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The space-borne missions have provided us with a wealth of high-qualityobservational 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 systematicerrors. To overcome this problem, an empirical correction has been introducedby Kjeldsen et al. (2008, ApJ, 683, L175) and is now commonly used for seismicinferences. Nevertheless, we still lack a physical justification allowing forthe quantification of the surface-effect corrections. We used a grid of thesesimulations computed with the CO$^5$BOLD code to model the outer layers ofsolar-like stars. Upper layers of the corresponding 1D standard models werethen replaced by the layers obtained from the horizontally averaged 3D models.The frequency differences between these patched models and the 1D standardmodels were then calculated using the adiabatic approximation and allowed us toconstrain the Kjeldsen et al. power law, as well as a Lorentzian formulation.We find that the surface effects on modal frequencies depend significantly onboth the effective temperature and the surface gravity. We further provide thevariation in the parameters related to the surface-effect corrections usingtheir power law as well as a Lorentzian formulation. Scaling relations betweenthese parameters and the elevation (related to the Mach number) is alsoprovided. The Lorentzian formulation is shown to be more robust for the wholefrequency spectrum, while the power law is not suitable for the frequencyshifts in the frequency range above $\nu_{\rm max}$.

Surface-effect corrections for solar-like oscillations using 3D hydrodynamical simulations