## 4 Prediction of Oil-Spill Pollution Area
Accordingly, the response surface model could be established by using source data of the oil spill, which were attained from the numerical calculation results of Section 3 shown in Table 3. In order to improve the accuracy of fitting function, the functional relationship among \(t\), \((\)X, Y\()\), \(C^{\prime}\), \(L\), and \(D\) was correlated by the fourth-order and sixth-order models. The design variable was \(t\) and the responses were X, Y, C', \(L\), and \(D\). The fourth-order and sixth-order models are shown in Equation (20). \[ \begin{array}{l}\mathrm{x}\ =\ 0.00000003t^{4}+0.0000006t^{3}-0.0027t^{2}+0.83 42t+94.618\\ \mathrm{y}\ =\ -0.0000000005t^{6}+0.00000003t^{5}-0.000008t^{4}+0.0009t^{3}-0.0498t^{2 }\\ \hskip 100.0 pt+1.1752\mathrm{t}-4.6404\\ \hskip 100.0 pt\mathrm{L}\ =\ 0.0000003t^{4}-0.0001t^{3}+0.0077t^{2}+1.2593t+97.143 \\ \hskip 100.0 pt\mathrm{D}\ =\ -0.0000001t^{4}+0.00002t^{3}-0.0291t+9.6553\\ \hskip 100.0 pt\mathrm{C}^{\prime}\ =\ -0.000000002t^{6}+0.000001t^{5}-0.0003t^{4}+0.0 243t^{3}-1.0821t^{2}+21.627t\\ \hskip 100.0 pt-67.344\end{array}  \tag{20}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Time(\(h\))/ \(h\)** & **Area(S)/\(m\)** & **Perimeter(C)/\(m\)** & **Centroid Central (Q)/Q/m** & **Coordinate Coordinate (Y)/m** & **Horizontal Maximum Distance (Q)/m** & **Vertical Maximum Distance (D)/m** \\ \hline  
\hline \end{tabular}
\end{table} 
Table 3: Oil-spill parameters extracted from oil-spill trajectory simulation. The minimum number of sample points depended on the model order and the number of input variables. A third-order response surface model could be constructed for the oil-spill area \(S\), for which the input variables were the perimeter \(C^{\prime}\), centroid position (X, Y), horizontal maximum distance \(L\) and maximum vertical distance \(D\). Thus, the number of variables \(M\) was 5. The minimum number of sample points must be equal to \((M+1)(M+2)/2+M\), which was 26 in this research. Therefore, the fitting equation of the oil-spill area \(S\) is shown in Equation (21). \[\begin{array}{rl}\mathrm{S}=165789.84&+109.3C^{\prime}-4672.69X-6503.4Y+1138.7L +973.86D+0.18C^{\prime 2}+36.87X^{2}\\ &-349.34Y^{2}+1.26L^{2}-5.86D^{2}+0.1C^{\prime}*X-14.07C^{\prime}*Y-0.7C^{ \prime}*L\\ &+0.72C^{\prime}*D+72.93X*Y  
Obviously, factor Y played a dominant role in both local effects and global effects for S. Therefore, it was essential to predict the oil-spill area by designated centroid coordinate Y in icy waters. Emergency treatment measures for the oil spill could then be organized without considering non-sensitivity factors \(L\) and \(C^{\prime}\).  
The reliability of this approximate model was evaluated by the multiple correlation coefficient \(R^{2}\) as given in Equation (22):  
\[R^{2}\ =\ 1-\sum_{i\ =\ 1}^{n^{\prime\prime}}(y_{i}-\hat{y}_{i})^{2}/\sum_{i\ =\ 1}^{n^{\prime\prime}}(y_{i}-\overline{y_{i}})^{2} \tag{22}\]  
The value of \(R^{2}\) was used to measure the degree of conformity between the approximate model and the sample points. \(R^{2}\) being close to 1.00 indicated that the approximate model had a relatively high degree of credibility. In this case, \(R^{2}\) was 0.9 and achieved the established threshold, which meant the approximate model could provide a relatively credible prediction of oil-spill diffusion. Figure 14: Local effects (gradients) for S.  
Figure 15: Global effects (average gradients) for S.