Next: - downward extrapolation Up: Alvarez and Artman: Wavefield Previous: Introduction

Overview of Mixed-domain Downward Extrapolation

In this section, we will briefly review, from the mathematical point of view, the - - - algorithm. This will serve as the starting point for the presentation of the new extrapolation method in the next section.

Let be the wavefield at depth step N in the -domain and let be the wavefield extrapolated from depth step N to depth step N+1 using reference velocity Vl. That is:

where is the depth of the Nth layer and kzl is given by the dispersion relation:
 (1)
with being the magnitude of the horizontal wavenumber vector. PSPI handles the difference between the true velocity and the reference velocity by interpolating the downward-continued wavefields in the - domain based on the difference between the reference velocities and the model velocity at each position. The interpolated wavefield is therefore given by:

where is an interpolation factor (), is the downward-continued wavefield in the - domain at the spatial location j, and nv is the number of reference velocities.

Extended split-step adds a correction before the interpolation, the so-called thin lens term'':

where V is the true model velocity. Depending on the choice and number of reference velocities, split-step can make significant improvements in accuracy compared to PSPI.

Other methods, such as pseudo-screen and Fourier finite-difference, increase the accuracy of the result by adding high-order spatial derivatives to the computation of the kzl term Biondi (2004); Huang et al. (1999); Ristow and Ruhl (1994); Xie and Wu (1999). The more accurate approximation of kzl relaxes the need for a large number of reference velocities such that with fewer reference velocities similar or even better accuracies can be obtained compared with split-step.

The last step in either of these methods is to Fourier transform the interpolated wavefield to the - space. This wavefield will then be the input to the propagation at the next depth step.

It should be clear that the accuracy of these methods, especially PSPI and extended split-step, is directly related to the accuracy of the wavefield interpolation and the number and choice of the reference velocities.

Next: - downward extrapolation Up: Alvarez and Artman: Wavefield Previous: Introduction
Stanford Exploration Project
5/3/2005