## 4 Numerical Results
### Application for Real-World Engineering Problems
#### 4.2.2 Problem 2: Sonic Boom Uncertainty Quantification  
The near-field and atmospheric propagation process involves several uncertain parameters that affect the ground noise prediction. Within the scope of this application, the effect of uncertainties in the atmospheric propagation process on the prediction of the ground noise is examined. The altitude-dependent temperature, relative humidity, and wind speeds in X and Y directions in the atmospheric propagation process all affect the aleatory uncertainties in the problem; the resulting uncertainty variable statistics are presented in Table 6. Statistics of uncertain variables are taken from the study of [PERSON] and et al. [31] and values are obtained by simplifying the atmosphere profiling defined in SBPW-II [54]. Figure 14: Comparison of probability density function (\(N_{lf}=70\), \(N_{hf}=40\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Mean (dB)** & \multicolumn{2}{c}{**Standard Dev.**} & \multicolumn{1}{c}{**Skewness**} & \multicolumn{1}{c}{**Kurtosis**} \\  & & **(dB)** & & \\ \hline Reference & 83.0656 & 0.8435 & \(-0.4968\) & 3.3274 \\ PCE & 82.9029 & 0.8354 & 0.0134 & 2.9538 \\ LRA & 83.0479 & 0.8493 & \(-0.4080\) & 3.2467 \\ MF-PCE & 83.0426 & 0.8060 & \(-0.4811\) & 4.3893 \\ MF-LRA & 83.0621 & 0.8478 & \(-0.5038\) & 3.3242 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of descriptive statistics of sonic boom loudness (\(N_{lf}=70\), \(N_{hf}=40\)). During the atmospheric propagation process, the change in the ground altitude and reflection factor values are considered as epistemic uncertainty parameters, and their statistics are shared in Table 7. The value ranges of these parameters are chosen to be compatible with previous low-boom UQ studies [28; 31]. As in the first sonic boom problem, the JWB geometry is used in this application. Sonic boom uncertainty quantification is performed using the near field pressure signature obtained from PANAIR for low-fidelity flow analysis and the SU2-Euler solver as high-fidelity flow analysis. A surrogate model is established with single- and multi-fidelity versions of the PCE and LRA methods using different numbers of high-fidelity analyses. Surrogate model accuracy metrics calculated at \(10^{5}\) test points are given in Figure 15. As observed from the metrics, an accurate surrogate model can be constructed for a 32-D engineering problem with the LRA method using 200 high-fidelity simulations results. With the MF-LRA method, it is observed that an accurate surrogate model is established using fewer high-fidelity simulations results, thanks to the information obtained from the low-fidelity model. PCE-based methods, on the other hand, require a large number of simulations results, because the number of unknown coefficients increases exponentially with problem size. Therefore, surrogate models with single and multi-fidelity PCE methods give erroneous results when few simulations results are used. When approximately 1000 simulations results are used with the PCE method, a moderately accurate surrogate model could be established. The PDF values are obtained by using the established surrogate models and shared in Figure 16a. Since the PCE and MF-PCE methods are highly erroneous, the results of the LRA, MF-LRA and MC are shared in Figure 16b in order to make a proper comparison. The MC result is calculated using \(10^{6}\) simulations results. As observed in Figure 16, the results obtained with the PCE and MF-PCE method are highly erroneous, while the LRA and MF-LRA results overlap with the MC results.