## 3 Results
### Model Development  
For the development of an accurate multiple regression model, the explanatory power of the independent variable for the dependent variables is extremely important. To develop a model for predicting turbidity from color changes in images, multiple regression equations were developed using a total of 502 data points, excluding low-reliability measurement results (LISST equipment transmission value less than 0.3), the data corresponding to RGB values severely fluctuating because of sediment injection, and the data that were reserved for verifying the developed regression equation (Figure 10). Unlike turbidity and concentration measurements performed at a specific point and time, there was a time difference between the turbidity and concentration values that were measured at the same time and the RGB values that were analyzed from the images because the point of capture varied depending on the visibility range of the images. In this study, the model development was performed after reflecting the corresponding time difference on the dependent variable because the experiment was performed under a highly constant flow condition (average flow velocity = 0.198 m/s) and there was a constant time difference (50 s) between the occurrence of the change in red band value and an increase in turbidity (Figure 10). To develop a multiple linear regression model, regression models  
Figure 10: Sections used for model development and verification, and the results reflecting the time difference between the image and turbidity measurement. Figure 9: Red value changes in the underwater images captured by Video 1 (at 18 m) and Video 2 (at 31.8 m). were developed using the five combinations of (R, G, B, Gray); (R, G, B); (R, G); (G, B); and (R, B) that were obtained from the nine pixels as independent variables with consideration of the time differences between the turbidity and explanatory variables. Table 1 shows the adjusted R-squared values. Among the developed regression models, the models that included both RGB and the gray scale showed the highest values and the regression equations that included the red band generally presented high values. In addition, the regression models developed using the data obtained from the pixels that were parallel to the concentration and turbidity measuring instruments among the nine pixels generally showed high adjusted R-squared values (* in Table 1). The regression models that were developed based on the pixels are as follows:  
\[\text{Turbidity}\;[\text{NTU}]=246.12+0.11\cdot\text{R}-0.36\cdot\text{G}-0.15 \cdot\text{B}-0.99\cdot\text{GR} \tag{3}\]  
\[\text{Turbidity}\;[\text{NTU}]=248.85-0.23\cdot\text{R}-0.69\cdot\text{G}-0.48 \cdot\text{B} \tag{4}\]  
\[\text{Turbidity}\;[\text{NTU}]=310.48+0.21\cdot\text{R}-1.73\cdot\text{G} \tag{5}\]  
\[\text{Turbidity}\;[\text{NTU}]=267.68-1.16\cdot\text{G}-0.25 \tag{6}\]  
\[\text{Turbidity}\;[\text{NTU}]=190.71-0.47\cdot\text{R}-0.77\cdot\text{B} \tag{7}\]  
For evaluating the developed regression models, the root mean squared error (RMSE) was calculated using the additional verification data using the following equation:  
\[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(x_{i}-y_{i}\right)^{2}} \tag{8}\]  
Figure 11 shows the results of verifying the five developed multiple regression models. All the developed models showed the RMSE values of 3.6-4.1 NTU and presented highly satisfactory prediction results. In the case of multiple linear regression, correlations among the independent variables that cause an incorrect interpretation of variables or degrade the prediction accuracy need to be examined. In general, if the variance inflation factor (VIF) value is 10 or higher, the variables are judged to be multicollinear. VIF is calculated using Equation (9), and the results of examining multicollinearity are shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{ \begin{tabular}{c} **Position** \\ **(x, y)** \\ \end{tabular}