  Modern ground-based telescopes rely on a technology called adaptive optics
(AO) in order to compensate for the loss of image quality caused by atmospheric
turbulence. Next-generation AO systems designed for a wide field of view
require a stable and high-resolution reconstruction of the refractive index
fluctuations in the atmosphere. By introducing a novel Bayesian method, we
address the problem of estimating an atmospheric turbulence strength profile
and reconstructing the refractive index fluctuations simultaneously, where we
only use wavefront measurements of incoming light from guide stars. Most
importantly, we demonstrate how this method can be used for model optimization
as well. We propose two different algorithms for solving the maximum a
posteriori estimate: the first approach is based on alternating minimization
and has the advantage of integrability into existing atmospheric tomography
methods. In the second approach, we formulate a convex non-differentiable
optimization problem, which is solved by an iterative thresholding method. This
approach clearly illustrates the underlying sparsity-enforcing mechanism for
the strength profile. By introducing a tuning/regularization parameter, an
automated model reduction of the layer structure of the atmosphere is achieved.
Using numerical simulations, we demonstrate the performance of our method in
practice.
