phoempglobal

    This program finds lunar-Lambert or Minnaert photometric functions to 
    approximate a more realistic but more complex Hapke (1981; 1984; 1986) 
    model.  At each of a range of phase angles, the simpler model is fit to 
    the Hapke model by adjusting its one parameter and its overall brightness 
    so that the sum-squared-residual between the two is minimized. Both the 
    parameter (which, for both types of simple model, mainly controls limb 
    darkening) and the brightness (normalized as an empirical phase curve) 
    are reported in a table against phase angle. 

    In this program, the fit is over a portion of the visible hemisphere of
    an idealized spherical and uniform planet, and a table of results vs phase 
    angle is output.  This type of fit is useful for building an empirical
    photometric function to be used in program photomet for normalizing
    images for mosaicking.  The companion program phoemplocal can be used
    to perform a single fit at fixed incidence/emission/phase geometry
    to determine the photometric parameter for photoclinometry with a 
    particular image. This program, phoempglobal, can also be used to
    support photoclinometry, by producing a table of fits at multiple phase
    angles that can be interpolated to give the parameters for use with
    any given image. In this application, the parameter EMAMAX should be
    set to a relatively small value that represents the typical range of
    surface slopes, and the fit will apply to images with vertical viewing.

    Because the program photomet (used to make photometric corrections to
    images) incorporates all the same atmospheric scattering models as
    phoempglobal, one would normally set ATMNAME=NONE and ADDOFFSET=NO to 
    obtain an empirical model for the surface alone, then apply the 
    atmospheric scattering parameters in photomet. Fitting with an 
    atmospheric model and ADDOFFSET=YES in phoempglobal is more useful for 
    application to photoclinometry where images are normally corrected by 
    subtracting a uniform haze estimate rather than by applying a full 
    atmospheric scattering model.

    For the original description of the fitting process and a useful 
    compilation of Hapke parameters from the scientific literature, see McEwen 
    (1991). The atmospheric model used in the fits is discussed by Kirk et al. 
    (2000, 2001).

    Example:  Mars

    The following Hapke parameters for Mars are from Johnson
    et al. (1999) for IMP data of Photometry Flats (soil) and may be 
    reasonably representative of Mars as a whole. Note that
    (HG1, HG2=1.0) is equivalent to (-HG1, HG2=0.0) 

    Band    WH     B0     HH    HG1    HG2 
    Red    0.52  0.025  0.170  0.213 1.000 
    Green  0.29  0.290  0.170  0.190 1.000 
    Blue   0.16  0.995  0.170  0.145 1.000 

    Kirk et al. (2000) found that Mars whole-disk limb-darkening data of 
    Thorpe (1973) are consistent with THETA=30, but results of Tanaka and 
    Davis (1988) based on matching photoclinometry of local areas to shadow 
    data are more consistent with THETA=20 when the domain of the fit is 
    restricted to small emission angles (=< 20 degrees).

    Values of the photometric parameters for the martian atmosphere, adopted 
    from Tomasko et al. (1999) are:
 
    Band  WHA     HGA 
    Red    0.95     0.68 
    Blue   0.76     0.78 

    Hapke, B. W., 1981. Bidirectional reflectance spectroscopy 
    1: Theory. J. Geophys. Res., pp. 86,3039-3054.

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

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

    Johnson, J. R., et al., 1999, Preliminary Results on Photometric 
    Properties of Materials at the Sagan Memorial Station, Mars, 
    J. Geophys. Res., 104, 8809. 

    Kirk, R. L., Thompson, K. T., Becker, T. L., and Lee, E. M., 
    2000. Photometric modelling for planetary cartography. 
    Lunar Planet. Sci., XXXI, Abstract #2025, Lunar 
    and Planetary Institute, Houston (CD-ROM).

    Kirk, R. L., Thompson, K. T., and Lee, E. M., 2001. 
    Photometry of the martian atmosphere:  An improved 
    practical model for cartography and photoclinometry. 
    Lunar Planet. Sci., XXXII, Abstract #1874, Lunar 
    and Planetary Institute, Houston (CD-ROM). 

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

    Tanaka, K. L., and and Davis, P. A., 1988, Tectonic History of 
    the Syria Planum Provice of Mars, J. Geophys. Res., 93, 14,893. 

    Thorpe, T. E., 1973, Mariner 9 Photometric Observations of Mars 
    from November 1971 through March 1972, Icarus, 20, 482.

    Tomasko, M. G., et al., 1999, Properties of Dust in the Martian 
    Atmosphere from the Imager on Mars Pathfinder, J. Geophys. Res., 
    104, 8987

    Programmer: Randolph Kirk, U.S.G.S., Flagstaff, AZ