The reciprocity properties of geometrical spreading

Abstract

Reciprocity is an important property of acoustic and elastic waves. In this work it is explicity verified that acoustic waves also satisfy the reciprocity theorem in a ray-geometric approximation. This is achieved by deriving a reciprocity relation for the geometric spreading. The analysis is based on integrating the equations of dynamic ray tracing from the source to a receiver and in the reverse direction. It is shown that for a point source the geometric spreading for rays travelling in opposite directions differs by a factor depending on the velocities at the endpoints of the ray. This factor depends on the number of dimensions that one considers. Since the equations of kinematic and dynamic ray tracing are the same for elastic waves and acoustic waves, the derived reciprocity relations for the geometrical spreading hold for elastic waves as well. The results obtained are used to correct some errors in the derivation of an averaging theorem by Snieder & Lomax (1996).