Wavefield extrapolation in pseudodepth domain

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## Anisotropic extrapolation

Velocity variation with angle in anisotropic media allows more choices for the mapping velocity in Equation 2. It is natural to use, but not limited to, the vertical velocity, , as .

The kinematics of a quasi-P wave in an anisotropic acoustic medium can be characterized by three parameters: P-wave velocity in the direction of the axis of symmetry , NMO velocity and anellipticity  (Alkhalifah, 2000), here and are Thomsen parameters (Thomsen, 1986).

The quasi-P wave motion in transversely isotropic media with vertical axis of symmetry (VTI) is described by the following first-order system (Duveneck and Bakker, 2011)

 (20)

where and are horizontal and vertical stresses, is the particle momentum.

In the domain, the wave equation is obtained by applying the chain rule 9 to 20, the resulting system of equations is

 (21)

Equation 20 has the second-order form

 (22)

Similarly, the second order form of 21 is

 (23)

In addition to the cost reduction, the vertical time axis also allows time processing in VTI media to be independent of the vertical velocity , which is usually unresolvable from surface seismic data.

In transversely isotropic media with tilted axis of symmetry (TTI), the symmetry plane and symmetry axis are rotated by tilt angle and azimuth . The two-way wave equation is obtained by substituting derivatives in Equation 20 by the following relations

 (24)

In domain, the TTI extrapolation equation is obtained by replacing each of the spatial derivatives on the right-hand side of Equation 24 by the expressions given by chain rule in Equation 9. The coordinate transformation does not affect stability of the extrapolation.

 Wavefield extrapolation in pseudodepth domain

Next: Implementation aspects Up: pseudodepth domain wave equation Previous: Isotropic extrapolations

2013-04-02