Seismic coherent states and ray geometrical spreading

Abstract

SUMMARY The coherent-state transform (CST) is essentially a Gaussian-windowed Fourier transform and it yields a combined slowness–position (p, x) domain representation of seismic wavefields. Several forms of the inverse CST exist and the set of ‘coherent states’ form an overcomplete basis for wave analysis, in many ways similar to modern wavelets. The asymptotic or ‘ray’ solution to the CST of the seismic wave equation involves a phase function S(p, x) that is complex due to the Gaussian decay. Hence one must consider complex rays, as well as a higher-dimensional phase space (p, x, ∂pS, ∂xS) corresponding to the extended configuration or base space (p, x). The initial conditions and geometrical spreading of these rays involve generalizations of standard procedures, exemplified by analysis of the coherent states excited by a ray theory incident wavefield. Surfaces of constant S are ‘tangential’ to the standard ray theory wavefront T(x) and the Maslov phase fronts given by the Legendre transformation of T. Hence the real rays of T (and the Maslov phase) are particular rays of S. The geometrical spreading of an individual coherent-state (CS) wavefield requires careful consideration. Although the transport equation involves divergence in the higher-dimensional base space (p, x), Smirnov's Lemma applied in this space still gives the solution. The incorporation of initial conditions is correspondingly intricate, but the final spreading function is better behaved (‘smoother’) than either a standard ray or Maslov theory amplitude. This provides a solution to the problem of pseudo-caustics. The task of finding a ‘KMAH index’ for each ray contributing to the inverse CST is simplified to choosing a complex square root that is smoothly connected between these rays (i.e. it does not fork). It is suggested that in practice only real rays are needed and then the method can be reduced to a smoothed form of the Maslov Snell wave sum representation. This is achieved by approximating the complex CS phase S(p, x) in the vicinity of the real rays. The result also then appears similar in form to a sum of Gaussian beams, although without reference to local ray-centred coordinates. Although the derivation involves some lengthy steps, the formulae finally obtained are easy to implement. Their effectiveness is demonstrated by a numerical example of interfering caustics and pseudo-caustics caused by two nearly plane wavefront segments joined at a triplication.