## 3 Highlighting an Optimal Criterion Based on Fourier Number
### Analysis of the Criterion
As each mode value provides particular information, it means that all modes after \(n_{\phi}\) do not contribute to the solution of the problem and are therefore non-significant for the reconstruction of heat sources. The comparison between Figure 5,d  
Figure 4: Representation of the 1D problem with 3 heat sources of intensity 0.5, 1 and 0.75 (a) and calculated temperature response at z = 0 (b). enable observing that there is a direct correlation between the significant number of singular values and the optimal number of discretizations along the in-depth direction. This optimal number \(n_{\phi}\) depends on the parameters of the model, such as the temporal discretization, linked to \(n_{t}\) and the Fourier number \(Fo_{z}\). A parametric study has been computed to determine \(n_{\phi}\) according to these parameters. The obtained results are illustrated in Figure 6a. It appears that the parameter \(n_{\phi}\) depends strongly on the Fourier number. The higher \(n_{\phi}\) is, the thinner the size is of the voxel in the depth. Therefore, for Fourier numbers below 0.01, as well  
Figure 5: Estimated source profile for a discretization of \(n=20\) (**a**). Estimated source profile for \(n=50\) (**b**). Logarithm representation of the singular values for a discretization of \(n=100\) (**c**) and error on the reconstructed sources as function of the number of singular values (**d**). Figure 6: Abacus of the number of singular values (i.e., \(z\) discretization number of nodes) as a function of the Fourier number and temperature response time step: for Dirac excitation (**a**) and for Heaviside excitation (**b**): representation in log-log scale. as high Fourier numbers, it will be very difficult to reconstruct the sources with a high resolution. At most, it will be possible to reconstruct 10 sources. For a given Fourier number, increasing the number of time steps \(n_{t}\) enables increasing the in-depth resolution (with the \(n_{\phi}\) parameter). However, this increase is made with a logarithmic rate, which means that a large number of temporal data will be necessary in order to gain a few numbers of discretization in depth. This abacus enables defining the optimal parameters (\(n_{t},n_{z},t_{f}...\)) in order to correctly reconstruct the heat sources. It is important to note that the criterion given in Figure 6a depends neither on the material nor on the thickness of sample. This criterion depends only on the Fourier number, which is universal, and on the number of time steps, which is numerical, for a Dirac thermal impulse response. To see the influence of temporal excitation, the same study has been performed with a continuous temporal heat source. The resulting thermal response is simply the temporal integral of the one resulting from Dirac given by Equation (7):  
\[\theta_{\text{Hvs}}(x,y,z,t)=\int_{t_{0}}^{t}\frac{Q(x^{\prime*},y^{\prime*},z ^{\prime*})}{8\rho C_{P}L_{x}L_{y}L_{z}}\frac{\exp\left(\frac{(x^{*}-x^{\prime* })^{2}}{4F\omega_{x}t^{*}}\right)}{\sqrt{\pi F\omega_{x}t^{*}}}\frac{\exp\left( \frac{(y^{*}-y^{\prime*})^{2}}{4F\omega_{y}t^{*}}\right)}{\sqrt{\pi F\omega_{z }t^{*}}}\frac{\exp\left(\frac{(z^{*}-z^{\prime*})^{2}}{4F\omega_{z}t^{*}} \right)}{\sqrt{\pi F\omega_{z}t^{*}}}d\tau \tag{20}\]  
The methodology used for the direct and inverse problem of the algorithm is the same as that for the Dirac excitation. However, the operator matrix \(\mathbf{I}\) is different. The results of the parametric study for Heaviside excitation are presented in Figure 6b. These results have the same tendency as those obtained with a Dirac response. Analysis of the Influence of Each Mode  
The criterion given in Figure 6 enables choosing the optimal discretization in depth (\(n_{z}=n_{\phi}\)). Once this parameter is defined, the operator matrix \(\mathbf{I}\), given by Equation (10), can be computed.