  Atmospheric tomography, the problem of reconstructing atmospheric turbulence
profiles from wavefront sensor measurements, is an integral part of many
adaptive optics systems used for enhancing the image quality of ground-based
telescopes. Singular-value and frame decompositions of the underlying
atmospheric tomography operator can reveal useful analytical information on
this inverse problem, as well as serve as the basis of efficient numerical
reconstruction algorithms. In this paper, we extend existing singular value
decompositions to more realistic Sobolev settings including weighted inner
products, and derive an explicit representation of a frame-based (approximate)
solution operator. These investigations form the basis of efficient numerical
solution methods, which we analyze via numerical simulations for the
challenging, real-world Adaptive Optics system of the Extremely Large Telescope
using the entirely MATLAB-based simulation tool MOST.
