  The space-borne missions have provided us with a wealth of high-quality
observational data that allows for seismic inferences of stellar interiors.
This requires the computation of precise and accurate theoretical frequencies,
but imperfect modeling of the uppermost stellar layers introduces systematic
errors. To overcome this problem, an empirical correction has been introduced
by Kjeldsen et al. (2008, ApJ, 683, L175) and is now commonly used for seismic
inferences. Nevertheless, we still lack a physical justification allowing for
the quantification of the surface-effect corrections. We used a grid of these
simulations computed with the CO$^5$BOLD code to model the outer layers of
solar-like stars. Upper layers of the corresponding 1D standard models were
then replaced by the layers obtained from the horizontally averaged 3D models.
The frequency differences between these patched models and the 1D standard
models were then calculated using the adiabatic approximation and allowed us to
constrain the Kjeldsen et al. power law, as well as a Lorentzian formulation.
We find that the surface effects on modal frequencies depend significantly on
both the effective temperature and the surface gravity. We further provide the
variation in the parameters related to the surface-effect corrections using
their power law as well as a Lorentzian formulation. Scaling relations between
these parameters and the elevation (related to the Mach number) is also
provided. The Lorentzian formulation is shown to be more robust for the whole
frequency spectrum, while the power law is not suitable for the frequency
shifts in the frequency range above $\nu_{\rm max}$.
