1. Kunbelka-Munk theory
This is the earlist model using a two-stream approximation
dIdz=−(k+s)I+sJdJdz=(k+s)J−sI
\begin{aligned}
&\frac{dI}{dz} = -(k+s)I+sJ\\
&\frac{dJ}{dz} = (k+s)J - sI
\end{aligned}
dzdI=−(k+s)I+sJdzdJ=(k+s)J−sI
Here, III and JJJ is downward and upward flux density, and kkk is obsorption coefficient, sss is back scattering coefficient, zzz is the metrical depth.
Another notation represent the K-M theory by
dE−dz=−aE−+σE+−dE+dz=−aE++σE−
\begin{aligned}
&\frac{dE^-}{dz} = -aE^-+\sigma E^+\\
&-\frac{dE^+}{dz} = -aE^+ + \sigma E^-
\end{aligned}
dzdE−=−aE−+σE+−dzdE+=−aE++σE−
Here, a=k+sa=k+sa=k+s is called attenuation coefficient, and σ\sigmaσ is backscattering coefficient.
2. Duntley equations
For considering the specular source like sun, we have Duntley equations.
dEsdz=−kEsdE−dz=s′Es−aE−+σE+dE+dz=−s′Es+aE+−σE−
\begin{aligned}
&\frac{dE_s}{dz} = -kE_s\\
&\frac{dE^-}{dz} = s'E_s -aE^-+\sigma E^+\\
&\frac{dE^+}{dz} = -s'E_s +aE^+ - \sigma E^-
\end{aligned}
dzdEs=−kEsdzdE−=s′Es−aE−+σE+dzdE+=−s′Es+aE+−σE−
Here, kkk is extinction coefficient for specular flux density, and s′s's′ is forward scatter coefficient for specular flux density, and sss is backward scatter coefficient for specualar flux density.
To now, these equations are not connected with canopy parameters, such as leaf area index.
3. Suit and SAIL model
Suit model is also Duntley equations, but the coefficients are directly expressed in biophysical parameters of the canopy. The coefficients of suit model only defined for horizontal and vertical leaves, SAIL model improved the Suit and its coefficients can be computed for any leaf inclination.
These two models are actually four-stream model, which is
Es/dz=−kEs,E−/dz=s′Es−aE−+σE+,E+/dz=−sEs−σE−+aE+,πIo+/dz=−wEs−vE−−v′E++KπIo+,πIo−/dz=w′Es+v′E−+vE+−KπIo−.
\begin{aligned}
& E_s/dz = -kE_s,\\
& E^-/dz=s'E_s-aE^-+\sigma E^+,\\
& E^+/dz=-sE_s-\sigma E^-+aE^+,\\
& \pi I_o^+/dz=-wE_s-vE^--v'E^++K \pi I_o^+,\\
& \pi I_o^-/dz=w'E_s+v'E^-+vE^+-K \pi I_o^-.\\
\end{aligned}
Es/dz=−kEs,E−/dz=s′Es−aE−+σE+,E+/dz=−sEs−σE−+aE+,πIo+/dz=−wEs−vE−−v′E++KπIo+,πIo−/dz=w′Es+v′E−+vE+−KπIo−.
The parameter are easy to understand and are same to the previous blog.