## 3 Implementation of the Model-Based Algorithm and the Experimental Setup
### Simulation Setup
Finally, the typical contrast-to-noise ratio (CNR) of a turbulence wedge is emulated, where the CNR is defined as  
\[\text{CNR}_{i}=\sqrt{\frac{(\left\langle I_{\text{lam}}\right\rangle_{i}- \left\langle I_{\text{wedge}}\right\rangle_{i})^{2}}{\sigma(I_{\text{lam}})_{ i}^{2}+\sigma(I_{\text{wedge}})_{i}^{2}}}, \tag{14}\]  
where \(\langle I_{\text{wedge}}\rangle_{i}\) and \(\langle I_{\text{lam}}\rangle_{i}\) are the mean pixel intensities in the turbulent region inside the wedge and the laminar region right outside of the turbulence wedge with index \(i\), respectively. Furthermore, \(\sigma(I_{\text{wedge}})_{i}\) and \(\sigma(I_{\text{lam}})_{i}\) denotes the spatial standard deviation of the pixel intensities of the turbulence wedge or its surrounding area. To realistically emulate real thermograms, the above mentioned features of real thermograms and turbulence wedges need to be quantified. For this reason, real thermograms of wind turbines in operation are analyzed. In the analysis, 14 different thermograms and 43 different turbulence wedges are included. All heights and widths are measured using the program ImageJ [16], while the CNR calculations are performed in Python. The loaded thermograms are normalized to a range of \([0,1]\). The results can be seen in Table 1, where the first section contains the values that are important for the modelling of a thermogram without turbulence wedges. The second section contains the characteristic values of the turbulence wedges. By adapting the mean values in Table 1 in the simulated thermograms, thermograms of rotor blades of in-service wind turbines are emulated realistically. The values used in the simulation can be seen in the rightmost column. Note that the values for \(\langle\sigma(I_{\text{turb}})\rangle\) and \(\langle\sigma(I_{\text{turb}})\rangle\) are identical in the simulation, because the marginal difference in the experimental results is neglected. Furthermore, the values of the heights \(h_{i}/h_{\text{max}}\) of the simulated turbulence wedges are drawn from a uniform distribution in the interval \([0.6,0.85]\) if not explicitly stated otherwise, which places the found mean value approximately in the middle of the range. The widths \(w_{i}\) of the simulated turbulence wedges are drawn from a normal distribution \(\mathcal{N}(\mu_{w}=h_{i}/3,\sigma_{w}^{2}=0.2)\), where \(\mu_{w}\) is the mean and \(\sigma_{w}^{2}\) is the variance of the normal distribution. By modifying the size of the turbulence wedges in the simulated thermograms in a small interval which is easily detectable by the algorithm, no size that is particularly well or poorly detected is chosen by chance. In order to characterize the dependency of the model-based wedge detection algorithm on the CNR value as well as the size of the turbulence wedges, Monte Carlo simulations are performed which utilize the simulated thermograms. With the results of the Monte Carlo \begin{table}
\begin{tabular}{c c c} \hline \hline
**Characteristic** & **Determined Value** & **Simulated Value** \\ \hline \(\langle I_{\text{lam}}\rangle\) & \(0.96\pm 0.01\) & \(0.96\) \\ \(\langle I_{\text{turb}}\rangle\) & \(0.75\pm 0.02\) & \(0.75\) \\ \(\langle\sigma(I_{\text{lam}})\rangle\) & \(0.010\pm 0.001\) & \(0.009\) \\ \(\langle\sigma(I_{\text{turb}})\rangle\) & \(0.009\pm 0.001\) & \(0.009\) \\ \(\left\langle h_{\text{grad}}/h_{\text{max}}\right\rangle\) & \(0.50\pm 0.07\) & \(0.50\) \\ \(\left\langle h_{\text{grad},0}/h_{\text{max}}\right\rangle\) & \(0.16\pm 0.04\) & \(0.15\) \\ \hline \(\langle h_{i}/h_{\text{max}}\rangle\) & \(0.73\pm 0.02\) & \([0.6,0.85]\) \\ \(\langle h_{i}/w_{i}\rangle\) & \(3.01\pm 0.07\) & \(3\) \\ \(\langle\text{CNR}_{i}\rangle\) & \(10.1\pm 1.1\) & \([2,20]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Important characteristics of the thermograms of rotor blades (first section) and the turbulent wedges (second section) determined in real thermograms of wind turbines in motion. All thermograms were normalized to a range of \(0\) to \(1\) in arbitrary intensity units.