Spectral aspects of the Gaussian beam method: reflection from a homogeneous half-space

Abstract

Summary. Use of paraxially approximated Gaussian beams continues to be actively pursued for construction of synthetic seismograms in complicated environments. How to select the beams in the stack remains a source of difficulty which has primarily been addressed by semi-heuristic considerations. In this paper, the classical example of line-source field reflection from a homogeneous half-space that can sustain a head wave is examined from a plane-wave spectral point of view. The individual beam fields are modelled exactly by the complex source point technique, which emphasizes the complex spectral content of these wave objects. The quality of the paraxial approximation of a typical reflected (Gaussian) beam characterized by different parameters is examined from this perspective, and is compared with uniform and non-uniform asymptotics generated from the exact beam field spectral integral. With this information as background, the reflected field for a real line-source is synthesized by beam superposition. Except for the immediate vicinity of the critical reflection angle, the well-known failure of narrow paraxial beams, no matter how densely stacked, to reproduce the head wave effects is shown to be due to the inadequate spectral content of these beams and not to the failure of beam stacking per se. When the rigorous solutions are used for the narrow-waist beams, even relatively few suffice to yield agreement with the exact solution. This circumstance emphasizes the importance of fully understanding the spectral implications of various beam stacking schemes.