## 4 Prediction of Oil-Spill Pollution Area  
Once an oil-spill accident has occurred in icy waters, it is nearly impossible to clean up the oil by mechanical technology. Thereby, a low-temperature oil-dispersant could be adopted as the treatment method. The diffusion process of oil dispersants could be negligible because they attach themselves onto the surface of oil film [32]. However, the presence of fast-moving or drifting ice in Arctic waters will affect both oil and oil-dispersants irregularly during the spreading process. Therefore, it will be difficult to control the dosage of oil dispersants and might lead to severe damage to the Arctic Ocean environment. Accordingly, the changing extent of the oil-contaminated area in icy waters should be predicted as soon as possible in order to introduce effective emergency treatment. For this purpose, approximate modeling technology utilizing a regression analysis to fit factors and responses was adopted. The RSM was used to establish an approximate model to depict the relationship between the responses for its benefits of simple mathematical expression, small calculation amount and fast convergence speed. The response surface in the form of polynomial function was used. The polynomial function response surface was verified with good fitting accuracy and calculation  
Figure 11: Oil spreading characteristics of the ice particle with 6 DOF movement at y = 0.25 m plane section. efficiency for the input-output relationship of most data structures. The relationship between factors and responses in the approximate modeling could be described by Equation (19). \[\hat{y}(x)\ =\ a+\sum_{i\ =\ 1}^{n^{\prime}}b_{i}x_{i}+\sum_{i\ =\ 1}^{n^{ \prime}}c_{i}x_{i}^{2}+\sum_{i\ =\ 1}^{n^{\prime}-1}\sum_{j>i}^{n^{\prime}}d_{ij}x_{i}x_{j}+\varepsilon \tag{19}\]  
The polynomial function was adopted by RSM to fit the approximate model of the design space. The procedure for constructing the approximate model of the oil spill area used a third-order RSM, which was listed as follows:  
1. The sample point \(x_{1}\) was determined to construct the approximate model in the design space using the experimental design methodology, where \(x_{i}\ =\ (x_{1},x_{2},x_{3},\ldots,x_{m})\), and \(m\) was the number of design variables. The data of sample points were the centroid coordinates (X, Y), area \(S\), perimeter C', horizontal maximum distance \(L\) and vertical maximum distance \(D\) of the contaminated area, which were extracted from the numerical results of oil spill diffusion at different times as presented in Section 3.4 (i.e., 40 s, 80 s, 120 s, 160 s, and 200 s). 2. A series of sample pairs of designated variables and responses \(\{(x_{i},y_{i})\}\) were computed by the model, where \(y_{i}\ =\ (y_{1},y_{2},y_{3},\ldots,y_{n})\) was the response of the oil-spill area, and \(n\) was the number of sample points designed for the test. 3. The regression analysis using RSM was conducted to obtain an approximate model of the oil-spill area. The multiple correlation coefficient was adopted to evaluate the credibility of the response surface model. If the requirements were met, this model could be applied to the subsequent optimization computation. Otherwise, the credibility of the model could be improved by changing the model type and increasing the sample points. The oil-spill area at a certain future time was predicted by the function that described the relationship between variables and responses. The appropriate equivalent weight of oil dispersants could be determined based on the oil-spill area to implement the emergency treatment. In order to shorten the calculation time, the prediction of the oil-spill pollution area was used as the continuation and supplement of the oil trajectory simulation.