## 2 Description of the Method through a 1D Example
### Reconstruction of Multiple Sources Using Bayesian Inference  
Previously, it was assumed that a unique source was present in the thickness. Then, the likelihood was computed using the response of a unique source at different depths. Now, if \(n\) potential sources are assumed in the thickness, the number of possible temperature fields \(N_{\Omega}\) to be used in the likelihoods calculation is as follows:  
\[N_{\Omega}=2^{n}. \tag{9}\]  
Then, following the same idea described previously, for each likelihood, one would have to compare the measured temperature with the \(2^{n}\) possible temperature fields. For example, for 25 sources, this corresponds to \(2^{25}\approx 10^{7}\) possibilities.  
To overcome this problem, Markov chain Monte-Carlo methods are commonly used [14; 16; 17; 18]. These stochastic methods allow for the simulation of a distribution via a Markov chain: a random source repartition is initially proposed to calculate the likelihood. Among its neighboring repartitions, the most probable one is retained. Step by step, the method converges toward the solution that corresponds to the data. However, these methods are cumbersome, as the number of tested cases remains high.  
To circumvent this, using the superposition principle of the thermal response is proposed, stemming from the linear characteristic of the problem. If \(\theta_{1}(t)\) and \(\theta_{2}(t)\) are, respectively, the resulting temperature field of sources \(S_{1}\) and \(S_{2}\), then the resulting temperature field of \(S_{1}+S_{2}\) is \(\theta(t)=\theta_{1}(t)+\theta_{2}(t)\). Moreover, the sources are detected from the one closest to the surface to the deepest one, allowing for sequential detection.  
The principle of the method is then to work by time steps; as soon as the position of a source is located \(\Omega(S_{1})\), its analytical contribution is subtracted from the measured temperature field \(\theta_{mes}\) using Equation (10):  
\[\theta^{\star}(t)=\theta_{mes}(t)-\mathbf{A}\Omega(S_{1}) \tag{10}\]  
The source detection then goes on using the corrected surface temperature \(\theta^{\star}(t)\) until the time interval has elapsed and all the sources have been detected. At the end of the algorithm, if all the sources have been correctly detected, the residual temperature \(R=\theta_{mes}-\sum_{i=1}^{N_{sources}}\mathbf{A}\Omega(S_{i})\) must be on the order of magnitude of the noise.  
Figure 5 illustrates the reconstructed sources obtained with the proposed algorithm starting from the noisy temperature for two different 1D examples. Figure 5a,b shows the results for the first example, whereas Figure 5c,d correspond to the second example. Graphs in Figure 5a,c represent the theoretical sources as well as their reconstructions with the Bayes algorithm. Graphs in Figure 5b,d simultaneously illustrate the theoretical, the noisy, and the residual temperatures at the end of the algorithm. If the algorithm fails in reconstructing a source, as is the case for the second example (Figure 5c), one can see that the residual temperature presents a discrepancy that provides information regarding the error (Figure 5d).  
By defining this residue using a floating window, as the time measurement is directly linked to the depth in the material, one can detect the position of the error and correct it in an iterative fashion until reaching a residue that is on the order of magnitude of the noise, as is the case in the graph in Figure 5b.