Verhoef对辐射传输模型的四流近似方法

Verhoef Four-stream theory

1. reflectance

The BRDF ${\rho }^{\prime }$ is defined as
$$dL(\Omega )={\rho }^{\prime }(\Omega ,{\Omega }^{\prime })I({\Omega }^{\prime })|{\mu }^{\prime }|d\Omega$$
For specular incident flux from direction ${\Omega }^{\prime }$ it becomes
$$I(\Omega )={\rho }^{\prime }({\Omega }^{\prime },\Omega )E({\Omega }^{\prime })$$
For Lambertian target, we have
$$M=\pi I$$
Here, $M$ is exitance. For ideal Lambertian (called White Lambertian reflector in verhoef’s book), $M=E=F$, thus the BRDF of this target is
$$\frac{F}{\pi }={\rho }^{\prime }F\to {\rho }^{\prime }=\frac{1}{\pi }$$
The radiance of arbitraty BRDF is
$$I(\Omega )={\int }_{2\pi -}{\rho }^{\prime }({\Omega }^{\prime },\Omega )|{\mu }^{\prime }|I({\Omega }^{\prime })d{\Omega }^{\prime }$$
The BRF $r_0$ is defined as
$$r_0=\frac{{\int }_{2\pi -}{\rho }^{\prime }({\Omega }^{\prime },\Omega )|{\mu }^{\prime }|I({\Omega }^{\prime })d{\Omega }^{\prime }}{\frac{1}{\pi }{\int }_{2\pi -}|{\mu }^{\prime }|I({\Omega }^{\prime })d{\Omega }^{\prime }}$$
So for spexular incident flux, using delta function, it could be writen as
$$r_{so}=\frac{{\rho }^{\prime }({\Omega }^{\prime },\Omega )|{\mu }^{\prime }|I({\Omega }^{\prime })}{|{\mu }^{\prime }|I({\Omega }^{\prime })\frac{1}{\pi }}=\frac{\pi I_0}{F_s^{\downarrow }}=\pi {\rho }^{\prime }$$
where $F_s^{\downarrow }=|{\mu }^{\prime }|I({\Omega }^{\prime })\frac{1}{\pi }$. And for a isopropic diffuse incident flux, we have
$$r_{do}=\frac{{\int }_{2\pi -}|{\mu }^{\prime }|{\rho }^{\prime }({\Omega }^{\prime },\Omega )d{\Omega }^{\prime }}{\frac{1}{\pi }{\int }_{2\pi -}|{\mu }^{\prime }|d{\Omega }^{\prime }}={\int }_{2\pi -}|{\mu }^{\prime }|{\rho }^{\prime }({\Omega }^{\prime },\Omega )d{\Omega }^{\prime }=\frac{\pi I_0}{F_d^{\downarrow }}$$
So using Eq. (7) we have the relationship of $r_{so}$ and $r_{do}$
$$r_{do}={\int }_{2\pi -}|{\mu }^{\prime }|{\rho }^{\prime }({\Omega }^{\prime },\Omega )d{\Omega }^{\prime }=\frac{1}{\pi }{\int }_{2\pi -}|{\mu }^{\prime }|r_{so}d{\Omega }^{\prime }$$
BHR (called diffuse reflectance in verhoef’s book) is defined as
$$r_d=\frac{{\int }_{2\pi +}I|\mu |d\Omega }{F^{\downarrow }}$$
For specular incident flux $r_d$ becomes $r_{sd}$, which is
$$r_{sd}=\frac{{\int }_{2\pi +}I({\Omega }^{\prime })|{\mu }^{\prime }|{\rho }^{\prime }({\Omega }^{\prime },\Omega )|\mu |d\Omega }{I({\Omega }^{\prime })|{\mu }^{\prime }|}={\int }_{2\pi +}{\rho }^{\prime }({\Omega }^{\prime },\Omega )|\mu |d\Omega$$
And we connect the Eq. (11) with Eq. (7), we found that
$$r_{sd}=\frac{1}{\pi }{\int }_{2\pi +}{r}_{so}|\mu |d\Omega$$
For isopropic diffuse incident flux, we could find that BHR is
$$\begin{array}{rl}{r}_{dd}& =\frac{{\int }_{2\pi +}I(\Omega )|\mu |d\Omega }{I}\\ & =\frac{\frac{1}{\pi }{\int }_{2\pi +}{\int }_{2\pi -}I({\Omega }^{\prime })\rho ({\Omega }^{\prime },\Omega )|{\mu }^{\prime }|d{\Omega }^{\prime }|\mu |d\Omega }{I}\\ & =\frac{1}{\pi }{\int }_{2\pi +}{\int }_{2\pi -}\rho ({\Omega }^{\prime },\Omega )|{\mu }^{\prime }|d{\Omega }^{\prime }|\mu |d\Omega \end{array}$$
If the incident flux is composed of a specular part $F_s$ and a diffuse part $F_d$, then the radiance $I$ of the surface with direction $\Omega$ is
$$\pi I(\Omega )=r_{so}{F}_s+r_{do}{F}_d$$

and we also have the approximation that
$$F^{\uparrow }=r_{sd}{F}_s+r_{dd}{F}^{\downarrow }$$
The Eq. (14) and Eq.(15) constitute the four-stream representation of reflectance of a surface.

Transmittance

Define the bidirectional transmittance distribution function (BTDF) as
$$dI(\Omega )={\tau }^{\prime }({\Omega }^{\prime },\Omega )I({\Omega }^{\prime })|{\mu }^{\prime }|d\Omega$$
So ${\tau }^{\prime }$ is very similar to BRDF, only different is ${\Omega }^{\prime }$ and $\Omega$ are in opposite hemishpheres. With similar defination of reflectance characters, we have
$$\begin{array}{rl}& {\tau }_{so}({\Omega }^{\prime },\Omega )=\pi {\tau }^{\prime }({\Omega }^{\prime },\Omega )\\ & {\tau }_{do}(\Omega )={\int }_{2\pi +}{\tau }^{\prime }({\Omega }^{\prime },\Omega )|{\mu }^{\prime }|d{\Omega }^{\prime }\\ & {\tau }_{sd}={\int }_{2\pi -}{\tau }^{\prime }({\Omega }^{\prime },\Omega )|\mu |d\Omega \\ & {\tau }_{dd}=\frac{1}{\pi }{\int }_{2\pi -}{\tau }_{do}|\mu |d\Omega \end{array}$$
They are direct-in-out, diffuse-in-direct-out, direct-in-diffuse-out, and diffuse-in-out transmittance. And if ${\Omega }^{\prime }=\Omega$, then the transmittance is called direct transmittance, for specular flux this is called ${\tau }_{ss}$ and for obervation direction this is called ${\tau }_{oo}$.

Four-Stream interaction with layers and surfaces

Now, we use $E$ for flux density (or irradiance). The four-stream radiative transfer equations for a layer is
$$\begin{array}{rl}& E_s(b)={\tau }_{ss}{E}_s(t),\\ & E^{-}(b)={\tau }_{sd}{E}_s(t)+{\tau }_{dd}{E}^{-}(t)+{\rho }_{dd}{E}^{+}(b),\\ & E^{+}(b)={\rho }_{sd}{E}_s(t)+{\rho }_{dd}{E}^{-}(t)+{\tau }_{dd}{E}^{+}(b),\\ & \pi I_o^{+}(t)={\rho }_{so}{E}_s(t)+{\rho }_{do}{E}^{-}t)+{\tau }_{do}{E}^{+}(b)+{\tau }_{oo}\pi I_o^{+}(b),\\ & \pi I_o^{-}(t)={\tau }_{so}{E}_s(t)+{\tau }_{do}{E}^{-}t)+{\rho }_{do}{E}^{+}(b)+{\tau }_{oo}\pi I_o^{-}(b).\end{array}$$
So it is easy to understand these equations, the only hard place is the last two equations, the factor $\pi$ looks strange. For this, we could divide the $\pi$ from both sides, then we find that the BRDFs or BRTF all become the BRF (BTF), so it looks better.

Then, the interaction with a surface is
$$\begin{array}{rl}& E^{+}(b)=r_{sd}{E}_s(b)+r_{dd}{E}^{-}(b)\\ & \pi I_0^{+}=r_{so}{E}_s(b)+r_{do}{E}^{-}(b)\end{array}$$
Then Eq. (20) and (21) discribe a surface and a layer beyund it. This process is discribed as the following figure: