冠层四流近似模型的发展历史

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)JsI
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.
在这里插入图片描述

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=sEsaE+σE+dzdE+=sEs+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=sEsaE+σE+,E+/dz=sEsσE+aE+,πIo+/dz=wEsvEvE++KπIo+,πIo/dz=wEs+vE+vE+KπIo.
The parameter are easy to understand and are same to the previous blog.

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