在辐射传输建模中分离非碰撞和碰撞辐射亮度

Seperation of uncollided and Collided Intensities

For one dimensional RTM of canopy, let the illumination is $F_{in}(L=0)=1$ and isotropic skylight is thus
$$\begin{array}{rl}& -\mu \frac{\partial }{\partial L}I(L,\Omega )+G(L,\Omega )I(L,\Omega )=\frac{1}{\pi }{\int }_{4\pi }d{\Omega }^{\prime }\Gamma ({\Omega }^{\prime },\Omega )I(L,{\Omega }^{\prime })\\ & I(L=0,\Omega )=\frac{f_{dir}}{|{\mu }_0|}\delta (\Omega ,{\Omega }_0)+\frac{1-f_{dir}}{\pi }\\ & I(L=L_H,\Omega )=\frac{1}{\pi }{\int }_{2\pi -}I(L=L_H,{\Omega }^{\prime }){\rho }_b({\Omega }^{\prime },\Omega )|{\mu }^{\prime }|d{\Omega }^{\prime }\end{array}$$
For numerical purposes, we seperate the uncollided radiation field from collided fields, that is
$$I(L,\Omega )=I^0(L,\Omega )+I^c(L,\Omega )$$
where $I^0$ specific intensity of uncollided photons and $I^c$ is the specific intensity of photons which experienced collisions with elements of host medium. Then ,we substitude the Eq. (2) into Eq. (1), we have:
$$\begin{array}{rl}& -\mu \frac{\partial }{\partial L}{I}^0(L,\Omega )+G(L,\Omega )I^0(L,\Omega )=0\\ & I^0(L=0,\Omega )=\frac{f_{dir}}{|{\mu }_0|}\delta (\Omega ,{\Omega }_0)+\frac{1-f_{dir}}{\pi },\mu <0\\ & I^0(L=L_H,\Omega )=\frac{1}{\pi }{\int }_{2\pi -}d{\Omega }^{\prime }{I}^0(L=L_H,{\Omega }^{\prime })|{\mu }^{\prime }|{\rho }_b({\Omega }^{\prime },\Omega ),\mu <0\end{array}$$
and coollided intensity
$$\begin{array}{rl}& -\mu \frac{\partial }{\partial L}{I}^C(L,\Omega )+G(L,\Omega )I^C(L,\Omega )=Q(L,\Omega )+S(L,\Omega )\\ & I^C(L=0,\Omega )=0,\mu <0\\ & I^C(L=L_H,\Omega )=\frac{1}{\pi }{\int }_{2\pi -}d{\Omega }^{\prime }{I}^C(L=L_H,{\Omega }^{\prime })|{\mu }^{\prime }|{\rho }_b({\Omega }^{\prime },\Omega ),\mu >0\end{array}$$
Here, $Q(L,\Omega )=\frac{1}{\pi }{\int }_{4\pi }\Gamma ({\Omega }^{\prime },\Omega )|\mu |I^0(L,{\Omega }^{\prime })d{\Omega }^{\prime }$ and $S(L,\Omega )=\frac{1}{\pi }{\int }_{4\pi }\Gamma ({\Omega }^{\prime },\Omega )|\mu |I^C(L,{\Omega }^{\prime })d{\Omega }^{\prime }$.

Uncollided problem

To solve Eq. (3), we have that
$$\begin{array}{rl}& \frac{1}{I^0(L,\Omega )}d{I}^0(L,\Omega )=G(L,\Omega )dL/\mu \\ & \to ln{I}^0(L,\Omega )=\frac{1}{\mu }{\int }_0^{L}G(L,\Omega )dL+c=GL+c\end{array}$$
Using the first boundary condtions , we have
$$c=ln{I}^0(L=0,\Omega )$$
Then we have
$$I^0(L,\Omega )=I^0(L=0,\Omega )e^{\frac{1}{\mu }GL},\mu >0$$
Using the second boundary conditions, we have
$$\begin{array}{rl}& ln{I}^0(L=L_H,\Omega )=G{L}_H+c\\ & \to c=ln{I}^0(r,\Omega )-G{L}_H\\ & \to ln{I}^0(L,\Omega )=\frac{1}{\mu }GL-G{L}_H+ln{I}^0(r,\Omega )\\ & \to I^0(r,\Omega )=I^0(r,\Omega )e^{\frac{1}{\mu }G(L-L_H)},\mu <0\end{array}$$
Sumarizing these equations, we have
$$\begin{array}{rl}& I^0(L,\Omega )=I^0(L=0,\Omega )P[\Omega ,(L-0)],\mu <0\\ & I^0(L,\Omega )=I^0(L=L_H,\Omega )P[\Omega ,(L_H-L)],\mu >0\end{array}$$
where
$$P[\Omega ,(L_2-L_1)]=exp[-\frac{1}{\mu }G(\Omega )(L_2-L_1)]$$
Then, Here, $I^0(L=0,\Omega )$ is given by boundary condition, and the upward intensity at the canopy lower bound is given by
$$I^0(L=L_H,\Omega )=\frac{1}{\pi }{\int }_{2\pi -}d{\Omega }^{\prime }{\rho }_s({\Omega }^{\prime },\Omega )I(L=L_H,{\Omega }^{\prime })|{\mu }^{\prime }|,\mu >0$$

First Collision Problem

Supposes the $S$ is very weak, we have
$$\begin{array}{rl}& -\mu \frac{\partial }{\partial L}{I}^1(L,\Omega )+G(\Omega )I^1(L,\Omega )=Q(L,\Omega )\\ & I^1(L=0,\Omega )=0\\ & I^1(L=L_H,\Omega )=\frac{1}{\pi }{\int }_{2\pi -}d{\Omega }^{\prime }{\rho }_s({\Omega }^{\prime },\Omega )|{\mu }^{\prime }|I^1(L=L_H,{\Omega }^{\prime }),\mu >0\end{array}$$
To solve this problem, we rewrite the Eq.(12) into
$$\frac{\partial I^1(L,\Omega )}{\partial L}-\frac{G(\Omega )I^1(L,\Omega )}{\mu }=-\frac{Q(L,\Omega )}{\mu }$$
Then using integral factor method:
$$F(L)=exp[\int G(\Omega )dL/\mu ]=e^{GL/\mu }$$
then we have
$$\begin{array}{r}{e}^{GL/\mu }{I}^1(L,\Omega )=-\frac{1}{\mu }\int e^{G{L}^{\prime }/\mu }Q(L^{\prime },\Omega )d{L}^{\prime }+c\end{array}$$
Using upper boundary condition, we have
$$c=\frac{1}{\mu }[\int e^{G{L}^{\prime }}Q(L^{\prime },\Omega )d{L}^{\prime }{]}_{L=0},\mu <0$$
Thus
$$\begin{array}{rl}{I}^1(L,\Omega )& =-\frac{1}{\mu }{\int }_0^L{e}^{\frac{G}{\mu }(L^{\prime }-L)}Q(L^{\prime },\Omega )d{L}^{\prime }\\ & =-\frac{1}{\mu }{\int }_0^L{e}^{-\frac{G}{\mu }(L-L^{\prime })}Q(L^{\prime },\Omega )d{L}^{\prime }\\ & =\frac{1}{|\mu |}{\int }_0^{L}Q(L^{\prime },\Omega )P[\Omega ,(L-L^{\prime })]d{L}^{\prime },\mu <0\end{array}$$
For lower bound condtion, we have
$$e^{G{L}_H}{I}^1(L=L_H,\Omega )+[\frac{1}{\mu }\int e^{G{L}^{\prime }/\mu }Q(L^{\prime },\Omega )d{L}^{\prime }{]}_{L=L_H}=c,\mu >0$$
Then we have
$$\begin{array}{rl}{e}^{GL/\mu }{I}^1(L,\Omega )=& -\frac{1}{\mu }\int e^{G{L}^{\prime }/\mu }Q(L^{\prime },\Omega )d{L}^{\prime }+e^{G{L}_H}{I}^1(L=L_H,\Omega )\\ & +[\frac{1}{\mu }\int e^{G{L}^{\prime }/\mu }Q(L^{\prime },\Omega )d{L}^{\prime }{]}_{L=L_H},\mu >0\end{array}$$
which is equal
$$\begin{array}{rl}& e^{GL/\mu }{I}^1(L,\Omega )=\frac{1}{|\mu |}{\int }_{L^{\prime }}^{L_H}{e}^{G{L}^{\prime }/\mu }Q(L^{\prime },\Omega )d{L}^{\prime }+e^{G{L}_H}{I}^1(L=L_H,\Omega ),\mu >0\\ & \to I^1(L,\Omega )=\frac{1}{|\mu |}{\int }_{L^{\prime }}^{L_H}{e}^{G(L^{\prime }-L)/\mu }Q(L^{\prime },\Omega )d{L}^{\prime }+e^{G(L_H-L)}{I}^1(L=L_H,\Omega ),\mu >0\end{array}$$
Combining the Eq. (17) and Eq. (20), we have:
$$I^1(L,\Omega )=\frac{1}{|\mu |}{\int }_0^{L}d{L}^{\prime }Q(L^{\prime },\Omega )P[\Omega ,L-L^{\prime }],\mu <0$$

$$I^1(L,\Omega )=\frac{1}{|\mu |}{\int }_0^{L_H}d{L}^{\prime }Q(L^{\prime },\Omega )P[\Omega ,L^{\prime }-L]+I^1(L=L_H,\Omega )P(\Omega ,L_H-L),\mu >0$$

Successive Orders of Scattering Approximation

Similarly, we have
$$\begin{array}{rl}& I^2(L,\Omega )=\frac{1}{|\mu |}{\int }_0^{L}d{L}^{\prime }{S}_1(L^{\prime },\Omega )P(\Omega ,L-L^{\prime }),\mu <0\\ & I^2(L,\Omega )=\frac{1}{|\mu |}{\int }_0^{L}d{L}^{\prime }{S}_1(L^{\prime },\Omega )P(\Omega ,L^{\prime }-L)+I^2(L=L_H,\Omega )P(\Omega ,L_H-L),\mu >0\end{array}$$

where
$$S_1(L,\Omega )=\frac{1}{|\mu |}{\int }_{4\pi }d{\Omega }^{\prime }\Gamma ({\Omega }^{\prime },\Omega )I^1(L,{\Omega }^{\prime })$$

Then, repeat this process,

$$\begin{array}{rl}& I^n(L,\Omega )=\frac{1}{|\mu |}{\int }_0^{L}d{L}^{\prime }{S}_n(L^{\prime },\Omega )P(\Omega ,L-L^{\prime }),\mu <0\\ & I^n(L,\Omega )=\frac{1}{|\mu |}{\int }_0^{L}d{L}^{\prime }{S}_n(L^{\prime },\Omega )P(\Omega ,L^{\prime }-L)+I^n(L=L_H,\Omega )P(\Omega ,L_H-L),\mu >0\end{array}$$
The total intensity $I$ and sources $S$ can be evalurated as
$$\begin{array}{rl}& I(L,\Omega )=I^0(L,\Omega )+\sum _{n=1}^{+\infty }{I}^n(L,\Omega )\\ & S(L,\Omega )=\sum _{n=1}^{+\infty }{S}_n(L,\Omega )\end{array}$$