  Here we describe a simple, efficient, and most importantly fully operational
point-spread-function(PSF)-reconstruction approach for laser-assisted ground
layer adaptive optics (GLAO) in the frame of the Multi Unit Spectroscopic
Explorer (MUSE) Wide Field Mode. Based on clear astrophysical requirements
derived by the MUSE team and using the functionality of the current ESO
Adaptive Optics Facility we aim to develop an operational PSF-reconstruction
(PSFR) algorithm and test it both in simulations and using on-sky data. The
PSFR approach is based on a Fourier description of the GLAO correction to which
the specific instrumental effects of MUSE Wide Field Mode (pixel size, internal
aberrations, etc.) have been added. It was first thoroughly validated with full
end-to-end simulations. Sensitivity to the main atmospheric and AO system
parameters was analysed and the code was re-optimised to account for the
sensitivity found. Finally, the optimised algorithm was tested and commissioned
using more than one year of on-sky MUSE data. We demonstrate with an on-sky
data analysis that our algorithm meets all the requirements imposed by the MUSE
scientists, namely an accuracy better than a few percent on the critical PSF
parameters including full width at half maximum and global PSF shape through
the kurtosis parameter of a Moffat function. The PSFR algorithm is publicly
available and is used routinely to assess the MUSE image quality for each
observation. It can be included in any post-processing activity which requires
knowledge of the PSF.
