Summary

We propose an interpolation algorithm for one-way wave equation migration, which can allow the migration to take much bigger extrapolation step than usual, and is still accurate up to the dip limit of the one way propagator. The proposed interpolation algorithm is frequency dependent and applicable to any one way extrapolator which works in the frequency domain. It requires that there be limited vertical variation in the velocity model within each extrapolation step. Numerical tests have validated our algorithm and showed significantly speed up of the migration with barely noticeable deterioration of the image quality.

Introduction

One way wave equation migration (OWEM) is beginning to dominate seismic depth imaging due to the rapid progression of computer power and its potential for imaging structures of great complexity. However, even for today''s computers, comprising thousands of processors, the computational burden remains challenging as studies increase in size and data density, e.g. with the advent of wide azimuth acquisition. Therefore improving computational efficiency without significant loss of image quality remains desirable. Here we propose an algorithm to speed up OWEM, in which we perform the wavefield extrapolation on a coarse grid, then interpolate the wavefield between extrapolation steps in order to apply the imaging condition on the originally-desired grid. This is potentially interesting because of the ability of some oneway propagators to extrapolate wavefields accurately, depending upon the velocity model, over steplengths that may be considerably greater than the desired sampling of the image. The idea has been investigated by Ng (2007) and Mi and Margrave (2001) with the implicit assumption that the wave propagates near-vertically. As a result, steeply-dipping events are not interpolated accurately. The algorithm which we propose here is accurate up to the dip limit of the one-way propagator, and still very fast. We have performed numerical tests in 2D, using the Fourier Finite Difference (FFD) (Ristow and Ruhl, 1994) propagator, in which we showed one example of increasing the extrapolation step to 3 times the output image sampling and then interpolate the wave field for the samples in between. The test show a speed up factor of at least 2 compared to extrapolating at the output sampling. In 3-D, we expect that the speed up factor will be greater. The interpolation scheme can be applied either before or after the wave field cross correlation (Ng, 2007) (But in any case before the summation over frequencies).

Method

In a laterally heterogeneous medium we cannot in general find the eigenvectors of the extrapolator analytically but under certain assumptions we may make some heuristic progress, while understanding that it is provisional and approximate. If the phase angle, ?, can be considered constant over the extrapolation depth step at a given lateral position, then, implicitly, we are locally approximating kz over the step from pz+n and pz ?z. Then, effectively using this approximated kz, we can interpolate the wavefield to any depth level from z to z+n ?z using equation (3) acting in thespace-frequency domain.

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