## 4 Results
### Verification of the Bayesian Approach  
As a first indicator for the success of the Bayesian inference, the posterior distribution (Markov chain) of the transition position \(p(A|x)\) for the simulated image with noise and an informative prior with a mean value of 361 px and a standard deviation of 3 px is visually inspected and shown in Figure 3.  
The Bayesian posterior distribution of the proposed transition positions resembles a normal distribution with a mean value of 361.07 px and a standard deviation of 4.66 px. Indeed, a normal distribution with mean and standard deviation similar to the prior is expected, because, due to the high quality of the prior, the previous knowledge dominates the measurement data. Repetitive measurements show that the considered informative  
Figure 2: Thermographic images: Simulated images with no (**a**) and with (**b**) added white Gaussian noise (AWGN). Measured thermograms with no (**c**) and with (**d**) added noise. prior enables the determination of the transition position (\(\hat{A}=361.07\) px) with a standard deviation of \(\sigma_{\hat{A}}=0.25\) px for a simulated image with a very low CNR of 1.1. Additionally, this standard deviation, which is achieved with previous knowledge, is lower than what is achieved with no previous knowledge (1.62 px). The result is not surprising because the prior itself contains the true transition position with low uncertainty. However, the result verifies the Bayesian approach as a method for including previous knowledge into the thermogram evaluation for the determination of the laminar-turbulent transition position on rotor blades on wind turbines. Next, the Bayesian method is evaluated with respect to the influence of the image quality, i.e., the CNR of the image. Repetitively, the transition position is determined with and without the use of previous knowledge for simulated images with different CNRs, which are created through a change in the standard deviation of the added Gaussian noise. The used prior is normally distributed with the mean at the true position and a standard deviation of 3 px. As the added Gaussian noise has zero mean and is symmetric, it does not add any systematic error, and the obtained standard deviation of the flow transition position shown in Figure 4 (orange dots) can be interpreted as the standard measurement uncertainty. The quality of the reference signal evaluation without the use of previous knowledge (blue dots) depends on the CNR of the image and the standard deviation decreases with increasing CNR so that \(\lim\limits_{\text{CNR}\rightarrow\infty}\sigma_{\hat{A}}=0\;px\) and \(\lim\limits_{CNR\to 0}\sigma_{\hat{A}}=\infty\). In the case of a low CNR \(<7\), previous knowledge (orange dots) clearly reduces the standard deviation \(\sigma_{\hat{A}}\). On the other hand, for high CNR values \(>7\), previous knowledge increases the standard deviation. This behavior is a common effect of the Bayesian inference for nonlinear models, which should vanish for high CNR values. To summarize, the prior knowledge dominates the measurement result for low CNR values, and the measurement data dominates for high CNR values. The use of previous knowledge is especially interesting, when it improves the measurement quality without dominating the result. Figure 4 shows that the measurement uncertainty is lowered for CNR values below 7, and the prior dominates when the CNR is below 3. Figure 3: Markov chain of the Bayesian analysis with at \(\mu_{\text{prior}}=361\) px and \(\sigma_{\text{prior}}=3\) px. Generally, the prior's quality is high when the mean value is close to the true value and when the prior has a small width (low uncertainty). In order to examine the prior influence on the posterior result, different priors are tested for a simulated image with CNR = 6.7, as shown in Figure 2b. All priors are normally distributed with the mean value at the true position so that only the standard deviation is varied.