Introduction

Mixed domain wavefield extrapolation methods can handle, to a large extent, spatial velocity variations by performing part of their computation in the $\omega$-$\mathbf{k}$ domain and part of the computation in the $\omega$-$\mathbf{x}$ domain. The best-known mixed domain methods are Phase Shift Plus Interpolation (PSPI) and extended split-step migration Biondi (2004); Gazdag and Sguazzero (1984); Stoffa et al. (1990). They downward continue the wavefield in the $\omega$-$\mathbf{k}$ domain at each depth step with a series of reference velocities and then interpolate the wavefields in the $\omega$-$\mathbf{x}$ domain possibly after a split-step correction.

There are three main potential sources of error in mixed-domain algorithms: (1) the choice of the reference velocities, (2) the correction to account for the difference between the model velocity and the reference velocity at each spatial location, and (3) the accuracy of the interpolation of the wavefields.

In this paper, we present an alternative to $\omega$-$\mathbf{x}$ - $\omega$-$\mathbf{k}$ downward extrapolation that performs each depth extrapolation completely in the $\omega$-$\mathbf{k}$ domain, yet can handle arbitrary spatial variation of the migration velocities. The proposed algorithm eliminates the need for choosing reference velocities and requires no FFT's at each depth extrapolation level. Making the algorithm efficient, however, is still a research issue.

In the next section, we will present a brief overview of the standard $\omega$-$\mathbf{x}$ - $\omega$-$\mathbf{k}$ methods without getting into any specific implementation details. In the following section, we present our method, and in the last section, we discuss some implementation issues.