A global surface mode can exist on a ''surface'' which is not a true discontinuity, but without sufficient dissipation it is not a normal mode and it decays in time via resonance absorption. The decay rate can be calculated analytically when the ''surface'' is thin. The goal of this paper is to numerically estimate the decay rate when the surface is not thin. We consider a cold plasma in a uniform magnetic field, and we take the density to vary linearly across the ''surface.'' In our linearized calculation, the global surface mode is driven in steady state by an antenna located in one of the uniform regions external to the surface. The frequency is a free parameter, and resonance curves are computed numerically without approximations. The widths of the resonance curves are used to estimate the free decay times of undriven surface modes, via the uncertainty principle, equation (1). This procedure agrees with the analytical results for a thin surface. The thin surface results are found to break down when kT≈0.3, where T is the surface thickness and k is the wave number along the surface. When the angle between k and B0 exceeds about 40 degrees, the decay rates show distinct maxima at kT≈0.5--1.0. When applied to the active solar corona, the decay rates are large enough to account for the coronal heating, but is should be kept in mind that the role of nonlinearity in resonance absorption is still undetermined. ¿ American Geophysical Union 1990