## 2 Description of the Method through a 1D Example
### Inverse Processing Method Based on Bayesian Inference  
In the probabilistic framework, the idea is to reconstruct the repartition of the probability \(\eta\) of finding a source at each position \(i\) knowing the temperature evolution at the surface \(\theta_{(j)}\), \(j=1...m\) using Bayesian inference based on the Bayes relations [13]:  
\[\eta(\Omega|\theta)=\frac{P(\theta|\Omega)\cdot P(\Omega)}{P(\theta)} \tag{1}\]  
where \(\eta(S|\theta)\) is called the posterior--the probability of source repartition \(S\) knowing the surface temperature \(\theta\)--\(P(\Omega)\) is the prior, \(P(\theta|\Omega)\) is the likelihood--the probability of obtaining \(\theta\) knowing \(S\)--and \(p(\theta)\) is the model evidence. In our case, the prior, which allows an _a priori_ knowledge of \(\eta\) to be taken into account, is assumed to be irrelevant; the probability of finding a source is the same for all depths. This could be changed, for example, in the case of defect detection in a composite, where we could assume that the defects are more likely to be present at the layer interfaces. Therefore, in our case, the posterior can be written as follows [14]:  
\[P(\Omega|\theta)\propto\exp\left(-\frac{1}{2\Gamma^{2}}||\theta-\theta_{forward }(\Omega)||_{2}^{2}\right). \tag{2}\]  
where \(\theta_{forward}(\Omega)\) is the surface temperature evolution depending on the source repartition \(\Omega\). This temperature can come from experimental observations made for different source repartitions, or can be computed for different \(\Omega\) from a model that can be analytical or numerical. The choice of this model--called forward or direct model as it is representative of the direct problem--brings some remarks: (i) the model used does not need to mimic perfectly the reality as it is used in a probabilistic framework and only the maximum probability is considered but it has to be pertinent enough to bring information on the sources position, (ii) in order to be used efficiently, \(\theta_{forward}\) has to be known for all the possible source repartition \(\Omega\), then, its computation has to be less time consuming as possible. Equation (2) literally writes: if the sources position is well represented by \(\Omega\) then \(\|\theta-\theta_{forward}(\Omega)\|_{2}\approx 0\) and then the probability of \(\Omega\), \(P(\Omega|\theta)\approx 1\); if not, \(\|\theta-\theta_{forward}(\Omega)\|_{2}>>1\) and then \(P(\Omega|\theta)\approx 0\). Then, in order to identify the source repartition, one has to identify the \(\Omega\) maximizing the likelihood in Equation (2). where \(t^{\star}=\frac{t}{t_{f}}\), \(z^{\star}=\frac{z}{L_{z}}\), \(z^{\prime}{}^{\star}=\frac{z^{\prime}}{L_{z}}\), \(L_{z}\) is the thickness of the sample and \(Fo_{z}=\frac{a_{z}t_{f}}{L_{z}^{2}}\) is the Fourier coefficient. Then, the evolution of the temperature at the surface is given by the following:  
\[\theta_{1D}(0,t^{\star})=\frac{1}{L_{z}\sqrt{\pi Fo_{z}t^{\star}}}\exp\left(- \frac{(z^{{}^{\prime}\star})^{2}}{4 Fo_{z}t^{\star}}\right) \tag{4}\]  
To compute the temperature evolution at the surface for any position of sources in the material thickness and then compute the likelihood of Equation (2), the problem is discretized in space and time. A source vector \(\mathbf{\Omega}\) is defined by discretizing the material thickness with \(n-1\) elements, and \(m\) time steps are defined between \(t_{0}\) and \(t_{f}\) to sample \(\mathbf{\theta}\), the measured thermal field at \(z=0\). Then, the reconstruction problem can be written as in Equation (5):  
\[\left(\mathbf{A}_{1D}^{(1)}\mathbf{A}_{1D}^{(2)}...\mathbf{A}_{1D}^{(n)} \right)\mathbf{\Omega}=\mathbf{\theta} \tag{5}\]  
where each point of the matrix \(\mathbf{A}\) is calculated with Equation (4) [11]. The choice of the thickness discretization requires some remarks:  
* The size of the discretization is directly correlated with the intensity of the source.