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The reflectance field is split into three main components which are unscattered by the leaves (0), single scattered 
(1) and multiple scattered (M) radiations (see Fig-1): 
Verstraete et al. [1990] have derived a physically based analytical model for predicting the bidirectional 
reflectance of a semi-infinite porous medium. The major value of this model is to provide a good and physically 
sounded representation of the hot-spot, but it suffers from the limitation that the effects of the underlying soil are 
not accounted for, which limits its application to the case of thick canopies only. We have adopted their 
analytical formulation of the single scattering to compute the first two components of the reflectance field 
(including the hot-spot effect of the canopy) and we have added a realistic lower boundary condition allowing us 
to consider vegetation canopies with a finite optical thickness. 
1.1.a - Single scattering by canopy elements — p 1 (Gq.G) 
The first order scattering plays a very important role in the canopy reflectance and, ideally, it should be expressed 
exactly for any type of phase function. In the case of a vegetation canopy (Fig-2), illuminated from the direction 
Q 0 (|ìq=cos (0q),<pq) by direct solar radiation, and observed from the direction G (p=cos (0),(p), [Verstraete et 
al., 1990] demonstrated: 
where T (Gp-G) is the area scattering phase function (bi-Lambertian) 
Tq (L) is the transmission of direct solar radiation through the canopy layers above the level L 
T (L) is the transmission of scattered radiation 
L is the leaf area index (0 < L < Lj). 
When the direction of observation is exactly the same as the direction of illumination (Q=-Qq), the transmission 
T (L) should be unity because any radiation able of penetrate down to the level L before being scattered must exit 
the canopy without further interaction: this is called the hot-spot effect 
Four different volumes can be defined to parameterize the joint transmission effect 
Vq is a cylindrical volume defined in the canopy by the lit circle (sun-fleck) in direction of illumination 
V is the cylindrical volume defined by the lit circle in the direction of observation 
Vi is the common volume of both cylinders in the canopy (where transmissivity is one) 
V 2 is the complement of V| (proportion of the volume Vq not in common with V). 
The transmission of the scattered radiation is functionally depending on the transmission of the incident rays 
(because the two optical paths share a common volume, Vj, free of scatterers in the canopy) and is only affected 
during its travel within volume V2. It can be written as follows: 
where the function that accounts for the joint transmission of the incoming and outgoing radiations is: 
p TOT (g 0 ,g) = p° (g 0 ,g) + p 1 («o- n > + p m W- 
L T 
where 
V 2 («0.1 
where—.. T „ _ 
— -1 if L < Lj (Lj = 2rA / Geo («0,«) is the depth at 
4 Lj if Li Lj which the two cylinders intersect) 
—- -1 ifL<Lj (Lj = 2rA / Geo («Q,«) i: 
4 Lj if L i Lj which the two cylinc 
3tt T 
And after some mathematical development, the singly scattered radiance is expressed by: 
p 1 (Gq.G) - r . M + r, (Cl) lunl r ( Q 0-* Q ) p v 
G (Gq) |i + G (G) lp 0 l 
T (Gq-'G) P v (Gq.G). 
G (Op) n + G(G) Ippl 
IHO 1 ^ 
,)' erf (vi)]* exp ( 3 ” 
A. 2 r G(G)A > 
3rt Geo (Gq,G) p J 
( G(G 0 )p + G(G)lp 0 l 
j 
ino 1 ^