## 2 Methodology of the Inversion
### Direct Problem
It enables calculating the analytical solution \(\overline{G_{z}}\) of \(G_{z}\) in the Laplace domain:  
\[\overline{G_{z}}(z,p)=\frac{\cosh\left(z\sqrt{p/a}\right)\cosh\left((L-z^{ \prime})\sqrt{p/a}\right)}{\lambda\sqrt{p/a}\sinh\left(L\sqrt{p/a}\right)} \tag{6}\]  
The previous Equation (6) can be inverted numerically to find its original in temporal space. The numerical algorithms, such as Gaver-Stehfest, or a return of Laplace by Fourier [10] or using [PERSON]'s Invlap algorithm, can indeed be used. For the inversion, several models, including those defined by Equations (3), (4), (5) or (6), can be used. To compare the influence of each model along the \(z\) direction, three different configurations of heat sources are tested. As the \(x\) and \(y\) direction are considered infinite, the heat sources considered on the examples are an infinite plane along these dimensions, located at a given depth in the sample, as illustrated by Figure 2a. The first example is computed considering this heat source at \(z=L_{z}/4\) (near the measuring surface). On the second example, the source is located at \(z=L_{z}/2\) and for the last one, it is located at \(z=3L_{z}/4\) (deep source). The resulting temperatures \(\theta(z=0,t)\), are calculated by three different models (finite, semi-infinite and infinite along \(z\)), measured at the surface, and given by (b), (c) and (d), respectively, in Figure 2. For the three examples, \(Q_{0}=0.01\) J\(\cdot\)m\({}^{-3}\), \(a=2.5\times 10^{-7}\) m\({}^{2}\cdot\)s\({}^{-1}\) and \(pC_{p}=10^{6}\) J\(\cdot\)K\({}^{-1}\cdot\)m\({}^{-3}\). The study is made between \(t_{0}=0\) and \(t_{f}=L_{z}^{2}/a\). Whatever the depth of the heat sources, it can be seen that for short times, the temperature responses calculated by the three different models are exactly the same. The differences occur only for the long times. In this paper, all the works were done at short times. Therefore, the model used is the infinite model, given by Equation (3), because it is more convenient (analytically). However, the choice of the model is not decisive: the methodology that leads to the results is the same, and any models or numerical simulations can be used for the inversion. By introducing the following dimensionless data \(t^{*}=\frac{t}{t_{f}}\), \(i^{*}=\frac{t^{\prime}}{L_{i}}\) (\(i=x,y\) or \(z\)), as well as the Fourier number \(Fo_{i}=\frac{a_{i}t_{f}}{L_{i}}\), where \(t_{f}\) corresponds to the time of the experiment and \(L_{i}\) to the considered length of the material, Equation (3) can be rewritten in dimensionless form as follows:  
\[\theta_{\text{3D}}(x^{*},y^{*},z^{*},t^{*})=\frac{Q(x^{\prime*},y^{*\prime*}, z^{\prime*})}{8\rho C_{p}L_{x}L_{y}L_{z}}\frac{\exp\left(\frac{(x^{*}-x^{\prime*}) ^{2}}{4 Fo_{x}t^{*}}\right)}{\sqrt{\pi Fo_{x}t^{*}}}\frac{\exp\left(\frac{(y^ {*}-y^{\prime*})^{2}}{4 Fo_{y}t^{*}}\right)}{\sqrt{\pi Fo_{y}t^{*}}}\frac{\exp \left(\frac{(z^{*}-z^{\prime*})^{2}}{4 Fo_{z}t^{*}}\right)}{\sqrt{\pi Fo_{z}t^ {*}}}. \tag{7}\]  
Then for a given shape of a source the resulting temperature will be obtained by:  
\[\theta(r^{*},t^{*})=\int_{\text{sample}}Q(r^{\prime*})\theta_{\text{3D}}(r^{* }-r^{\prime*},t^{*})dr^{\prime}. \tag{8}\]  
In the case of the 3D problem, the whole material is discretized and each point of discretization is the representation of a voxel considered as a potential source, with a level of energy \(Q(x_{0},y_{0},z_{0})=Q_{0}\Omega(x_{0},y_{0},z_{0})\), which corresponds to the mean of the source energy on the defined voxel. The size of the voxels depends on the number of discretizations along each direction. The problem is then written in matrix form, given by Equation (9). The resulting surface temperature at \(z=0\) is a 2D field as a function of time. For the writing of the problem, each \(\theta(i)\) is a vector containing the 2D temperature reshaped into a 1D transient temperature:  
\[\theta_{\text{Dirac}}=\frac{Q(r^{\prime})}{\rho C_{p}}G_{3D}\left(r,t/r^{\prime}, t^{\prime}\right), \tag{1}\]0 Scheme of the sample along the \(z\) direction (**a**).