3-D crosswell transmissions: Paraxial ray solutions and reciprocity paradox

Abstract

Transmission of seismic waves through a 3-D earth model is of fundamental importance in seismology. If the model consists of many layers separated by curved interfaces, the only feasible solution to the transmitted waves is the one given by the geometrical optics approximation. Transmitted rays, transmitted wavefield, and the first Fresnel zone associated with a transmission point can be expressed by four 2 × 2 constant matrices constituting the 4 × 4 linearized ray transformation matrix. Generally, the ray transformation matrix can be constructed by dynamic ray tracing. However, if the layers are homogeneous, it can be formulated in closed form by using elementary vector calculus and coordinate transformations. Using the symplecticity of the ray transformation matrix, transmissions in opposite directions can be formulated by the ray transformation matrix for only one direction, and a reciprocity relation can be established. After the decomposition theorem for the ray transformation matrix, the transmitted wavefield in a model with many curved interfaces can be computed in a cascaded way. Using the [Formula: see text]‐matrix decomposition theorem, the normalized geometrical spreading factor can be expressed by means of the area of the first Fresnel zone of a transmission point. If seismic waves propagate through a locally spherical interface, the reciprocity relation may not hold. Using ray theory, this fact is shown by formulating the transmitted wavefield with the two principal radii of curvature of the transmitted wavefront at the transmission point under consideration. Using wave theory, this fact is shown by analyzing the Debye integral with the method of stationary phase.