Corrections for critical rays in 2-D seismic modelling

Abstract

The usual high-frequency (ω) ray method of modelling reflections from smooth boundaries does not account for interference arising near critical angles of incidence. When the reflector is effectively convex towards the incident-wave side, the ray method fails (i) in a transition zone of width O(ω-1/2) separating regions of partial and total ray reflection, (ii) for the transmission in a zone along the reflector having thickness O(ω-1/2) and where a whispering-gallery is formed by multiply-reflected turning waves, and (iii) for turning rays refracted back into the first medium at distances from the critical point less than O(mω-1/4), where m is the number of turns. The interference is a local effect and approximate analytic waveforms can be found for laterally varying media. The 2-D acoustic case is described here. The fields near the point of critical incidence and about the critically reflected ray are found in a way similar to the grazing ray solution described in an earlier paper. However, through the critical ray problem the first two terms of the asymptotic expansion must be considered. The whispering-gallery is obtained by exploiting work on modes by Buldyrev, Lewis, Ludwig and others. The modal solution is then matched with the solution near the critical point to determine the initial modal amplitudes. The individual ray contributions separate from the modes as the waves progress along the boundary. The modal form of solution is needed, even if one wishes to demonstrate only the matching with ray theory for the primary refracted/turning wave near the critical point. This is because the transition zone for the refraction is much wider than that for the reflected wave. Though it is more complicated than the earlier grazing ray solution, the critical-ray diffracted wavefield is still continued way from the boundary using ray coordinates. Hence, it is not difficult to provide ray tracing programs with corrections to ray synthetics. Some preliminary numerical examples are presented to indicate the waveforms and potential significance of these corrections. They are small in comparison with the total reflection, but they are very significant in comparison with the weaker refracted waves. Even where ray theory is (just) valid, for practical reasons one may wish to avoid tracing multiple reflections close to a numerically specified boundary. The boundary-layer formulae provide a more ‘natural’ solution to both the theoretical and numerical problems for the refracted part of the field.