## 3 Highlighting an Optimal Criterion Based on Fourier Number
### 3D Generalisation  
For the 3D problem, all Fourier numbers (\(Fo_{x}\), \(Fo_{y}\) and \(Fo_{z}\)) have to be considered. Since the \(x\) and \(y\) directions are symmetrical, the analysis of the problem represented by Figure 8 is sufficient to have access to the 3D criterion. In this study, the heat sources \(\Omega\) are distributed on the plane \(\pi(x=0)\), perpendicular to the surface (a). This plane is meshed along the \(y\) and \(z\) directions, with \(n_{y}\) and \(n_{z}\) discretization points, respectively (b).  
Figure 7: Standard error calculated between the reconstructed source and the actual source depending on the number of modes used for reconstruction (**a**). Illustration of the reconstructions obtained by taking the first 15 modes (**b**), first 27 modes (**c**) and the optimal number of modes 28 in this case (**d**).  
As explained in Section 3.1, \(n_{z}\) is limited and can be determined via the analysis of the SVD of the matrix operator. In (a) is illustrated the singular values obtained for \(Fo_{z}=0.1\), \(Fo_{y}=1\times 10^{-3}\) and \(L_{y}=2L_{z}\). Other parameters remain unchanged. The disposition of singular values is peculiar: they are disposed in decreasing steps. Each step contains exactly \(n_{y}\) singular values. As in the 1D problem, the last singular values reach a plateau and are then non-significant, and the number of steps corresponds to the maximal number \(n_{z}\) of discretizations. This distribution of singular values can be explained by the block repetitiveness in the operator matrix \(\mathbf{I}\). The limitation therefore does not come from \(n_{y}\) but from \(n_{z}\).  
Two principal observations can be made. First, the evolution of the criterion in the function of \(Fo_{z}\) is exactly the same as in the 1D problem. Second, contrary to what we could have expected due to separability of Equation (7), the evolution of the criterion with \(Fo_{y}\) is not the same as that with \(Fo_{z}\). This evolution is represented in Figure 9b, for \(Fo_{z}\) chosen at 0.1. In Figure 9c, the same criterion is represented, but for a number of time steps set at \(n_{t}=300\), with varying \(Fo_{y}\) and \(Fo_{z}\).  
Figure 8: Schema of problem in 3D with a plane perpendicular to the measured surface (**a**) and parameters of the plane discretization (**b**).  
Figure 9: Singular values of the 3D matrix I of Equation (12) (**a**), abacus along the \(y\) direction for the Fourier number equal to 0.1 in the \(z\) direction (**b**) and abacus between Fourier numbers along the \(y\) and \(z\) directions for a number of time steps of the temperature response equal to 300 (**c**).  
The 3D reconstruction is optimal for the smallest \(Fo_{y}\) and a reasonable number of time steps. The more \(Fo_{y}\) increases, the less the discretization can be thin. Physically, a high \(Fo_{y}\) means a fast diffusion along the \(y\) direction. Therefore, the temperature at the surface due to a buried point source will be largely spread for high \(Fo_{y}\). It can be compared to blurred information: the sources can be estimated with a low discretization. In contrast, for a small \(Fo_{y}\), the surface temperature will be more focused because diffusion along the \(y\) direction is almost non-existent. In this case, heat sources can be estimated with a thin discretization.  
Thus, optimal reconstruction of 3D buried heat sources is obtained for small Fourier numbers along the \(x\) and \(y\) directions and a Fourier number between 0.01 and 1 along the \(z\) direction.