Parameters

Coordinate system and notation

OceanLight.jl is formulated in the spherical system $\hat{\zeta} = (\theta,\phi)$, where polar angle $\theta$ is measured from the direction of $\hat{z}$ and the azimuthal ange $\phi$ is measured positive counter clockwise from $\hat{x}$, when looking toward the origin along $\hat{z}$. Let $\hat{\xi}$ denoted a unit vector pointing in the desired direction, when $\hat{\xi}=\left(\xi_{x},\xi_{y},\xi_{z}\right)$, and becasue $\hat{\xi}$ is of unit length, its component satisfy $\hat{\xi}_{x}^{2}+\hat{\xi}_{y}^{2}+\hat{\xi}_{z}^{2}=1$. ^[1] Therefore,

\[\hat{\xi} = \begin{bmatrix} \sin(\theta)\cos(\phi)\\ \sin(\theta)\sin(\phi)\\ \cos(\theta) \end{bmatrix}\]

To simplify the term above, we simplify $\hat{\xi}$ by using the cosine parameter.

\[\hat{\xi} = \begin{bmatrix}\mu_{x}\\ \mu_{y}\\ \mu_{z} \end{bmatrix} = \begin{bmatrix}\sin(\theta)\cos(\phi)\\ \sin(\theta)\sin(\phi)\\ \cos(\theta) \end{bmatrix} \]

Local coordinate system

When we calculate for the scattering direction, our result is in the local coordination system $(\hat(\theta),\hat(\phi),\hat(r))$, when radial unit vector $\hat(r)$ is the same initial direction of photons before scattering $\hat{\xi}$, the azimuthal unit vector $\hat(\phi)$ is defined by the cross product of the ocean coordinate system $\hat{z}$ and the incident vector's direction $\hat{\phi}=\frac{\hat{z}\times\hat{r}}{|\hat{z}\times\hat{r}|}$, and polar unit vector is given by $\hat{\theta}=\hat{\phi}\times\hat{r}$. Therefore, the unit vector of the scattered direction of photons ${\hat{\xi}_{s}}$ can be described in the local coordination system $(\hat(\theta),\hat(\phi),\hat(r))$ as,

\[\hat{\xi_(s)} = \begin{bmatrix} \sin(\theta_{s})\cos(\phi_{s})\\ \sin(\theta_{s})\sin(\phi_{s})\\ \cos(\theta_{s}) \end{bmatrix}\]

when, $\theta_{s}$ and $\phi_{s}$ is polar angle and azimuthal angle in relative to the local coordinate system $(\hat(\theta),\hat(\phi),\hat(r))$.

To change from the local coordinate system to the cartesian coordination in the global system, we multiply $\hat{\xi_(s)}$ by the basis of our local coordinate system.

\[\begin{bmatrix} \mu'_{x}\\ \mu'_{y}\\ \mu'_{z} \end{bmatrix} = \begin{bmatrix}\hat{\theta} & \hat{\phi} & \hat{r} \end{bmatrix}\begin{bmatrix}\hat{\xi_(s)}\end{bmatrix}\]

And, after we do the cross product and substitute $\hat{\xi_(s)}$.

\[\begin{bmatrix} \mu'_{x}\\ \mu'_{y}\\ \mu'_{z} \end{bmatrix} = \begin{bmatrix}\frac{\mu_{x}\mu_{z}}{\sqrt{1-\mu_{z}^2}}&-\frac{\mu_{y}}{\sqrt{1-\mu_{z}^2}}&\mu_{x}\\\frac{\mu_{y}\mu_{z}}{\sqrt{1-\mu_{z}^2}}&\frac{\mu_{x}}{\sqrt{1-\mu_{z}^2}}&\mu_{y}\\-\sqrt{1-\mu_{z}^2}&0&\mu_{z} \end{bmatrix}\begin{bmatrix} \sin(\theta_{s})\cos(\phi_{s})\\ \sin(\theta_{s})\sin(\phi_{s})\\ \cos(\theta_{s}) \end{bmatrix}\]

Reference