## 3 Sensitivity  
In order to compare sensitivities to different parameters, we define the sensitivity of the surface temperature to a given parameter \(p\) as:  
\[Sp=p\frac{\partial T}{\partial p},\ \ p=w,\ h,\ d,\ \varphi \tag{12}\]  
Figure 4: (**a**) Surface temperature \(OX\)-profiles calculated at the end of a \(\tau\) = 1 s burst, produced by a square (\(w\) = \(h\) = 2 mm) heat source emitting a flux of 10 kW/m\({}^{2}\) at an angle of \(\varphi\) = 20\({}^{\circ}\), for three different depths: \(d\) = 0.001, 0.2, and 0.6 mm. (**b**) \(OX\)-profiles calculated at the end of three bursts (\(\tau\) = 0.1, 1, 10 s) for a heat source of the same dimensions, orientation, and flux as in (**a**) buried \(d\) = 0.01 mm. In the following, we analyze the sensitivity of the surface temperature to the parameters characterizing the geometry and location of the heat source, namely, the width \(w\), the height \(h\), the depth \(d\), and the angle \(\varphi\). For the sensitivity analysis, we consider a standard squared heat source of dimensions \(w=h=1\) mm. In order to highlight the influence of the burst duration \(\tau\), we consider a challenging case: a vertical heat source (\(\varphi=90^{\circ}\)), buried below the surface \(d=0.5\) mm, and excited with a short \(\tau=0.5\) s burst (\(\mu_{\tau}=0.51\) mm). Figure 5a-d displays the sensitivities \(Sw\), \(Sh\), \(Sd\), and \(S\varphi\), calculated at \(t=\tau=0.5\) s, for \(\varPhi=40\) kW/m\({}^{2}\) (total emitted power 40 mW as in Figures 2-4). As can be seen, the highest sensitivity corresponds to the width, as the width information is available at the shallowest positions of the crack. The maximum values of this sensitivity appear at the location of the lateral ends of the square. The second most sensitive parameter is the angle which, for \(\varphi=90^{\circ}\), exhibits a bi-modal shape in the direction of the inclination (\(OX\)-axis): as the angle increases, the temperature increases for \(x\leq 0\) and decreases at positive values of the \(x\) coordinate. Figure 5: Sensitivity of the surface temperature distribution at the end of the burst calculated for a \(\tau=0.5\) s burst and a squared (\(w=h=1\) mm) vertical (\(\varphi=90^{\circ}\)) heat source buried \(d=0.5\) mm below the surface, emitting a flux of 40 kW/m\({}^{2}\). (**a**) Sensitivity to the width \(Sw\), (**b**) sensitivity to the height \(Sh\), (**c**) sensitivity to the depth \(Sd\), and (**d**) sensitivity to the angle \(S\varphi\). (**e**) Sensitivity of the temperature evolution at the origin to the height \(Sh\) (red) and to the depth \(Sd\) (brown). (**f**) \(Sd\) (blue) and \(Sh\) (red) along the \(OX\)-axis for a heat source of the same dimensions and depth and \(\tau\), but making an angle of \(\varphi\) = \(30^{\circ}\) with the surface. The next parameter in decreasing sensitivity order is the depth, which is negative as the temperature at the surface decreases for increasing depth. The height of the heat source is the less sensitive parameter, as it corresponds to changes at positions located at the maximum depth. For the particular situation analyzed, the deepest end of the heat source is buried \(d_{max}=1.5\) mm below the surface, whereas the thermal diffusion length associated to the end of the burst is \(\mu_{\tau}=0.51\) mm, as mentioned above. This is the reason why the values of \(Sh\) are so low. Very interestingly, the positions where the maximum sensitivities occur for the different parameters are well separated spatially, except for the height and the depth, for which the maximum sensitivity is located around the origin, in this configuration. Accordingly, we can conclude that the width \(w\) and angle \(\varphi\) are decorrelated with each other and also with height \(h\) and depth \(d\), but these two might be correlated, as far as the surface temperature at the end of the burst is concerned.