  Adaptive optics (AO) is a technology in modern ground-based optical
telescopes to compensate the wavefront distortions caused by atmospheric
turbulence. One method that allows to retrieve information about the atmosphere
from telescope data is so-called SLODAR, where the atmospheric turbulence
profile is estimated based on correlation data of Shack--Hartmann wavefront
measurements. This approach relies on a layered Kolmogorov turbulence model. In
this article, we propose a novel extension of the SLODAR concept by including a
general non-Kolmogorov turbulence layer close to the ground with an unknown
power spectral density. We prove that the joint estimation problem of the
turbulence profile above ground simultaneously with the unknown power spectral
density at the ground is ill-posed and propose three numerical reconstruction
methods. We demonstrate by numerical simulations that our methods lead to
substantial improvements in the turbulence profile reconstruction, compared to
standard SLODAR-type approach. Also, our methods can accurately locate local
perturbations in non-Kolmogorov power spectral densities.
