The multiple-scattering theory of K. Lumme and E. Bowell (1981) was criticized by B. Hapke by stating, in particular, that energy is not conserved. It is shown that Hapke's treatment is, in this respect, inferior to that of Lumme and Bowell, and itself violates the principal concepts of radiative transfer theory. Hapke's additinal claim that, in Lumme and Bowell's work, the reflectance tends to zero at the limb is also refuted. Comment is made on the deduction of surface physical properties by modeling photometric observations

Bowell, E.

Lumme, K.

English

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13 f.
A NOTE ON HAPKE'S "BIDIRECTIONAL REFLECTANCE SPECTROSCOPY.
3. CORRECTION FOR MACROSCOPIC ROUGHNESS"
KARI LUMME
Observatory and Astrophysics Laboratory, University of Helsinki,
Tahtitorninmaki, SF-00130 Helsinki 13, Finland
and
EDWARD BOWELL
Lowell Observatory, Mars Hill Road, 1400 West, Flagstaff, Arizona 86001
r
rNaSA-CR-176965) A N01E ON H f i P K E ' S N86-29739
BIDIBECTIONAL BEFI-ECTAHCE SPECTECSCOPY. .PART
3: CORRECTION FOE MACROSCOPIC fiOUGHNESS
(Helsinki Univ . ) 13 p CSCL' 03A Dnclas
63/89 43322
Manuscript totals 13 pages, including one table.
https://ntrs.nasa.gov/search.jsp?R=19860020267 2020-03-20T14:33:32+00:00Z
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RUNNING PAGE HEAD: NOTE ON HAPKE'S PHOTOMETRIC MODEL
SEND PROOFS TO: Dr. Edward Bowell
Lowell Observatory
Mars Hill Road, 1 MOO West
Flagstaff, Arizona 86001
Page 3
ABSTRACT
B. Hapke (1984, Icarus 59, 11-59) has criticized the
multiple-scattering theory of K. Lumme and E. Bowell (1981, Astron. J.
86, 1691-1704) by stating, in particular, that energy is not
conserved. It is shown that Hapke's treatment is, in this respect,
inferior to that of Lumme and Bowell, and itself violates the
principal concepts of radiative transfer theory. Hapke's additional
claim that, in Lumme and Bowell's work, the reflectance tends to zero
at the limb is also refuted. Comment is made on the deduction of
surface physical properties by modelling photometric observations.
1. INTRODUCTION
There has been considerable work on modelling the scattering of
visible light in planetary regoliths. The most prominent feature in
the phase curve of an atmosphereless body is the non-linear surge in
brightness as the solar phase angle tends to zero. The first
quantitative explanation of this effect was published almost a century
ago by Seeliger (1887), whose "mutual shadowing" mechanism is still
held to be valid. However, when applied to photometric observations of
atmosphereless bodies, it has been realized that mutual shadowing
alone is insufficient to explain the entire observed backscattered
flux. To better match the observational data on the Moon, Hapke (1966)
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introduced the concept of surface roughness. Later, Lumrae and Bowell
(198la, 198lb; hereafter LB) devised a radiative transfer model in
which the effects of both mutual shadowing and surface roughness were
taken into account. In addition, their model allowed for multiple
scattering between regolith particles, which is particularly important
for high-albedo surfaces. The way in which all three of these
phenomena combine has been a matter of dispute for some time. Hapke
(1982, 1984) has strongly criticized LB, asserting that the singly and
multiply scattered components of the radiation field in a regolith
must be affected by surface roughness in the same way or else energy
will not be conserved. In a reply to Hapke's (1982) contention, Lumme
and Bowell (1982) pointed out that the contrary situation obtains:
namely, that the emergent flux would be underestimated if Hapke were
correct. We show below, in a quantitative way, that such is indeed the
case.
We also comment on a second criticism by Hapke (1984) of LB's
modelling of the effects of surface roughness: that, contrary to
observation, the reflectance tends to zero at the limb. Yet other
disagreements that Hapke has with us, which were stated in his 1982
abstract but not elaborated on in his 1984 paper, we assume were
answered by our short published reply (Lumme and Bowell, 1982) and by
subsequent private discussions; we do, however, discuss the question
of deducing the physical nature of a surface from light-scattering
models.
Page 5
2. CONSERVATION OF ENERGY
The calculation of radiation transport in a coherent scattering
medium is a problem of the greatest difficulty unless simplifying
assumptions are made. In the classical sense, radiative transfer
involves light scattering in an infinite, horizontally homogeneous,
plane-parallel medium in which the individual scatterers are in the
far field (e.g., Chandrasekhar, 1960). However, when planetary
regoliths are considered, two problems arise: first, the scattering
medium is no longer horizontally homogeneous because of surface
roughness; second, the individual scatterers (that is, particles) are
not in the far field but rather touch each other. Horizontal
inhomogeneity may be dealt with statistically, since, although locally
disturbed, the observed surface has much greater dimensions than the
roughness and is smooth on a large scale. Problems associated with the
contiguity of particles are probably not serious since individual
particles are thought, on average, to be much larger Q10 urn) than the
wavelength .of light.
At the limit of geometric optics, the radiation field I can be
divided into two components: one consisting of light scattered only
once, termed the singly scattered component II; the other comprising
light scattered twice or more, termed the multiply scattered component
IM- While Ij is certainly affected by horizontal inhomogeneity
(roughness) everywhere in the observed area, the effect on IM is not
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at all clear a priori. It is here that there is sharp disagreement
between the treatments of Hapke (1984) and LB. Hapke claims that both
It and IM are affected equally since multiple scattering can occur
only within a small surface element and not on the larger scale, as
for example between surface elements. In contrast, it is assumed by LB
that only II is affected by roughness and that IM is calculable using
classical radiative transfer theory. It is qualitatively evident that
Hapke's treatment underestimates the total emergent flux because a
compojient of IM is ignored. Thus, Hapke incompletely applies the
concept of multiply scattered light which, by definition, comprises
all light scattered more than once, regardless of the mechanism. In
other words, Hapke considers that IM "remembers" the direction of
incidence (as does Ij), while we believe that 1^ is subject to
random-walk behavior. Naturally, LB's assumption that IM may be
treated by classical means is itself an approximation, although it is
likely that deviations from the (unknown) correct directions of
emergence of multiply scattered rays are random rather than
systematic.
To verify quantitatively our statements regarding energy
conservation in the Hapke and LB models, we have computed the Bond
albedo A as a function of the rms surface slope 6" (in Hapke's
notation) for conservative scattering; that is, when the
single-scattering albedo w0 = 1.0. Obviously, A = 1 = pq, where p is
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the geometric albedo and q is the phase integral, which is related to
the phase function $ by the well-known relationship
ir
q = 2 $(a)sln a da,
0
where a is the phase angle. For convenience, we further assume that
the single-particle phase function is isotropic, that is, independent
of the phase angle, although that assumption is not required. All the
necessary equations are given by LB and Hapke (1984). From LB, we use
Eqs. (25), (46), (47), and (49) in Paper I, and Eqs. (23)-(26) in
Paper II, setting x = 0, D = 0.37, p = 1.17, .5J0 = 1 .0, and g = 0.
Here, x is the contribution to the surface density by particles too
small to cast shadows; D and p are, respectively, the volume density
and surface roughness, with chosen values equal to averages derived
from a large variety of atraosphereless bodies; and g is the asymmetry
factor, the zero value being a consequence of the assumed equality of
forward- and backward scattering for single particles [this assumption
appears to be borne out by the results of modelling photometric data
on the zodiacal cloud (Lumme and Bowell, 1985)]. From Hapke (1984), we
use Eqs. (53) to (55)* and (65) to (67).
*We note that Eq. (54) may be compared to its original form, Eq. (8) of
Hapke (1963). The symbol g denotes the compaction parameter in the
earlier paper and phase angle in the later one. We also note a
misprint in the last term of Eq. (54).
Page 8
The result of this comparison is given in Table I. Bond albedos A
and geometric albedos p have been computed as a function of the rms
surface slope "5" (= tan"1 p) by means of a six-point Gaussian
integration scheme. It can be seen from Table I that in both cases
there are deviations from the nominal value A = 1.0, but that the
deviations resulting from Hapke's model are much larger.
3. SURFACE BRIGHTNESS AND THE ROUGHNESS CORRECTION
As the second example of a "serious error" by LB, Hapke (198*0
claims that the roughness correction "makes the reflectance approach
zero at the limb." The general expression for the surface brightness I
in our work is given by Eq. (39) of Paper I*. In refutation of Hapke's
claim, we note that, at the limb, where the geometric quantity E,
(defined by Eq. (10) Paper I) tends to infinity, the surface
brightness tends to 1-q, q being the fraction of the surface occupied
by holes. Thus, unless the surface is entirely saturated with
holes—presumably"a physically unrealistic situation—the intensity at
the limb is finite. The reduction by a factor 1-q affects the
well-known Lommel-Seeliger spike in the brightness near the limb and
*There is an obvious misprint in Eq. (39): the left member should be II
rather than I0.
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at least for the Moon, is not observed. Parenthetically, we note that
Eq. (40) of Paper I, an approximation of Eq. (39), is to be used only
for studies of the integrated brightness and not for the calculation
of surface brightness.
4. DERIVATION OF PHYSICAL PARAMETERS FROM PHOTOMETRY
Hapke (1984) has also asserted that "while it may be possible to
fit some astronomical data to [Lumme and Bowell's] theory, the
relations between the deduced photometric parameters and actual
surface properties of the body are unclear and are likely to be
seriously in error." Since Hapke was unspecific in his criticism, it
is difficult to know quite what he has in mind. However, we believe,
along with Hapke, that modelled optical properties may not necessarily
represent the physical nature of regoliths. It is obvious, for
example, that the modelled volume density may overestimate the true
volume density because of voids inside particles that play little or
no part in the scattering of light; that the whole-disk optical
properties may give no clue as to the heterogeneous nature of a
surface on a small scale; and that, if particles in a surface have
some preferential alignment, the physical nature of the surface could
be erroneously deduced even though the photometric data were
adequately modelled. However, we do believe that, in general,
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differences in the modelled optical properties are indicative of
differences in the physical natures of the surfaces involved. In this
regard, we think our inferences concerning Callisto (LB, Paper II) are
useful, as is one of our basic conclusions in LB, Paper II that,
except for albedo, the global optical properties of most
atmosphereless bodies differ by only modest amounts.
We also believe that the average numerical values for the
roughness (.p"-= 1.17) and volume density (D = 0.37) obtained by LB are
physically plausible; whereas values of the counterparts of these
parameters derived by Hapke are not. The surface roughness has been
discussed above in section II. For the volume density in regoliths,
Hapke derives values of his parameter h that imply 0.03 <• D < 0.1.
Direct, in situ measurements are, of course, lacking, though
laboratory measurements on lunar fines indicate D = 0.4 (from Birkebak
et al., 1971; and Greene et al., 1971),-and for two particulate
terrestrial samples D = 0.28 and 0.42 (Lumme et al., 1980). We find it
hard to understand how planetary regoliths can be almost as porous as
Saturn's rings, for which the modelled D. = 0.02 (Lumme et al., 1983).
\
REFERENCES
Birkebak, R. C., C. J. Cremers, and J. P. Dawson (1971). Spectral
directional reflectance of lunar fines as a function of bulk
density. In Proc. Second Lunar Sci. Conf. 3, 2197-2202.
Page 11
Chandrasekhar, S. (1960). Radiative Transfer. Dover, New York.
Greene, C. H., D. Pye, H. J. Stevens, D. E. Rase, and H. F. Kay
(1981). Compositions, homogeneity, densities, and thermal history
of lunar glass particles. In Proc. Second Lunar Sci. Conf. 3_,
2049-2055.
Hapke, B. (1963). A theoretical photometric function for the lunar
surface. J. Geophys. Res. 68, 4571-4586.
Hapke, B. (1966). An improved lunar theoretical photometric function.
Astron. J. 7J_, 333-339.
Hapke, B. (1982). The Lumme-Bowel1 photometric parameters: Reality or
fantasy? Bull. Amer. Astron. Soc. 14, 726.
Hapke, B. (1984). Bidirectional reflectance spectroscopy. 3.
Correction for macroscopic roughness. Icarus 59, 41-59.
Lumme, K., and E. Bowell (1981 a). Radiative transfer in the surfaces
of atmosphereless bodies. I. Theory. Astron. J. 86, 1694-1704
[Paper I].
Lumme, K., and E. Bowell (I98lb). Radiative transfer in the surfaces
of atmosphereless bodies. II. Interpretation of phase curves.
Astron. J. 86, 1705-1721 [Paper II].
Lumme, K., and E. Bowell (1982). A reply to Hapke's criticism of the
Lumme-Bowell photometric theory. Bull. Amer. Astron. Soc. 14, 726.
Lumme, K., and E. Bowell (1985). Photometric properties of zodiacal
light particles. Icarus 62, 54-71.
Page 12
Lumme, K., W. M. Irvine, and L. W. Esposito (1983). Theoretical
interpretation of the ground-based photometry of Saturn's B ring.
Icarus 53, 174-184.
Lumme, K., E. Bowell, and B. Zellner (1980). Interpretation of
laboratory sample photometry by means of a generalized radiative
transfer theory. Lunar and Planetary Science XI, 637-639. Lunar
and Planetary Institute, Tucson.
Seeliger, H. (1887). Zur Theorie der Beobachtung der grossen Planeten,
insbesondere des Saturn. Abb. Bayer. Akad. Wiss. Math. Naturwiss.
Kl. 16, 405-516.
Page 13
TABLE I.
Comparison between Lumme and Bowell's (1981 a) and
Hapke's (198*0 model for the case of conservative (u>0 = 1)
and isotropic (g = 0) scattering.
F
20°
30
MO
50
60
Lumme
A
0.97
0.94
0.93
0.92
0.90
and Bowell
P
0.81
0.81
0.81
0.81
0.81
Hapke
A P
0.88 0.66
0.78 0.61
0.67 0.57
0.56 0.52
0.45 0.48


A note on Hapke's bidirectional reflectance spectroscopy. Part 3: Correction for macroscopic roughness

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