685
In the above equation, A denotes the leaf area density [m^/m^] and r [m] is the radius of the sun-flecks on the
illuminated leaf.
1.1.b - Uncollided radiation (first order reflectance from the soil) — (QQ,i2)
The second component of the reflectance is the product of the soil albedo by the transmission of direct solar and
scattered radiations. It represents the contribution due to photons that have not undergone any interaction in the
canopy, and which are reflected by the soil. This component also includes the canopy hot-spot effect, which we
derived analytically from Verstraete et al. [ 1990]:
1.1.c - Multiply scattered radiation — (ììq)
For the multiple scattering component of the reflectance, the two-stream approximation, which is based on two
integrated fluxes (one upward and one downward), has been used for semi-infinite canopies; in the context of our
present development, we chose a (numerical) one-angle DOM, which is reasonably fast and accurate.
Shultis J. K. & R. B. Myneni [1988] and others (e.g. Marshak A. L. [1989], Knyazikhin Y. et al. [1993])
have developed a variant of this DOM (finite differences, exact kernel) to solve the canopy transport equation (and
its azimuthal average) numerically. In their method, photons are restricted to travel in a finite set of discrete
directions (the coordinates of the Gaussian quadrature used to compute the integrals numerically), and the
derivatives (with respect to the optical depth) are approximated by a finite difference scheme.
In order to save computer time, especially for inversion purposes, we reduced the one-dimensional transport
equation to a one-angle problem and assumed isotropic scattering, that is:
and boundary conditions are specified through a Lambertian soil at the bottom, and a direct solar irradiance
incident at the top (in direction (to):
the multiply scattered intensity 1^ (i.e. photons which have been scattered two or more times in the finite
canopy):
The equations used to represent and compute these three terms are given explicitly in the appendix, together with
the appropriate boundary conditions. Finally the multiply scattered radiation exiting at the top of the canopy is
integrated over the azimuth angle, which leads to an approximation for the corresponding upward flux:
As indicated by the above equation, the quantity p M (Dq) is kept constant for all the exiting angles to be
considered over the upper hemisphere.
1.2 - Leaf properties
p0 (îîq,Q) = Rs Tq (Lj) T (Lj) = Rs exp -
G (Ap) G (fl) v 2 (Qq.Q.L t )
Ipol + p V (Q,L T )
The above equation was solved by separating the uncollided intensity I® and the first scattered intensity I* from
I (L,p) = 1° (L,p) + I 1 (L,|i) + I M (L,p).
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