The complete radiation field pattern of a vertical Hertzian dipole antenna on or above a lossless or low-loss dielectric half-space is studied using a rigorous Sommerfeld formalism. The reflected fields in the air above the interface and the subsurface fields transmitted into the dielectric are computed by numerical integration of the Sommerfeld integrals. Furthermore, to facilitate the physical interpretation of these results, a detailed asymptotic saddle-point integration method analysis is presented, which includes terms that vary in magnitude with the second power of the inverse distance from the dipole. It is shown that the second-order field constituents are dominant at the interface, where the first-order geometrical fields vanish. These second-order terms comprise an evanescent wave propagating along the interface in the upper half-space and a lateral wave, also known as the head wave, which propagates in the subsurface along the direction of the critical angle. The two waves only exist between two cones whose half-angles are equal to the critical angle, and their interference with the geometrical-optics fields determines the radiation pattern for elevation angles near the horizon. The far zone surface fields on either side of the interface comprise two second order waves that propagate along the interface, one with the phase velocity in the air, and the other with the phase velocity in the dielectric. Away from the interface, the leading field components vary with the first power of the inverse distance, which explains the sharp dip in the field pattern at the interface-a phenomenon known as the interface pattern extinction. Another distinctive phenomenon, observed in the subsurface field pattern, is the rippling that occurs in the angular range between the critical angle cone and the interface. The asymptotic analysis has shown that this pattern scalloping results from the interference of the lateral wave with the geometrical-optics spherical wave.