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and the point exactly below the observer) and the 'mirror meridian' (the great circle
orthogonal to the photometric equator on which
θ
i
=
θ
e
). The intensity values are
then extracted in photometric longitude (the angle measured along the photometric
equator) and photometric latitude (the angle measured along the mirror meridian).
According to McEwen (
1991
), the opposition effect parameters
B
0
and
h
have a
significant influence on the observed reflectance only for phase angles below about
5
◦
, where it is not favourable to apply photoclinometric techniques also with respect
to illumination and viewing geometry. As it tends to be difficult to adapt the lunar-
Lambert function to the Hapke function near the limb of the planetary disk, values of
θ
e
>
70
◦
+
α/
9 are not taken into account for the parameter estimation. This is no
restriction, because photoclinometric methods are preferentially applied for small
values of
θ
e
, corresponding to a viewing direction approximately perpendicular to
the surface.
The behaviour of
L(α)
is illustrated by McEwen (
1991
) as a sequence of refer-
ence diagrams for different values of the single-scattering albedo
w
, different pa-
rameter values of the function
p(α)
, and macroscopic roughness values between 0
◦
and 50
◦
. For low-albedo surfaces with
w
0
.
1 and macroscopic roughness values
θ
larger than about 10
◦
, the shape of the angular scattering function has only a mi-
nor influence on
L(α)
. For small phase angles,
L(α)
is always close to 1, while
it approaches zero for phase angles larger than about 140
◦
. For intermediate phase
angles, the behaviour of
L(α)
strongly depends on the value of
θ
, where in the prac-
tically relevant range for values of
θ
between 0
◦
and 30
◦
,
L(α)
monotonously de-
creases with increasing
θ
. As a consequence, the reflectance behaviour becomes in-
creasingly Lambertian for increasing phase angles and
θ
values (cf. Fig.
8.1
). For the
lunar surface, the behaviour of
L(α)
can also be expressed as a third-order polyno-
mial in
α
(McEwen,
1996
). The lunar-Lambert reflectance function with
L(α)
as es-
tablished by McEwen (
1991
) is used in Sects.
8.2
and
8.4
for the three-dimensional
reconstruction of lunar craters and lunar volcanic features.
≈
8.2 Three-Dimensional Reconstruction of Lunar Impact Craters
8.2.1 Shadow-Based Measurement of Crater Depth
The depth of a crater can be estimated based on the length of the shadow cast by the
crater rim, when the shadow ends near the crater centre (Pike,
1988
). A method gen-
erally applicable to images acquired under oblique viewing conditions is described
by Pike (
1988
) (cf. Appendix A by Clow and Pike therein), who utilises it for mea-
suring the depths of simple bowl-shaped craters and the heights of their rims above
the surrounding surface in Mariner 10 images of the planet Mercury and obtains a
depth-diameter relation for Mercurian craters.
A sketch of depth-diameter relations for impact craters on Mercury, the Moon,
Mars, and the Earth established by Pike (
1980
) is shown in Fig.
8.3
. It has been
observed by Pike (
1980
,
1988
) and also in previous classical works, e.g. by Wood
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