The Fresnel volume and transmitted waves

TLDR: In this article, the authors introduce the physics of the Fresnel volume and present a solution of the wave equation that accounts for the band limitation of waves, which is a special case of the finite-frequency wave theory in the limit of infinite frequency.

Abstract: In seismic imaging experiments, it is common to use a geometric ray theory that is an asymptotic solution of the wave equation in the high-frequency limit. Consequently, it is assumed that waves propagate along infinitely narrow lines through space, called rays, that join the source and receiver. In reality, recorded waves have a finite-frequency content. The band limitation of waves implies that the propagation of waves is extended to a finite volume of space around the geometrical ray path. This volume is called the Fresnel volume. In this tutorial, we introduce the physics of the Fresnel volume and we present a solution of the wave equation that accounts for the band limitation of waves. The finite-frequency wave theory specifies sensitivity kernels that linearly relate the traveltime and amplitude of band-limited transmitted and reflected waves to slowness variations in the earth. The Fresnel zone and the finite-frequency sensitivity kernels are closely connected through the concept of constructive interference of waves. The finite-frequency wave theory leads to the counterintuitive result that a pointlike velocity perturbation placed on the geometric ray in three dimensions does not cause a perturbation of the phase of the wavefield. Also, it turns out that Fermat’s theorem in the context of geometric ray theory is a special case of the finite-frequency wave theory in the limit of infinite frequency. Last, we address the misconception that the width of the Fresnel volume limits the resolution in imaging experiments.