## 4 Soil Parameter Updating
### Response-Surface Model
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Run Order**} & \multicolumn{3}{c}{**Input Factors**} & \multicolumn{6}{c}{**Numerical Responses (Bending Strain \(\times\) 10\({}^{-6}\))**} \\ \cline{2-11}  & \(\gamma\) & \(\Phi^{\prime}\) & K & S1 & S2 & S3 & S4 & S5 & S6 & S7 \\ \hline
1 & 1550.00 & 45.20 & 550.00 & 19.91 & 23.56 & \(-\)0.36 & 31.33 & 66.31 & 51.50 & \(-\)77.01 \\
2 & 1550.00 & 36.80 & 550.00 & 18.45 & 22.36 & \(-\)0.33 & 29.55 & 63.03 & 49.50 & \(-\)74.32 \\
3 & 1550.00 & 41.00 & 100.00 & 27.69 & 30.08 & \(-\)1.24 & 43.91 & 89.28 & 66.26 & \(-\)91.90 \\
4 & 1800.00 & 41.00 & 550.00 & 20.44 & 23.13 & \(-\)2.05 & 31.60 & 67.98 & 52.18 & \(-\)86.90 \\
5 & 1550.00 & 41.00 & 550.00 & 19.16 & 22.94 & \(-\)0.35 & 30.40 & 64.59 & 50.44 & \(-\)75.64 \\
6 & 1300.00 & 41.00 & 550.00 & 13.70 & 20.14 & 0.76 & 30.24 & 62.18 & 49.44 & \(-\)64.25 \\
7 & 1550.00 & 41.00 & 1000.00 & 14.35 & 18.46 & \(-\)0.05 & 22.32 & 49.55 & 40.49 & \(-\)64.46 \\
8 & 1396.91 & 38.43 & 274.43 & 19.35 & 24.49 & 0.04 & 36.96 & 75.13 & 57.74 & \(-\)76.20 \\
9 & 1703.09 & 43.57 & 274.43 & 25.54 & 27.81 & \(-\)1.89 & 39.09 & 81.50 & 60.85 & \(-\)93.46 \\
10 & 1703.09 & 38.43 & 274.43 & 24.29 & 26.90 & \(-\)1.92 & 38.49 & 80.23 & 60.15 & \(-\)92.27 \\
11 & 1396.91 & 43.57 & 274.43 & 19.82 & 24.91 & 0.03 & 37.75 & 76.54 & 58.63 & \(-\)77.08 \\
12 & 1703.09 & 38.43 & 825.57 & 17.10 & 20.41 & \(-\)0.78 & 24.63 & 54.73 & 43.80 & \(-\)73.16 \\
13 & 1396.91 & 38.43 & 825.57 & 13.23 & 18.44 & 0.49 & 24.38 & 52.40 & 42.67 & \(-\)61.51 \\
14 & 1550.00 & 41.00 & 550.00 & 19.16 & 22.94 & \(-\)0.35 & 30.40 & 64.59 & 50.44 & \(-\)75.64 \\
15 & 1396.91 & 43.57 & 825.57 & 13.91 & 19.16 & 0.51 & 25.67 & 54.74 & 44.20 & \(-\)63.21 \\
16 & 1703.09 & 43.57 & 825.57 & 18.22 & 21.26 & \(-\)0.89 & 25.78 & 56.97 & 45.10 & \(-\)75.54 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design of experiments and corresponding numerical responses. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Coefficient**} & \multicolumn{6}{c}{**Coefficient Values for Each Target Equation**} \\ \cline{2-9}  & **S1** & **S2** & **S3** & **S4** & **S5** & **S6** & **S7** \\ \hline \(u_{0}\) & 19.182 & 22.941 & \(-\)0.3531 & 30.405 & 64.609 & 50.454 & \(-\)75.643 \\ \(u_{1}\) & 3.655 & 1.744 & \(-\)1.3645 & 0.6661 & 2.949 & 1.3645 & \(-\)11.44 \\ \(u_{2}\) & 0.724 & 0.595 & \(-\)0.0144 & 0.8238 & 1.545 & 0.9437 & \(-\)1.2922 \\ \(u_{3}\) & \(-\)5.920 & \(-\)5.365 & 0.6143 & \(-\)10.66 & \(-\)19.527 & \(-\)12.69 & 13.52 \\ \(u_{11}\) & \(-\)2.193 & \(-\)1.324 & \(-\)0.2795 & 0.494 & 0.425 & 0.325 & 0.08 \\ \(u_{22}\) & \(-\)0.083 & \(-\)0.003 & 0.0165 & 0.015 & 0.017 & 0.015 & \hypothesis. A lower \(p\)-value increases the significance. If the \(p\)-value is less than 0.05, then the lack of fit can be taken to be insignificant, and a \(p\)-value greater than 0.1 renders the lack of fit significant [44]. Table 4 depicts that \(\gamma\), \(\Phi^{\prime}\), K, and \(\gamma*\gamma\) were significant model terms for all seven responses. In ANOVA, the goodness of fit (R\({}^{2}\)) should not be less than 95% [45]. Table 5 shows that the goodness of fit (R\({}^{2}\)) was greater than 95% for all the responses and closely resembled the adjusted goodness of fit (Adj. R\({}^{2}\)). The S-value represents the interval between the fitted values and the data values in the units of the output responses. Residual plots also examine the fitness of the model. Figure 7 depicts the residual plots for all seven responses. It expresses the outline of the response data in the design order. Normal probability plots provide an expression of the normal distribution of the residuals. Supposing that the normal probability plot forms a straight line, the model could be termed to be adequate [46]. The normal probability plots for all seven responses formed straight lines, so the model was adequate. The residual versus fit plot expresses how the model achieves its target.