  Spin-dependent scattering from magnetic impurities inside a superconductor
gives rise to Yu-Shiba-Rusinov (YSR) states within the superconducting gap. As
such, YSR states have been very successfully modeled with an effective
scattering potential (Kondo impurity model). Using a scanning tunneling
microscope, we exploit the proximity of the tip to a surface impurity and its
influence on the YSR state to make a quantitative connection between the YSR
state energy and the impurity-substrate hybridization. We corroborate the
coupling between impurity and substrate as a key energy scale for surface
derived YSR states using the Anderson impurity model in the mean field
approximation, which accurately explains our observations. The model allows to
decide on which side of the quantum phase transition the system resides based
on additional conductance measurements. We propose that the Anderson impurity
model is much more appropriate to describe YSR states from impurities on a
superconducting surface than the Kondo impurity model, which is more
appropriate for impurities inside a superconductor. We thus provide a first
step towards a more quantitative comparison of experimental data with fully
correlated calculations based on the Anderson impurity model.
