1. Kunbelka-Munk theory
This is the earlist model using a two-stream approximation
d
I
d
z
=
−
(
k
+
s
)
I
+
s
J
d
J
d
z
=
(
k
+
s
)
J
−
s
I
\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,
I
I
I and
J
J
J is downward and upward flux density, and
k
k
k is obsorption coefficient,
s
s
s is back scattering coefficient,
z
z
z is the metrical depth.
Another notation represent the K-M theory by
d
E
−
d
z
=
−
a
E
−
+
σ
E
+
−
d
E
+
d
z
=
−
a
E
+
+
σ
E
−
\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
+
s
a=k+s
a=k+s is called attenuation coefficient, and
σ
\sigma
σ is backscattering coefficient.
2. Duntley equations
For considering the specular source like sun, we have Duntley equations.
d
E
s
d
z
=
−
k
E
s
d
E
−
d
z
=
s
′
E
s
−
a
E
−
+
σ
E
+
d
E
+
d
z
=
−
s
′
E
s
+
a
E
+
−
σ
E
−
\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,
k
k
k is extinction coefficient for specular flux density, and
s
′
s'
s′ is forward scatter coefficient for specular flux density, and
s
s
s 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
E
s
/
d
z
=
−
k
E
s
,
E
−
/
d
z
=
s
′
E
s
−
a
E
−
+
σ
E
+
,
E
+
/
d
z
=
−
s
E
s
−
σ
E
−
+
a
E
+
,
π
I
o
+
/
d
z
=
−
w
E
s
−
v
E
−
−
v
′
E
+
+
K
π
I
o
+
,
π
I
o
−
/
d
z
=
w
′
E
s
+
v
′
E
−
+
v
E
+
−
K
π
I
o
−
.
\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.
