Introduction

Seismic imaging is a two-step process: velocity estimation and migration. As the velocity function becomes more complex, the two steps become more and more dependent on each other. In complex depth-imaging problems, velocity estimation and migration are applied iteratively in a loop. To assure that this iterative imaging process converges to a satisfactory model, it is crucial that the migration and the velocity estimation are consistent with each other.

Kirchhoff migration often fails in complex areas, such as sub-salt, because the wavefield is severely distorted by lateral velocity variations, and thus complex multipathing occurs. As the shortcomings of Kirchhoff migration have become apparent O'Brien and Etgen (1998), there has been a renewal of interest in wave-equation migration and the development of computationally efficient 3-D prestack depth-migration methods based on the wave equation Biondi and Palacharla (1996); Biondi (1997); Mosher et al. (1997). However, there has been no corresponding progress in the development of migration velocity analysis (MVA) methods that can be used in conjunction with wave-equation migration.

In this paper, we propose a method that aims to fill this gap and that, at least in principle, can be used in conjunction with any downward-continuation migration method. In particular, we have been applying our new methodology to downward continuation based on the Double Square Root equation in two dimensions Claerbout (1985); Popovici (1996); Yilmaz (1979) and on common-azimuth continuation in three dimensions Biondi and Palacharla (1996).

As for migration, wave-equation MVA is intrinsically more robust than ray-based MVA, because it avoids the well-known problems that rays encounter when the velocity model is complex and has sharp boundaries. The transmission kinematic component of the finite-frequency wave propagation is mostly sensitive to smooth variations in the velocity model. Consequently, wave-equation MVA produces smooth velocity updates and is therefore stable. In most cases, no smoothing constraints are needed to assure stability in the inversion. In contrast, ray-based methods require strong smoothing constraints to avoid quick divergence.

Our method is closer to conventional MVA than other wave-equation methods that have been proposed to estimate the background velocity model Bunks et al. (1995); Fogues et al. (1998); Noble et al. (1991), because it tries to maximize the quality of the migrated image rather than to match the recorded data. In this respect, our method is related to differential semblance optimization (DSO) Symes and Carazzone (1991) and multiple migration fitting Chavent and Jacewitz (1995). However, in contrast to these two methods, our method has the advantage of exploiting the power of residual prestack migration to speed up the convergence.