Although the observational data for both model fitting and spacecraft instrument comparisons are radiometric measurements, the development of coefficients and operation of the lunar irradiance model is done in dimensionless reflectance. This eliminates complications arising from the solar spectral structure, and allows interpolation of model results along the lunar reflectance spectrum, which is smooth, with only broad, shallow absorption features.
For development of the lunar irradiance model analytic form and determination of the model coefficients, the ROLO observational data are converted to disk-equivalent reflectance. The ROLO metadata table contains the computed sums of pixels on the lunar disk, including the unilluminated portion, in instrument units for each ROLO lunar image. These pixel sums IΣ are converted to exoatmospheric irradiances I′ by:
where CL is the radiance calibration coefficient, Cext is the extinction correction, and Ωp is the solid angle of one pixel. For both model development and spacecraft observation comparisons, irradiance values are corrected to standard distances for Sun-Moon (1 AU) and Moon-observer (384,400 km). For a band k, the distance-corrected irradiance Ik is given by:
where DM-V is the Moon-observer (viewer) distance in km and DS-M is the Sun-Moon distance in AU. The conversion to reflectance Ak is:
where ΩM is the solid angle of the Moon (6.4236x10^-5 sr) and Ek is the solar irradiance at the effective wavelength of band k, both at standard distances. This conversion involves a solar spectral irradiance model, which may have significant uncertainties in some wavelength regions. However, the direct dependence on solar model cancels to first order as long as the same model is used in going from irradiance to reflectance and back.
The reflectances Ak, along with the corresponding observational geometry parameters, are the quantities that populate the lunar model.
Model Form and Development of Coefficients
The analytic expression for the lunar disk-equivalent reflectance was developed empirically, choosing a form that minimized correlations among the fit residuals. Fitting is done in the natural logarithm of disk-equivalent reflectance Ak:
where g is the absolute phase angle, θ and φ are the selenographic latitude and longitude of the observer, and Φ is the selenographic longitude of the Sun.
The first polynomial represents the basic photometric function dependence upon phase angle, neglecting any opposition effect. The second polynomial approximates the asymmetry of the surface of the Moon that is illuminated, primarily the distribution of maria and highlands. The c-coefficient terms account for the face of the Moon that is actually observed (topocentric libration), with a consideration of how that is illuminated. The form of the last three terms, all non-linear in g, is strictly empirical. The first two represent the opposition effect and the last one simply addresses a correlation seen in the residuals.
The ROLO data selected for fitting are constrained to 1.55° < g < 97 °, and the requirement that all images used be part of complete 32-filter sequences. Data were weighted based on nightly observing conditions. Two iterations of a least-squares fitting process were applied using the above form, but with all the non-linear terms set to zero. After the first fit, data with residuals greater than 3 standard deviations of the residual average were removed. After the second fit, any point with residual >0.25 was removed. This process leaves about 1200 observations for each band. Then a fitting process that handles both the linear and non-linear terms in multiple steps was applied, resulting in 8 coefficients that are constant over wavelength (4 for libration and the 4 opposition effect parameters) and 10 additional coefficients for each band.
The coefficients for model version 311g (2004 April) have been published. The 320 wavelength-dependent coefficients are available in ASCII text format for personal use by interested scientific personnel; contact the Project Scientist, Tom Stone.
The mean absolute fit residual over all bands is 0.0096 in ln A, or about 1%. This is a measure of the model's capability for predicting the variation in irradiance due to geometric effects of phase and libration. Although the processed ROLO data span about 1/4 of the 18.6-year libration repeat cycle, libration coverage is sufficient for a satisfactory fit and predictive capability. The phase plot below shows the lunar model results (in red) and observational data (in white) for 1234 data points in the ROLO 555nm band. The phase range is 90 degrees before Full Moon to 90 degrees after. The model deviations from a smooth phase curve show the effects of libration. Model studies have determined that libration effects are responsible for up to 7% of the total irradiance signal over the full range of libration angles.
The lunar model version 311g produces reflectance spectra that exhibit band-to-band deviations not characteristic of the lunar reflectance, which is generally smooth. A correction was developed to adjust the model outputs to a form representing a mix of Apollo-16 sample laboratory spectra. These are shown in the plot below. The sample “mix” spectrum (95% soil, 5% breccia) is scaled to the model outputs (“ROLO”) and fitted with a linear function of wavelength.
1 Astronomical Journal 129, 2887-2901 (2005)
U.S. Department of the Interior | U.S. Geological Survey | U.S.G.S. Astrogeology
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