Photometry of the Martian Atmosphere: An Improved Practical Model for Cartography and Photoclinometry

Introduction: This abstract describes recent work on approximate modeling of radiative transfer in the martian atmosphere with applications to the photometric normalization of images for cartographic products and for quantifying and improving the accuracy of topographic models obtained by photoclinometry, also known as shape-from-shading. One year ago, we described [1] a major effort to improve the photometric modeling and correction software in the USGS Integrated Software for Imagers and Spectrometers (ISIS) system. Motivated by the desire to make photometrically corrected image mosaics of Mars [2, 3] and the Galilean satellites [4, 5], we created a greatly enhanced new program called PHOTOMET to do photometric normalization of images in which we • greatly increased the variety of surface scattering models • implemented more realistic yet reasonably efficient models of atmospheric scattering • allowed normalization of albedo variations, topographic shading, or a mixture (i.e., normalization of albedo at low incidence angles blending smoothly to normalization of topography near the terminator) • modularized the software so photometric functions, atmospheric scattering models, and normalization options could be combined freely In addition, we provided a tool called SHADOW_TAU for estimating the atmospheric optical depth in an image from the calibrated reflectance values of a shadow and of a nearby level region. The normal albedo of the shadow and level areas is assumed to be the same and is estimated simultaneously with optical depth. The results are model-dependent, but SHADOW_TAU uses the same model as PHOTOMET so, the estimated optical depth is appropriate for use in the latter program. On the whole, our new software yielded substantial improvements in the photometric normalization of images of airless bodies [5] but the results for Mars, on which atmospheric scattering of light contributes significantly to the images, were less successful. The albedo and optical depth values we obtained from shadow measurements appeared reasonable individually, but examination of the results for several hundred Viking Orbiter images covering the entire planet revealed that albedo and optical depth were each correlated with both the incidence and phase angles [1]. Applying the software (and optical depth values) to normalize the images of the global mosaic made it plain that this was not the result of a sampling effect, but was caused by a shortcoming in the models. Adjacent images with different illumination (in particular, with large or small versus intermediate phase angles) did not match in brightness and contrast after normalization. To produce a cosmetically acceptable mosaic, we resorted to a least-squares adjustment of brightness and contrast of each image sequence to minimize variations between sequences [2]. An obvious candidate for the weakness of our atmospheric model was its failure to take into account the extremely forwardscattering character of martian atmospheric particles [6, 7, 8]. The analytic scattering model, which is described below, has been developed only for isotropic and weakly anisotropic (one-term Legendre polynomial for the single-particle phase function) scattering. As we pointed out [1], forward scattering would result in increased scattering onto the surface at low phase angles and increased scattering to the camera at large phase angles, corresponding to what our model interprets as larger albedos in these situations. We have therefore extended the analytic model by adding an approximate correction for arbitrary phase function in the singly scattered light only. Analytic Atmospheric Scattering Model: Our models attempt to balance the need for a relatively simple, fast calculation of the additive and multiplicative effects of atmospheric scattering with the desire for physical realism. We use Chandrasekhar’s [9] solution of the “planetary problem” of a slab atmosphere over a reflective surface (§72). The derivation strictly applies only to a Lambert surface because it takes account of the fraction of incident photons reflected from the surface but not their angular distribution. We therefore approximate the actual surface photometric model by numerically calculating the albedo of an “equivalent Lambert model. ” For the term involving surface illumination of sunlight, we use the hemispheric albedo Ah, which is averaged over emission angle, whereas for surface illumination by light from the atmosphere we use the bihemispheric albedo Ab, which is averaged over both incidence and emission angles. In addition, we correct the overall result by assuming that the light that passes through atmosphere completely unscattered is reflected according to the detailed surface photometric model rather than by Lambert’s (We do not apply the detailed surface law to light scattered in the atmosphere before hitting the surface; this would require calculation of and averaging over the detailed angular distribution of sky light, which would be prohibitively slow. This effect will be unimportant if the surface is nearly Lambertian or if the sky light is only weakly directional.) Our atmospheric scattering model can be written in the form