## 4 Image Analyses
### Sunbeam Analyses
#### 4.2.2 Acquisition of Coordinates on the Pond Surface from Images and Data Preprocessing
In order to simulate the incidence of sunbeams into the pond, the reflection points at the pond bottom and scattering points inside the pond were scanned, with fixing the incident point of sunbeam of (x\({}^{\prime}\)K, y\({}^{\prime}\)K) in the absolute coordinate system obtained from the incident point of (x\({}_{K}\), y\({}_{K}\)) in the relative coordinate system. We calculated the RGB values of sunbeam trajectories only when the incident point coordinates T\({}_{2}\)(x\({}_{2}\), y\({}_{2}\), 0) on the pond surface satisfied the following limitation expressions for optical paths with small \(\Delta\)a:  
\[ \begin{array}{l}\mathrm{x}^{\prime}_{K}-\Delta\mathrm{a}\leq\mathrm{x}_{2} <\mathrm{x}^{\prime}_{K}+\Delta\mathrm{a},\\ \mathrm{y}^{\prime}_{K}-\Delta\mathrm{a}\leq\mathrm{y}_{2}<\mathrm{y}^{\prime} _{K}+\Delta\mathrm{a}.\end{array}  \tag{47}\]  
Next, the integral of the theoretical values along the perpendicular axis to the axial center were obtained by the following equation:  
\[ \begin{array}{l}\mathrm{R}(\mathrm{X}_{0},\mathrm{Y}_{0})=\sum\limits_{x_{ 3},y_{3}}\frac{2558\left(x_{3},y_{3}\right)}{\sqrt{2\pi\alpha}}\exp\left[- \frac{\left\{\left(x_{0}-x_{3}\right)^{2}+\left(\mathrm{Y}_{0}-y_{3}\right)^ {2}\right\}}{2\sigma^{2}}\right],\\ \mathrm{G}(\mathrm{X}_{0},\mathrm{Y}_{0})=\sum\limits_{x_{3},y_{3}}\frac{255 \left(\mathrm{G}(\mathrm{x}_{3},y_{3})\right)}{\sqrt{2\pi\alpha}}\exp\left[- \frac{\left\{\left(x_{0}-x_{3}\right)^{2}+\left(\mathrm{Y}_{0}-y_{3}\right)^ {2}\right\}}{2\sigma^{2}}\right],\\ \mathrm{B}(\mathrm{X}_{0},\mathrm{Y}_{0})=\sum\limits_{x_{3},y_{3}}\frac{25 \left(\mathrm{G}(\mathrm{x}_{3},y_{3})\right)}{\sqrt{2\pi\alpha}}\exp\left[- \frac{\left\{\left(x_{0}-x_{3}\right)^{2}+\left(\mathrm{Y}_{0}-y_{3}\right)^ {2}\right\}}{2\sigma^{2}}\right],\end{array}  \tag{48}\]  
Figure 18: Displays of integrated RGB data for image analyses. The x-coordinate on the horizontal axis was depicted in the relative coordinate system. where (x\({}_{3}\), y\({}_{3}\)) indicates the coordinates of the emission points of lights on the pond surface corresponding to the scanned reflection points at the bottom or scattering points inside the pond. Definitions of the RGB values, R(x\({}_{3}\), y\({}_{3}\)) etc., are described below. The above equations defined a one-dimensional distribution of integrated RGB values on an axial center of the light trajectory of the sunbeam. These equations were valid because of the following two reasons. The numerical calculations of the sunbeams were done under the optical path limitations, as described above. Here, the light emission point coordinates T\({}_{3}\)(x\({}_{3}\), y\({}_{3}\), 0) on the pond surface were located almost on a straight line along the axial center. In addition, the emission light intensities on the surface had the 2D point symmetrical distributions around T\({}_{3}\)(x\({}_{3}\), y\({}_{3}\), 0). From Equations (31), (39) and (41), the following equations hold:  
\[ \begin{array}{l}\mathrm{P}_{\mathrm{Irr}}^{\prime\prime}\left(\mathrm{x}_{3 },\mathrm{y}_{3},\mathrm{\lambda}_{i}\right)=\frac{\mathrm{A}\mathrm{x} \mathrm{x}\mathrm{y}}{\mathrm{n}_{\mathrm{Iz}}}\cdot\mathrm{P}_{\mathrm{Irr}} \left(\theta,\mathrm{\phi},\mathrm{x}_{1},\mathrm{y}_{1},\mathrm{z}_{1}, \mathrm{\lambda}_{i}\right),\\ \mathrm{P}_{\mathrm{Irr}}^{\prime\prime}\mathrm{E}\left(\mathrm{x}_{3}, \mathrm{y}_{3},\mathrm{\lambda}_{i}\right)=\mathrm{A}\mathrm{x}\mathrm{x} \mathrm{y}\cdot\mathrm{P}_{\mathrm{ES}}\left(\theta,\mathrm{\phi},\mathrm{x} _{n},\mathrm{y}_{n},\mathrm{z}_{n},\mathrm{\lambda}_{i}\right),\\ \mathrm{P}_{\mathrm{Mie}}^{\prime\prime}\left(\mathrm{x}_{3},\mathrm{y}_{3}, \mathrm{\lambda}_{i}\right)=\mathrm{A}\mathrm{x}\mathrm{x}\mathrm{y}\cdot \mathrm{P}_{\mathrm{Mie}}\left(\theta,\mathrm{\phi},\mathrm{x}_{n},\mathrm{y} _{n},\mathrm{z}_{n},\mathrm{\lambda}_{i}\right).\end{array}