Exact and approximate underground wavefront shape determination from an above ground GPR source

Waves radiated from a ground penetrating radar in air above soil refract at the planar ground surface, and change from a circular to a flattened shape resembling a hyperbola. It is useful to predict analytically the shape of this wavefront in the underground half space. The exact wavefront shape is derived, along with a hyperbolic approximation with specific parameters depending on the dielectric constant of the half space, ε´ (= n²), the height of the source, h, and p, the propagation time from source to wavefront multiplied by the wave velocity in the half space.