introduction

It is well known that wavefield-continuation-based migration methods are better able to image areas affected by multi-pathing and are more effective in handling complex wave behavior than ray-tracing-based Kirchhoff methods. However, wave-equation depth migration such as phase-shift plus interpolation Gazdag and Sguazzero (1984), extended split-step Fourier Kessinger (1992); Stoffa et al. (1990), split-step double square root Popovici (1996), or Fourier finite difference plus interpolation Biondi (2002) require multiple reference velocities for wavefield extrapolation through laterally inhomogeneous velocity models. With multiple reference velocities higher angles can be accurately handled and the quality of the image can be improved, but the cost of the migration is increased in proportion to the number of reference velocities used.

When it comes to anisotropic media, the vertical wavenumber k_z becomes a function of three anisotropic parameters (v, $\delta$ and $\varepsilon$ (or $\eta$)). To extrapolate wavefields in Fourier domain, we must choose not only multiple reference velocities, but also multiple $\delta$s and $\varepsilon$s (or $\eta$s). The computational cost subsequently increases significantly. The conventional method Baumstein and Anderson (2003); Rousseau (1997) for selecting those three parameters based on the geometric distribution of the input model is first to uniformly sample the vertical velocity v at each depth level, then for each vertical velocity, to uniformly sample a range of $\delta$s and $\eta$s (or $\varepsilon$s) from their minima to their maxima. The main drawback of this method is that it may undersample the parameters at places where the lateral variations are significant, or it may oversample the parameters at places where the model is smooth. To get a better result it may require a large number of reference parameters, which is impractical for migrating large data sets.

Clapp (2004) borrowed the idea of quantization from the field of electrical engineering, and used the 1D Lloyd's algorithm to select reference velocities for isotropic migration. He demonstrated that reference velocities selected by Lloyd's algorithm can produce higher-quality images while using fewer reference velocities. In fact, the 1D Lloyd's algorithm Lloyd (1982) is a special case of the clustering method, which tries to find the global optimum solutions according to some statistical criteria. Reference-velocity selection based on the statistical distribution of the velocity model is receiving increasing attention Bagaini et al. (1995); Geiger and Margrave (2005). In the companion paper, Clapp (2006) extends the 1D Lloyd's algorithm to multi-dimensional case, and in this paper we use the 3D version of Lloyd's algorithm to select the reference anisotropic parameters for wavefield extrapolation in laterally varying VTI media. We show that, by incorporating the 3D Lloyd's algorithm, we greatly reduce the computational cost and obtain high-quality images.