## 4 Fittings of Synthetic Data  
In this section we present fittings of Equations (10) and (11) to synthetic \(T\)(x,0,0,\(\tau\)), \(T\)(0,y,0,\(\tau\)) and \(T\)(0,0,0,\(t\)) data calculated for rectangular heat sources of different orientations in a thermal isolator (\(D\) = 0.13 mm\({}^{2}\)/s, \(K\) = 0.5 W/mK) with added Gaussian noise with zero mean and normal distribution. The fitting parameters are \(\Phi\), \(w\), \(h\), \(d\), and \(\varphi\). We analyze the effect of the signal to noise ratio in the data and of the duration of the burst on the accuracy of the retrieved values of the five fitting parameters. We start considering a standard \(w\) = \(h\) = 1 mm square heat source, buried \(d\) = 0.5 mm below the surface at an intermediate inclination of \(\varphi\) = 45\({}^{\circ}\). The maximum depth of this heat source is \(d_{max}\) = 1.2 mm. In Figure 8 we present synthetic data calculated for three bursts \(\tau\) = 0.5, 2, and 10 s, corresponding to thermal diffusion lengths of \(\mu_{\tau}\) = 0.51, 1, and 2.3 mm respectively. In order to separate the effect of the noise in the data from the mere effect of the duration of the burst \(\tau\), all data sets feature the same signal to noise ratio, SNR = 60 at \(T\)(0,0,0,\(\tau\)), i.e., \(T\)(0,0,0,\(\tau\)) = 3 K and noise of 50 mK. As can be seen, for a SNR of 60 the fittings provide sensible values of the geometrical parameters defining the crack in the whole burst range, which confirms the adequacy of the selected information to characterize the heat source. However, the quality of the results does depend on the SNR. As an illustration, in Figure 9 we present best fit parameters and standard deviations corresponding to the same heat source as in Figure 8 (\(w\) = \(h\) = 1 mm, \(d\) = 0.5 mm, \(\varphi\) = 45\({}^{\circ}\)), excited with a \(\tau\) = 5 s burst, obtained from data featuring different values of the SNR. For this \(\tau\), the thermal diffusion length associated to the end of the burst, \(\mu_{\tau}\) = 1.6 mm, largely covers the maximum depth of the heat source \(d_{max}\) = 1.2 mm. In all cases, both the accuracy but especially the precision of the estimated parameters decrease with decreasing SNR. However, the plots show different responses of the fitting parameters to noise. If we focus on the estimated \(w\), \(h\), and \(d\) (note that the vertical scales on Figure 9a-c span the same parameter range), the height \(h\) is the parameter most affected by noise, with an uncertainty that varies from 20% to 1% for SNRs ranging between 15 and 150. The depth and the width exhibit similar relative uncertainties, although the accuracy is higher for the depth estimation in all SNRs. This might be due to the effect of the relatively long burst, that washes out the signature of the width. Finally, regarding the estimation of the angle (Figure 9d), the accuracy is very stable but the increase in uncertainty with decreasing SNR is the most pronounced. As a summary, the results show that even for SNR as low as 15, the accuracy of the retrieved parameters is better than 10%. These general trends are similar for other depths, with reduced uncertainty and increased accuracy for shallower heat sources. Let us analyze now the effect of the angle of the heat source on the quality of the retrieved parameters. In Figure 10, we present values of the retrieved \(w\), \(h\), \(d\), and \(\varphi\) for the same standard heat source \(w=h=1\) mm, buried \(d=0.5\) mm below the surface, as a function of the inclination \(\varphi\), also excited with a \(\tau=5\) s burst. The signal to noise ratio of the data is SNR = 60. Figure 8: (**a**) \(T\)(\(x\),0,0,\(\tau\)), (**b**) \(T\)(0,\(y\),0,\(\tau\)), and (**c**) \(T\)(0,0,0,\(t\)) corresponding to a square heat source (\(w=h=1\) mm) buried \(d=0.5\) mm, and making an angle \(\varphi=45^{\circ}\) with the surface calculated for three bursts \(\tau=0.5\), \(2\), and \(10\) s, with added noise (symbols) and fittings of Equations (10) and (11) (solid lines). SNR = 60 in all cases.