Time migration velocity analysis by velocity continuation

Introduction

Migration velocity analysis is a routine part of prestack time migration applications. It serves both as a tool for velocity estimation (Deregowski, 1990) and as a tool for optimal stacking of migrated seismic sections prior to modeling zero-offset data for depth migration (Kim et al., 1997). In the most common form, migration velocity analysis amounts to residual moveout correction on CIP (common image point) gathers. However, in the case of dipping reflectors, this correction does not provide optimal focusing of reflection energy, since it does not account for lateral movement of reflectors caused by the change in migration velocity. In other words, different points on a stacking hyperbola in a CIP gather can correspond to different reflection points at the actual reflector. The situation is similar to that of the conventional normal moveout (NMO) velocity analysis, where the reflection point dispersal problem is usually overcome with the help of dip moveout (Hale, 1991; Deregowski, 1986). An analogous correction is required for optimal focusing in the post-migration domain. In this paper, I propose and test velocity continuation as a method of migration velocity analysis. The method enhances the conventional residual moveout correction by taking into account lateral movements of migrated reflection events.

Velocity continuation is a process of transforming time migrated images according to the changes in migration velocity. This process has wave-like properties, which have been described in earlier papers (Fomel, 2003,1994,1997). Hubral et al. (1996) and Schleicher et al. (1997) use the term image waves to describe a similar concept. Adler (2002,2000) generalizes the velocity continuation approach for the case of variable background velocities, using the term Kirchhoff image propagation. Although the velocity continuation concept is tailored for time migration, it finds important applications in depth migration velocity analysis by recursive methods (Vaillant et al., 2000; Biondi and Sava, 1999).

Applying velocity continuation to migration velocity analysis involves the following steps:

1. prestack common-offset (and common-azimuth) migration - to generate the initial data for continuation,
2. velocity continuation with stacking and semblance analysis across different offsets - to transform the offset data dimension into the velocity dimension,
3. picking the optimal velocity and slicing through the migrated data volume - to generate an optimally focused image.

The first step transforms the data to the image space. The regularity of this space can be exploited for devising efficient algorithms for the next two steps. The idea of slicing through the velocity space goes back to the work of Shurtleff (1984), Fowler (1988,1984), and Mikulich and Hale (1992). While the previous slicing methods constructed the velocity space by repeated migration with different velocities, velocity continuation navigates directly in the migration velocity space without returning to the original data. This leads to both more efficient algorithms and a better understanding of the theoretical continuation properties (Fomel, 2003).

In this paper, I demonstrate all three steps, using both synthetic data and a North Sea dataset. I introduce and exemplify two methods for the efficient practical implementation of velocity continuation: the finite-difference method and the Fourier spectral method. The Fourier method is recommended as optimal in terms of the accuracy versus efficiency trade-off. Although all the examples in this paper are two-dimensional, the method easily extends to 3-D under the assumption of common-azimuth geometry (one oriented offset). More investigation may be required to extend the method to the multi-azimuth case.

It is also important to note that although the velocity continuation result could be achieved in principle by using prestack residual migration in Kirchhoff (Etgen, 1990) or frequency-wavenumber (Stolt, 1996) formulation, the first is inferior in efficiency, and the second is not convenient for the conventional velocity analysis across different offsets, because it mixes them in the Fourier domain (Sava, 2000). Fourier-domain angle-gather analysis (Sava et al., 2001; Sava and Fomel, 2003) opens new possibilities for the future development of the Fourier-domain velocity continuation. New insights into the possibility of extending the method to depth migration can follow from the work of Adler (2002).

Time migration velocity analysis by velocity continuation