## 3 Generalization of the Bayesian Approach to 3D Source Reconstruction  
In this part, the idea is to go from the 1D problem to the 3D one, where the objective is to reconstruct the 3D thermal source(s) from the 2D surface temperature measurement. To do so, one can reproduce the same scheme as in Section 2.1. In the 3D orthotropic problem, each heat source diffuses in all directions, \(x,y\), and \(z\). The temperature evolution can then be written as follows:  
\[\theta_{\text{3D}}(x^{*},y^{*},z^{*},t^{*})=\frac{Q(x^{*s},y^{*s},z^{*s})}{L_{ x}L_{y}L_{z}/\rho C_{p}}\frac{\exp\left(-\frac{(x^{*}-x^{*s})^{2}}{4F_{\text{$ \sigma_{x}$}}t^{*}}\right)}{\sqrt{\pi F_{\text{$\sigma_{x}$}}t^{*}}}\frac{\exp \left(-\frac{(y^{*}-y^{*})^{2}}{4F_{\text{$\sigma_{y}$}}t^{*}}\right)}{\sqrt{ \pi F_{\text{$\sigma_{z}$}}t^{*}}}\frac{\exp\left(-\frac{(z^{*}-z^{\prime s})^{ 2}}{4F_{\text{$\sigma_{z}$}}t^{*}}\right)}{\sqrt{\pi F_{\text{$\sigma_{z}$}}t ^{*}}} \tag{11}\]  
with \(F_{\text{$\sigma_{x}$}}\), \(F_{\text{$\sigma_{y}$}}\), and \(F_{\text{$\sigma_{z}$}}\) as the Fourier coefficients in each direction. The 3D volume can then be discretized in \(n_{x}\times n_{y}\times n_{z}\) nodes, each one being a possible source whose density represents the density of energy in the elementary volume surrounding it. The surface temperature is also discretized in space and time. Finally, the 3D problem is given by the following:  
\[\left( \begin{array}{cc}\mathbf{A}_{\text{3D}}^{(1)}&\mathbf{A}_{\text{3D}}^ {(2)}\cdots&\mathbf{A}_{\text{3D}}^{(n)}\end{array} \right)\mathbf{\Omega}= \mathbf{\theta} \tag{12}\]  
In this case, the classical inversion method to reconstruct the sources is not tractable. Concerning the Bayesian inference, one could reproduce the same scheme as in Part 1 by looking for the probability repartition of sources in the 3D volume by calculating the likelihood \(P(\eta(x,y,z)|\theta(x,y))\). Nevertheless, the size of the problem can be prohibitive when the size of the volume and the surface become larger. To bypass this complexity, treating the 2D temperature field as \(n_{x}\times n_{y}\) independent 1D problems is proposed. Indeed, looking at the temperature only above the source allows us to limit the quantity  
Figure 5: 1D example of three distinct heat sources: (**a**) theoretical and reconstructed distribution of the sources, (**b**) theoretical, noisy, and residual resulting temperature at the surface. 1D example of an extended deep heat source: (**c**) theoretical and reconstructed distribution of the source, (**d**) theoretical, noisy, and residual resulting temperature at the surface. of data to process without loosing much information as almost all the information is contained in it. However, to take into account the 3D component and the interaction between the sources, in this case, once a source is detected, its whole 3D analytical effect given by Equation (11) is subtracted from the measured temperature field \(\theta_{mes}(x,y)\):  
\[\theta^{*}(x,y,t)=\theta_{mes}(x,y,t)-\mathbf{A}_{3D}\Omega_{S_{1}}(x,y,t) \tag{13}\]  
As an example, let us consider a problem of 3 point sources \(p_{1}(5,3,27)\), \(p_{2}(11,11,11)\) and \(p_{3}(18,17,20)\) placed in a volume discretized in \(n_{x}=20\), \(n_{y}=20\), and \(n_{z}=30\), as illustrated in Figure 6.  
Figure 7 illustrates different steps of the reconstruction algorithm. Figure 7a represents the 400 measured temperature as function of time, which constitutes the time evolution of the surface temperature \(T_{surf}(x,y,t)\). Figure 7b represents the points where the probability of finding a source is superior to 99%. As explained earlier, the first sources that are detected are the ones closer to the surface. Therefore, the first source that will be detected for this example should be \(p_{1}\), then \(p_{3}\), and finally \(p_{2}\). In Figure 7b, the closest point to the surface is indeed placed at \(x=5\), \(y=3\), and \(z=27\), where the probability almost reaches 1. This point corresponds exactly to \(p_{1}\).