## 3 Layer Thickness Determination
### Unknown Thermal Properties and Issues with All-at-Once Optimization  
The approach (Equation (25)) seems unusable for real-world applications as the thermal properties of every coating layer must be known. For example, in the automotive industry, paint mixtures are changed occasionally, and it does not make sense to perform a complex thermal analysis each time. In general, what is available for the calibration of any nondestructive testing device are different samples of a fixed multi-layered coating setup with varying but known coating layer thicknesses. These are typically measured either in a destructive (e.g., cross-sectional images under the microscope, etc.) or non-destructive (e.g., laser triangulation, X-ray, etc.) way. Whether this coating layer thickness information of every layer in every sample is gathered before or after the frequency scans and phase angle data collection with an infrared camera depends on the method in use. However, it is worth noting that this process only needs to be performed once for each coating system setup. In the following, a concept to identify the needed thermal properties in order to determine \((L_{1},\ldots,L_{n})\) of such coating systems is presented. As a first idea, one can replace the functional from Equation (25) with  
\[\min_{(\mathbf{p},\mathbf{L})}||F_{n}(\mathbf{p},\mathbf{L})-\mathbf{\varphi}_{\mathbf{\hat{T}},meas} ||_{2}^{2}, \tag{26}\]  
which represents an all-at-once approach, i.e., all unknown parameters are determined simultaneously. Unfortunately, this approach shows the following ambiguity issue: For every \(j=1,\ldots,n\), terms of the form  
\[\exp(-2\sigma_{j}L_{j})=\exp(-2(1+i)\sqrt{\frac{\omega}{2\alpha_{j}}}L_{j})\]  
imply  
\[\frac{L_{j}}{\sqrt{\tilde{\alpha}}j}=\frac{\tilde{c}\cdot L_{j}}{\sqrt{\tilde {c}^{2}\cdot\tilde{\alpha}_{j}}}\text{ for }\tilde{c}>0,\]  
i.e.,  
\[||F_{n}(\mathbf{p},\mathbf{L})-\mathbf{\varphi}_{\mathbf{\hat{T}},meas}||_{2}=||F_{n}(\tilde{ c}^{2}\mathbf{p},\tilde{c}\mathbf{L})-\mathbf{\varphi}_{\mathbf{\hat{T}},meas}||_{2},\]  
since the thermal contrast between the layers and, thus, the reflection coefficients are identical. Hence, it is necessary to decouple the determination of \(\mathbf{p}\in\mathbb{R}^{2n+1}_{+}\) and \(\mathbf{L}\in\mathbb{R}^{n}_{+}\). This can be done by utilizing the sample data in a clever way. Assume that \(k\in\mathbb{N}\) samples (\(n\) coating layers with different but known \(L_{j}:=(L_{1,j},\ldots,L_{n,j})^{T}\) for \(j=1,\ldots,k\)) are available for the calibration process. Modulation of every sample surface by the frequencies \(\mathbf{\omega}\in\mathbb{R}^{m}_{+}\) leads to a total amount of \(m\times k\) phase angle values, i.e.,  
\[\left(\mathbf{\varphi}_{\mathbf{\hat{T}},meas}\right)_{j=1,\ldots,k}.\]For \(1<k_{1}<k\), the entire batch of sample data is divided in two sets  
\[\underbrace{S_{1}=\left(\mathbf{\varphi}_{\mathbf{\hat{T}},\text{meas}_{j}}\right)_{j=1, \ldots,k_{1}}}_{\text{Training or calibration set}}\text{ and }\underbrace{S_{2}=\left(\mathbf{\varphi}_{\mathbf{\hat{T}},\text{meas}_{j}}\right)_{j= k_{1}+1,\ldots,k}}_{\text{Test or confirmation set}} \tag{27}\]  
and proceeded with the following steps. STEP 1: Determine thermal properties from \(S_{1}\), i.e.,  
\[\bar{\mathbf{p}}:=\operatorname*{arg\,min}_{\mathbf{p}\in\mathbb{R}^{n+1}_{+}}\sum_{j= 1}^{k_{1}}||F_{n}(\mathbf{p},\mathbf{L}_{j})-\mathbf{\varphi}_{\mathbf{\hat{T}},\text{meas}_{j }}||_{2}^{2}.\]  
STEP 2: Feed the thermal properties from STEP 1 into the objective functional and determine the coating layer thicknesses of sample \(j\) by  
\[\bar{\mathbf{L}}_{j}:=\operatorname*{arg\,min}_{\mathbf{L}\in\mathbb{R}^{n}_{+}}||F_{ n}(\bar{\mathbf{p}},\mathbf{L})-\mathbf{\varphi}_{\mathbf{\hat{T}},\text{meas}_{j}}||_{2}^{2}\]  
for \(j=k_{1}+1,\ldots,k\).