## 4 Results
### Validation of the Bayesian Approach  
In order to validate the findings from the study with the simulated images, the measured thermogram is now considered and superposed with noise (Figure 2d). Figure 6 compares the standard deviation \(\sigma_{\hat{A}}\) of determining the laminar-transition position on real measurement data with and without previous knowledge depending on the image quality. The standard deviation is lower with the use of previous knowledge for low CNRs, and the crossover point is at a CNR below 7, similar to Figure 4. The similarity validates the Bayesian inference as a method to lower the uncertainty of determining the position of the laminar-turbulent transition position on a noisy image. Figure 7 compares the standard deviation \(\sigma_{\hat{A}}\) of determining the laminar-turbulent transition position on real measurement data with and without previous knowledge depending on standard deviation \(\sigma_{\text{prior}}\) of the prior. In Figure 7 the standard deviation \(\sigma_{\hat{A}}\) is lower for determining the laminar-turbulent transition position with the use of pre-knowledge when the standard deviation of the prior is \(\sigma_{\text{prior}}\leq 3\). This validates the results from Figure 5 and, hence, validates the Bayesian inference as a method to lower the uncertainty when determining the position of the laminar-turbulent transition with an informative prior. Figure 6: Standard deviation \(\sigma_{\hat{A}}\) for determining the transition position on thermographic images superposed with Gaussian noise for different CNR values. Summarizing and combining both findings from Figures 6 and 7, the transition position uncertainty is lowered for images with a CNR below 7 and a normally distributed prior with a standard deviation smaller or equal to 3 px. If the prior has less quality or the image has a higher CNR, the Bayesian inference increases the uncertainty and should not be used. To compare the findings from Figure 7 with the expectations from estimation theory, the same data is shown again in Figure 8, but in a normalized manner. The ordinate is normalized by the standard deviation \(\sigma_{\hat{A}\text{ measurement}}\) of the detected transition position without previous knowledge. The abscissa is normalized by the characteristic prior standard deviation \(\widetilde{\sigma}_{\text{prior}}\), which is the point of intersection between the theoretical transition position standard deviation in case of a dominant previous knowledge (orange crosses: linear extrapolation through 0 and the first data point with minimal variance of the prior) and the transition position standard deviation with no previous knowledge (blue dots). For the studied example, \(\widetilde{\sigma}_{\text{prior}}=5.4\) px, which means that this prior is as informative as the measurement data. Hence, for a low standard deviation of the prior smaller than \(\widetilde{\sigma}_{\text{prior}}\), the prior information should dominate the measurement result and the standard deviation of the detected transition position asymptotically should attain the standard deviation of the prior (orange dotted line). This theoretical expectation agrees with the experimental data. For a high standard deviation of the prior larger than \(\widetilde{\sigma}_{\text{prior}}\), where the information from the measurement data dominates, the standard deviation of the determined transition position should asymptotically reach \(\sigma_{\hat{A}\text{ measurement}}\), i.e., the same value as for the non-Bayesian evaluation. Specifically, the normalized standard deviation of the transition position is expected to obey the relation [33]  
\[\frac{\sigma_{\hat{A}}}{\sigma_{\hat{A}\text{ measurement}}}=\frac{1}{\sqrt{\left(\frac{\widetilde{\sigma}_{\text{prior}}}{ \sigma_{\text{prior}}}\right)^{2}+1}}, \tag{17}\]  
This is shown as a solid orange line. However, the experimental results of the implemented Bayesian evaluation do not follow the theoretical expectation for an increasing width of the prior.