## 4 Results and Discussion
### Accuracy of Interpolation
As in the interpolation with the 2 nd-degree response surface model, the interpolation with the 3 rd-degree response surface model can interpolate the profile well. Obviously, the 3 rd-degree response surface model reconstructs the boundary layer profile better than that with the 2 nd-degree response surface model as the error appeared in the region around the probe height of 20 mm-30 mm in Figure 9 is not seen here. The maximum discrepancy between the reference and compared cases in their interpolated profiles was found to be in the order of \(10^{-7}\%\) for \(C_{p}\) and less than 2.6% for \(U\)-velocity. Accounting for the results shown in Figures 9 and 10, the 3 rd-degree response surface model can interpolate the entire boundary layer profile better than the 2 nd-degree response surface model. Figure 11a presents the uncertainty in the interpolation with the 3 rd-degree response surface model for all other flight conditions. Each data point expressing an average difference between the profiles obtained at two flights was derived as shown in Figure 9. Average differences with regard  
Figure 10: Comparison of profiles for smooth wall cases at two different flights (where ‘Ref’ denotes a condition with \(M_{\infty}=0.65\), \(\alpha=2.68\lx@math@degree\), \(s^{+}=23.8\), and ‘Com’ denotes a condition with \(M_{\infty}=0.65\), \(\alpha=2.59\lx@math@degree\), \(s^{+}=23.0\)): (**a**) \(C_{p}\); (**b**) \(U\)-velocity. Third-degree response surface model with two variables: \(x_{1}=P_{0\infty}\) and \(x_{2}=M_{\infty}\) was employed. Figure 9: Comparison of profiles for smooth wall cases at two different flights (where ‘Ref’ denotes a condition with \(M_{\infty}=0.65\), \(\alpha=2.68\lx@math@degree\), \(s^{+}=23.8\), and ‘Com’ denotes a condition with \(M_{\infty}=0.65\), \(\alpha=2.59\lx@math@degree\), \(s^{+}=23.0\)): (**a**) \(C_{p}\); (**b**) \(U\)-velocity. Second-egree response surface model with two variables: \(x_{1}=P_{0\infty}\) and \(x_{2}=M_{\infty}\) was employed. to \(U\)-velocity not using the interpolation and using the interpolation for all pitot-probes between the reference case and compared case are presented. As seen from average differences for data not using the interpolation and using the interpolation in Figure 11, error level is reduced from 0.84% to 0.39% by implementing the interpolation. The maximum error after interpolation is 1.8% and the average error level is approximately 0.39%. The maximum error level is then considered to be cumulative error by implementing the interpolation. Note that the total uncertainty could contain uncertainty in the freestream condition, and will be discussed in the next section. As a comparison, the 2 nd-degree response surface model yielded the maximum error level of approximately 1.8% and the average error level of approximately 0.54%, as shown in Figure 11a. The 3 rd-degree response surface model was, therefore, employed in this study due to a reduced error level compared to the 2 nd-degree response surface model. As the average error value appeared to scatter to some extent against \(s^{+}\) value since the available dataset was limited, Figure 11b presents a histogram of the average error in \(U\)-velocity. A normal distribution is superimposed with the mean and standard deviation values obtained from the error values plotted in Figure 11a for \(U\)-velocity. The distribution is normalized by its maximum value so that the peak value shows unity. Approximately 68.3% of the error population appears within an error value of 0.39%, indicating that only a small population could have a large error.