffusive low-pass filtering profile (damping). The surface energy absorption part is also present in the ESD of finite thickness opaque samples. However, the interface interacted thermal-wave part of the ESD in these samples (both back and front interfaces) contribute to the resulting diffusion-wave frequency spectrum. In this sense, the delayed portion of the ESD (subtractive part) becomes a function of sample thickness and can be used as a sensitive parameter to distinguish various thicknesses, Figure 3b. Figure 3a,b suggest that, as the sample thickness decreases, the subtractive mode energy density increases. The energy increase is more pronounced at lower frequencies as low frequency components are less damped during their propagation. Figure 3c shows the simulated TWR/cross-correlation output signals for 100 \(\upmu\)m-thick and semi-infinite steel samples. It can be observed that, as the steel sample thickness increases from 100 \(\upmu\)m to infinity, the cross-correlation peak delay time (\(\tau_{p}\)) decreases from 83 ms to 39 ms, as magnified in the figure inset. In addition, since the radiometric signal amplitude decreases as the sample thickness increases, the actual cross-correlation peak values are significantly different between these samples (32% difference). The theoretical TWR cross-correlation peak delay times as a function of sample thickness are plotted in Figure 3d (solid line). Simulations suggest that the TWR output peak delay time decreases as the thickness increases. This is due to the fact that increasing the sample thickness dampens the contribution of thermal-waves from depths other than the surface, such as confined thermal-waves at the steel-air back interface. As a result, the ESD of the TWR signal becomes more similar to that of the semi-infinite sample and the TWR cross-correlation peak delay time approaches that of the semi-infinite sample (dotted line), as expected. The cross-correlation peak delay time of a semi-infinite sample has a non-zero value (39 ms) due to the well-known time shift of the surface temperature oscillation with respect to the incident thermal-wave flux [49]. The dashed line in Figure 3d represents the theoretical variation of the TWR subtractive mode cross-correlation \(\tau_{p}\) with sample thickness. This simulation suggests the capability of TWR in resolving delayed contributions from much deeper regions in the subtractive mode.