Echoing yesterday's use of a target to evaluate a turbulence field in (CS / Imaging With Nature: Turbulence Aided Lucky Imaging), one element of importance is to understand the type of sparsity that can be found in that field. If sparsity or approximate sparsity is not there, then compressive sensing can be of no use to determine the transfer function of interest. As it so happens, turbulence, a much studied subject , clearly show some sparsity in computations:
The different eddies follow a power law first discovered by Kolmogorov.
With the advent of wavelets, there have been several studies at using wavelets to study and compute geophysical flows. The point being that a turbulence field can be found to be sparse using the right basis and is therefore clearly a phenomena that can be probed using the set up mentioned yesterday. Let me mention one more thing, reference  is focused on getting better images through turbulence whereas the purpose of this set-up is to image the turbulence itself (think cheap microburst detection on airplanes, or monitoring of large swatch of the atmosphere) while playing with the target being imaged.
Good reads include  and .
 Turbulence in nature and in the laboratory, Z. Warhaft
 Imaging through turbulence using compressive coherence sensing by Ashwin Wagadarikar, Daniel L. Marks, Kerkil Choi, Ryoichi Horisaki, and David Brady. / Ashwin Wagadarikar's thesis entitled: Compressive Spectral and Coherence Imaging. Turbulence, and Fluid Mechanics in General, Cosma Shalizi
 The Nobel Prize Winner as Neglected Genius