    Modeling of pseudo-acoustic P-waves in orthorhombic media with a lowrank approximation  Next: Dispersion Relation for Orthorhombic Up: Theory Previous: Theory

## Acoustic Wave Extrapolation

The acoustic wave equation is widely used in forward seismic modeling and reverse-time migration (Bednar, 2005; Etgen et al., 2009): (1)

where is the seismic pressure wavefield and is the wave propagation velocity.

Assuming the model is homogeneous , after a Fourier transform in space, we get the following explicit expression in the wavenumber domain: (2)

where (3)

Equation 2 has the following analytical solution: (4)

which leads to the well-known second-order time-marching scheme (Etgen, 1989; Soubaras and Zhang, 2008) :  (5)

Equation 5 provides a very accurate and efficient solution in the case of a constant-velocity medium with the aid of FFTs. When the seismic wave velocity varies in the medium, equation 5 turns into a reasonable approximation by replacing with , and taking small time steps, . However, FFTs can no longer be applied directly to evaluate the inverse Fourier transform, because a space-wavenumber mixed-domain term appears in the integral operation: (6)

As a result, a straightforward numerical implementation of wave extrapolation in a variable velocity medium with mixed-domain matrix 6 will increase the cost from to , the original cost for the homogeneous case, in which is the total size of the three-dimensional space grid. A number of numerical methods (Fomel et al., 2010; Du et al., 2010; Song et al., 2013,2011; Song and Fomel, 2011; Etgen and Brandsberg-Dahl, 2009; Liu et al., 2009; Zhang and Zhang, 2009; Fomel et al., 2012) have been proposed to overcome this mixed-domain problem.

In the case of orthorhombic acoustic modeling, we derive a new phase operator to replace of the isotropic model. We describe the details in the next section.    Modeling of pseudo-acoustic P-waves in orthorhombic media with a lowrank approximation  Next: Dispersion Relation for Orthorhombic Up: Theory Previous: Theory

2013-07-26