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Fig. 8.2
Image of the lunar
disk, acquired under a phase
angle of
α
=
59
.
6
◦
. No bright
rim at the limb of the disk is
apparent
account—cf. Fig.
8.2
for a real image of the lunar disk, in which no bright rim is
visible at the limb. The bright rim disappears more and more for increasing values
of
θ
, and for the phase angle configuration shown in Fig.
8.1
, the observed inten-
sity increasingly resembles that of a Lambertian surface. Most planetary surfaces
display macroscopic roughness values between 10
◦
and 30
◦
(Hapke,
1984
; Helfen-
stein,
1988
; Veverka et al.,
1988
; Warell,
2004
).
It is not straightforward, however, to apply the reflectance function introduced
by Hapke (
1981
,
1984
,
1986
) to three-dimensional surface reconstruction using
intensity-based methods. According to McEwen (
1991
), a reflectance function that
fairly well describes the reflectance behaviour of dark regolith surfaces is of the
form
cos
θ
i
cos
θ
i
+
R
L
(θ
i
,θ
e
,α)
=
f(α)
,
(8.5)
cos
θ
e
with the phase angle-dependent factor
f(α)
. For constant
f(α)
,(
8.5
) corresponds
to the Lommel-Seeliger reflectance law (Lohse and Heipke,
2004
). For large phase
angles approaching 180
◦
, the reflectance of the lunar surface becomes increasingly
Lambertian. McEwen (
1991
) proposes the lunar-Lambert function, which is a com-
bination of the Lommel-Seeliger and the Lambertian reflectance function according
to
ρ
2
L(α)
L(α)
cos
θ
i
cos
θ
e
+
1
cos
θ
i
cos
θ
i
+
R
LL
(θ
i
,θ
e
,α)
=
−
(8.6)
with
ρ
as an albedo parameter that also absorbs quantities specific to the image
acquisition process and
L(α)
as an empirical phase angle-dependent parameter
(cf. Sect.
3.2.1
for the corresponding bidirectional reflectance distribution function
(BRDF)). The formulation in (
8.6
) has the advantage that it can be used directly
for ratio-based photoclinometry and shape from shading methods as described in
Sect.
3.3.2.1
, as the surface albedo
ρ
then cancels out for each image pixel.
To obtain a function
L(α)
that provides a realistic description of the reflectance
behaviour of the lunar surface, McEwen (
1991
) computes intensity profiles for the
'photometric equator' (the great circle on the planetary sphere which contains the
point on the sphere in which the Sun is located in the zenith, i.e. the 'subsolar point',
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