Reflection of a video pulse from a caustic

The problem of radio sounding of a smoothly inhomogeneous medium with broadband pulses is considered. The need for non-destructive investigation of subsurface structures of the Earth has led to the construction of special type radar systems called Ground Penetration Radar. Most of these systems use a short pulse consisting of only a few oscillations as a probing signal. The presence of sharp boundaries of the medium properties (dielectric permittivity) in the medium under study leads to a mirror reflection of the signal, without distortion of the pulse shape, which greatly simplifies the interpretation of the probing results. However, in most cases, the wave propagation medium is smoothly inhomogeneous. In this case, the most common and adequate method for solving wave equations is based on the approximation of geometric optics. Quite often, a family of geometric-optical rays in a smoothly inhomogeneous medium has an envelope called a caustic. Strictly speaking, the geometric optics approximation is not applicable on the caustic, but there is an asymptotic representation of the wave field that is well consistent with this approximation. In this situation, the propagation of short pulses is fundamentally different from the propagation of monochromatic radiation. On the caustic, the phase of the wave loses a quarter of its period, which leads to a fundamental change in the Fourier spectrum of the wave radiation. As a result, the shape of the probing signal undergoes significant modifications upon reflection from the caustic. The pulse is blurred and does not have well-defined boundaries, which makes it difficult to interpret the results of the probe. In addition, the presence of a sharp gradient in the amplitude profile of the probe signal can lead to a significant increase in the amplitude of the reflected signal. If the probe signal has the most significant amplitude changes at the leading edge, the radiation delay time is determined by the first maximum in the amplitude of the reflected signal.