Martian phase function: Modeling the visible to near-infrared surface photometric function using HST-WFPC2 data
Introduction
Accurate photometric characterization of a surface is required for a number of important applications in planetary science. For example, in order to accurately compare observations acquired under varying geometries and lighting conditions (e.g., data acquired from surface landers/rovers compared to data acquired from orbital platforms; Arvidson et al., 2004, Greeley et al., 2004, Poulet et al., 2004, Pinet et al., 2005) knowledge of the photometric properties of the surface are necessary. With the continued development of more complex atmospheric radiative transfer models, accurate surface photometric properties are becoming increasingly critical. For example, analysis of Mars Global Surveyor (MGS) data for atmospheric characterization explicitly requires assumptions about the scattering properties of the surface (e.g., albedo, phase function; Smith et al., 2001, Wolff et al., 1999, Wolff and Clancy, 2003). Photoclinometry studies of Mars (e.g., Herkenhoff and Murray, 1990, Kirk et al., 2003, Soderblom and Kirk, 2003) derive digital elevation models of the martian surface near the resolution of the imaging system using photometric properties of the surface.
Estimates of the visible to near infrared martian phase function date back to as far as 1865 by Zöllner (de Vaucouleurs, 1968). More recent photometric studies of the surface materials of Mars using telescopic data (e.g., de Vaucouleurs, 1968, O'Leary and Rea, 1968, Binder and Jones, 1972, Thorpe, 1982, de Grenier and Pinet, 1995, Vdovichenko et al., 1997, Bell III et al., 1999, Erard, 2000) and spacecraft data (e.g., Guinness, 1981, Arvidson et al., 1989, Johnson et al., 1999, Johnson et al., 2006) have yielded information regarding the photometric properties of the martian surface. For example, O'Leary and Rea (1968) reported a strong opposition surge, which varied inversely with albedo and moderate limb darkening. de Grenier and Pinet (1995) found a linear relationship between this limb darkening and albedo in the near infrared but found no such relationship in the visible. Those authors attributed at least some of this variation to increased atmospheric scattering contributions at the longer wavelengths. Bell et al. (1999) reported normal albedos and phase coefficients for various regions of interest and albedo units and found little correlation between phase coefficient and normal albedo at 1042 nm.
These properties can in turn be related to surface properties such as macroscopic surface roughness, particle size, and particle packing density. The angular width of the opposition effect has been shown to be related to surface porosity and the size distribution of the scattering particles (Helfenstein and Veverka, 1987; Hapke, 1986, Hapke, 1993). The magnitude of the opposition effect has been shown to be related to particle opacity (Hapke, 1986, Hapke, 1993, Domingue et al., 1997) and/or the composition and particle microstructure (Helfenstein et al., 1997). Macroscopic surface roughness is thought to represent the average surface facet tilts for all surfaces (e.g., Hapke, 1984, Hapke, 1993, Helfenstein and Shepard, 1999, Helfenstein and Shepard, 2003, Cord et al., 2004). The shape of the single-particle phase function has been shown to be related to the particle shape and internal scattering (McGuire and Hapke, 1995, Hartman et al., 1996). Finally single-particle scattering albedo has been shown to be related to composition (index of refraction) and the effective particle size (Hapke, 1981, Hapke, 1993, Guinness et al., 1997).
The goal of this work is to develop a simple yet accurate model of the martian surface photometric function over a wide range of phase angles applicable to telescopic, orbital, and landed surface imaging and spectroscopic observations. To this end the data presented in this paper were fit with five different photometric functions testing four different binning criteria. Section 2 discusses the data reduction and the various binning methods. The photometric functions considered are presented in Section 3. Sections 4 Results, 5 Conclusions and implications present and discuss the results of the various model fits.
Section snippets
Data
Data presented in this paper were acquired using the Hubble Space Telescope (HST) Wide Field/Planetary Camera 2 (WFPC2) instrument (Burrows, 1995) between 1996 and 2005 as part of a long-term observing campaign of Mars. Data were collected under HST General Observer (GO) programs 6852, 6741, 8391, 8152, 8577, 9268, and 9738, and Director's Discretionary (DD) programs 10065 and 10770 (see Table 1). While these data were acquired through many WFPC2 filters, this study utilizes data acquired
Photometric models
The goal of this work is to develop a simple yet accurate model of the martian surface photometric function over a wide range of phase angles applicable to telescopic, orbital, and landed surface imaging and spectroscopic observations. Thus, we considered four general classes of photometric functions when modeling our observations: Lambert, Minnaert, lunar–Lambert, and Hapke. These models describe the variations in as a function of the viewing geometry expressed as the cosine of the
Results
HST WFPC2 data acquired at 502, 673, 953, and 1042 nm were sorted into Viking bolometric albedo bins, TES bolometric albedo bins, TES thermal inertia bins, and historic geographic units (shown in Fig. 2, Fig. 3, Fig. 4, Fig. 5) and were fit to five photometric functions described in detail above. Fig. 10 shows the goodness of fit for fits to these scattering functions plotted against the different binning criteria. For all five scattering models the best fits were achieved by binning the
Conclusions and implications
We have modeled the scattering properties of the martian surface using a variety of photometric functions for a variety of surface types. Our preferred photometric model is the empirical lunar–Lambert function given in Eq. (7), fit to three surface regions defined by TES bolometric albedo as low-, moderate-, and high-albedo surfaces, and valid under periods of low atmospheric opacity () and for phase angles of . The modeled parameters for these three surface types, listed by
Acknowledgements
We thank the TES Team for calculating and publishing dust opacities and Eldar Noe Dobrea for his help in retrieving these data for each of our observation dates. We thank Jeffrey Johnson (USGS) and John Hillier (Grays Harbor College) for their valuable reviews of this manuscript. Funding for this research was provided by grants from the NASA Planetary Geology and Geophysics Program (NNG04G163G) and the Space Telescope Science Institute. This work is based on observations with the NASA/ESA
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Currently at Space Astrophysics Laboratory, Caltech, MS 405-47, Pasadena, CA 91125, USA.