pho_emp_global Documentation

PHO_EMP_GLOBAL - Fit empirical photometric functions to Hapke
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 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 pho_emp_local 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.

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).

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

Output file contains fits

NONE

Single scattering albedo of
surface particle

0.52

Magnitude of opposition surge

0.0

Opposition surge width h

0.0

Macroscopic surface roughness

8.0

Type of single particle
phase function

HENYEY-GREENSTEIN

Henyey-Greenstein asymmetry
parameter for single particle
for PHASEFUNC=HENYEY-GREENSTEIN

0.213

2nd Henyey-Greenstein parameter
controls mix of +HG1, -HG1
components for PHASEFUNC=
HENYEY-GREENSTEIN

1.0

Coefficient of P1(cos(phase))
in single particle phase
function for PHASEFUNC=LEGENDRE

0.0

Coefficient of P2(cos(phase))
in single particle phase
function for PHASEFUNC=LEGENDRE

0.0

Include atmosphere in model?

NO

Normal atmospheric optical depth

0.5

Single-scattering albedo of
atmospheric particles.

0.9

Coeff of atmospheric particle
Henyey-Greenstein phase fn.

0.7

Atmospheric shell thickness
normalized to planet radius.
Default 0.003 is for Mars.

0.003

Order of approximation in
atmospheric scatter model

2ND

Type of photometric function
to fit (lunar-lambert, Minnaert)

LUNAR-LAMBERT

Allow additive offset in
fit?  (used with DOATM)

NO

Minimum incidence angle
for fit

0.0

Maximum incidence angle
for fit

89.0

Minimum emission angle
for fit

0.0

Maximum emission angle
for fit

70.0

Fraction of phase angle
to add to maximum
emission angle

0.111111111111

Minimum phase angle for
output table

0.0

Maximum phase angle for
output table

170.0

Number of phase angles

18

Note for output file

""

ADDITIONAL NOTES:

Parm	Description
TO	This output is an ASCII file containing a header recording all parameter values including the user NOTE, followed by a table with one row for each phase angle. Columns give the phase angle, best-fit limb darkening parameter, best-fit brightness both in absolute units and relative to the zero-phase model, and RMS residual to the fit.
WH	Single-scattering albedo of surface particles. See Hapke (1981). Not to be confused with albedo WHA of the atmospheric particles.
B0	Magnitude of the opposition effect for the surface. See Hapke (1984).
HH	Width parameter for the opposition effect for the surface See Hapke (1984).
THETA	"Macroscopic roughness" of the surface as it affects the photometric behavior. This is the RMS slope at scales larger than the distance photons penetrate the surface but smaller than a pixel. See Hapke (1986).
HG1	Asymmetry parameter used in the Henyey-Greenstein model for the scattering phase function of single particles in the surface, used if PHASEFUNC=HENYEY-GREENSTEIN. See Hapke (1981). The two-parameter Henyey-Greenstein function is P(phase) = (1-HG2) * (1-HG1**2)/(1+HG1**2+2*HG1*COS(PHASE))**1.5 + HG2 * (1-HG1**2)/(1+HG1**2-2*HG1*COS(PHASE))**1.5
HG2	Second parameter of the two-parameter Henyey-Greenstein model for the scattering phase function of single particles in the surface, used if PHASEFUNC=HENYEY-GREENSTEIN. This parameter controls a the proportions in a linear mixture of ordinary Heneyey- Greenstein phase functions with asymmetry parameters equal to +HG1 and -HG1. See HG1 for the full formula.
BH	When PHASEFUNC=LEGENDRE, a two-term Legendre polynomial is used for the scattering phase function of single particles in the surface P(phase) = 1 + BH * P1(COS(PHASE)) + CH * P2(COS(PHASE))
CH	When PHASEFUNC=LEGENDRE, a two-term Legendre polynomial is used for the scattering phase function of single particles in the surface P(phase) = 1 + BH * P1(COS(PHASE)) + CH * P2(COS(PHASE))
DOATM	If YES, an atmospheric scattering model will be applied in addition to the surface Hapke model. This atmospheric model uses a Hapke-like approach of combining an anisotropic model for multiple scattering with an isotropic model (one parameter Henyey-Greenstein) for single scattering. The atmospheric scattering both attenuates the surface signal and adds its own contribution to the radiance. If DOATM= YES it therefore makes sense to also set IORD=YES so that the additive contribution of the atmosphere will be modeled by an additive constant in the fit.
TAU	Normal optical depth of atmosphere.
WHA	Single-scattering albedo of atmospheric particles. Not to be confused with albedo WH of the surface particles.
HGA	Henyey-Greenestein asymmetry parameter for atmospheric particle phase function, Not to be confused with corresponding parameter HG1 for the surface particles.
IATM	Order of approximation used in the isotropic part of the Atmospheric scattering model. The second-order model (IORD=2) is slightly slower but more accurate and is preferred.
IEMP	Determines which empirical photometric function will be fitted to the Hapke model. Choices are Lunar-Lambert FUNC=B(phase) * ((1-L)*u0 + 2*L*u0/(u0+u)) Minnaert FUNC=B(phase) * u0**K(phase) * u**(K(phase)-1) The tables of B and L or B and K can be used with the lunar-Lambert empirical or Minnaert empirical photometric functions in the photometric normalization program photomet. Output from this program will have to be reformatted as described in photomet.pdf
IORD	If YES, the fit to the Hapke model will include not only the empirical photometric function (lunar-Lambert or Minnaert) but a constant offset at each phase angle. This term represents additive atmospheric scattering, so IORD=YES is intended to be used with DOATM=YES.
INCMIN	Minimum incidence angle included in the fit. The empirical photometric function will be fitted to the Hapke model over a portion of the visible hemisphere of an idealized planet, with INCMIN =< incidence angle =< INCMAX EMAMIN =< emission angle =< EMAMAX + EMAMAX_PCOEFF * phase INCMIN and EMAMIN are normally set to 0 and INCMAX and EMAMAX to values approaching 90 to exclude only limited regions near the limb and terminator from the fit. EMAMAX_PCOEFF allows the emission angle limit to increase slightly at high phase angles, because otherwise the region of fit becomes very small.
INCMAX	Maximum incidence angle included in the fit. See INCMIN for details. for fit
EMAMIN	Minimum emission angle included in the fit. See INCMIN for details. for fit
EMAMAX	Maximum emission angle included in the fit. See INCMIN for details.
EMAMAX_PCOEFF	EMAMAX_PCOEFF allows the range of emission angles included in the fit to increase slightly at high phase angles, because otherwise the region of fit becomes very small. See INCMIN for details.
PHMIN	Minimum phase angle at which a fit will be performed, corresponding to the first row of the output table.
PHMAX	Maximum phase angle at which a fit will be performed, corresponding to the last row of the output table.
NPH	Number of phase angles at which a fit will be performed, equal to the number of rows in the output table.
NOTE	User note that will be included in the header of the output file. Maximum 80 characters.