Wavefield extrapolation in pseudodepth domain |

The Sampling of the axis should be small enough to avoid wavefield aliasing in the domain, for example

(25) |

where is the minimum velocity in the model and is the maximum frequency of the wave. Accordingly, the number of samples representing the axis should be chosen to cover the largest expected value,

(26) |

The mapping velocity is often chosen as a slightly smoothed version of true velocity . This is because the domain wave equation involves a differentiation of , for example the first equation in 15.

For second-order wave equations, the operators on the right-hand side can become significantly complicated in the domain, such as Equation 23. Thus, it is more convenient to code up its first-order form 21. For consistency, we will extrapolate the wavefields using the first-order form for all the examples in this paper. Thus, the time derivatives in these equations are approximated by central differences,

(27) |

and the spatial derivatives are approximated using the Fourier pseudospectral approuch (Carcione et al., 2002; Gazdag, 1981), as follows

(28) |

where superscript indicate time steps, is the spatial Fourier transform in the direction. The change of the vertical axis from to does not affect the stability condition. For both isotropic and VTI extrapolations, the same time-step is used in both the Cartesian and domains.

Wavefield extrapolation in pseudodepth domain |

2013-04-02