## 3 Sensitivity
However, if we look at the sensitivities of the temperature evolution at the origin \(T(0,0,0,t)\) to the height and the depth (Figure 5e), we notice that the maximum sensitivities to \(h\) and \(d\) occur at different instants and thus these parameters are not correlated in the timing-graph at the origin. It is also remarkable that, because the burst is short (\(\mu_{\tau}<d_{max}\)), as already mentioned, the sensitivity of the spatial information to \(h\) is very low (Figure 5b), whereas the timing-graph at the origin features a much higher sensitivity, which appears after the end of the burst. It is worth noting that the previously mentioned correlation between \(Sh\) and \(Sd\), only occurs for this challenging case of a vertical crack. In Figure 5f, we show \(OX\) profiles of \(Sd\) and \(Sh\) calculated for the same \(w=h=1\) mm crack buried \(d=0.5\) mm below the surface and excited with the same burst, \(\tau=0.5\) s, but making an angle of \(\varphi=30^{\circ}\) with the surface. The displacement of the maxima of the curves clearly demonstrates the lack of degeneracy between the two magnitudes. Moreover, a comparison between the sensitivity values of \(Sd\) and \(Sh\) in Figure 5f, for \(\varphi=30^{\circ}\), and in Figure 5b,c for \(\varphi=90^{\circ}\), illustrates the fact that the sensitivities to the lateral dimensions and depth of the heat source increase when the inclination decreases. Finally, it is worth noting that, regarding the thermogram at the end of the burst, \(T(x,y,0,\tau)\), \(OX\)- and \(OY\)- axes summarize all relevant sensitivity information on the four parameters: \(OX\)-axis on \(Sw\), \(OY\)-axis on \(S\varphi\), and both \(OX\)- and \(OY\)-axes, on \(Sh\) and \(Sd\). According to this sensitivity analysis, we can conclude that gathering the temperature profiles obtained at the end of the burst along \(OX\)- and \(OY\)-axes (\(T(x,0,0,\tau)\) and \(T(0,y,0,\tau)\)) and the timing-graph at the origin (\(T(0,0,0,t)\)), we combine spatial and temporal information for which the sought parameters \(w\), \(d\), \(h\), and \(\varphi\) are not correlated and thus we should be able to retrieve the geometrical characteristics of the crack univocally. Furthermore, this reduction of data enables a computationally efficient fitting and thus, a fast characterization of the heat source. We focus now on the sensitivity to the angle. The definition of sensitivity given by Equation (12) was useful to compare sensitivities to different parameters (Figure 5). However, in order to compare the sensitivity of the surface temperature to the angle for different orientations of the heat source, the definition is not appropriate because the product of the derivative by the actual angle penalizes low angles. In the limit, for a horizontal heat source, the definition gives zero sensitivity, which is nonsense. Accordingly, in order to compare the sensitivities to the angle for different angles, we introduce a new definition:  
\[S^{\prime}=\frac{\partial T}{\partial\varphi} \tag{13}\]  
In Figure 6a, we represent \(S^{\prime}\) along the \(OX\)-axis at the end of a \(\tau=5\) s burst (\(\mu_{\tau}=1.6\) mm) for the same standard heat source (\(w=h=1\) mm) buried \(d=0.5\) mm below the surface emitting a flux of \(\Phi=40\) kW/m\({}^{2}\), and several inclinations, namely, \(\varphi=0^{\circ}\), \(10^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\). For \(\varphi=90^{\circ}\), the maximum depth of the heat source is \(d_{max}=1.5\) mm, of the same order as \(\mu_{\tau}\). The sensitivity is maximum for the horizontal heat source, \(\varphi=0^{\circ}\) and minimum for the vertical source, \(\varphi=90^{\circ}\), with a relationship of \(10/1\). However, if the duration of the burst corresponds to a thermal diffusion length \(\mu_{\tau}\) significantly shorter than \(d_{max}\), then the maximum to minimum sensitivity relationship is much larger.