## 4 Results
### Characterization
To investigate the systematic and random error concerning the size of the turbulence wedge, the relative deviation of the height, the width and the area of the wedge are assessed. The relative deviation for a variable \(X\) is defined as  
\[F_{X}=\frac{X_{i}-\bar{X}_{i}}{\bar{X}_{i}}, \tag{17}\]  
where \(X\) can be the height \(h\), the width \(w\) or the area \(A\) to define the average relative deviations \(F_{h}\), \(F_{w}\) and \(F_{A}\). Figure 11 shows the mean relative deviations of the heights (Figure 11a, left), the widths (Figures 11b, left) and the areas (Figure 11c, left) over different CNR values. In general, the lower the CNR, the larger the deviations. Furthermore, while the height of the wedge is consistently underestimated, the width is overestimated for all CNR values. The largest deviations occur at \(\text{CNR}<4\), which is a CNR value not often found in real thermograms as it is outside the \(1\,\sigma\) interval. As a result, the turbulence wedge areas are underestimated throughout the investigated CNR range. The plots on the right hand side of each figure show the standard deviation of the relative deviation, which is a measure of the random error. For the width, the random error is slightly larger than the systematic error by a factor of about 1.5 for \(\text{CNR}>4\). However, for the height and the area, the random error is of about the same magnitude as the systematic error. Therefore, the results should be corrected by the systematic error in the future. Currently, the height and area of the wedge are underestimated on average by maximally about 10% for CNR values that are common in thermograms, \(4\leq\text{CNR}\leq 16\). Figure 10: (**Left**): Mean position error \(\left\langle\frac{\Delta x}{\overline{w}}\right\rangle\) normalized by the average wedge width (see Equation (16)) for \(n=100\) thermograms for different CNR values. The error bar indicates the standard deviation of the mean. The gray region marks realistic CNR values commonly found in real thermograms. (**Right**): The standard deviation of \(\frac{\Delta x}{\left\langle\overline{w}\right\rangle}\) in percent, which represents the random error. To further investigate how the height of the turbulence wedge influences the wedge detection algorithm, a thermogram of the size \((140\times 140)\) pixels2 with a single turbulence wedge with \(\text{CNR}=11\) is investigated, and the height \(\tilde{h}\) of the wedge was changed systematically from \(\tilde{h}=0.1\cdot h_{\text{max}}\) to \(\tilde{h}=h_{\text{max}}\) in 15 steps. For each wedge height, \(n=50\) thermograms were investigated. The width of the wedge was drawn anew for each thermogram from \(\mathcal{N}(\mu=\tilde{h}/3,\sigma^{2}=0.2)\). In Figure 11(a), \(\langle F_{N}\rangle\) is shown, while Figure 11(b) shows the position error, and Figure 11(c) shows the relative deviations of the wedge's size and area \(\langle F_{h}\rangle\), \(\langle F_{w}\rangle\) and \(\langle F_{A}\rangle\) over the relative true wedge heights \(\tilde{h}/h_{\text{max}}\). The gray region in each figure indicates relative heights commonly found in real thermograms, adapted from Table 1.  
Footnote 2: The \(\langle F_{N}\rangle\) is plotted over the relative wedge heights \(\tilde{h}/h_{\text{max}}\). For a relative height \(\tilde{h}/h_{\text{max}}<0.2\), no wedge is detected by the algorithm because it is too small. Therefore, no information about the positions or the areas is available and the thermograms are excluded from the analysis for the determination of the position error and the relative  
Figure 11: **Left**: Mean relative deviation of **(a)** the wedges’ heights \(\langle F_{h}\rangle\), the wedges’ widths \(\langle F_{w}\rangle\) **(b)** and wedges’ areas \(\langle F_{A}\rangle\) **(c)** for each CNR value with \(n=100\) trials with randomly sized wedges. The error bar indicates the standard deviation of the mean, and the gray region shows realistic CNR values found in real thermograms. **Right**: Standard deviation of all respective mean relative deviations in percent.