Previous methods developed for computation of the attenuation function for a homogeneous spherical earth are generalized for an arbitrary surface impedance. The power series and small-curvature formula developed for short distances can be used for such surfaces without modification. Care must be taken, however, to include properly the trapped surface wave that may arise in the formulation. In general, it is found that the inductive type of boundary enhances the total field but, if the resistive part is small, rather strong interference takes place between the trapped surface wave and the usual ground wave. These effects are illustrated for a surface-based vertical electric dipole for frequencies from 1 to 30 MHz.