## 2 Materials and Methods
### The Bayesian Approach  
The studied approach to detect and localize the laminar-turbulent transition is the Bayesian inference method and the use of previous knowledge. A general description of the method can be found in many textbooks as for example in \"A Guide to Bayesian Inference for Regression Problems\" [27] by [PERSON] et al. as a manual from seven national metrology institutes to make Bayesian inference applicable to various purposes. The fundamental principle is to treat the measurand, here, the transition position \(A\), not as an unknown constant, but to act on the assumption of a probability distribution. Accordingly, a Markov chain was created to simulate the distribution of the transition position. This Markov chain Monte-Carlo simulation (MCMC), first introduced by [PERSON] et al. [28], iteratively proposes a new transition position \(a_{\text{new}}\) through a slightly changed copy of only the last link \(a_{\text{last}}\) of the Markov chain. The Markov chain is built upon the Bayesian idea and the posterior probability \(p(a|x)\), which is the key element of the Bayesian theory, is calculated for both \(a_{\text{new}}\) and \(a_{\text{last}}\) and describes the probability for the transition to be at the position \(a\), given measured temperature values \(x\) along the chord axis position. The vector  
\[x=[x_{1},x_{2},x_{3},\ldots x_{c}] \tag{3}\]  
contains the measured temperatures with noise along the chord axis position \(y=1,\ \ldots,\ c\), where \(c\) is the chord length in pixels. Only if the quotient  
\[\frac{p(a_{\text{new}}|x)}{p(a_{\text{last}}|x)}>\text{rand}(0,1) \tag{4}\]  
according to [28] exceeds a random number between 0 and 1, the proposed transition position is appended to the Markov chain. This ensures that a proposed transition position \(a_{\text{new}}\) is surely accepted when its probability given the measured values \(x\) is higher than the probability of the last link in the chain \(a_{\text{last}}\) and possibly accepted when the new link is less probable. As a result, the Markov chain resembles the posterior probability of the transition position. The calculation of the posterior probability is as follows:  
\[p(a|x)=\frac{p(x|a)p(a)}{p(x)} \tag{5}\]  
entails calculation of the evidence \(p(x)\), and the nominator as a product of the likelihood \(p(x|a)\) and the previous knowledge \(p(a)\). Previous experiments provide previous knowledge in the form of a probability distribution \(p(A)\), and \(p(a)\) is an abbreviation for \(p(A=a)\). Firstly, the evidence or denominator of Equation (5) is defined through  
\[p(x)=\int p(x|a)p(a)da \tag{6}\]  
can be neglected in this approach because it is independent of the transition position \(a\) and its usual role as a normalization constant of the posterior probability is not necessary, as it cancels out in Equation (4). Secondly, the nominator \(p(x|a)p(a)\) is piecewise linearized along the finite number of chord axis positions \(y\) so that  
\[p(x|a)p(a)=\prod_{y=0}^{c}p\big{(}x_{y}|a\big{)}p(a). \tag{7}\]Next, the random variable \(a\) is transformed via  
\[a=g(T,y)=f^{-1}(T,y), \tag{8}\]  
where \(f^{-1}(T,\ y)\) is the inverse function of \(f\) that is defined in Equation (9). The best guess of the local temperature \(T\) is determined via a least square regression for Equation (2) with the regression parameters \(\alpha_{\rm fit},\ \beta_{\rm fit},\ \sigma_{\rm fit}\), but with the proposed transition \(a\) instead of \(A_{\rm fit}\):  
\[T=f(y,a)=\alpha_{\rm fit}\cdot\text{erf}(\frac{y-a}{\sqrt{2}}\sigma_{\rm fit}) +\beta_{\rm fit}. \tag{9}\]  
The value of \(a\) causes a shift of the fit function \(f(y,\ A_{\rm fit})\) along the chord axis so that the turning point is at \(y=a\) (and not at \(y=A_{\rm fit}\)).