Determination of the influence zone for surface wave paths

An approximate description of the zone of influence around the propagation path for a surface wave is provided by investigating the Fresnel zones for the frequency range of interest. The influence zone about surface wave paths, over which surface waves are coherent in phase, is identified as approximately one-third of the width of the first Fresnel zone. A technique called Fresnel-area ray tracing (FRT) for surface waves has been used to estimate this region around the ray path for each frequency. The FRT technique is developed by combining two standard ray tracing methods, i.e. kinematic ray tracing (KRT) and dynamic ray tracing (DRT). To obtain the exact Fresnel area in a laterally heterogeneous structure would require the solution of a large number of KRT equations. In contrast, the FRT approach requires just a few ray tracing calculations. In the first step, the trajectory of the surface wave is computed by solving the KRT system for the phase-velocity distribution at the required frequency. In the next step, the behaviour of rays in the zone surrounding the KRT path is calculated by solving the DRT system twice; once from the source to the receiver and once more from the receiver to the source along the same trajectory. Finally, combining the solutions of these ray tracing systems using paraxial ray theory, the Fresnel area around a central ray can be estimated. Using FRT, stationary-phase fields can be constructed around a central ray path in a laterally heterogeneous structure. The influence zone around the ray path is then estimated from the stationary-phase function with simple assumptions concerning the perturbed wavefield. The estimate of the influence zone can be efficiently calculated in laterally heterogeneous structure by using the FRT technique, and allows an extension of current methods of surface wave analysis, which have commonly been based on geometrical ray theory and on the approximation of great-circle propagation. This approach allows the treatment of finite-width rays as well as deviations in propagation from the great circle induced by moderate lateral heterogeneity as revealed by recent tomography models. Such finite-width rays should be of major benefit in enhancing ray-based surface wave tomography.