Seperation of uncollided and Collided Intensities
For one dimensional RTM of canopy, let the illumination is Fin(L=0)=1F_{in}(L=0)=1Fin(L=0)=1 and isotropic skylight is thus
−μ∂∂LI(L,Ω)+G(L,Ω)I(L,Ω)=1π∫4πdΩ′Γ(Ω′,Ω)I(L,Ω′)I(L=0,Ω)=fdir∣μ0∣δ(Ω,Ω0)+1−fdirπI(L=LH,Ω)=1π∫2π−I(L=LH,Ω′)ρb(Ω′,Ω)∣μ′∣dΩ′
\begin{aligned}
& -\mu\frac{\partial }{\partial L}I(L,\Omega)+G(L,\Omega)I(L,\Omega)=\frac{1}{\pi}\int_{4\pi}d\Omega'\Gamma(\Omega',\Omega)I(L,\Omega')\\
&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')\rho_b(\Omega',\Omega)|\mu'|d\Omega'
\end{aligned}
−μ∂L∂I(L,Ω)+G(L,Ω)I(L,Ω)=π1∫4πdΩ′Γ(Ω′,Ω)I(L,Ω′)I(L=0,Ω)=∣μ0∣fdirδ(Ω,Ω0)+π1−fdirI(L=LH,Ω)=π1∫2π−I(L=LH,Ω′)ρb(Ω′,Ω)∣μ′∣dΩ′
For numerical purposes, we seperate the uncollided radiation field from collided fields, that is
I(L,Ω)=I0(L,Ω)+Ic(L,Ω)
I(L,\Omega)=I^0(L,\Omega)+I^c(L,\Omega)
I(L,Ω)=I0(L,Ω)+Ic(L,Ω)
where I0I^0I0 specific intensity of uncollided photons and IcI^cIc 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:
−μ∂∂LI0(L,Ω)+G(L,Ω)I0(L,Ω)=0I0(L=0,Ω)=fdir∣μ0∣δ(Ω,Ω0)+1−fdirπ,μ<0I0(L=LH,Ω)=1π∫2π−dΩ′I0(L=LH,Ω′)∣μ′∣ρb(Ω′,Ω),μ<0
\begin{aligned}
& -\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'I^0(L=L_H,\Omega')|\mu'|\rho_b(\Omega',\Omega),\mu<0
\end{aligned}
−μ∂L∂I0(L,Ω)+G(L,Ω)I0(L,Ω)=0I0(L=0,Ω)=∣μ0∣fdirδ(Ω,Ω0)+π1−fdir,μ<0I0(L=LH,Ω)=π1∫2π−dΩ′I0(L=LH,Ω′)∣μ′∣ρb(Ω′,Ω),μ<0
and coollided intensity
−μ∂∂LIC(L,Ω)+G(L,Ω)IC(L,Ω)=Q(L,Ω)+S(L,Ω)IC(L=0,Ω)=0,μ<0IC(L=LH,Ω)=1π∫2π−dΩ′IC(L=LH,Ω′)∣μ′∣ρb(Ω′,Ω),μ>0
\begin{aligned}
& -\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'I^C(L=L_H,\Omega')|\mu'|\rho_b(\Omega',\Omega), \mu>0
\end{aligned}
−μ∂L∂IC(L,Ω)+G(L,Ω)IC(L,Ω)=Q(L,Ω)+S(L,Ω)IC(L=0,Ω)=0,μ<0IC(L=LH,Ω)=π1∫2π−dΩ′IC(L=LH,Ω′)∣μ′∣ρb(Ω′,Ω),μ>0
Here, Q(L,Ω)=1π∫4πΓ(Ω′,Ω)∣μ∣I0(L,Ω′)dΩ′Q(L,\Omega)=\frac{1}{\pi} \int_{4\pi}\Gamma(\Omega',\Omega)|\mu|I^0(L,\Omega')d\Omega'Q(L,Ω)=π1∫4πΓ(Ω′,Ω)∣μ∣I0(L,Ω′)dΩ′ and S(L,Ω)=1π∫4πΓ(Ω′,Ω)∣μ∣IC(L,Ω′)dΩ′S(L,\Omega)=\frac{1}{\pi} \int_{4\pi}\Gamma(\Omega',\Omega)|\mu|I^C(L,\Omega')d\Omega'S(L,Ω)=π1∫4πΓ(Ω′,Ω)∣μ∣IC(L,Ω′)dΩ′.
Uncollided problem
To solve Eq. (3), we have that
1I0(L,Ω)dI0(L,Ω)=G(L,Ω)dL/μ→lnI0(L,Ω)=1μ∫0LG(L,Ω)dL+c=GL+c
\begin{aligned}
&\frac{1}{I^0(L,\Omega)}d I^0(L,\Omega) = G(L,\Omega) d L/\mu\\
&\rightarrow lnI^0(L,\Omega) = \frac{1}{\mu}\int_0^LG(L,\Omega)dL + c=GL+c
\end{aligned}
I0(L,Ω)1dI0(L,Ω)=G(L,Ω)dL/μ→lnI0(L,Ω)=μ1∫0LG(L,Ω)dL+c=GL+c
Using the first boundary condtions , we have
c=lnI0(L=0,Ω)
c=lnI^0(L=0,\Omega)
c=lnI0(L=0,Ω)
Then we have
I0(L,Ω)=I0(L=0,Ω)e1μGL,μ>0
I^0(L,\Omega)=I^0(L=0,\Omega)e^{\frac{1}{\mu}GL}, \mu >0
I0(L,Ω)=I0(L=0,Ω)eμ1GL,μ>0
Using the second boundary conditions, we have
lnI0(L=LH,Ω)=GLH+c→c=lnI0(r,Ω)−GLH→lnI0(L,Ω)=1μGL−GLH+lnI0(r,Ω)→I0(r,Ω)=I0(r,Ω)e1μG(L−LH),μ<0
\begin{aligned}
&lnI^0(L=L_H,\Omega) = GL_H +c\\
&\rightarrow c=lnI^0(r,\Omega)-GL_H\\
&\rightarrow lnI^0(L,\Omega)=\frac{1}{\mu}GL-GL_H + lnI^0(r,\Omega)\\
&\rightarrow I^0(r,\Omega)= I^0(r,\Omega)e^{\frac{1}{\mu}G(L-L_H)}, \mu <0
\end{aligned}
lnI0(L=LH,Ω)=GLH+c→c=lnI0(r,Ω)−GLH→lnI0(L,Ω)=μ1GL−GLH+lnI0(r,Ω)→I0(r,Ω)=I0(r,Ω)eμ1G(L−LH),μ<0
Sumarizing these equations, we have
I0(L,Ω)=I0(L=0,Ω)P[Ω,(L−0)],μ<0I0(L,Ω)=I0(L=LH,Ω)P[Ω,(LH−L)],μ>0
\begin{aligned}
&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{aligned}
I0(L,Ω)=I0(L=0,Ω)P[Ω,(L−0)],μ<0I0(L,Ω)=I0(L=LH,Ω)P[Ω,(LH−L)],μ>0
where
P[Ω,(L2−L1)]=exp[−1μG(Ω)(L2−L1)]
P[\Omega,(L_2-L_1)]=exp[-\frac{1}{\mu}G(\Omega)(L_2-L_1)]
P[Ω,(L2−L1)]=exp[−μ1G(Ω)(L2−L1)]
Then, Here, I0(L=0,Ω)I^0(L=0,\Omega)I0(L=0,Ω) is given by boundary condition, and the upward intensity at the canopy lower bound is given by
I0(L=LH,Ω)=1π∫2π−dΩ′ρs(Ω′,Ω)I(L=LH,Ω′)∣μ′∣,μ>0
I^0(L=L_H,\Omega)=\frac{1}{\pi}\int_{2\pi-}d\Omega'\rho_s(\Omega',\Omega)I(L=L_H,\Omega')|\mu'|, \mu>0
I0(L=LH,Ω)=π1∫2π−dΩ′ρs(Ω′,Ω)I(L=LH,Ω′)∣μ′∣,μ>0
First Collision Problem
Supposes the SSS is very weak, we have
−μ∂∂LI1(L,Ω)+G(Ω)I1(L,Ω)=Q(L,Ω)I1(L=0,Ω)=0I1(L=LH,Ω)=1π∫2π−dΩ′ρs(Ω′,Ω)∣μ′∣I1(L=LH,Ω′),μ>0
\begin{aligned}
&-\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'\rho_s(\Omega',\Omega)|\mu'|I^1(L=L_H,\Omega'), \mu>0
\end{aligned}
−μ∂L∂I1(L,Ω)+G(Ω)I1(L,Ω)=Q(L,Ω)I1(L=0,Ω)=0I1(L=LH,Ω)=π1∫2π−dΩ′ρs(Ω′,Ω)∣μ′∣I1(L=LH,Ω′),μ>0
To solve this problem, we rewrite the Eq.(12) into
∂I1(L,Ω)∂L−G(Ω)I1(L,Ω)μ=−Q(L,Ω)μ
\frac{\partial I^1(L,\Omega)}{\partial L}-\frac{G(\Omega)I^1(L,\Omega)}{\mu}=-\frac{Q(L,\Omega)}{\mu}
∂L∂I1(L,Ω)−μG(Ω)I1(L,Ω)=−μQ(L,Ω)
Then using integral factor method:
F(L)=exp[∫G(Ω)dL/μ]=eGL/μ
F(L)=exp[\int G(\Omega)dL/\mu]=e^{GL/\mu}
F(L)=exp[∫G(Ω)dL/μ]=eGL/μ
then we have
eGL/μI1(L,Ω)=−1μ∫eGL′/μQ(L′,Ω)dL′+c
\begin{aligned}
e^{GL/\mu}I^1(L,\Omega)=-\frac{1}{\mu}\int e^{GL'/\mu}Q(L',\Omega)dL'+c
\end{aligned}
eGL/μI1(L,Ω)=−μ1∫eGL′/μQ(L′,Ω)dL′+c
Using upper boundary condition, we have
c=1μ[∫eGL′Q(L′,Ω)dL′]L=0,μ<0
c=\frac{1}{\mu}[\int e^{GL'}Q(L',\Omega)dL']_{L=0}, \mu <0
c=μ1[∫eGL′Q(L′,Ω)dL′]L=0,μ<0
Thus
I1(L,Ω)=−1μ∫0LeGμ(L′−L)Q(L′,Ω)dL′=−1μ∫0Le−Gμ(L−L′)Q(L′,Ω)dL′=1∣μ∣∫0LQ(L′,Ω)P[Ω,(L−L′)]dL′,μ<0
\begin{aligned}
I^1(L,\Omega)&=-\frac{1}{\mu}\int_0^L e^{\frac{G}{\mu}(L'-L)}Q(L',\Omega)dL'\\
&=-\frac{1}{\mu}\int_0^L e^{-\frac{G}{\mu}(L-L')}Q(L',\Omega)dL'\\
&=\frac{1}{|\mu|}\int_0^LQ(L',\Omega)P[\Omega,(L-L')]dL', \mu <0
\end{aligned}
I1(L,Ω)=−μ1∫0LeμG(L′−L)Q(L′,Ω)dL′=−μ1∫0Le−μG(L−L′)Q(L′,Ω)dL′=∣μ∣1∫0LQ(L′,Ω)P[Ω,(L−L′)]dL′,μ<0
For lower bound condtion, we have
eGLHI1(L=LH,Ω)+[1μ∫eGL′/μQ(L′,Ω)dL′]L=LH=c,μ>0
e^{GL_H}I^1(L=L_H,\Omega) + [\frac{1}{\mu}\int e^{GL'/\mu}Q(L',\Omega)dL']_{L=L_H} = c, \mu >0
eGLHI1(L=LH,Ω)+[μ1∫eGL′/μQ(L′,Ω)dL′]L=LH=c,μ>0
Then we have
eGL/μI1(L,Ω)=−1μ∫eGL′/μQ(L′,Ω)dL′+eGLHI1(L=LH,Ω)+[1μ∫eGL′/μQ(L′,Ω)dL′]L=LH,μ>0
\begin{aligned}
e^{GL/\mu}I^1(L,\Omega)=&-\frac{1}{\mu}\int e^{GL'/\mu}Q(L',\Omega)dL'+ e^{GL_H}I^1(L=L_H,\Omega) \\&+ [\frac{1}{\mu}\int e^{GL'/\mu}Q(L',\Omega)dL']_{L=L_H}, \mu > 0
\end{aligned}
eGL/μI1(L,Ω)=−μ1∫eGL′/μQ(L′,Ω)dL′+eGLHI1(L=LH,Ω)+[μ1∫eGL′/μQ(L′,Ω)dL′]L=LH,μ>0
which is equal
eGL/μI1(L,Ω)=1∣μ∣∫L′LHeGL′/μQ(L′,Ω)dL′+eGLHI1(L=LH,Ω),μ>0→I1(L,Ω)=1∣μ∣∫L′LHeG(L′−L)/μQ(L′,Ω)dL′+eG(LH−L)I1(L=LH,Ω),μ>0
\begin{aligned}
&e^{GL/\mu}I^1(L,\Omega)=\frac{1}{|\mu|}\int_{L'}^{L_H}e^{GL'/\mu}Q(L',\Omega)dL'+e^{GL_H}I^1(L=L_H,\Omega), \mu >0\\
&\rightarrow I^1(L,\Omega)=\frac{1}{|\mu|}\int_{L'}^{L_H}e^{G(L'-L)/\mu}Q(L',\Omega)dL'+e^{G(L_H-L)}I^1(L=L_H,\Omega), \mu >0
\end{aligned}
eGL/μI1(L,Ω)=∣μ∣1∫L′LHeGL′/μQ(L′,Ω)dL′+eGLHI1(L=LH,Ω),μ>0→I1(L,Ω)=∣μ∣1∫L′LHeG(L′−L)/μQ(L′,Ω)dL′+eG(LH−L)I1(L=LH,Ω),μ>0
Combining the Eq. (17) and Eq. (20), we have:
I1(L,Ω)=1∣μ∣∫0LdL′Q(L′,Ω)P[Ω,L−L′],μ<0
I^1(L,\Omega)=\frac{1}{|\mu|}\int_0^LdL'Q(L',\Omega)P[\Omega,L-L'], \mu<0
I1(L,Ω)=∣μ∣1∫0LdL′Q(L′,Ω)P[Ω,L−L′],μ<0
I1(L,Ω)=1∣μ∣∫0LHdL′Q(L′,Ω)P[Ω,L′−L]+I1(L=LH,Ω)P(Ω,LH−L),μ>0 I^1(L,\Omega)=\frac{1}{|\mu|}\int_0^{L_H}dL'Q(L',\Omega)P[\Omega,L'-L] + I^1(L=L_H, \Omega)P(\Omega,L_H-L), \mu>0 I1(L,Ω)=∣μ∣1∫0LHdL′Q(L′,Ω)P[Ω,L′−L]+I1(L=LH,Ω)P(Ω,LH−L),μ>0
Successive Orders of Scattering Approximation
Similarly, we have
I2(L,Ω)=1∣μ∣∫0LdL′S1(L′,Ω)P(Ω,L−L′),μ<0I2(L,Ω)=1∣μ∣∫0LdL′S1(L′,Ω)P(Ω,L′−L)+I2(L=LH,Ω)P(Ω,LH−L),μ>0
\begin{aligned}
&I^2(L,\Omega)=\frac{1}{|\mu|}\int_0^LdL'S_1(L',\Omega)P(\Omega,L-L'), \mu<0\\
&I^2(L,\Omega)=\frac{1}{|\mu|}\int_0^LdL'S_1(L',\Omega)P(\Omega,L'-L) + I^2(L=L_H,\Omega)P(\Omega,L_H-L), \mu >0
\end{aligned}
I2(L,Ω)=∣μ∣1∫0LdL′S1(L′,Ω)P(Ω,L−L′),μ<0I2(L,Ω)=∣μ∣1∫0LdL′S1(L′,Ω)P(Ω,L′−L)+I2(L=LH,Ω)P(Ω,LH−L),μ>0
where
S1(L,Ω)=1∣μ∣∫4πdΩ′Γ(Ω′,Ω)I1(L,Ω′)
S_1(L,\Omega)=\frac{1}{|\mu|}\int_{4\pi}d\Omega'\Gamma(\Omega',\Omega)I^1(L,\Omega')
S1(L,Ω)=∣μ∣1∫4πdΩ′Γ(Ω′,Ω)I1(L,Ω′)
Then, repeat this process,
In(L,Ω)=1∣μ∣∫0LdL′Sn(L′,Ω)P(Ω,L−L′),μ<0In(L,Ω)=1∣μ∣∫0LdL′Sn(L′,Ω)P(Ω,L′−L)+In(L=LH,Ω)P(Ω,LH−L),μ>0
\begin{aligned}
&I^n(L,\Omega)=\frac{1}{|\mu|}\int_0^LdL'S_n(L',\Omega)P(\Omega,L-L'), \mu<0\\
&I^n(L,\Omega)=\frac{1}{|\mu|}\int_0^LdL'S_n(L',\Omega)P(\Omega,L'-L) + I^n(L=L_H,\Omega)P(\Omega,L_H-L), \mu >0
\end{aligned}
In(L,Ω)=∣μ∣1∫0LdL′Sn(L′,Ω)P(Ω,L−L′),μ<0In(L,Ω)=∣μ∣1∫0LdL′Sn(L′,Ω)P(Ω,L′−L)+In(L=LH,Ω)P(Ω,LH−L),μ>0
The total intensity III and sources SSS can be evalurated as
I(L,Ω)=I0(L,Ω)+∑n=1+∞In(L,Ω)S(L,Ω)=∑n=1+∞Sn(L,Ω)
\begin{aligned}
&I(L,\Omega)=I^0(L,\Omega)+\sum_{n=1}^{+\infin}I^n(L,\Omega)\\
&S(L,\Omega)=\sum_{n=1}^{+\infin}S_n(L,\Omega)
\end{aligned}
I(L,Ω)=I0(L,Ω)+n=1∑+∞In(L,Ω)S(L,Ω)=n=1∑+∞Sn(L,Ω)