INTRODUCTION

When the eikonal equation is solved for modeling wave propagation, the seismic traveltimes are assumed to be frequency independent Bleistein (1984); Cervený (1987). This assumption does not always hold for reflection seismic data; its range of validity is limited by the the rate of variation of the wave amplitudes, compared with the frequency of the wavefield Beydoun (1985); Woodward (1990). In particular, when the variations in the velocity function have wavelengths similar to the wavelength of the propagating waves, the traveltimes are not frequency independent and cannot be evaluated by simply solving the conventional eikonal equation. Media where this phenomenon happens are called ``dispersive'', because wave components having different frequencies, or wavelengths, propagate along different ``wave-paths'', and thus the wavefronts are dispersed Woodward (1990). In dispersive media, modeling by solving the conventional eikonal equation not only causes errors in the computed traveltimes, but it also causes errors in the evaluation of the directions of propagation of seismic energy. All frequencies are erroneously assumed to travel along the wave-path corresponding to infinite frequency; that is the ``ray paths''. Often, in such media, the solution of the eikonal shows some ``pathological'' behavior such as caustics and shadow zones, while the solution of the full wave equation, with band-limited sources, is well behaved.

On the other hand, the modeling of wave propagation by computing traveltimes with raytracing Bleistein (1984); Cervený (1987), or with finite differences, VanTrier and Symes (1991); Vidale (1988), or with traveltime minimization Moser (1989) is very attractive because it is far less expensive than solving the full wave equation. Further, there are many applications, for example seismic tomography and Kirchhoff migration, where raytracing is preferred over wave-equation modeling because its results are easier to interpret and to use. The different components of the wavefield, and the effects of different parts of the model on the wavefield, are more naturally separated in a ray solution than in a wave solution.

In this paper, I present a method for approximating the solution of a frequency-dependent version of the eikonal equation, also called ``extended eikonal'' by Zhu 1988, `hypereikonal'' by Beydoun 1985, and ``exact' eikonal by Foreman 1989. The frequency dependent eikonal equation can be directly derived from the wave equation, via the Helmholtz equation, and does not rely on high-frequency assumptions. Its solution, together with the solution of the transport equation, provides an exact solution to the modeling problem. The numerical evaluation of the exact solution of the frequency-dependent eikonal is difficult, and thus I propose to compute an approximate solution. My method extrapolates in frequency the phase slowness by solving a non-linear partial differential equation. The extrapolation starts from infinite frequency, where the phase slowness is equal to the medium slowness, and moves down the frequency axis. Having computed the phase-slowness function, I solve the conventional eikonal using the computed phase-slowness function as the equivalent slowness of the medium. The method, although more expensive than simple raytracing, is still considerably cheaper than full-waveform modeling. It also retains many of the other attractive features of raytracing regarding the usability of the results. The direct application of my method for improving the accuracy of Kirchhoff migration should be straightforward, but it should also possible to apply it to seismic tomography.

In the first section I derive the frequency-dependent eikonal equation from the wave equation. In the second section I describe the slowness-extrapolation method for solving the ``new'' eikonal. Finally, in the third section I show the results of some preliminary numerical experiments using the proposed method, and compare the results with wave-equation modeling.