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Icarus

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Application of multiple photometric models to disk-resolved measurements of Mercury's surface: Insights into Mercury's regolith characteristics

Deborah L. Domingue a'*, Brett W. Denevib, Scott L. Murchie b, Christopher D. Hashc

aPlanetary Science Institute, 1700 E. Fort Lowell, Suite 106, Tucson, AZ 85719-2395, USA b The Johns Hopkins University Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, MD 20723, USA cApplied Coherent Technology Corporation, Herndon, VA 20170, USA

ARTICLE INFO ABSTRACT

Photometric analyses are used to standardize images obtained at a variety of illumination and viewing conditions to a common geometry for the construction of maps or mosaics and for comparison with spectral measurements acquired in the laboratory. Many models exist that can be used to model photometric behavior. Two of the most commonly use models, those of Hapke and Kaasalainen-Shkuratov, are compared for their ability to standardize MESSENGER images of Mercury. Analysis of the modeling results shows that photometric corrections using the Kaasalainen-Shkuratov model provides significantly less contrast between images acquired at large differences in emission angle. The contrast seen between images acquired at large differences in either incidence and phase angle is smaller with the Hapke model based corrections, but not significantly better than that provided by the Kaasalainen-Shkuratov model. Photometric studies are also used to infer scattering properties of the surface regolith. The quantitative correlation between photometric model parameters and surface properties is questionable, but laboratory studies do indicate general correlations and trends between parameters and sample properties that allow for comparisons between surfaces based on photometric modeling. Based on comparisons with the Moon and several asteroids that have been observed by spacecraft, the photometric analyses presented here are interpreted to indicate that Mercury's regolith is smoother on micrometer scales and has a narrower particle size distribution with a lower mean particle size than lunar regolith. Grain structures of regolith particles from Mercury are inferred to be different than those of the Moon or those asteroids observed to date. Mercury's regolith may contain a component compositionally distinct from lunar regolith.

© 2016 The Authors. Published by Elsevier Inc. This is an open access article under the CCBY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

CrossMark

Article history: Received 29 July 2015 Revised 24 November 2015 Accepted 30 November 2015 Available online 8 January 2016

Keywords:

Mercury

Regoliths

Spectrophotometry

1. Introduction

Photometry is defined here as the variation in reflection as a function of lighting geometry, specifically the incidence angle of incoming irradiance (from the surface normal), the emission angle of outgoing radiance (from the surface normal), and the phase angle (the angle between the incident and reflected rays of light). Variations in reflectance are influenced by the properties of the reflecting surface, and in the case of rocky planetary bodies, properties of the surface regolith. Models of spectrophotometric behavior (photometric behavior as a function of wavelength) attempt to predict the scattering properties of regolith, which are affected by texture and composition. With knowledge of the

* Corresponding author at: 400 Teresa Marie Ct., Bel Air, MD 21015, USA. E-mail address: domingue@psi.edu (D.L. Domingue).

scattering properties, these models are used to predict reflectance at a given illumination and viewing geometry. However, commonly it is the inverse problem that interests planetary scientists: with no a priori knowledge of the regolith scattering properties (1) can a model accurately predict how the surface reflect lights at an unmeasured geometry given knowledge of how it reflects light for a subset of possible incidence and emission angles, and (2) can regolith scattering properties be derived by modeling photometric observations that only partially cover possible illumination and viewing geometries?

A model that can accurately predict (within 2-5% relative accuracy) the reflectance of a surface at an unmeasured geometry, based on measurements that cover only a subset of possible incidence and emission angle values, is invaluable for standardizing imaging data to a common illumination and viewing geometry. This ''photometric standardization" or ''photometric correction"

http://dx.doi.org/10.1016/j.icarus.2015.11.040 0019-1035/© 2016 The Authors. Published by Elsevier Inc.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.8 -1.6 -1.4 -1.2 1

0.8 -0.6 -0.4 0.2 0

SHOE width parameter

0.3 0.4 0.5 filling factor

5 4.5 4 3.5 3 2.5 2 1.5 1

SHOE width parameter

• 0.25 fill

■ 0.5 fill

»•0.75 fill

kS—<a-

-■--■--„--•,- Vr f r -,,--,,--,,--,,-ir 11-1 ,r ir 11- ir If" if 0 50 100 150 200

radius largest particle/radius smallest particle

Fig. 1. (a) The variation in the shadow-hiding opposition effect width as a function of regolith filling factor (1-porosity), assuming a regolith comprised of equant particles larger than the wavelength of the observing light with a narrow size distribution. (b) The variation in the shadow-hiding opposition effect width as a function of the ratio of the radius of the largest to smallest sized particles for a range of filling factor values (black, solid line: / = 0.25; gray, dashed line: / = 0.5; black, dotted line: / = 0.75).

to a standard illumination and viewing geometry enables the construction of reflectance maps from images taken at varying geometries, the comparison of surface spectral reflectance from one region to another observed under different geometries, and interpretation of composition based on laboratory measurements taken at geometries different from the planetary observations.

A model that can accurately translate a set of reflectance measurements acquired at different geometries into a prediction of regolith physical properties provides a tool for understanding the structure and evolution of the regolith. These properties include, but are not limited to, single-scattering albedo (ratio of amount of light scattered to the amount of light both scattered and absorbed), grain size and shape, porosity, and surface roughness. Measurement of such properties would enable comparisons of regolith across the surface of an object, correlation of regolith properties with geologic terrains and processes, and comparison of regolith between Solar System bodies.

The structure of planetary regoliths vary on multiple spatial scales, from geologic units of meters to kilometers in scale to grains and clumps of grains on the order of micrometers to centimeters in size. The optical characteristics of the regolith material also strongly affect the reflective properties of the regolith and may

vary within and between grains (Shkuratov et al., 2011). These characteristics include, but are not limited to, complex indices of refraction, inclusions (providing non-uniformity in scattering and absorption, affecting the scattering mean free path and direction), and grain size (affecting the scattering mean free path and direction). Photometric models that attempt to correlate photometric properties with regolith properties are thus inevitably complex and contain numerous parameters, making the uniqueness of the modeling solution difficult to assess. Empirical formulas with fewer parameters are therefore usually used when the goal is to determine a photometric correction and not to decipher the properties of the regolith. However such empirical formulas may not be more accurate.

Using images acquired by the MErcury Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER) spacecraft's Mercury Dual Imaging System (MDIS), the MESSENGER project has produced and delivered to the Planetary Data System (PDS) a global eight-color mosaic (Domingue et al., 2011, 2015). Although the images in the mosaic were photometrically corrected, there are obvious residuals in images acquired at large incidence and emission angles (Domingue et al., 2015). Therefore in this paper we investigate: (1) which models provide a photometric correction

CBOE width parameter

transport mean free path (^m)

Fig. 2. The variation in the coherent-backscatter opposition effect width as a function of transport mean free path (micrometers) for a variety of wavelengths of light (black, solid line: 0.3 im; black, dashed line: 0.5 im; gray, solid line: 1.0 im; back, dotted line: 2.0 im).

Table 1

Hapke model parameter descriptions.

Parameter Value range Relation to regolith

w 0 to 1 Volume-averaged single scattering albedo, the ratio of the amount of light scattered

single-scattering albedo to the amount of light both scattered and absorbed

b -1 to 1 Henyey-Greenstein function parameter that governs the relative amplitudes of

single-particle scattering function amplitude forward and backward scattering

c -1 to 1 Henyey-Greenstein function parameter that governs the partition of scattered light

single-particle scattering function partition coefficient into forward or backward directions

/ 0 to 0.75 Volume fraction within the regolith occupied by grains

filling factor (1 - porosity)

bs0 0 to 1 Amplitude of the opposition effect due to the shadow-hiding mechanism; a single

SHOE amplitude scattering mechanism. Note: in many applications where the terms for coherent backscatter are omitted, this parameter is allowed to exceed unity to account for both mechanisms contributing to the opposition surge

hs a Width of the opposition effect due to shadow-hiding; related to grain size

SHOE width distribution and filling factor within the scattering volume

bc0 0 to 1 Amplitude of the opposition effect due to coherent backscatter; a multiple scattering

CBOE amplitude mechanism

hc b Width of the opposition effect due to coherent backscatter, related to the wavelength

CBOE width of incident light and the transport mean free path within the scattering volume

e 0° to 40° Mean slope angle, averaged over a size range bounded on the upper end by the

surface roughness angular resolution of the detector, and the lower end radiative scattering; typically 100-1000 im (Hapke, 2012a)

a See Fig. 1. b See Fig. 2.

MESSENGER mosaics with the least relative error, and (2) regolith properties predicted by each model for the average surface of Mercury. Section 2 reviews each of these models, their usefulness in providing photometric corrections for image mosaics, and how well they predict regolith characteristics.

2. The photometric models

Several photometric models and variants of them have been used to predict reflectance of planetary surfaces as a function of geometry. The most commonly used model is that of Hapke (1981, 1984, 1986, 1993, 2002, 2008, 2012a), which is based on geometric optics and the equations of radiative transfer. It incorporates expressions and parameters to account for surface roughness, porosity, grain scattering properties, and both mechanisms proposed to create the opposition surge, i.e., the steep increase in

reflectance at small phase angles (the shadow-hiding opposition effect, or SHOE, and coherent backscatter opposition effect, or CBOE). Domingue et al. (2015) used an early version of this model (Hapke, 1981, 1984,1986) to provide a photometric correction for the 8-color mosaic (Merc-G8CM) with mixed results. The

Table 2

Phase function descriptions.

Phase functiona f (a) Parameters Relation to regolith

e-11a+me-12a 1+m Related to opposition surge width

l2 Related to surface roughness

m Related to opposition surge amplitude

e-1a l Related to surface roughness

a All functions listed here are empirical functions.

Table 3

Disk function descriptions.

Disk function Parameters Parameter description3

Dls None Not applicable

Dm Cl Empirical

Dm k or k(f) Empirical

Dlslm L(f) Empirical

Da V Empirical

Das None Not applicable

dasii g Semi-empirical

a See Shkuratov et al. (2011) and Schröder et al. (2013) for parameter descriptions.

photometric correction is seen to break down for images taken at high incidence (>70°) or high emission (>70°) angle.

Another model that has been applied recently to planetary image mosaics is that of Kaasalainen et al. (2001) and Shkuratov et al. (2011). This model (hereafter referred to as the Kaasalai-nen-Shkuratov model) has been used to provide photometric corrections to Dawn images of Vesta (Schröder et al., 2013) and to describe telescopic observations of the Moon (Shkuratov et al., 2011). A strength of this model is its separation of effects due to phase angle from those due to incidence and emission angles (which are affected by local topography). Shkuratov et al. (2011) compared applications of the Hapke and Kaasalainen-Shkuratov models to disk-resolved lunar observations and demonstrated a better correlation between Kaasalainen-Shkuratov model parameters with geologic units than Hapke model parameters.

2.1. Hapke model

The full current Hapke model includes SHOE and CBOE opposition surge mechanisms, porosity, and large-scale surface roughness, and has up to nine parameters linked theoretically to regolith characteristics (e.g. Hapke, 2012a). This section reviews the equations that compose the Hapke model, and the model parameters that link photometric measurements with regolith properties. For the full derivation of these equations consult the original works by Hapke (1981, 1984, 1986, 1993, 2002, 2008, 2012a).

The Hapke equation is given as

r(i, e, a ) = K

-{[p(a)[1 + ßscßs(a)]]

4p loe + le

H(ß0e/K)H(ße/K) - 1]}[1 + BcoBc(a)]S(i, e, a, h),

where r(i, e, a) is reflectance at incidence angle i, emission angle e, and phase angle a, i0e and ,ue, are modified cosines of the incidence and emission angles for surface roughness respectively, K is a porosity term dependent on the filling factor (amount of regolith volume filled by material, inverse to porosity) /, given by

- ln(1 - 1.209/2/3) 1.209/2/3 '

w is the volume-averaged single-scattering albedo, and p(a) is the single-particle scattering function. A commonly used expression for p(a) is the Henyey-Greenstein function, given by

p(a ) = -

(1 - c)(1 - b2)

(1 - 2b cos(a)

-b2)3/2

c(1 - b2)

(1 + 2b cos(a)

-b2)3/2'

where c is the parameter indicating partition between forward and backward scattering, and b is the amplitude of the scattering component. BS0 is the SHOE amplitude and the SHOE term BS is given by

Bs(a) =

1+£tan S

where hs is the angular half-width of the SHOE peak (in radians). This physical mechanism for the opposition surge depends on the particle size distribution within the regolith in addition to the porosity of the regolith. The porosity, q, is related to the filling factor by q = 1 - If some assumptions are made on the particle size distribution, then the SHOE width parameter can be expressed as a function of /, thus decreasing the number of model parameters. Hapke (2012a) showed that if the particles within the regolith can be assumed to be larger than the wavelength of observing light, equant, and the size distribution is narrow, then hs, can be related to the filling factor by

hs = 3T ="0-3102/1/3ln(1 - 1.209/2/3). (5)

However, it has been shown that for regoliths that are the product of comminution or grinding by meteorite impacts, the particle size distribution is best described by a power law (McKay et al., 1974; Bhattacharya et al., 1975). In this case, Hapke (2012a) shows that the relation between hs and / can be described by

= f^l) K/ _ M 8 1 ln(a,/as) ~

-0.3102р3/1/3 ln (1 - 1.209/2/3) ln (ai/as)

where al and as are the radii of the largest and smallest particles, respectively. Using one of these expressions for hs eliminates one independent parameter. Conversely, if / can be derived from the porosity correction, K, then an estimate of the particle size distribution can be derived from hs. The variation in the value of hs with / for both expressions given above is shown in Fig. 1.

BC0 is CBOE amplitude and the CBOE term BC is approximated as

1 + 1.42K

(1 +1 tan(f))2

1 - exp -1.42K (£■ tan (f

1 tan (f)

where hc is the CBOE width (radians). The width of the CBOE opposition is related to both the wavelength of reflected light that is being modeled (k) and the transport mean free path (KT, the average distance a wave travels before its direction is randomized), and is given by

4лЛт

as shown in Fig. 2 for a variety of observational wavelengths. The H function is approximated by the equation

H(x/K) =

1 + 2x/K 1 + 2yx/K'

where y iW1 - w and x represents either i0e or ie. The large-scale roughness expression, S(i, e, a, 6) where h is a measure of the average surface tilt or surface roughness, and the modified cosines of the incident and emission angles (i0e and ie, respectively) due to roughness are given for the case where i < e by

S(i,e; W,ff) =

ge(e) g0e(i) 1 - f (W)+f (W)V(h) Щ

loe = V(h)

le = V(h)

cos(i)+sin(i) tan(6)

cos(e) + sin(e) tan(6)

cos(W)E2 (e) + sin2(W/2)E2(i)

2 - E1 (e)-(W/n)E1(i)

E2(e)-sin2(W/2)E2(i)

2 - E1(e)-(W/p)E1(i)

(11a) (11b)

and for the case where i > e by

Fig. 3. The (A) Beethoven, (B) Rembrandt, and (C) Matabei photometric regions. The rectangles represent the areas sampled to provide the photometric measurements needed to derive a photometric correction.

S(i, e, w -Ir

ge(e) g0e(i) 1 - f (W) +f (W)Z(ê) fe]

l0e = X(h)

le = Z(h)

cos(i) + sin(i) tan(0)

E2(i)-sin2(W/2)E2(e)

cos(e) + sin(e) tan(0)

2 - E1(i)-(W/n)E1(e)

cos(W)E2(i) + sin2(W/2)E2(e)

2 - E1(i)-(W/n)E1 (e)

(13a) (13b)

where the terms independent of photometric angles are: 1

X(h) =

(1 + p tan2(0))1/2'

g0e(i) = Z(h)

ge(e) = Z(ö)

cos(i) + sin(i) tan(0) cos(e) + sin(e) tan(0)

2 - E1 (i)

2 - E1 (e)

Ei(x) = exp E2(x) = exp

- p cot(0) cot(x)

- p cot2(0)cot2(x)

(17a) (17b)

f (W) = exp and

-2tan( W

Table 4

Hapke Basic model parameters.

Filter Wavelength (nm) w b c в

Domingue Hapke Basic Domingue Hapke Basic Domingue Hapke Basic Domingue Hapke Basic

et al. (2015) et al. (2015) et al. (2015) et al. (2015)

F 433.2 0.2111 0.1514 0.3341 0.1551 0.6248 0.1261 26.4272 14.6013

C 479.9 0.2308 0.1697 0.3248 0.1474 0.6135 0.1062 26.0306 14.7452

D 558.9 0.2589 0.1974 0.3147 0.1365 0.6025 0.0818 25.7300 14.7801

E 628.8 0.2809 0.2187 0.3094 0.1292 0.5983 0.0704 25.7043 14.6528

G 748.7 0.3129 0.2491 0.3057 0.1223 0.5990 0.0729 25.9386 14.2707

L 828.4 0.3299 0.2655 0.3051 0.1220 0.6021 0.0902 26.1120 14.0295

J 898.8 0.3417 0.2778 0.3045 0.1248 0.6046 0.1160 26.1607 13.9099

I 996.2 0.3542 0.2921 0.3018 0.1339 0.6052 0.1679 25.8967 14.0090

Error bars for Hapke Basic model solutions: w = ±0.01, b = ±0.01, c = ±0.01, в = ±1.

Table 5

H2012 model parameters (no opposition surge).

Filter Wavelength (nm) w b c 0 k

F 433.2 0.1933 0.2707 0.1408 13.9951 1.0407

C 479.9 0.2156 0.2679 0.1275 14.0164 1.0076

D 558.9 0.2480 0.2539 0.0954 12.9653 1.0051

E 628.8 0.2720 0.2465 0.0844 13.0377 1.0247

G 748.7 0.3055 0.2520 0.1153 12.9677 1.0252

L 828.4 0.3243 0.2589 0.1468 12.0256 1.0171

J 898.8 0.3396 0.2603 0.1628 11.9891 1.0330

I 996.2 0.3607 0.2600 0.1623 12.9991 1.0026

Error bars: w = ±0.01, b = ±0.01, c = ±0.01, B = ±0.01, k = ±0.01, 0 = ±1.

cos(W) =

cos(a) - cos(i) cos(e) sin(i) sin(e) '

where W is the azimuth angle (the angle between the projections on to the surface of the incidence and emission rays). The resulting 9 parameters are listed in Table 1, including their range of values and their purported relationship to regolith properties.

Depending on the data set being modeled, various simplifications to the equations may be used. For example, in modeling data whose phase angle range does not encompass the opposition effect (phase angles >20°), commonly both opposition expressions are set to unity (equivalent to BS0 and BC0 equal to zero). In analysis of laboratory data, surface roughness is assumed to be negligible (0 = 0°)

and the expressions for it are removed (S(i,e, W, 0) = 1,

l0e = l0 ' le = 1).

2.1.1. History for providing photometric corrections

Typically mosaics constructed from images taken at different illumination and viewing geometries are corrected to standard incidence (i), emission (e), and phase (a) angle values of i = 30°, e = 0°, a = 30°, the standard geometry for laboratory measurements of minerals and planetary samples. Hapke's model has been used to model a photometric correction to images of many Solar System objects, including the Moon (Sato et al., 2014), Mercury (Domingue et al., 2011, 2015), asteroids (Domingue and Hapke, 1989; Helfenstein et al., 1994, 1996; Murchie et al., 2002; Lederer et al., 2005, 2008; Hillier et al., 2011; Li et al., 2013; Spjuth et al., 2012; Maoumzadeh et al., 2015), and many planetary satellites (Simonelli and Veverka, 1986; Helfenstein et al., 1988, 1991; Domingue et al., 1991, 1995; Skypeck et al., 1991; Domingue and Hapke, 1992; Verbiscer and Veverka, 1992, 1994; Hillier et al., 1994; Domingue and Verbiscer, 1997; Simonelli et al., 1998; Hendrix et al., 2005; Verbiscer et al., 2005; Ciarniello et al., 2011; Fraeman et al., 2012). The form of the model used has depended on the state of the development of the model at the time of application, image coverage of the range of plausible i, e, and a values, and coverage of the opposition surge.

Single Scattering Albedo

Single Particle Scattering Function Paction Parameter C

3 0.25

600 700 800

Wavelength |nm|

Surface Roughness

-f".....f.....

600 700 800

Wavelngth (nm)

■Domingue et al. (2015) • H2012 - no Opposition

500 600 700 800 900 1001

Wavelength |nm|

Single Particle Scattering Funrtion Amplitute Parameter D

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Wavelength |nm|

•O-Hapke Basic —H2012 - w/Opposition

Fig. 5. Comparison of Hapke model parameter values for the modeling solutions from Domingue et al. (2015), the Hapke Basic model, the H2012 with no opposition, and the H2012 model with opposition parameter values from disk-integrated modeling. The parameters compared are the single scattering albedo (A), the surface roughness parameter (B), the single particle scattering function partition (C) and amplitude (D) parameters.

Table 6

Coherent backscatter amplitude.

Wavelength (nm) hc

433.2 0.5968

479.9 0.6612

558.9 0.7700

628.8 0.8663

748.7 1.0315

828.4 1.1413

898.8 1.2383

996.2 1.3724

Table 7

H2012 model parameters (with opposition parameters).

Filter Wavelength w b c g k

F 433.2 0.1816 0.2305 0.1205 13.9882 1.0110

C 479.9 0.1989 0.2286 0.1183 14.0397 1.0368

D 558.9 0.2271 0.2126 0.0735 13.9160 1.0269

E 628.8 0.2506 0.1977 0.0460 14.0914 1.0327

G 748.7 0.2861 0.19079 0.0430 12.9217 1.0167

L 828.4 0.3053 0.2001 0.0767 12.0621 0.9957

J 898.8 0.3187 0.2096 0.1301 12.9737 1.0233

I 996.2 0.3308 0.20008 0.1081 12.9982 1.0224

Error bars for H2012 model solutions ;: w = ±0.01, b = ±0.01, c = ±0.01, g = ±0.01,

k = ±0.01, e = ±1.

Table 8

Kaasalainen-Shkuratov model solutions (parameter free disk function).

Wavelength (nm) KS1 KS2

An l An i

433.2 0.0597 0.6207 0.0750 0.7173

479.9 0.0704 0.6276 0.08473 0.6954

558.9 0.0813 0.6151 0.0961 0.6677

628.8 0.0874 0.6045 0.1036 0.6507

748.7 0.0971 0.5697 0.1161 0.6304

828.4 0.1043 0.5512 0.1244 0.6180

898.8 0.1097 0.5615 0.1301 0.6043

996.2 0.1101 0.5249 0.1300 0.5762

Comparisons with exact solutions of the radiative transfer

equations (Mishchenko et al., 1999; Cheng and Domingue, 2000;

Hapke et al., 2009; Zhang and Voss, 2011) show that Hapke's model provides approximations within <10%, though the accuracy depends on the form of the Hapke model used. Many laboratory studies have examined the relationship between Hapke model parameters and characteristics of laboratory samples (Kamei and Nakamura, 2002; Cord et al., 2003; Gunderson et al., 2005; Shkuratov et al., 2005, 2007; Shepard and Helfenstein, 2007; Hapke et al., 2009; Helfenstein and Shepard, 2011; Souchon et al., 2011; Ciarniello et al., 2014) and yielded mixed results in the ability of the Hapke model to accurately replicate reflectance measurements (discussion of prediction of sample properties from model parameters is provided in the next section). For example, Gunderson et al. (2005) analyzed the photometry of the JSC-1 lunar soil simulant using Hapke's (2002) model (Hapke, 2002, hereafter referred to as H2002, which includes expressions for SHOE and CBOE but does not explicitly account for porosity). H2002 provided an excellent fit to those lunar soil simulant reflectance measurements acquired at fixed incidence and variable emission angles. However, the model performed poorly in extrapolating outside the range of measured angles to reflectances at fixed, near-zero phase and varying incidence angles. Gunderson et al. (2005) found no solution to H2002 that simultaneously fit both groups of measurements. In contrast, Shepard and Helfenstein (2007) and

Souchon et al. (2011), using the same version of Hapke's model (H2002), found that the model could be well matched to their laboratory-based photometric measurements of multiple, well characterized samples.

Additional tests of Hapke's model have included examinations of versions of this model that assumed isotropic multiple scattering (the isotropic multiple scattering approximation or IMSA model) and anisotropic multiple scattering (the anisotropic multiple scattering approximation or AMSA model). Using idealized artificial soils consisting of spherical glass beads with known optical constants, particle sizes, and compaction states, Hapke et al. (2009) demonstrated that the IMSA model provides a good match to the photometric measurements of these samples. In this case there was no macroscopic roughness, no complex particle shapes, and no shadow-hiding opposition effect (because the particles were high-albedo). Under such conditions the model accurately predicts photometric behavior (Hapke et al., 2009; Helfenstein and Shepard, 2011). Using a Monte Carlo ray-tracing method, Ciarniello et al. (2014) examined and compared the IMSA, the AMSA, and the Hapke (2008) model (hereafter referred to as H2008), which explicitly incorporates porosity. H2008 was found to best describe particulate media with arbitrary porosities and anisotropic scattering outside of the opposition regime, unless the material is strongly forward scattering. Using a subset of the samples from Shepard and Helfenstein (2007), Helfenstein and Shepard (2011) examined a version of H2008 that excluded an expression for surface roughness. They found that H2008 accurately described the data, especially for low- and moderate-albedo samples (Helfenstein and Shepard, 2011).

2.1.2. Does the Hapke model accurately predict regolith properties?

Laboratory tests of correlations between Hapke model parameters and sample characteristics have also had mixed results. Shkuratov et al. (2007) used nephelometer measurements to determine particle phase functions of samples for which they also measured photometric reflectance. The nephelometer suspended particles in air, allowing a measure of the angular distribution of scattered light from individual particles. Samples created from the same type of particles under known compaction states where then measured using a photopolarimeter (which measures the polarization of reflected light as a function of illumination and viewing angles). Using H2002 they found that the particle phase function predicted by the modeling of the photopolarimeter measurements did not match the nephelometer-measured phase functions. Shepard and Helfenstein (2007) applied H2002 to well characterized laboratory samples and found no correlation between sample properties and the model parameters. They ascribe the poor correlation to the inability of the model to adequately account for discrete or particulate media (as opposed to a continuous slab) and the effects of porosity (Shepard and Helfenstein, 2007). Helfenstein and Shepard (2011) re-applied H2008, which more explicitly incorporates porosity, to a subset of the Shepard and Helfenstein (2007) laboratory samples. They found that the porosity correction improves the fidelity of the fits to low- and moderate-albedo samples and provides a more reliable estimate of sample porosity (Helfenstein and Shepard, 2011). A separate test of H2002 by Souchon et al. (2011) found qualitative agreement between the observed microstructure of natural grains and the physical interpretation of the model-derived particle phase function, in the case where measurements that include opposition measurements or large incidence or emission angle are excluded.

Recall that the surface roughness parameter, Q, is defined in principle at a scale that depends on the spatial resolution of the detector (Hapke, 2012a). Using computer generated fractal surfaces Shepard and Campbell (1998) determined that regardless of

Normal Albedo

400 500 600 700 800

Wavelength (nm)

-©- KS1 •€>• KS2 -O KS3 KS4 -Ж- KS5 KS6

Phase Function Parameters

400 500 600 700 800 Wavelength (nm)

KS1 KS2 -O- KS3 • * • KS4

KS5 —в— KS6 - m1 KS6 - m2 KS6 - m

Disk Function Parameter

400 500 600 700 800

Wavelength (nm)

KS5 KS6

Fig. 6. Comparison of the Kaasalainen-Shkuratov model parameter values. The parameters compared are (A) the normal albedo, AN, (B) the phase function parameters (ij, i2, and m), and (C) the disk function parameters (o, k, g).

the detector, the scale dominating photometric roughness is the Moon is thought to be ~100 im (Helfenstein and Shepard, 1999). smallest scale at which well-defined shadows exist, which for the Tests using Apollo Close-up Stereo Images found the best match

Table 9

Kaasalainen-Shkuratov model solutions (single parameter disk functions).

Wavelength (nm)

433.2 479.9 558.9 628.8

748.7 828.4

898.8 996.2

0.0700 0.0797 0.0911 0.0986 0.1111 0.1194 0.1251 0.1250

0.6363 0.6219 0.5976 0.5800 0.5628 0.5570 0.5494 0.5200

0.6293 0.6277 0.6186 0.6228 0.6424 0.6369 0.6172 0.6303

0.0702 0.0795 0.0910 0.0989 0.1110 0.1191 0.1254 0.1300

0.6573 0.6449 0.6234 0.6063 0.5856 0.5782 0.5746 0.5699

0.6651 0.6599 0.6601 0.6650 0.6697 0.6653 0.6598 0.6700

0.07098 0.08219 0.0944 0.1014 0.1117 0.1191 0.1253 0.1300

0.67672 0.69072 0.6644 0.6338 0.6071 0.5969 0.5845 0.5859

0.6803 0.7322 0.7126 0.7739 0.7697 0.6766 0.7208 0.7690

Table 10

Kaasalainen-Shkuratov model solutions (KS6).

Wavelength (nm) An I1 l2 m g

433.2 0.0914 1.15975 0.9489 0.6375 0.6957

479.9 0.0984 1.0311 1.0120 0.2410 0.7052

558.9 0.1098 0.9767 0.9910 0.1992 0.7087

628.8 0.1192 0.9642 0.9472 0.4837 0.7029

748.7 0.1333 0.9152 0.9227 0.9574 0.6862

828.4 0.1411 0.9046 0.9195 0.9898 0.6796

898.8 0.1466 0.9281 0.8994 0.7987 0.6826

996.2 0.1519 0.8902 0.8733 0.3779 0.7093

between lunar photometric roughness and lunar regolith relief at submillimeter-size scales (Helfenstein and Shepard, 1999; Goguen et al., 2010).

While there is evidence that at least some Hapke model parameters are qualitatively related to the regolith physical properties, a quantitative correlation with such properties has not been firmly established. Based on the work of Helfenstein and Shepard (2011) and studies summarized above, it appears that H2008 provides the closest quantitative correlation between model parameters and laboratory reflectance measurements of regolith simulants. H2008 may also provide a basis for identifying regions having similar regolith characteristics. This study applies two versions of the H2008 model, where the components for surface roughness are included. The applied models are described in Hapke (2012a) and are labeled accordingly in Section 4.1.

2.2. Kaasalainen-Shkuratov model

A simpler, more empirical class of model than that of Hapke is the Kaasalainen-Shkuratov (KS) model, in which the dependence of reflectance on phase angle is explicitly decoupled from the dependence on incidence and emission angles and thus from topography (Kaasalainen et al., 2001; Shkuratov et al., 2011; Schröder et al., 2013). The generalized form of the KS model is given by

pr(a , i, e, k) = Aeq(a, k)D(a , i, e, k) ,

where Aeq (a, k) is described in the literature as the equigonal albedo (dependent on phase angle and wavelength only) and D(a , i,e, k) is the disk function (where the dependence on i and e, which are affected by local topography, is expressed). Various expressions have been provided for the equigonal albedo. Shkuratov et al. (2011) define the equigonal albedo as

Aeq(a ,k) =A(a0 , k)f (a ,k) ,

where a = a0 = 0 is the phase angle at opposition, A(a0 , k) is the absolute apparent albedo (also called the normal albedo, which is a constant unlike Hapke's (1981) definition of normal albedo depending on topography (Shkuratov et al., 2011; Schröder et al., 2013)), and f (a ,k) is the phase function (normalized to unity at

a0). The phase function allows for the examination of the shadow-hiding effects controlled by regolith structure (Shkuratov et al., 2011). An empirical formula for the phase function was suggested by Akimov (1988b), and Shkuratov (1983) showed that it could be derived from simplified theoretical considerations. This form is given by

f (a) =

e-11a + me-12a

where i2 is associated with surface roughness, and i and m describe the width and amplitude of the opposition surge, respectively (Shkuratov et al., 2011). Schröder et al. (2013) used a form of this phase function, f (a) = e~<"a, to model Vesta photometric observations, with opposition parameters set to zero because the Vesta imaging did not contain opposition measurements.

In studies of the Moon, Velikodsky et al. (2011) suggested an alternate function for the lunar phase function of the form

Aeq (a)= A:e-11a + A2e-12a + A3e-13a

where at a = 0, A1 + A2 + A3 = Aeq(0)= AN = normal albedo. Shkuratov et al. (2011) relate the parameters of this form to various physical aspects of the surface. For example, the first term, A] e-1>a, (which has the maximum value of the exponent) approximately describes the components of the opposition surge due to a combination of the shadow-hiding effect (Hapke, 1986), coherent-backscatter effect (Shkuratov, 1988; Hapke, 2002), lensing effect (Shkuratov, 1983; Trowbridge, 1984), and the fractality of the surface (Shkuratov and Helfenstein, 2001a, 2001b). The second term, A2e-l2a, models shadow-hiding effects and the single-particle scattering behavior of the regolith material, surface albedo, and incoherent multiple scattering between regolith particles (Shkuratov et al., 2011). The third term, A3e-l3a, (which has the minimal value of the exponent) describes shadowing effects due to surface topography (Shkuratov et al., 2011).

Wu et al. (2013) used an empirical phase function of the form

f(a) = b0e ba + a0 + a1a + a2a2 + a3a3 + a4a4

in conjunction with a Lommel-Seeliger disk function to derive a photometric correction for the Chang' E-1 Interference Imaging Spectrometer (IIM) observations of the Moon.

There are several disk functions that can be incorporated within this model. Among the better known are the Lommel-Seeliger (Dls), Lambert (DL), and Minnaert (DM) functions. Each is normalized to unity at i = e = a = 0. The Lommel-Seeliger disk function is given by

Dis = -

2 cos i cos i + cos e '

This form of the disk function has been used for the single-scattering component of radiative transfer in particulate media (Hapke, 1981, 1993, 2012a). This disk function, however, predicts strong limb brightening at large phase angles (Shkuratov et al.,

Wavelength (nm) Kaasalainen-Shkuratov Model Chi Values В

Wavelength (nm)

Fig. 7. Comparisons of the chi values for the (a) Hapke model solutions and the (b) Kaasalainen-Shkuratov model solutions.

2011; Schroder et al., 2013), which is seen in many models but is not typically observed (Hapke, 1984; Shkuratov et al., 2011). An example of a single-parameter disk function is the Minnaert function given by

Dm = (cos i)k(cos e)k-1,

where for the lunar case the parameter k depends on phase angle

k(a) =

(10 a +1) 2 '

where a is in units of degrees (Helfenstein and Veverka, 1987; Shkuratov et al., 2011).

An alternate single-parameter disk function is a combination of the Lommel-Seeliger and the Lambert (DL = cos i) functions and is labeled here as the LSL function,

Dlsl = ci

2 cos i

cos i cos e

(1 - cl ) cos i,

that contains a single, variable, parameter c. A variant on the combination of the Lommel-Seeliger and Lambert functions has been applied to the Moon (McEwen, 1991, 1996; McEwen et al., 1998; Shkuratov et al., 2011), where the disk function is provided by

Dlslm = l(a)

2 cos (a - y) cos(a - y) + cos y

(1 - L(a)) cos b cos(a - y), (29)

where b and y are the photometric latitude and longitude, respectively, and are related to the incidence and emission angles by

cos i = cos b cos(a - y), and

(27) cos e = cos b cos y.

Note: photometric latitude and longitude are based on a coordinate system where the equator is defined as the great circle containing both the subsolar and subobserver points; the central meridian contains the subobserver point and positive longitude is to the east. McEwen (1996) suggested the following expression for L(a) for the Moon:

(28) L(a) = 1 + A1a + A2a2 + Аза3

based on analysis of Galileo lunar images, where A] = -1.9 x 10 2, A2 = 2.42 x 10-4, A3 = -1.4 x 10-6, and a is in units of degrees. McEwen (1996) and McEwen et al. (1998) coupled this disk function with a Hapke-model based phase function from Helfenstein et al. (1994) to photometrically correct the Clementine UVVIS

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

• « il

D.L. Domingue et al./Icarus 268 (2016) 172-203 Reflectance Ratio: Hapke Basic

o 1.4 V 2 1.2 v

£ 1 TO

S 0.8 2 0.6

0.4 0.2 0

30 40 50 60 70 incidence angle (deg)

Reflectance Ratio: KS-3

'Sölls

s° es?*®''»'® »si

30 40 50 60 70 incidence angle (deg)

Reflectance Ratio: Hapke Basic

Reflectance Ratio: KS-3

ra 1.2

Iff* Ä * OO # . • A » A

№№-

30 40 50 60 70 emission angle (deg)

ra 1.2

(U 0.6

w Mfeg-'krf

30 40 50 60 70 emission angle (deg)

Fig. 8. Example of the reflectance ratio values as a function of incidence (top graphs) and emission (bottom graphs) for the Hapke Basic model (left column) and the KS3 model (right column). The spread in ratio values clusters about unity in both model examples, thought the Hapke Basic model trends away from unity at large emission angles more than the KS3 model.

images. Schroder et al. (2013) compare the value of L(a) for the Moon with data and model values of ct for Vesta.

Based on Akimov's (Akimov, 1979, 1988b) own photometric lunar observations, he proposed this empirical expression for the lunar disk function:

Da = cos g)(cos fl" (CQS/a -J^i-f1^" , (32)

V2y (1 - (sin §) J cos J

where a is expressed in radians. The Akimov disk function (DA) is normalized to unity at the mirror point (i = e = a/2) and does not

Median Reflectance Ratio

1.03 1.02 1.01 1

0.99 0.98 0.97 0.96 0.95 0.94

//•Y\ •7IPs v^

A\\ //•' ¿A \

/1 \\ \ \\ /*,' , ! Vi \ h:// TfS&at^Q

fÎL^z/ ^ \ N Obs/Hapke Basic // «n- 0bS/H2012 Nn OPP

f •0-0bs/H2012

Obs/KS2

1 A Obs/KS3 Obs/KS5

Obs/KS6

400 500 600 700 800

Wavelength (nm)

Fig. 9. Variations in the median reflectance ratio as a function of wavelength for each of the model solutions applied to the MESSENGER photometric data.

incidence angle subdivision

40<i<60

40<i<60

40<i<60

emission angle subdivision

40<e<60

40<e<60

40<e<60

Hapke H2012-Basic No Opp H2012 KS2

incidence angle subdivision

40<i<60

40<i<60

40<i<60

emission angle subdivision

40<e<60

40<e<60

40<e<60

Hapke H2012-Basic No Opp H2012 KS2

.......

>10% 8-10% 6-8% 4-6% 2-4% <2%

Fig. 10. These charts map the reflectance ratio analysis results. In both charts, each row represents one of the incidence and emission angle subdivisions and each column represents one of the photometric models. Color coded are the residual value bins for that subdivision and model. The analysis represented in chart (A) excludes the residuals from the 749-nm filter, and the analysis represented in chart (B) excludes the residuals from the 749-nm filter and the filter with the highest residual.

incidence angle subdivision

phase angle sub-division

H2012-NO

Hapke Basic Opp H2012 KS2

incidence angle subdivision

phase angle sub-division

H2012-NO

Hapke Basic Opp H2012 KS2

Fig. 11. These charts map the reflectance ratio analysis results. In both charts, each row represents one of the incidence and phase angle subdivisions and each column represents one of the photometric models. Color coded are the residual value bins for that subdivision and model. The analysis represented in chart (A) excludes the residuals from the 749-nm filter, and the analysis represented in chart (B) excludes the residuals from the 749-nm filter and the filter with the highest residual.

predict any limb brightening predicted by the Lommel-Seeliger disk function. The Akimov function has one parameter, v, which for a <90° is equal to 0.16 and 0.31 for the maria and highlands, respectively (Akimov, 1979, 1988b; Shkuratov et al., 2011). Akimov (1976, 1988a) and Shkuratov et al. (1994, 2003) derived a theoretical version of this disk function that is parameter free, given by

DAs = cos (- cos

(cos b)'

a/(p-a)

cos у

Shkuratov et al. (2011) provided a semi-empirical formula based on the above theoretical expression that also uses a single parameter, g, to model the disk function of the Moon:

Fig. 12. These mosaics of the Caloris Basin were constructed using the Hapke Basic model (top) and the KS3 model (bottom) to provide the photometric correction to standard illumination and viewing geometries of 30°, 0°, 30° in incidence, emission, and phase angles, respectively. Both mosaics are constructed from the 1000-nm filter images and stretched to the same dynamical range. The red box is the region shown and examined in Fig. 13.

r p / a^ (cos b)ga/(p-a) DASU = cos y cos [— (y - ¿P cosy ' (34)

where g equals 0.34 and 0.52 for the lunar maria and highlands, respectively (Akimov et al., 1999, 2000; Shkuratov et al., 2011).

The aforementioned phase and disk functions are summarized in Tables 2 and 3, respectively. For the derivations of these disk and phase functions the reader is referred to Shkuratov et al. (2011) and references therein.

2.2.1. History for providing photometric corrections

The KS model has recently been used as an alternate to the Hapke models for photometric corrections to build mosaics from planetary images acquired under varying incidence, emission, and phase angles. Modeling Vesta images using a phase function defined by

Aeq(a) = Ciai, (35)

(where C0 = AN) and the DASn disk function, Schröder et al. (2013) found solutions to the KS model that accurately describe Vesta's measured reflectance. Using the above phase function they normalized measured reflectances to common photometric angles prior to

building a global mosaic of Vesta's surface. Multiplying this corrected mosaic by a telescopically derived visual normal albedo, Schröder et al. (2013) constructed a visual albedo map of Vesta's surface.

2.2.2. Does the Kaasalainen-Shkuratov model accurately predict regolith properties?

The KS model has not yet been extensively compared with laboratory measurements, but two recent applications to photometric observations are notable. First, using the simplified Akimov expression for the phase function (Aeq(a) =ANe~la) and the DASII disk function, Schröder et al. (2013) identified and compared terrains with steeper or shallower than average photometric slopes. Maps of phase function parameters (AN and l) show that large-scale albedo variations appear to be correlated with compositional differences (Schröder et al., 2013). Regions of steeper phase curves are associated with steep slopes on crater walls and faults and with young impact crater ejecta, suggesting photometric slope is associated with regolith roughness on both macro and micro scales (Schröder et al., 2013). Second, using a simplified version of the Hapke model which excludes the expressions for CBOE and porosity, but includes a simple particle scattering function (p(a) = 1 + cos a) and the term for surface roughness, Shkuratov

Ï! 40

Angle Variations

—incidence ^-emission phase

1180 Pixel Position

Model Comparisons

-H-430

-H-750

■H-1000

-KS-430

-KS-750

■KS-1000

1180 Pixel Position

Fig. 13. The sub-region defined in Fig. 12, is shown from the Hapke mosaic (top) and the KS mosaic (bottom). The angle excursions across the transect shown in the mosaics are displayed in the center graph (incidence angle: thin line, emission angle: double line, phase angle: thick line). The bottom graph compares the reflectance variations across the trace for the three wavelengths used to construct the color mosaic (red: 1000-nm, green: 750-nm, blue: 430-nm). Image boundaries occur at pixel positions 984,1142, 1196, and 1393.

Fig. 14. These mosaics were constructed using the Hapke Basic model (top) and the KS3 model (bottom) to provide the photometric correction to standard illumination and viewing geometries of 30°, 0°, 30° in incidence, emission, and phase angles, respectively. Both mosaics are constructed from the 1000-nm filter images and stretched to the same dynamical range. The red box is the region shown and examined in Fig. 15.

et al. (2011) modeled lunar disk-resolved observations of Velikodsky et al. (2011) and mapped the Hapke model parameter values. With the exception of the single scattering albedo, w, none of the Hapke parameters exhibited correlation with geologic units. However applying the KS model with the Akimov phase function and the DAS disk function, Shkuratov et al. (2011) showed that KS model parameters AN, m, i1, and i2 exhibit strong correlations with geologic units. Although this comparison reveals the potential of KS parameters to correlate with regolith properties, it does not invalidate the full form of the more complicated Hapke model, as more recent work has shown strong correlations between Hapke model parameters and geologic terrains (e.g., Sato et al., 2014).

There has been no laboratory testing of the correlations between Kaasalainen-Shkuratov model parameters and the physical properties of regolith samples.

3. MESSENGER image data

MDIS contains both a narrow-angle camera (NAC) and a wide-angle camera (WAC). The WAC includes a 12-position filter wheel with eleven narrow-band spectral filters and one broadband filter. Eight of the narrow spectral filters were utilized during orbit to globally image the surface. Details regarding the global color imag-

ing sequences can be found in Domingue et al. (2015) and Murchie et al. (2015). The work presented in this section focuses on the radiometric calibration and the imaging sequences used to derive the photometric correction.

3.1. Radiometric calibration

The conversion from raw digital number (DN) to radiance values (Wm~2sr_1 im_1) for each pixel within an image frame is described by Hawkins et al. (2007, 2009) and is reviewed by Domingue et al. (2011) and Murchie et al. (2015). The most current radiometric calibration is described in Hash (2008).

During Mercury orbit the WAC responsivity changed in association with an event on or about 24 May 2011 when MESSENGER first approached one of the planet's ''hot poles" near spacecraft periapsis (Keller et al., 2013). This event has been interpreted as the result of deposition and subsequent loss of a contaminant on the WAC optical surfaces, and a correction has been incorporated into the radiometric calibration to account for this event by treating the responsivity as time-variable (Keller et al., 2013; Domingue et al., 2015; Murchie et al., 2015). Details of the correction have been described by Keller et al. (2013), and the resulting radiometric calibration, as described in the PDS calibration document, is now given by:

UxyfTth,_ Lin[DN(x ,y, f ,T ,t ,b ,MET ) — Dk(x ,y, T ,t ,b , MET)—Sm(x ,y ,t ,b)]

L(x,y J,',t,b)- (— ; — ^ , (36)

{Flat(x, y, f ,b)*t * yfi}

where L(x y f T t b is the radiance of a pixel located at column x and row y, taken in filter f with charged coupled device (CCD) temperature T, binning mode b, and exposure time t. DN(x y f T t b MET is the pixel raw digital number for an image taken at mission elapsed time (MET). Dk(x,y, T, t, b, MET) is the dark level, Sm(x y t b is the scene-dependent frame transfer smear, Flat(x,y,f,b) is the non-uniformity or 'flat-field' correction, Resp(f b T is the responsivity which relates the dark-, flat-, and smear-corrected DN per unit exposure time to radiance. Correct(f MET is the time-variable correction to the responsivity as described by Keller et al. (2013) and the PDS calibration document. Lin is a function that corrects small nonlinearities in the detector response. Unlike the study by Domingue et al. (2015), which limited the data modeled to determine a photometric correction to those acquired prior to the contamination event, this study includes all observations acquired of the photometric target regions through 26 April 2013.

3.2. Photometric sequences

MESSENGER's orbital campaigns included repeated eight-filter imaging of selected targets, covering two regions of the surface repeatedly while maximizing the range of photometric (incidence, emission, and phase) angles at which reflectance was sampled, with angular resolution of 5°. One region is near Beethoven basin (between 8-50°S, 200-270°E, hereafter referred to as the Beethoven region) and the other near Rembrandt basin (between 17-44°S, 55-99°E, hereafter referred to as the Rembrandt region). The Domingue et al. (2015) analysis included only data from the Beethoven region; this study includes data from both. Data from twenty small, uniform spots within each of these regions, selected based on low topographic relief, lack of bright rays, and moderate albedo, (Fig. 3a and b) were extracted for photometric analysis.

The photometric campaign was planned to maximize the coverage in incidence and emission angle space; hence many images were acquired at off-nadir geometries. However images acquired as part of Merc-G8CM were at a near-nadir geometry at the lowest solar incidence angle values available for the latitudes of each image, thus representing a section of the incidence and emission angle space not well sampled in the photometric campaign. To derive a photometric correction more appropriate to Merc-G8CM geometries, Domingue et al. (2015) analyzed a region of Merc-G8CM that was imaged over as broad as possible a range of incidence and emission angle values (between 3-40°S, 296-346°E), near Matabei (Fig. 3c). Ten areas from it, selected based on low topographic relief, lack of bright rays, and moderate albedo, are also included in this study. An example of the photometric angle coverage provided by each region is displayed in Fig. 4.

4. Modeling methodology and application

Each photometric model applied to the MDIS photometric data was fit using a least squares grid-search to find the best value of each model's parameters to describe the data set. The grid-search algorithm minimized the value of chi, v, defined by

v 'y \\J (rmeasured — rmodeO =N,

where N is the number of measurements, rmeasured is the measured reflectance, and rmodel is the model predicted reflectance. All parameters in each model were varied simultaneously. The smallest grid increment was 0.01 for all parameters except 0, where the smallest grid value was 1. The resulting parameter values were then analyzed as a function of wavelength, and a polynomial was fit to each parameter as a function of wavelength to remove artifacts between filters due to round-off errors. The parameter values are presented to eight figures to keep round-off errors from reintroducing wavelength-dependent artifacts into the photometric corrections. The formal errors listed are based on the grid increment values.

4.1. Hapke model application and results

Three versions of the Hapke model were applied to the MDIS photometric series described above. The first model, hereafter cited as the Hapke Basic model, was used by Domingue et al. (2015) to derive a photometric correction for the initial version of Merc-G8CM delivered to the PDS in February 2013, and is expressed as:

r(i, e, a ) =

4p l0e + le

— 1]}S(i, e, a,„„

{[P(a ) [1 + Bs0 Bs(a ) ]]+ [H(ioe) HA)

which is equivalent to Eq. (1) with K = 1 and BC0 = 0. Here, we reap-plied the Hapke Basic model using the values of the opposition parameters (BS0 = 3.086, hs = 0.090) derived by Domingue et al. (2015) from modeling the disk-integrated data described in Domingue et al. (2010). The resulting parameter values are listed in Table 4, and differ from values obtained by Domingue et al. (2015). The values from this study were used to photometrically correct a 3-color mosaic of northern latitudes delivered to the PDS in March 2014 and the updated Merc-G8CM redelivered in September 2014.

The second model, hereafter cited as H2012-NoOpp, is Eq. (1) with both the SHOE and CBOE opposition expressions set to unity (BS0 = 0 and BC0 = 0), because there are no measurements within the opposition surge in the disk-resolved photometric data. The resulting parameter values are listed in Table 5, and compared with the results from the Hapke Basic model in Fig. 5.

The third model, hereafter cited as H2012, is the full Hapke model described in Section 2.1. Opposition parameter values were from the 588.9-nm disk-integrated observations analyzed by Domingue et al. (2010). We refit the disk-integrated data using the following expression for the disk-integrated reflectance, which includes both opposition expressions:

/(a, w, 0) = Kr(a, 0 ) /(a, w, 0 = 0 ),

/(a, w, 0 = 0 ) =

(1 + C )

■{[1 + Bs0Bs(a ) ]p(a ) — 1}

/a\ /a\ , / /a\

[1 — r°ll[1 — sin © tan © ln (cot (4))] 4r0 /sin(a ) + (p — a ) cos(a )

1 + Bc0Be (a )],

W;; S * : %

. ' - —

ir '.. I I' • r T'

-H , .V * w-C ... '

r | | J,» • ~

Angle Variations

bo 0) o

g 0.07 re tt 4)

'S o.oe ce

incidence emission phase

100 120 140 160 180 200 220

Pixel Postion Model Comparisons

M 1/t A

\ i \ N

Vy/^ —7 vh vy\

Y^J V » V v.

H-430 H-750 H-1000 KS-430 KS-750 KS_1000 1

Pixel Position

Fig. 15. The sub-region defined in Fig. 14, is shown from the Hapke mosaic (top) and the KS mosaic (bottom). The angle excursions across the transect shown in the mosaics are displayed in the center graph (incidence angle: thin line, emission angle: double line, phase angle: thick line). The bottom graph compares the reflectance variations across the trace for the three wavelengths used to construct the color mosaic (red: 1000-nm, green: 750-nm, blue: 430-nm). Image boundary occurs at pixel position 156.

(1 — c)

(1 + c):

c = (1 — w) 1/2,

Ap0 = {^8 [p(a = 0) (1 + Bs0) — 1] + (0.49r0 + 0.19r2) }(1 + BCT),

where Kr(a; 00 is the surface roughness correction from Hapke (1984, 2012a). Fitting the disk-integrated data yielded values of 0.85, 0.07, 0.02, and 0.77 for Bs0; hs; BC0; and hc, respectively. Modeling disk-resolved data then assumed that Bs0; hs; and BC0 are independent of wavelength. Assuming the transport mean free path is independent of wavelength, hc was calculated for each wavelength using Eq. (8); values are listed in Table 6. Parameter values derived from modeling the combined Beethoven, Rembrandt, and Mosaic regions are listed in Table 7, and compared with the results from the application of the other two versions of the Hapke model in Fig. 5. Interpretations of these results in terms of global surface properties are discussed in Section 6.1.

4.2. Kaasalainen-Shkuratov model application and results

Several combinations of the phase and disk functions in the KS model were applied. The first five used a simplified version of the phase function, A1, with opposition terms omitted because the MDIS data do not cover the opposition region, expressed as A1 = ANe—al where AN = A(a0, k). The first five simplified KS models are thus defined as:

KS1 = ANe

2 cos i cos i + cos e,

KS2 - ^cos g) cos [-—- (c — 2)] 2 cos i

a\ ] (cos b )

a/(p—a)

KS3 = ANe

cos i + cos e

(1 — cl ) cos i

KS4 = ANe—ai(cos i ) k(cos e ) ■ KS5 = ANe—ai cos g)

c—2)1

a\ ] (cos b)

ga/(p—a)

Fig. 16. These mosaics were constructed using the Hapke Basic model (top) and the KS3 model (bottom) to provide the photometric correction to standard illumination and viewing geometries of 30°, 0°, 30° in incidence, emission, and phase angles, respectively. Both mosaics are constructed from the 1000-nm filter images and stretched to the same dynamical range. The red box is the region shown and examined in Fig. 17.

w' Tjfl

40 35 30 25

t/i <u 0) Ë0 20 0J

15 10 5

Angle Variations

-incidence

^-emission —phase

870 880 890 900 910 920 930 940 950 960 Pixel Position

Model Comparisons

-H-430

-H-750

-H-1000

-KS-430

-KS-750

-KS-1000

910 920 930 Pixel Position

Fig. 17. The sub-region defined in Fig. 16, is shown from the Hapke mosaic (top) and the KS mosaic (bottom). The angle excursions across the transect shown in the mosaics are displayed in the center graph (incidence angle: thin line, emission angle: double line, phase angle: thick line). The bottom graph compares the reflectance variations across the trace for the three wavelengths used to construct the color mosaic (red: 1000-nm, green: 750-nm, blue: 430-nm). Image boundary occurs at pixel position 918.

The sixth and final version of the KS model incorporated opposition terms in the phase function, A2, where A2 = ANe 111+™ 12a, so:

KSe = An-

e-"ia + me_12a

/a\ r p cos — cos -

\2J Lp - a

(cos b)

ga/(p-a)

Both the KS1 and KS2 model have parameter-free disk functions; the only variables are for the phase function and include normal albedo, AN, and the surface roughness term i. Best-fit values for both are given in Table 8, and compared with results from other KS models in Fig. 6. KS models 3-5 have the same phase function, but incorporate disk functions with single parameters defined in Section 2.2. The values for the disk function parameters are listed in Table 9 and compared between models in Fig. 6. The final KS model has five parameters, four for the phase function expression and one for the disk function. They are listed in Table 10, and compared with other KS model parameters in Fig. 6. Possible interpretations in terms of global surface properties are discussed in Section 6.2.

4.3. Quality of fit analysis

Three sets of quality of fit tests were performed with the modeling results. The first utilized chi values from the model solution derivations, in which each model was fit using the grid search method to minimize the value of chi. The second test is analysis of the reflectance ratio, defined as the ratio of the measured to model-predicted reflectance at a given geometry. A perfect fit would produce a reflectance ratio of unity. The final test examines seams between images acquired with large differences in incidence, emission, or phase angle, to ascertain if one model provides a more seamless mosaic than the other.

4.3.1. Chi fit analysis

Chi values of the three Hapke model solutions (Fig. 7a) shows that no one Hapke model is a significantly better descriptor of the data. In contrast, chi values of the six Kaasalainen-Shkuratov (Fig. 7b) model solutions shows that four of the KS models are better descriptors of reflectance behavior than the other two KS models, and these four have comparable chi values. The KS2, KS3, KS5, and KS6, models and all three Hapke models are further analyzed using the reflectance ratio method.

Fig. 18. These mosaics were constructed using the Hapke Basic model (top) and the KS3 model (bottom) to provide the photometric correction to standard illumination and viewing geometries of 30°, 0°, 30° in incidence, emission, and phase angles, respectively. Both mosaics are constructed from the 1000-nm filter images and stretched to the same dynamical range. The red box is the region shown and examined in Fig. 19.

k_ M 40

Angle Variations

-incidence -emission -phase

1200 1205 1210 1215 1220 1225 1230 1235 1240 1245 1250 1255 Pixel Position

Model Variations

• H-430 ■H-750 ■H-1000

■ KS-430

■ KS-750 ■KS-1000

1200 1205 1210 1215 1220 1225 1230 1235 1240 1245 1250 1255 Pixel Position

Fig. 19. The sub-region defined in Fig. 18, is shown from the Hapke mosaic (left) and the KS mosaic (right). The angle excursions across the transect shown in the mosaics are displayed in the center graph (incidence angle: thin line, emission angle: double line, phase angle: thick line). The contrasts between images are comparable to that seen in the emission angle cases. The bottom graph compares the reflectance variations across the trace for the three wavelengths used to construct the color mosaic (red: 1000-nm, green: 750-nm, blue: 430-nm). Image boundary occurs at pixel position 1222.

Fig. 20. Comparison of the global 8-color mosaic corrected using the Hapke Basic model (top) and the KS3 model (bottom) for the photometric correction. The red, green, and blue channels are provided by the 1000-nm, 750-nm, and 430-nm filter mosaics, respectively. Arrows indicate examples of differences between the two mosaics.

4.3.2. Reflectance ratio analysis

Reflectance ratios were calculated using the trended model parameter values for each measured spot in the photometric data sets. Examples of these ratio values versus incidence and emission angle are shown in Fig. 8. The Hapke model example (Hapke Basic) has values for the ratio that cluster around unity. Similarly, the KS model example (KS3) also has ratio values that cluster about unity, though the trend with emission angle departs more from unity in the Hapke model example than the KS model example. To simplify the comparison the median values in the ratios as a function of wavelength was examined.

For each photometric model the median reflectance ratio was calculated at each wavelength (Fig. 9). None of the seven models clearly describes the photometric measurements better than the remainder, with many of the reflectance ratio values falling within

2% of a perfect solution. However the reflectance ratio at 749 nm is systematically lower than those at surrounding wavelengths in all model solutions. Most of the reflectance ratios are within 2% of unity, and all but one value is within 5% of unity. The worst fit (a ratio of 0.942) between the measurements and a model occurs at 430 nm with the H2012-No0pp model. The maximum departure from unity across all wavelengths ranges from 4.9% to 5.8% for the Hapke models and from 2.7% to 4.6% for the KS models; with the lowest maximum departure (indicating a better correspondence between model and measurement) in the KS2 model.

The next step in this analysis was to subdivide the entire data set based on incidence angle (i < 40°, 40° 6 i < 60°, i p 60°) and emission angle (e < 40°, 40° 6 e < 60°, e P 60°). The reflectance ratio was calculated using the trended model parameter values for each data point (see Fig. 8) and reexamined as a function of

60 80 Phase Angle (deg.)

m cn cc

60 80 100 Phase Angle (deg.)

Fig. 21. These graphs plot the ratio of the reflectance at 996.2 nm to the reflectance at 433.2 nm (gray circles) as a function of phase angle for two areas sampled in the Beethoven photometric region. The solid line shows a linear trend through the data sets. For comparisons the Hapke Basic model predicted ratio (solid small circles) and the KS3 model predicted ratio (small gray diamonds) are also shown. The models use the same set of incidence, emission, and phase angles as present in the data set.

£ 0.4

Single Scattering Albedo

0 fft ♦

■ V -м--®

1100 1300"о

Wavelength (nm)

•Hapke Basic H2012 - no Opposition • H2012 - w/Opposition Eros (Domingue et al. 2002) Eros (Clark et al. 2002) Eros (Li et al. 2004) •Itokawa (Lederer et al. 2008) Vesta (Li et al. 2013) disk-integrated Vesta (Li et al. 2013) disk-resolved Gaspra (Helfenstein et al. 1994) Ida (Helfenstein et al. 1996) Dactyl (Helfenstein et al. 1996) Mathilde (Clark et al. 1999) Deimos (Thomas et al. 1996) Average S (Helfenstein & Veverka 1989) Average С (Helfenstein & Veverka 1989) Moon (Helfenstein & Veverka 1987) Moon (Hartman & Domingue 1998) Moon - mare (Sato et al. 2014) Moon - highlands (Sato et al. 2014) Moon - general (Sato et al. 2014)

Fig. 22. Comparisons of the volume-averaged single scattering albedo for Mercury from the three Hapke model solutions to single scattering albedos derived from similar Hapke modeling efforts for asteroids and the Moon.

incidence and emission angle subdivision. The median reflectance ratio was then calculated at each wavelength for each model within each subdivision. No single model was found to outperform the others across all incidence and emission angle subdivisions.

To determine which models perform better within incidence and emission angle subdivisions, reflectance ratio values were binned according to quality of fit (with bins defined as having residuals >10%, 8-10%, 6-8%, 4-6%, 2-4%, and <2%). 749-nm reflectance ratio values were excluded because they are commonly anomalous within all modeling results. This is commensurate with the performance of this filter in comparison to the others after the contamination event (Keller et al., 2013), and thus a calibration artifact is suspected (see Fig. 9). For a given model, spectral reflectance ratio values for each incidence and emission subdivision were examined to find the highest median residual for any wavelength. For each model, each photometric subdivision was assigned the quality of fit for its worst wavelength. The results are illustrated in Fig. 10a. The same classification was performed a second time excluding reflectance ratio at both 749 nm and at the worst of the remaining wavelengths. Those results are illustrated in Fig. 10b.

In Fig. 10a, the only models that have no incidence and emission angle subdivisions with median reflectance ratio residuals >10%

are the KS3 and KS5 models. All models have subdivisions with median reflectance ratio residuals >8%, but only the Hapke Basic model has a single subdivision with a median reflectance ratio residual >8%. All other models have multiple subdivisions with a median reflectance ratio residual above 8%. Only the KS3 model has a photometric subdivision with a median reflectance ratio residual <2%. This model also has the largest number of subdivisions with a residual <4%. The three Hapke models perform somewhat worse, with the largest number of subdivisions having residuals <6%. Of the three Hapke models, the Hapke Basic model has the lowest number of subdivisions with residuals >8% and the largest number of subdivisions with residuals <6%. Of the KS models, KS3 has the lowest number of subdivisions with residuals >8% and the largest number of subdivisions with residuals <6%. For both the Hapke Basic and KS3 models, the worst residuals are in the e >60 subdivision.

Most MESSENGER MDIS mapping campaigns were designed to acquire images with emission angles minimized within the spacecraft pointing restrictions, resulting in phase angle varying with incidence angle. For this reason it is informative to examine the quality of fit as a function of phase angle. The data were divided into the incidence angle subdivisions as above, and subdivided

Absorption

400 500 600 700 800

Wavelength (nm)

Fig. 23. Comparison of the scattering coefficient for the optically active portion of Mercury's regolith from the Hapke Basic (solid line), H2012-No0pp (dotted line), and H2012 (dashed line) models.

again based on phase angle (a < 40°, 40° 6 a < 60°, 60° 6 a < 90°, 90° 6 a < 110°, a p 110°). Reflectance ratios were calculated using trended model parameter values and methodology as described above for the incidence and emission angle analysis. Results were again classified by median residual for all

wavelengths, first excluding 749-nm reflectance and then again excluding reflectance at both 749 nm and the worst remaining wavelength. Results are illustrated in Fig. 11a and b.

Fig. 11a shows that all models have at least one photometric subdivision with median reflectance ratio residual >10%, but the

Hapke Basic

H2012 No Opposition

Phase angle (deg.)

Phase angle (deg.)

Single Particle Scattering Function

100 150

Phase angle (deg.)

50 100 150

Phase angle (deg.)

"Hapke Basic •H2012 NoOpp - H2012 w/Opp

Fig. 24. These graphs display the single particle scattering function at each wavelength for the three Hapke models used in this study. The median value across all wavelengths for each model is displayed and compared in the lower right graph.

Table 11

Single scattering albedo values for comparable objects to Mercury.

Object k (nm) w Reference

Eros 550 0.43 Domingue et al. (2002) (solution 3)

Eros 550 0.33 Li et al. (2004)

Eros 950 0.42 Clark et al. (2002) (nominal case)

Itokawa 360 0.53 Lederer et al. (2008)

Itokawa 440 0.66 Lederer et al. (2008)

Itokawa 550 0.7 Lederer et al. (2008)

Itokawa 640 0.71 Lederer et al. (2008)

Itokawa 790 0.73 Lederer et al. (2008)

Itokawa 1260 0.69 Lederer et al. (2008)

Itokawa 1600 0.61 Lederer et al. (2008)

Itokawa 2220 0.58 Lederer et al. (2008)

Gaspra 560 0.36 Helfenstein et al. (1994)

Ida 560 0.218 Helfenstein et al. (1996)

Dactyl 560 0.211 Helfenstein et al. (1996)

Average S-type 550 0.23 Helfenstein and Veverka (1989)

Mathilde 700 0.035 Clark et al. (1999)

Average C-type 550 0.037 Helfenstein and Veverka (1989)

Vesta 700 0.424 Li et al. (2004) (case 4, disk-integrated)

Vesta 700 0.491 Li et al. (2004) (case 3, disk-resolved)

Deimos 540 0.079 Thomas et al. (1996)

Moon 550 0.21 Helfenstein and Veverka (1987)

Moon 550 0.37 Hartman and Domingue (1998)

Moon - mare 320 0.09 Sato et al. (2014)a

Moon - mare 360 0.12 Sato et al. (2014)a

Moon - mare 420 0.14 Sato et al. (2014)a

Moon - mare 570 0.2 Sato et al. (2014)a

Moon - mare 600 0.22 Sato et al. (2014)a

Moon - mare 650 0.24 Sato et al. (2014)a

Moon - mare 690 0.26 Sato et al. (2014)a

Moon - highlands 320 0.175 Sato et al. (2014)a

Moon - highlands 360 0.21 Sato et al. (2014)a

Moon - highlands 420 0.25 Sato et al. (2014)a

Moon - highlands 570 0.36 Sato et al. (2014)a

Moon - highlands 600 0.39 Sato et al. (2014)a

Moon - highlands 650 0.41 Sato et al. (2014)a

Moon - highlands 690 0.44 Sato et al. (2014)a

Moon - general 320 0.16 Sato et al. (2014)a

Moon - general 360 0.19 Sato et al. (2014)a

Moon - general 420 0.21 Sato et al. (2014)a

Moon - general 570 0.33 Sato et al. (2014)a

Moon - general 600 0.36 Sato et al. (2014)a

Moon - general 650 0.39 Sato et al. (2014)a

Moon - general 690 0.41 Sato et al. (2014)a

a Values extrapolated from Fig. 17 in Sato et al. (2014).

KS3 model has the fewest subdivisions (1) with a residual >8%. The KS2 and KS5 models are the only models having photometric subdivisions with residuals <2%. Of the Hapke models, the Hapke Basic has the largest number of subdivisions with residuals <4% and the lowest number of subdivisions with residuals >10%. Similarly, of the KS models, KS3 model has the largest number of subdivisions with residuals <4% and the smallest number of subdivisions with residuals >10%.

Based on the above results, the version of the Hapke model that best describes the MESSENGER disk-resolved photometry is the Hapke Basic model. Similarly, the version of the KS model that best describes the data set is KS3. Both were then applied to sections of the Merc-G8CM as a final test of the quality of fit.

4.3.3. Mosaic application test

The Merc-G8CM is divided into 54 subquadrants, a number of which contain large discontinuities in photometric geometries. Several such subquadrants were reconstructed in all 8 wavelengths using both Hapke Basic and KS3 photometric models to standardize corrected reflectance at incidence, emission, and phase angle values of 30°, 0°, and 30°, respectively. The mosaics that used the Hapke Basic correction (hereafter referred to as the Hapke mosaics)

were compared to those using the KS3 correction (hereafter referred to as the KS mosaics).

In the northern quadrants there are several seams separating areas covered with large differences in emission angle, most noteworthy in Caloris (Fig. 12, showing the 1000 nm mosaics stretched using the same dynamic range). The most noticeable discontinuities in corrected reflectance with large differences in emission angles are the Hapke mosaic. Quantitative comparisons of both corrections at all wavelengths along a transect through the discontinuities are shown in Fig. 13. The reflectance discontinuity at different emission angles is apparent at all wavelengths.

In the equatorial quadrants there are differences observed across image boundaries that are related to variations in all three photometric angles. Variations across seams with large differences in incidence angle are examined in Figs. 14 and 15, using 1000-nm wavelength images stretched to the same dynamical range. In this case the seam has low contrast as compared to that corresponding to contrasting emission angle values, and the contrast is larger in the KS mosaics (Fig. 15) than the Hapke mosaics.

Variations across seams with large differences in phase angle values are examined in Figs. 16 and 17, again using 1000-nm images stretched to the same dynamic range. As with incidence angle, the seam is low-contrast as compared to where there are large differences in emission angle. The contrast between adjacent images is larger in the KS mosaics (Fig. 17) than in the Hapke mosaics, though the magnitude of the seams is comparable to that in the contrasting incidence angle case, and much smaller than where there are contrasting emission angles.

The southern quadrants also have examples where there are seams or contrasts between adjacent images within the mosaic. The above examples are cases where adjacent images were acquired under variations in either incidence, emission, or phase angle values. In contrast, the 1000-nm subquadrant shown in Figs. 18 and 19 represents a condition where a seam is created by a combination of different emission and phase angle values. The residual in corrected reflectance along the transect through the discontinuities (Fig. 19) is comparable to those seen in the cases where the adjacent images were acquired under different emission angles, except that the contrast is larger in the KS than the Hapke mosaic (Fig. 19) at all wavelengths. Other examples were investigated where there are variations in both incidence and phase angles, and the scale of the residuals and the relative performance of the models is consistent with the case of variation mainly in incidence angle.

5. MESSENGER 8-color mosaic

The Merc-G8CM mosaic, produced alternatively using the Hapke Basic model and the KS3 model, are shown in Fig. 20 displaying 1000-, 750-, and 430-nm mosaics in the red, green, and blue channels, respectively. Both mosaics have been stretched to the same dynamic range to facilitate comparisons. The top mosaic uses the Hapke Basic model and was delivered to the PDS (September 2014); the bottom mosaic uses the KS3 model. The KS3 model based photometric correction will be applied in the construction of the final color mosaics to be delivered by the MESSENGER project to the PDS in March 2016. This photometric correction was chosen based on the performance comparisons presented in this study: (1) more photometric subdivisions with reflectance ratios closest to unity, (2) less contrast between images acquired at large differences in emission angle, (3) the contrast between images acquired at large differences in either incidence or phase angle is not significantly larger than the contrast using the Hapke model. The incidence, emission, and phase angle value ranges from 5.83°

Single Particle Scattering Functions

■Hapke Basic -H2012 - w/Opp ■Ida (Helfenstein et al. 1996) •Median Itokawa (Lederer et al. 2008) •Vesta (Li et al. 2013) disk-integrated ■Deimos (Thomas et al. 1996) Median lunar highlands (Sato et al. 2014)

80 100 120 Phase angle (deg.)

--H2012 -NoOpp

Eros (Li etal. 2004) ^—Gaspra (Helfenstein et al. 1994) • • • "Mathilde (Clark et al. 1999) — — Vesta (Li et al. 2013) disk-resolved Median lunar mare (Sato et al. 2014) Average Moon (Sato et al. 2014)

Fig. 25. Comparison of the single particle scattering functions from the three Hapke model solutions for Mercury with those of S-type asteroids (Eros, Ida, Gaspra, and Itokawa), C-type asteroid (Mathilde), Vesta, Mars satellite Deimos, and the lunar surface (mare, highlands, and the average Moon).

Fig. 26. Comparison of single particle scattering function parameter values from Mercury (this paper), Eros (Clark et al., 2002; Domingue et al., 2002; Li et al., 2004), Gaspra (Helfenstein et al., 1994), Ida (Helfenstein et al., 1996), Mathilde (Clark et al., 1999), and the Moon (Sato et al., 2014) with the laboratory particles of McGuire and Hapke (1995) and the hockey stick relation from Hapke (2012b). The inset graph displays the details within the black box of the main graph, where the Mercury results can be better discerned.

to 87.82°, 0.85° to 66.41°, and 23.42° to 86.66°, respectively, across these mosaics (see Fig. 4 from Domingue et al., 2015).

In order to further examine the importance of the photometric standardization, and the differences between the Hapke Basic and KS3 model based corrections, we examine the phenomena of phase reddening. Comparisons are made of the color ratio between the near-infrared at 996.2 nm and the near-ultraviolet at 433.2 nm.

This ratio is a proxy for the spectral slope between these two wavelength regions, and the variation in this ratio as a function of phase angle is displayed in Fig. 21 for two example areas from the Beethoven photometric region. Color ratios are often used to examine space-weathering properties and to determine regolith maturity in both lunar and asteroid regoliths (Lucey et al., 1995, 1998, 2000; Staid and Pieters, 2000; Murchie et al., 2002; Gillis-Davis

et al., 2006; Blewett et al., 2011). This examination emphasizes the importance of photometric standardizations prior to interpreting color ratio results. The ratio of near-infrared to near-ultraviolet reflectance is plotted against phase angle for the observations in each sample area. The corresponding ratios derived from the model predicted reflectance is also shown. Note that the color ratio predicted by the Hapke Basic model turns down at higher phase angle values compared to that predicted by the KS3 model. There is sufficient scatter in the data set, such that it is impossible to discern if the data supports the presence or absence of such a down turn at high phase angles.

6. Photometric analysis of Mercury's surface properties

Photometric modeling provides insight into the surface characteristics of Mercury's regolith (1) by examining the model parameters' theoretical relation to surface properties, and (2) by comparing model parameter values from this study with corresponding parameter values from studies of other Solar System objects, especially the Moon.

6.1. Hapke model interpretations

The parameters from the Hapke models can be grouped into two categories: (1) single scattering albedo and the single particle scattering function, which are supposed to be correlated to properties of single particles within the regolith, and (2) the opposition parameters, the porosity term, and the surface roughness parameter, which are supposed to be related to the relationships between regolith particles.

6.1.1. Single scattering albedo

Single scattering albedo (w) in the Hapke model is volume-averaged and defined as the ratio of the scattering to extinction coefficients. The extinction coefficient is the sum of the scattering and absorption coefficients, so a smaller value of w equates to a more absorbing medium. The spectrum of w (Fig. 22) is featureless like the reflectance spectrum of Mercury's surface with low values indicative of dark materials that are less absorbing at longer wavelengths. The absorption coefficient, shown in Fig. 23 as a function of wavelength, indicates that a 430-nm photon has an 80-85% probability of being absorbed, whereas a 996.2-nm photon has a

64-71% probability of being absorbed. This is a constraint on the composition of the optically active regolith.

Comparisons of Mercury's single scattering albedo to those of various asteroids and the Moon derived using Hapke's model and covering the same wavelength range are shown in Fig. 22 and Table 11. Mercury's w is similar to some S-type asteroids (Ida, Dactyl, and average S-types) but lower than others (Eros, Itokawa, and Gaspra), and brighter than for either Deimos or C-type asteroids (Mathilde and average C-types). Mercury's w is similar to the lunar value derived by Helfenstein and Veverka (1987) and is lower than that derived by Hartman and Domingue (1998). Comparisons with the lunar w values derived by Sato et al. (2014) show that Mercury's surface has lower w values than the lunar highlands and the average or general lunar surface. Mercury has comparable to slightly higher w values than the lunar mare (Sato et al., 2014). This implies that on average Mercury's regolith is more absorbing than the lunar highlands or the average lunar regolith. This is consist with the findings of Denevi and Robinson (2008) that the regolith of Mercury contains an additional darkening agent.

6.1.2. Single particle scattering function

The single particle scattering function represents the probability distribution for the direction in which a regolith particle will scatter light, which depends on the physical and compositional structure of the particles. Structures such as inclusions, cracks, and grain boundaries serve as scattering centers. Some compositions vary in transparency with wavelength, creating wavelength dependencies to the scattering probabilities. Single particle scattering functions predicted by each of the Hapke models applied to the MESSENGER Mercury measurements are shown in Fig. 24. All three Hapke models predict backward scattering regolith particles, probably due to the paucity of observations in the forward scattering direction (a > 150°). None exhibit a strong dependency on wavelength.

Fig. 25 compares median single particle scattering functions across all wavelengths from the three models with a representative set of asteroid and lunar single particle scattering functions. All of the objects exhibit predominantly backward scattering functions; in most cases this is due to the paucity of measurements in the forward scattering direction, including the Mercury measurements. The scattering function of Mercury is most similar to those of Eros (Li et al., 2004) and Mathilde (Clark et al., 1999), which display identical scattering functions, Vesta (Li et al., 2013), Gaspra

Surface Roughness

d 45-1—

Fig. 27. Comparison of Mercury's surface roughness with those of other Solar System bodies modeled using the Hapke Basic model.

(Helfenstein et al., 1994), and the lunar mare (Sato et al., 2014). The surface of Eros has been shown to contain areas of fine-grained regolith (Robinson et al., 2001; Cheng et al., 2002). The surface of Vesta has been suggested to contain a well-mixed fine-grained component (Pieters et al., 2012), and the lunar surface is also known to consist of a fine-grained regolith component. The similarities with these objects suggests that Mercury's surface is also has a significant fine-grained component.

McGuire and Hapke (1995) and Hapke (2012b) examined relations between particle characteristics and scattering functions, using a two-parameter Henyey-Greenstein function form, as used in this study. Hapke (2012b) refined this relationship by including additional laboratory and planetary modeling data to define a correlation between the Henyey-Greenstein function parameters, noted as the ''hockey stick relation". These results are compared with results from Hapke modeling efforts for various Solar System bodies in Fig. 26, with a caveat that several objects in this comparison set were modeled with a single term Henyey-Greenstein (two of the Eros data points, Gaspra, Ida, and Mathilde, and one of the lunar data points). Regardless, some inferences can be drawn from these comparisons. Mercury's regolith behaves as expected for particles with a high density of internal scatterers, possibly higher than lunar regolith particles, consistent with a highly space weathered surface (Hapke, 2001; Domingue et al., 2014) or one incorporating extremely fine-grained opaques (Murchie et al., 2015), which has been postulated to explain the darkening agent present in Mercury's regolith compared to the lunar surface (Denevi and Robinson, 2008; Robinson et al., 2008; Braden and Robinson, 2013).

6.1.3. Opposition and porosity terms

The disk-resolved observations modeled in this study do not include any measurements acquired within the opposition effect region of incidence, emission, and phase angle values. Measures of the characteristics of Mercury's global opposition effect are available only in ground-based observations of Mercury (Mallama et al., 2002), and are not further discussed in this study.

6.1.4. Surface roughness

Hapke (2012a) describes the possible physical significance of the surface roughness parameter, Q, as the mean surface slope averaged over all scales between an upper limit of detector resolution (in this case, kilometers) and a lower limit of several times the mean particle separation (~100-1000 im). Fractal analysis suggests (Shepard and Campbell, 1998) roughness is dominated by the scale at which well-defined shadows exist, or approximately 100 im for the Moon (Helfenstein and Shepard, 1999). The surface roughness derived from the Hapke models applied to the Mercury 8-color observations are compared in Fig. 27 to values derived for other Solar System objects. This comparison suggests that Mercury's surface, at the 100-p.m scale, is smoother than the surfaces of the Moon or asteroids that have been observed by spacecraft.

6.2. Kaasalainen-Shkuratov model interpretations

The correlations between KS model parameter values and surface physical characteristics are not clear. General relationships for the phase function portion of the models are outlined in Table 1; Table 3 shows that the disk functions have no established correlation with surface properties. Phase function parameter relationships with opposition surge and surface roughness are both general. KS models have been applied to the Moon (Shkuratov et al., 2011) and asteroids (Schroder et al., 2013; Li et al., 2013), so some comparisons with these objects can be made with the parameter values derived for Mercury in this study.

6.2.1. Comparison with the Moon

Shkuratov et al. (2011) applied the KS6 model to ground-based observations of the Moon (Velikodsky et al., 2011) at an effective wavelength of 603 nm. The disk function parameter g was assumed to be 0.5, similar to the 0.52 value for the lunar highlands rather than the 0.34 value for the maria derived by Akimov et al. (1999, 2000). The resulting lunar phase function parameter values were 0.14 for the normal albedo (AN), 10 for opposition width parameter i1, 0.07 for surface roughness parameter i2, and 2 for opposition amplitude parameter m. This is in comparison to Mercury values at 628.8 nm of 0.12, 0.96, 0.95, 0.48, and 0.70 for AN, 11, i2, m, and g, respectively. It should be noted that the lunar data contained observations within the opposition region whereas the Mercury data did not, so the opposition related parameters (i1 and m) are poorly constrained for Mercury. The difference in normal albedo is consistent with the observation that immature materials on Mercury are 30-50% lower in reflectance than corresponding materials on the Moon, suggesting the presence of a darkening component within Mercury's regolith compared to the Moon (Denevi and Robinson, 2008; Robinson et al., 2008; Braden and Robinson, 2013; Murchie et al., 2015).

The comparison also implies different surface roughness properties of the lunar and Mercury surfaces, with the model returning a higher roughness parameter value for Mercury compared to the Moon. However, it is not clear if higher roughness parameter (i2) values correspond to higher or lower surface roughnesses, as the term in the phase function, e~1l2a, increases with decreasing values of i2. Shkuratov et al. (2011) do not provide any mathematical correlation between surface roughness and the i2 parameter.

6.2.2. Comparison with Vesta

Schröder et al. (2013) used a set of KS models to examine global and regional photometric properties of Vesta's surface. In their global analysis they used a phase function of the form

Aeq(a, = £Ciai, (46)

where C0 is normal albedo (AN). They applied different disk functions (Dl, Dm, Das, and Dasii) where the parameters for the DL, DM, and Dasii functions vary linearly with phase angle. In this study's application to Mercury the phase function applied was exponential and the disk function parameters constants. The derived globally averaged normal albedo at 700 nm for Vesta ranged from 0.292 to 0.301, depending on the disk function used. These values are much higher than the 0.09-0.13 values derived for Mercury at 748.7 nm, commensurate with a darkening agent present in Mercury's rego-lith (Denevi and Robinson, 2008; Robinson et al., 2008; Braden and Robinson, 2013; Murchie et al., 2015).

In a similar study, Li et al. (2013) used an exponential phase function of the form:

Aeq(a, k)= An 100:4ba, (47)

where ß is the phase slope parameter in mag/deg, and the DM disk function where the disk function parameter, k, varies linearly with phase angle. The normal albedo values ranged from 0.21 to 0.32 over 400 to 1000 nm wavelength, consistent with results from Schröder et al. (2013).

6.3. Implications of photometric modeling results

Some tests of the Hapke model demonstrate qualitative agreement between model parameters and sample characteristics (Gunderson et al., 2005; Souchon et al., 2011; Helfenstein and Shepard, 2011), but the results depend on the form of the model used and the range of angles sampled (e.g., excluding large inci-

dence and emission angles, measurements within the opposition surge). Other tests have shown poor quantitative correlation between Hapke model parameters and sample properties (Shkuratov et al., 2007; Shepard and Helfenstein, 2007). Using samples of an idealized soil comprised of spherical glass beads with known optical constants, particle sizes, and compaction states, Hapke et al. (2009) demonstrated that the Hapke model can predict scattering behavior, consistent with underlying assumptions in the model: that the surface is comprised of equant particles much larger than a wavelength of light (Hapke, 2012a). Although these assumptions may sound valid for the surface of a regolith generated by meteorite impact (comminution), space weathering processes create scattering centers on scales comparable to a wavelength of light (Domingue et al., 2014 and references therein). There has been no unambiguous proof of correlations between Hapke model parameters and physical properties of natural soils in their natural environment (whose surface roughness and porosity are difficult to reproduce in the laboratory). At best, intercomparisons between different planetary surfaces can provide insight into contrasting properties, but the parameters derived from the Hapke model should not be interpreted as absolute measures of surface characteristics.

The KS model has not been tested as thoroughly as the Hapke model in terms of correlating parameters with specific surface characteristics. Thus the relation between surfaces fit by differing model parameters must also be interpreted only in a comparative sense.

6.4. Mercury's surface properties

Implications of the results from these photometric models can be better understood by comparing surfaces among objects that have been similarly studied and placed in the context of what is known regarding the processes that form these surfaces and that are still active. Asteroids that have been imaged by spacecraft have been analyzed using either the Hapke or KS model have mostly been S-type asteroids, with the exception Mathilde, which is C-type, and Vesta, a V-type. The lowest spatial resolutions were attained for those visited by flybys (Mathilde, Ida, and Gaspra). Higher resolutions are available for Itokawa and Eros; both have been imaged by orbiting spacecraft. Their surfaces are densely populated by boulders, though some areas exhibit smooth, finegrained ponds. Vesta shows a highly cratered surface. Pieters et al. (2012) argued that the maturation processes on Vesta's surface have produced a highly mixed fine-grained regolith component. The lunar surface also exhibits large variations in regolith properties. There are areas on the lunar surface that show regolith mixed with boulders and other places where bedrock is exposed. Samples from the lunar surface and Itokawa contain fine-grained components in addition to the coarser grained components seen in the imaging data. This variety is also exemplified in images of Mercury's surface, which shows landform types not yet seen on either asteroids or the Moon. This evidence shows that the regolith on any Solar System object is highly variable, and that a "global" characterization of the regolith cannot capture the diversity in regolith properties, only, at best, an average.

An obvious comparison that can be made between bodies is of ''surface roughness". KS modeling results yield a difference in the surface roughness between the Moon and Mercury. However it is not clear if higher values of the KS roughness parameters indicate a rougher or smoother surface. The Hapke modeling results show a difference in the surface roughness between Mercury, the Moon, and the subset of asteroids examined; with Mercury displaying the smallest roughness parameter.

Recall that surface roughness represents unresolved topography, and at least for the Hapke model is thought to be most affected by smaller scales (hundreds of micrometers; Shepard

and Campbell, 1998; Helfenstein and Shepard, 1999). For all objects, the modeling utilized data with resolutions on the order of kilometers or larger (especially in those studies that only examined globally averaged reflectances). The comparatively high surface roughness value for Itokawa is consistent with the extremely blocky nature that dominates the global characteristics of its surface. Note that the roughness differences seen between Eros and Itokawa are commensurate with the evaluation of the ''blockiness" of their surfaces by Ernst et al. (2015). Schroder et al. (2013) demonstrated that areas of low roughness on Vesta are associated with exposed bedrock whereas high roughness is correlated with regions associated with mass wasting deposits and ejecta, signaling that even in the KS modeling results the surface roughness may be governed by the micrometer scale.

If the Hapke model derived surface roughness is dominated by the micrometer scale, this implies that the regolith on Mercury is smoother, on this scale, then either the Moon or many asteroids. This implies that the regolith maturation processes tend to smooth surfaces on micrometer scales, assuming Mercury is more highly space weathered (e.g. Domingue et al., 2014). The overall lower surface roughness of Mercury's surface may be consistent with a larger flux and higher velocity of impactors than on the Moon (Cintala, 1992). Mercury's surface has been more highly comminuted and melted, and most of its surface may be glass (Cintala, 1992). Space weathering processes may have produced a surface that is more amorphous, with a larger component of agglutinates, and finer-grained than the lunar surface (Domingue et al., 2014 and references therein). The less crystalline structure may lend to smoother grains. Agglutinates bond grains within a glass matrix, possibly reducing the angularity of regolith grains. Fine-grains in an electrostatic environment could produce ''fairy-castle" structures (Hapke and Van Horn, 1963; Hapke, 2012a), creating a porous surface with potentially more angular or rougher properties. The balance ofthese processes may thus lead to a smoother surface on the hundreds-of-microns scale on Mercury than on the Moon. Alternatively, the lower surface roughness value than for Itokawa and Eros may simply indicate that the regolith is less blocky.

There are additional boundaries that can be placed on comparative regolith particle properties by examining the single particle scattering functions from the Hapke modeling, and single scattering and normal albedos from the Hapke and KS models respectively. Laboratory studies of the single particle scattering function (McGuire and Hapke, 1995; Hapke, 2012b) suggest that the particles within Mercury's regolith may have a high density of internal scatterers, i.e., boundaries between optical properties including but not limited to vesicles, fractures, inclusions, or phase. An agglutinate would be expected to have a particularly rich population of scattering centers that would include many of these examples in one grain. The lunar work by Sato et al. (2014) attributes the differences in the single particle scattering function between the lunar mare and highlands to the amount and composition of the agglutinate and sub-micron phase iron (npFe). The backward scattering nature of the highlands is attributed to the formation of agglutinates from high-albedo materials and a smaller amount of npFe than the mare (Sato et al., 2014). The more forward scattering nature of the mare compared to the highlands is ascribed to a higher content of an opaque (ilmenite) and higher quantities of npFe (Sato et al., 2014). The hypothesis is that the higher npFe in the mare agglutinates reduces the agglutinate's backward scattering more than the silicate forward scattering (Sato et al., 2014). The scattering behavior for Mercury most closely resembles that of the mare than the highlands, with Mercury displaying an even higher component of forward scattering, though the forward scattering is not well constrained in the Mercury data set. This implies that the sub-micron phase material on Mercury is more abundant than on the lunar surface.

Comparison of the normal albedo derived from the KS model results indicate that Mercury has a lower single scattering and normal albedo than either the Moon or Vesta. The single scattering albedo is most comparable to the lunar mare results of Sato et al. (2014). These differences are consistent with compositional differences in addition to differences in the structures of regolith particles between these objects, implying that Mercury has some compositionally distinct components.

7. Conclusions

Two photometric models, and variations within these models, were examined for producing the photometric correction for the final color image mosaics to be supplied to the PDS by the MESSENGER project in March 2016. Each set of models was tested numerically and visually. The KS3 model was selected based on the following performance results:

• greater number of incidence, emission, and phase angle subdivisions with reflectance ratio values nearest unity,

• significantly less contrast along seams between images acquired at large differences in emission angle than obtained using the Hapke model,

• less contrast along seams between images acquired in regions of geologic interest, such as Caloris Basin,

• the contrast along seams between images acquired a large differences in either incidence or phase angle is not significantly larger than the contrast using the Hapke model.

Other results that are of interest to note is that the Hapke Basic model performs as well, and in this case better, than the more complicated versions of Hapke's model. This indicates that the basic, core equations, dominate the reflectance predictions, and the modifications for the opposition surge and porosity require data within specific incidence, emission, and phase angle regimes to improve the modeling capabilities. For example, without detailed measure of the opposition region (measures that include variation in incidence and emission within low, <5-10°, phase) the simple assumption of a shadow-hiding source is sufficient to model the reflectance properties of the surface to within the accuracy of the model.

A second interesting result is the photometric angle region in which the KS models out perform the Hapke model. The Hapke model performed worst at large emission angles, where the KS model provided a significantly better solution. At large incidence or phase angles, the difference between the KS and Hapke model solutions were much smaller.

Some conclusions can be drawn regarding the properties of Mercury's regolith based on photometric modeling. These properties are based on comparing modeling results from this study of Mercury with modeling results of similar analyses of asteroids and the lunar surface since laboratory testing of the models show only evidence of qualitative correlations between parameters and surface properties, and then only under restricted ranges of incidence and emission angle. The properties derived from the photometric analyses presented in this study are globally averaged properties.

• On micrometer scales Mercury is smoother than the lunar and asteroid surfaces, and is consistent with a less blocky regolith.

• The physical structure of the regolith grains on Mercury is different than those on the lunar and asteroid surfaces, commensurate with a larger abundance of sub-microscopic materials (such as extremely fine grained opaques and/or nanophase space-weathering products).

• Mercury's regolith contains at least one compositional component distinct from the lunar regolith.

Acknowledgments

The authors would like to thank the MESSENGER Mission Operations Team and the MDIS Instrument Operations team, without whose efforts this research would not be possible. The MESSENGER project is supported by the NASA Discovery Program under contracts NASW-00002 to the Carnegie Institution of Washington and NAS5-97271 to The Johns Hopkins University Applied Physics Laboratory. The authors would like to thank Jang Yang Li, Stefan Schröder, and Yasuhiro Yokota for their thoughtful review and insightful comments.

References

Akimov, L.A., 1976. Influence of mesorelief on the brightness distribution over a

planetary disk. Sov. Astron. 19 (3), 385-388. Akimov, L.A., 1979. Brightness distribution over the lunar and planetary disks.

Astron. Zh. 56, 412-418. Akimov, L.A., 1988a. Light reflection by the Moon, I. Kinem. Phys. Celest. Bodies 4 (2), 3-10.

Akimov, L.A., 1988b. Light reflection by the Moon, II. Kinem. Phys. Celest. Bodies 4 (2), 10-16.

Akimov, L.A., Velikodsky, Y.I., Korokhin, V.V., 1999. Dependence of lunar highland brightness on photometric latitude. Kinem. Phys. Celest. Bodies 15 (4), 232236.

Akimov, L.A., Velikodsky, Y.I., Korokhin, V.V., 2000. Dependence of latitudinal brightness distribution over the lunar disk on albedo and surface roughness. Kinem. Phys. Celest. Bodies 16 (2), 137-141. Bhattacharya, S. et al., 1975. Lunar regolith and gas rich meteorites: Characterization based on particle tracks and grain size distribution. Lunar Planet. Sci. 6, 3509-3526. Blewett, D.T. et al., 2011. Lunar swirls: Examining crustal magnetic anomalies and space weathering trends. J. Geophys. Res. 116, E02002. http://dx.doi.org/ 10.1029/2010JE003656. Braden, S.E., Robinson, M.S., 2013. Relative rates of optical maturation of regolith on Mercury and the Moon. J. Geophys. Res.: Planets 118,1903-1914. http://dx.doi. org/10.002/jgre20143. Cheng, A.F., Domingue, D.L., 2000. Radiative transfer models for light scattering

from planetary surfaces. J. Geophys. Res. 105, 9477-9482. Cheng, A.F. et al., 2002. Ponded deposits on Asteroid 433 Eros. Meteorit. Planet. Sci. 37, 1095-1105.

Ciarniello, M. et al., 2011. Hapke modeling of Rhea surface properties through

Cassini-VIMS spectra. Icarus 214 (2), 541-555. Ciarniello, M. et al., 2014. A test of Hapke's model by means of Monte Carlo ray-

tracing. Icarus 237, 293-305. Cintala, M.J., 1992. Impact-induced thermal effects in the lunar and mercurian

regoliths. J. Geophys. Res. 97, 947-973. Clark, B.E. et al., 1999. NEAR photometry of Asteroid 253 Mathilde. Icarus 140,53-65. Clark, B.E. et al., 2002. NEAR infrared spectrometer photometry of Asteroid 433 Eros.

Icarus 155 (1), 189-204. Cord, A.M. et al., 2003. Planetary regolith surface analogs: Optimized determination of Hapke parameters using multi-angular spectro-imaging laboratory data. Icarus 165, 414-427. Denevi, B.W., Robinson, M.R., 2008. Mercury's albedo from Mariner 10: Implications

of the presence of ferrous iron. Icarus 197 (1), 239-246. Domingue, D., Hapke, B., 1989. Fitting theoretical photometric functions to asteroid

phase curves. Icarus 78, 330-336. Domingue, D., Hapke, B., 1992. Disk-resolved photometric analysis of European

terrains. Icarus 99 (1), 70-81. Domingue, D., Verbiscer, A., 1997. Re-analysis of the solar phase curves of the icy

Galilean satellites. Icarus 128 (1), 49-74. Domingue, D.L. et al., 1991. Europa's phase curve - Implications for surface

structure. Icarus 90, 30-42. Domingue, D.L., Lockwood, G.W., Thompson, D.T., 1995. Surface textural properties of icy satellites: A comparison between Europa and Rhea. Icarus 115 (2), 228-249. Domingue, D. et al., 2002. Disk-integrated photometry of 433 Eros. Icarus 155,205219.

Domingue, D.L. et al., 2010. Whole-disk spectrophotometric properties of Mercury: Synthesis of MESSENGER and ground-based observations. Icarus 209,101-124. http://dx.doi.org/10.1016/jicarus.2010.02.022. Domingue, D.L. et al., 2011. Photometric correction of Mercury's global color

mosaic. Planet. Space Sci. 59,1873-1887. Domingue, D.L. et al., 2014. Mercury's weather-beaten surface: Understanding Mercury in the context of lunar and asteroidal space weathering studies. Space Sci. Rev. 181, 121-214. Domingue, D.L. et al., 2015. Mercury's global color mosaic: An update from

messenger's orbital observations. Icarus 257, 477-488. Ernst, C.M. et al., 2015. Evaluating small body landing hazards due to blocks. Lunar Planet. Sci. 46, 2095.

Fraeman, A.A. et al., 2012. Analysis of disk-resolved OMEGA and CRISM spectral observations of Phobos and Deimos. J. Geophys. Res. 117 CiteID E00J15.

Gillis-Davis, J.J., Lucey, P.G., Hawke, B. Ray, 2006. Testing the relation between UV-vis color and TiO2 content of the lunar maria. Geochim. Cosmochim. Acta 70, 6074-6102.

Goguen, J.D. et al., 2010. A new look at photometry of the Moon. Icarus 208, 548557.

Gunderson, K., Thomas, N., Whitby, J.A., 2005. First measurements with the Physikalisches Institut Radiometric Experiment (PHIRE). Planet. Space Sci. 54, 1046-1056.

Hapke, B., 1981. Bidirectional reflectance spectroscopy. 1. Theory. J. Geophys. Res. 68, 4571-4586.

Hapke, B., 1984. Bidirectional reflectance spectroscopy 3. Correction for macroscopic roughness. Icarus 59, 41-59.

Hapke, B., 1986. Bidirectional reflectance spectroscopy 4. The extinction coefficient and the opposition effect. Icarus 67, 264-280.

Hapke, B., 1993. Theory of Reflectance and Emittance Spectroscopy. Cambridge University Press, NY, 455pp.

Hapke, B., 2001. Space weathering from Mercury to the asteroid belt. J. Geophys. Res. 10 (E5), 10039-10073.

Hapke, B., 2002. Bidirectional reflectance spectroscopy. 5. The coherent backscatter opposition effect and anisotropic scattering. Icarus 157, 523-534.

Hapke, B., 2008. Bidirectional reflectance spectroscopy. 6. Effects of porosity. Icarus 195, 918-926.

Hapke, B., 2012a. Theory of Reflectance and Emittance Spectroscopy, second ed. Cambridge University Press, NY, p. 513pp.

Hapke, B., 2012b. Bidirectional reflectance spectroscopy 7. The single particle phase function hockey stick relation. Icarus 221,1079-1083.

Hapke, B., Van Horn, H., 1963. Photometric studies of complex surfaces with applications to the Moon. J. Geophys. Res. 68, 4545-4570.

Hapke, B. et al., 2009. A quantitative test of the ability of models based on the equation of radiative transfer to predict the bidirectional reflectance of a well-characterized medium. Icarus 199, 210-218.

Hartman, B., Domingue, D., 1998. Scattering of light by individual particles and the implications for models of planetary surfaces. Icarus 131, 421-448.

Hash, C., 2008. MESSENGER MDIS Calibrated (CDR) Data E/V/H V1.0. NASA Planetary Data System.

Hawkins III, S.E. et al., 2007. The Mercury Dual Imaging System on the MESSENGER spacecraft. Space Sci. Rev. 131, 247-338.

Hawkins III, S.E. et al., 2009. In-flight performance of MESSENGER's Mercury Dual Imaging System. In: Hoover, B. et al. (Eds.), Instruments and Methods for Astrobiology and Planetary Missions XII. SPIE Proceedings 7441, Paper 74410Z. SPIE, Bellingham, WA, 12pp.

Helfenstein, P., Shepard, M.K., 1999. Submillimeter-scale topography of the lunar regolith. Icarus 141,107-131.

Helfenstein, P., Shepard, M.K., 2011. Testing the Hapke photometric model: Improved inversion and the porosity correction. Icarus 215, 83-100.

Helfenstein, P., Veverka, J., 1987. Photometric properties of lunar terrans derived from Hapke's equation. Icarus 72, 342-357.

Helfenstein, P., Veverka, J., 1989. Physical characterization of asteroid surfaces from photometric analysis. In: Binzel, R., Gehrels, T., Matthews, M.S. (Eds.), Asteroids II. Univ. of Arizona Press, Tucson, pp. 557-593.

Helfenstein, P., Veverka, J., Thomas, P.C., 1988. Uranus satellites - Hapke parameters from Voyager disk-integrated photometry. Icarus 74, 231-239.

Helfenstein, P. et al., 1991. Oberon - Color photometry from Voyager and its geological implications. Icarus 90, 14-29.

Helfenstein, P. et al., 1994. Galileo photometry of Asteroid 951 Gaspra. Icarus 107,37-60.

Helfenstein, P. et al., 1996. Galileo photometry of Asteroid 243 Ida. Icarus 120 (1), 4865.

Hendrix, A.R., Domingue, D.L., King, K., 2005. The icy Galilean satellites: Ultraviolet phase curve analysis. Icarus 173 (1), 29-49.

Hillier, J. et al., 1994. Photometric diversity of terrains on Triton. Icarus 109 (2), 296312.

Hillier, J.K., Bauer, J.M., Buratti, B.J., 2011. Photometric modeling of Asteroid 5535 Annefrank form Stardust observations. Icarus 221 (1), 546-552.

Kaasalainen, M., Torppa, J., Muinonen, K., 2001. Optimization methods for asteroid lightcurve inversion. Icarus 153, 37-51.

Kamei, A., Nakamura, A.M., 2002. Laboratory study of the bidirectional reflectance of powdered surfaces: On the asymmetry parameter of asteroid photometric data. Icarus 156, 551-561.

Keller, M.R. et al., 2013. Time-dependent calibration of MESSENGER's wide-angle camera following a contamination event. Lunar Planet. Sci. 44. Abstract 2489.

Lederer, S.M. et al., 2005. Physical characteristics of Hayabusa target Asteroid 25143 Itokawa. Icarus 173 (1), 153-165.

Lederer, S.L. et al., 2008. The 2004 Las Campanas/Lowell Observatory campaign II. Surface properties of Hayabusa target Asteroid 25143 Itokawa inferred from Hapke modeling. Earth Planets Space 60, 49-59.

Li, J.-Y., A'Hearn, M.F., McFadden, L.A., 2004. Photometric analysis of Eros from NEAR data. Icarus 172 (2), 415-431.

Li, J.-Y. et al., 2013. Global photometric properties of Asteroid (4) Vesta observed with Dawn Framing Camera. Icarus 226,1252-1274.

Lucey, P.G., Taylor, G.J., Malaret, E., 1995. Abundance and distribution of iron on the Moon. Science 268,1150-1153.

Lucey, P.G., Blewett, D.T., Hawke, B.R., 1998. Mapping the FeO and TiO2 content of the lunar surface with multispectral imaging. J. Geophys. Res. 103 (E2), 3679-3699.

Lucey, P.G., Blewett, D.T., Jollif, B.L., 2000. Lunar iron and titanium abundance algorithms based on final processing of Clementine UVVIS data. J. Geophys. Res. 105 (E8), 20297-20306.

Mallama, A., Wang, D., Howard, R.A., 2002. Photometry of Mercury from SOHO/ LASCO and Earth: The phase function from 2 to 170°. Icarus 155, 253-264.

Maoumzadeh, N. et al., 2015. Photometric analysis of Asteroid (21) Lutetia from Rosetta-OSIRIS images. Icarus 257, 239-250.

McEwen, A.S., 1991. Photometric functions for photoclinometry and other applications. Icarus 92, 298-311.

McEwen, A.S., 1996. A precise lunar photometric function. Lunar Planet. Sci. 27, 841-842.

McEwen, A.S. et al., 1998. Summary of radiometric calibration and photometric normalization steps for the Clementine UVVIS images. Lunar Planet. Sci. 29.

McGuire, A., Hapke, B., 1995. An experimental study of light scattering by large irregular particles. Icarus 113,134-155.

McKay, D.S., Fruland, R.M., Heiken, G.H., 1974. Grain size and the evolution of lunar soils. Proceedings of the Fifth Lunar Conference (supplement 5, Geochim. Cosmochim. Acta) 1, pp. 887-906.

Mishchenko, M.I. et al., 1999. Bidirectional reflectance of flat, optically thick particulate layers: An efficient radiative transfer solution and applications to snow and soil surfaces. J. Quant. Spectrosc. Radiat. Trans. 63, 409-432.

Murchie, S. et al., 2002. Color variations on Eros from NEAR multispectral imaging. Icarus 155, 145-168.

Murchie, S.L. et al., 2015. Orbital multispectral mapping of Mercury with the MESSENGER Mercury Dual Imaging System: Evidence for the origins of plains units and low-reflectance material. Icarus 254, 287-305.

Pieters, C.M. et al., 2012. Distinctive space weathering on Vesta from regolith mixing processes. Nature 491 (November), 79-82.

Robinson, M.S. et al., 2001. The nature of ponded deposits on Eros. Nature 413 (6854), 396-400.

Robinson, M.S. et al., 2008. Reflectance and color variations on Mercury: Regolith processes and compositional heterogeneity. Science 321, 66-69.

Sato, H. et al., 2014. Resolved Hapke parameter maps of the Moon. J. Geophys. Res.: Planets 119 (8), 1775-1805.

Schröder, S.E. et al., 2013. Resolved photometry of Vesta reveals physical properties of crater regolith. Planet. Space Sci. 85, 198-213.

Shepard, M.K., Campbell, R., 1998. Shadows on a planetary surface and implications for photometric roughness. Icarus 134, 279-291.

Shepard, M.K., Helfenstein, P., 2007. A test of the Hapke photometric model. J. Geophys. Res. 112, E03001. http://dx.doi.org/10.1029/2005JE002625.

Shkuratov, Y.G., 1983. A model of the opposition effect in the brightness of airless cosmic bodies. Sov. Astron. 27 (5), 581-583.

Shkuratov, Y., 1988. Diffraction model of the brightness surge of complex structure surfaces. Kinem. Phys. Celest. Bodies 4, 33-39.

Shkuratov, Y.G., Helfenstein, P., 2001a. Shadow-hiding and coherent-backscatter opposition effects and the quasi-fractal structure of regolith: I. Theory. Icarus 152, 96-116.

Shkuratov, Y.G., Helfenstein, P., 2001b. The opposition effect and the quasi-fractal structure of regolith: I. Theory. Icarus 152 (1), 96-116.

Shkuratov, Y. et al., 1994. Principle of perturbation invariance in photometry of atmosphereless celestial bodies. Icarus 109,168-190.

Shkuratov, Yu.G., Petrov, D.V., Videnn, G., 2003. Classical photometry of pre-fractal surfaces. J. Opt. Soc. Am. 20 (11), 2081-2092.

Shkuratov, Y.G. et al., 2005. Interpreting photometry of regolith-like surfaces with different topographies: Shadowing and multiple scattering. Icarus 173, 3-15.

Shkuratov, Y. et al., 2007. Photometry and polarimetry of particulate surfaces and aerosol particles over a wide range of phase angles. J. Quant. Spectrosc. Radiat. Trans. 106, 487-508.

Shkuratov, Y. et al., 2011. Optical measurements of the Moon as a tool to study its surface. Planet. Space Sci. 59, 1326-1371.

Simonelli, D.P., Veverka, J., 1986. Phase curves of materials on Io - Interpretation in terms of Hapke's function. Icarus 68, 503-521.

Simonelli, D.P. et al., 1998. Photometric properties of PHOBOS surface materials from Viking images. Icarus 131 (1), 52-77.

Skypeck, A. et al., 1991. The photometric roughness of Ariel is not unusual. Icarus 90, 181-183.

Souchon, A.L. et al., 2011. An experimental study of Hapke's modeling of natural granular surface samples. Icarus 215, 313-331.

Spjuth, S. et al., 2012. Disk-resolved photometry of Asteroid (2867) Steins. Icarus 221 (2), 1101-1118.

Staid, M.I., Pieters, C.M., 2000. Integrated spectral analysis of mare soils and craters: Applications to eastern nearside basalts. Icarus 45, 122-139.

Thomas, P.C. et al., 1996. The surface of Deimos: Contribution of materials and processes to its unique appearance. Icarus 123, 536-556.

Trowbridge, T.S., 1984. Rough-surface retroreflection by focusing and shadowing below a randomly undulating interface. J. Opt. Soc. Am. A 1 (10), 1019-1027.

Velikodsky, Y.I. et al., 2011. New Earth-based absolute photometry of the Moon. Icarus 214, 30-45.

Verbiscer, A., Veverka, J., 1992. Mimas - Photometric roughness and albedo map. Icarus 99 (1), 63-69.

Verbiscer, A.J., Veverka, J., 1994. A photometric study of Enceladus. Icarus 110 (1), 155-164.

Verbiscer, A.J., French, R.G., McGhee, C.A., 2005. The opposition surge of Enceladus: HST observations 338-1022 nm. Icarus 173 (1), 66-83.

Wu, Y. et al., 2013. Photometric correction and in-flight calibration of Chang' E-1 Interference Imaging Spectrometer (IIM) data. Icarus 222, 283-295.

Zhang, H., Voss, K.J., 2011. On Hapke photometric model predictions on reflectance of closely packed particulate surface. Icarus 215, 27-33.