6S User Guide Version 3, November 2006
138
DESCRIPTION OF THE SUBROUTINES USED
TO COMPUTE GROUND BRDF
6S User Guide Version 3, November 2006
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SUBROUTINE HAPKALBE
Function: To calculate the spherical albedo using the BRDF computed by Hapke’s (1981)
model.
Description: The target spherical albedo s is equal to the flux reflected by the target divided
by the incoming flux from an isotropic source. It is defined as:
∫
∫
π
π
θθθ
θθθθ
=
2/
0
sss
2/
0
ssss
d)sin()cos(
d)sin()cos()(a
s
where a(
s
) is the directional albedo for a parallel solar beam, given by
∫∫
∫∫
ππ
ππ
φθθθ
φθθθφ−φ=φθθρ
=θ
2
0
2/
0
vvv
2
0
2/
0
vvvvsvs
s
dd)sin()cos(
dd)sin()cos(),,(
)(a
with the bidirectional reflectance ),,(
vs
φ
θ
θρ generated by the user's inputs (see HAPRBRDF).
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SUBROUTINE IAPIALBE
Function: Same as HAPKALBE but for a BRDF from the subroutine IAPIBRDF.
SUBROUTINE MINNALBE
Function: Same as HAPKALBE but for a BRDF from the subroutine MINNBRDF.
SUBROUTINE OCEALBE (and GLITALBE)
Function: Same as HAPKALBE but for a BRDF from the subroutine OCEABRDF.
SUBROUTINE RAHMALBE
Function: Same as HAPKALBE but for a BRDF from the subroutine RAHMBRDF.
SUBROUTINE ROUJALBE
Function: Same as HAPKALBE but for a BRDF from the subroutine ROUJBRDF.
SUBROUTINE VERSALBE
Function: Same as HAPKALBE but for a BRDF from the subroutine VERSBRDF.
SUBROUTINE WALTALBE
Function: Same as HAPKALBE but for a BRDF from the subroutine WALTBRDF.
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SUBROUTINE BRDFGRID
Function: To generate a BRDF following the user's inputs.
Description: The user enters the value of for the Sun at a given sun zenith angle
s
for the
view zenith angle
v
ranging from 0° to 80° by steps of 10° and equal to 85°, and for the view
azimuth angle
v
φ ranging from 0° to 360° by steps of 30°. The user does the same for the Sun
which would be at
v
. In addition, the spherical albedo of the surface and observed reflectance in
the selected geometry ),,,(
vsvs
φφθθρ need to be specified.
Parameters:
1. for
s
the user has to enter ),(
vv
φθρ :
(0°,0°), (10°,0°), ... (80°,0°), (85°,0°)
(0°,30°), (10°,30°), ... (80°,30°), (85°,30°)
...
(0°,360°), (10°,360°), ... (80°,360°), (85°,360°)
2. for
s
=
v
the user has to enter ),(
vv
φ
θρ :
(0°,0°), (10°,0°), ... (80°,0°), (85°,0°)
(0°,30°), (10°,30°), ... (80°,30°), (85°,30°)
...
(0°,360°), (10°,360°), ... (80°,360°), (85°,360°)
3. the spherical albedo of the surface
4. the observed reflectance in the selected geometry ),,,(
vsvs
φ
φ
θ
θ
ρ
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SUBROUTINE HAPKBRDF
Function: To generate a BRDF following Hapke's (1981) model.
Description (from Pinty & Verstraete, 1991): From the fundamental principles of radiative
transfer theory, Hapke (1981) derived an analytical equation for the bidirectional reflectance
function of a medium composed of dimensionless particles. The singly scattered radiance is
derived exactly, whereas the multiply scattered radiance is evaluated from a two-stream
approximation, assuming that the scatterers making up the surface are isotropic. The
bidirectional reflectance of a surface illuminated by the sun from a direction ),(
ss
φθ , observed
from a direction ),(
vv
φθ , and normalized with respect to the reflectance of a perfectly reflecting
Lambertian surface under the same condition is given by
[]
{}
1)(H)(H)g(P)g(B1
1
4
),,,(
vs
vs
vvss
−µµ++
µ+µ
ω
=φθφθρ ,
where
is the average single scattering albedo of medium particles,
)cos(
ss
θ=µ and )cos(
vv
θ=µ ,
g is the phase angle between the incoming and outgoing rays, defined as
)cos()sin()sin()cos()cos()gcos(
vsvsvs
φ
−
φ
θ
θ
+
θ
θ= ,
B(g) is a backscattering function that accounts for the hot spot effect:
[]
)2/gtan()h/1(1)0(P
)0(S
)g(B
+ω
= ,
with the amplitude and width of the hot spot S(0) and h,
P(g) is the average phase function of medium particles, computed here by the
Heyney and
Greenstein
's function:
)gcos(21
1
)g(P
2
2
Θ+Θ+
Θ−
=
with the asymmetry factor ranging from -1 (backward scattering) to +1 (forward scattering),
and H(µ) is a function to account for multiple scattering:
µω−+
µ
+
=µ
2/1
)1(21
21
)(H.
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