## 4 Experimental Validation of a Volumetric Hot Wire Warmed by the Joule Effect
At each time step, a probability is calculated on each point of the sample using the theoretical model given by Equation (16):  
\[\theta_{th}(t)=\int_{t_{0}}^{t}\left[\frac{Q}{\rho C_{p}}\frac{\exp\left(-\frac {L_{z}^{2}}{4\pi t}\right)}{\sqrt[3]{\pi a\tau}}\right]d\tau \tag{16}\]  
where \(L_{z}\) (m) is the position at the depth where the thermal source is located, and \(Q\) (W) is the intensity (power) tested (W\(\cdot\)m\({}^{-3}\)). Equation (16) corresponds to the analytical solution of the thermal response at the level measured at the surface just above a source located at depth \(L_{z}\) for a 3D model. **Hypothesis 1**. : _For this study, it is assumed that the intensities of the thermal sources generated by the wire are identical and constant at all points._  
Under this hypothesis, the problem can be written in such a way as to search only for the position of the thermal sources (and not their intensities). To do this, we normalize the temperature fields, the measured one (spatio-temporal) and the theoretical one given by the Equation (16) to have an evolution between 0 and 1. The time interval is between \(t=0\) and \(t_{f}=2\) s. All temperature values are weighted by the maximum temperature value indicated at time \(t_{f}\). It is important to note that the temperature profiles in each pixel are conserved. The discretization of the sample along the \(x\) and \(y\) directions is given by the IR camera (pixel of 290 \(\upmu\)m \(\times\) 290 \(\upmu\)m), which leads to \(n_{x}=126\), \(n_{y}=72\). The discretization along the \(z\) direction is defined as \(n_{z}=16\)[11; 12]. The probabilistic approach reconstruction algorithm is then applied, and the solution obtained is illustrated in Figure 10, representing the probability of the presence of the source at each point of the sample. The solution given by the program is a 3D matrix, which represents the volume of the sample, where each point of the matrix contains the probability (>98%) of the presence of a source at this point. The depths of the sources are well found with this method on average; the wire is reconstructed on the plane parallel to the surface, in the center of the plate. Regarding the reconstruction in the \(x\) and \(y\) directions, the wire is rebuilt thicker than it actually is, especially in areas where the temperature variation is greater. This error is due to the fact that the algorithm considers a constant intensity at any point of the wire. It can also be noted in Figure 10a that in these areas, the wire is reconstructed at a slightly higher position. Finally, an important point to highlight is the execution time of the program. The calculation, run on a laptop computer (the least powerful of the computers tested) lasted 8.35 s. To compare, in [12], the proposed reconstruction method based on SVD and regularization allows estimating the sources for the same experimental problem. Even if a reduction of the study domain is applied, the execution  
Figure 10: Wire reconstruction using the Bayesian inference method: (**a**) profile view, (**b**) front view. time of the program is particularly long, more than 6 h are necessary to reconstruct the same sources. The Bayesian approach is therefore definitely faster than a deterministic approach.