Topographic Power Laws and Shape Model Estimation

Abstract:

Estimation of the shape (topography) of the rocky bodies of the Solar System is essential to the understanding of crustal composition, internal structure, and thermal evolution. Topography obeys an empirical power law as a function of spherical harmonic degree n, whereby the degree variance V(n) is roughly proportional to 1/(n(n+1)). First observed for the Earth, this law has been documented for Mars, Mercury, Moon and Venus. This law is similar to Kaula’s law for gravity potential, but with the caveat that gravity predicted from topography must also consider density variations and partial compensation by relief on the crust-mantle interface. Neither power law can be rigorously deduced but the latter is often used a priori as a constraint for regularization of the problem of estimation of coefficients.

With the resolution afforded by laser altimetry and satellite-satellite tracking over airless bodies it is feasible to examine gravity and topography to degree 100 and beyond with confidence. However, in the case of topography obtained from sparse occultation measurements, radar ranging, and/or flyby altimetry, even to achieve estimates of low-degree coefficients it is necessary to expand solutions to a maximum degree that the data do not support on a global basis. As a result, the use of a prior constraint is required to avoid the instability of least-squares methods. An example from the occultation measurements of the southern hemisphere of Mercury combined with the northern hemisphere altimetric measurements by the Mercury Laser Altimeter illustrates how the spectral power behavior is a critical constraint on the solution and its error estimation. A simulation of the topography that could be obtained by the Europa Flyby Mission with a laser altimeter demonstrates the ability to extract critical shape parameters from limited observations.