## 4 Image Analyses
### Image Analyses without Solar Shading by Tree Leaves above the Pond Surface
#### 4.1.2 Outline of Calculation Method of RGB Theoretical Values  
As cited in [PERSON] et al. (2016), the solar altitude \(\theta\) and azimuth angle \(\phi\) were calculated from the date and time, and the latitude and longitude of the observation site by [PERSON]'s method. The pond bottom irregular reflection points T\({}_{1}\) or the scattering points T\({}_{5}\) in the pond were scanned at intervals with \(\Delta\)x and \(\Delta\)y for the former cases and with \(\Delta\)x, \(\Delta\)y and \(\Delta\)z for the latter cases. The light incident points on the pond surface were obtained from these coordinates and from the direction vector of the solar radiation, using the above angles, \(\theta\) and \(\phi\) of the solar radiation, by the method described in [PERSON] et al. (2016). Next, from the coordinates of observation point H and scanning points T\({}_{1}\) or T\({}_{5}\), the biquadratic equation of [PERSON] et al. (2016) for a square of b\({}_{12}\) of the unit direction vector \(\mathbf{B_{1}}\), here \(\mathbf{B_{1}}=(b_{1x},b_{1y},b_{12})\), of the light flux in water after irregular reflection or scattering was numerically solved by the DKA method. Because b\({}_{12}\) is a positive square root of the positive solution of the biquadratic equation, we could obtain it satisfying 0 \(<\) b\({}_{12}\) \(<\) 1. From thus obtained b\({}_{12}\), b\({}_{1x}\) and b\({}_{1y}\) were determined according to [PERSON] et al. (2016) and the paths of lights after the reflection or scattering were determined so as to give the pond surface emission points T\({}_{3}\). Then parameters such as the cosine of the angle related to the path were determined, and in the case of scattering, the scattering angles and phase function values were obtained. The calculations of colors as RGB values from the model formulas are given below:  
1. In the case of irregular reflection at the pond bottom: * step 1. Loop: Scanning the lake bottom T\({}_{1}\)(x\({}_{1}\), y\({}_{1}\), z\({}_{1}\)) at the intervals of \(\Delta\)x and \(\Delta\)y; * step 2. Determination of the light incident point T\({}_{2}\)(x\({}_{2}\), y\({}_{2}\), 0) on the pond surface and of the emission point T\({}_{3}\)(x\({}_{3}\), y\({}_{3}\), 0) from the surface to obtain L\({}_{1}\) and L\({}_{2}\) in Equation (31); * step 3. Calculations of the other factors determining the light intensities. Figure 13: Original pictures, standardized pictures, and displays of utilized color elements. step 4. Determination of \(\mathrm{P_{Irr}(\theta,\,\phi,\,x_{1},\,y_{1},\,z_{1},\,\lambda_{i})}\);
* step 5. Loop: Scanning of the pond surface (x, y);
* step 6. Calculation of \(\mathrm{P^{\prime}_{Irr}(x,\,y,\,\lambda_{i})}\);
* End of loop of step 5;
* End of loop of step 1;
* In the case of water density fluctuation scattering or in the case of Mie scattering:
* step 1. Loop: Scanning the in-lake scattering point \(\mathrm{T_{5}(x_{n},\,y_{n},\,z_{n})}\) at the intervals of \(\Delta x\), \(\Delta y\) and \(\Delta z\);
* step 2. Determination of the light incident point \(\mathrm{T_{2}(x_{2},\,y_{2},\,0)}\) on the pond surface and of the emission point \(\mathrm{T_{3}(x_{3},\,y_{3},\,0)}\) from the surface to obtain \(\mathrm{L_{3}}\) and \(\mathrm{L_{4}}\) in Equations (39) and (41);
* step 3. Calculations of the other factors determining the light intensities, e.g., scattering angles included in the estimations of phase functions;
* step 4. Determination of \(\mathrm{P_{ES}(\theta,\,\phi,\,x_{n},\,y_{n},\,z_{n},\,\lambda_{i})}\) or \(\mathrm{P_{Mie}(\theta,\,\phi,\,x_{n},\,y_{n},\,z_{n},\,\lambda_{i})}\);
* step 5. Loop: Scanning of the pond surface (x, y);
* step 6. Calculation of \(\mathrm{P^{\prime}_{ES}(x,\,y,\,\lambda_{i})}\) or \(\mathrm{P^{\prime}_{Mie}(x,\,y,\,\lambda_{i})}\);
* End of loop of step 5;
* End of loop of step 1;
* Calculation of RGB values:
* step 1.