## 3 Data Reduction Methodologies
### Application of Response Surface Methodology (RSM) to Flight Data
\[C_{p_{com}}=f(x_{1},\;x_{2},\;\ldots,\;x_{n})+\varepsilon \tag{9}\]  
Here, \(x_{1},\;x_{2},\;\ldots\) are variables obtained by flight experiments such as \(\alpha\), \(Re_{u\infty}\), \(\beta\), \(U_{\infty}\), \(T_{0\infty}\), and \(P_{0\infty}\). Each variable is independent of each other. \(\varepsilon\) is the uncertainty arising from reconstructing \surface. It should be noted that each variable would be non-dimensionalized prior to making a fitting curve using Equation (10). The cross terms in Equation (10) appear to introduce the curvature of the response surface since the fitting curve for this case would not be linear. 5. Coefficients appearing in Equation (10) can be obtained by expressing Equation (10) in matrix notation as follows: \[C_{p_{fit}}=\textbf{X}*\textbf{{A}}+\varepsilon=\left( \begin{array}{ccc} \textbf{1}&\cdots&\textbf{{x}}_{\textbf{1},\textbf{{k}}}\\ \vdots&\ddots&\vdots\\ \textbf{1}&\cdots&\textbf{{x}}_{\textbf{n},\textbf{{k}}}\end{array} \right) \left( \begin{array}{c}\textbf{{a}}_{\textbf{1}}\\ \vdots\\ \textbf{{a}}_{\textbf{n}}\end{array} \right)+\left( \begin{array}{c} \varepsilon_{\textbf{1}}\\ \vdots\\ \varepsilon_{\textbf{n}}\end{array} \right).\] (11) where \(C_{p_{fit}}\) is a vector of the parameter being reconstructed by the fitting dataset for interpolation, \(X\) is the matrix of values of the design variables at each flight condition, \(A\) is the vector of tuning parameters, and \(\varepsilon\) is a vector of errors inherent in the curve fitting using the response surface model (equivalent to the one expressed in Equation (10)). \(n\) is the total number of experimental cases and \(k\) is the total number of variables. \(A\) can be estimated using the least-square error method as: \[\textbf{{A}}=\left(\textbf{{X}}^{T}*\textbf{{X}}\right)^{-1}\textbf{{X}}^{T}\] (12)
6. The interpolated pressure coefficient for compared case \(C_{p_{fit_{\text{corr}}}\left(x_{1},\,x_{2},\,...,\,x_{N}\right)}\), which is interpolated for equivalent flight conditions of the reference case \(\left(x_{1}\text{-}x_{N}\right)\,\text{\text{\text{\text{\text{\text{ \text{\text{\text{\text{\text{\text{\text  
Figure 6 presents the relationship between each variable and \(C_{p}\) at the outmost probe height (probe number 45). \(C_{p}\) was created using Equation (4). A representative curve fit is also superimposed in each figure. The measured value at the outmost probe height was considered to represent the freestream value. As seen in Figure 6, each variable has a certain sensitivity on \(C_{p}\). Here, flight data obtained at the premised fight Mach number of 0.70 are plotted as an example. Note that the aforementioned RSM will be applied to flight test conditions at a single premised flight for the Mach number cases. Note that other probe height data showed a similar trend to that at probe number 45. A weak correlation is shown against \(C_{p}\) for the angle of attack, yaw angle, and temperatures. In contrast, the flight Mach number, pressures, and unit Reynolds number have a certain linear sensitivity. In order to further investigate the reasons behind those observed behaviors of variables, some governing parameters responsible for reconstructing the pressure coefficient are explored. The primary parameter of the pressure coefficient is again shown below. \[C_{p,\ at\ each\ height}=\frac{P_{pitol,\ at\ each\ height}-P_{\infty}}{\frac{1}{2} \cdot\gamma P_{\infty}\cdot M_{\infty}^{2}\cdot\left(1+\frac{\gamma-1}{2}M_{ \infty}^{2}\right)^{-\frac{2}{\gamma-1}}}=\frac{\text{{Measured\ Value}}}{ \text{{Flight\ Data}}} \tag{16}\]  
Here, the numerator is the measured data and the denominator is the flight data obtained by the ADS. Since the pitot pressure is normalized by the flight data, the flight Mach number and total atmospheric pressure that appear in the denominator would have an influential impact on the scaling of \(C_{p}\).