Numeric implementation of wave-equation migration velocity analysis operators |

For the case of the phase-shift operation in media with lateral
slowness variation, the mixed-domain solution involves forward and
inverse Fourier transforms (denoted fFT and iFT in our algorithms)
which can be implemented efficiently using standard Fast Fourier
Transform algorithms. The numeric implementation is summarized in the
following table:

In this chart, denotes the component of the depth wavenumber and denotes the component of the depth wavenumber. An example of mixed-domain implementation is the Split-Step Fourier (SSF) method, where represents the SSR equation computed with a constant reference slowness , and represents a space-domain correction (Stoffa et al., 1990).

Based on the equation 27, the derivative of the depth
wavenumber relative to slowness is

The linearized scattering operator can also be implemented in a
mixed-domain by expanding the square-root from relation
A-1 using a Taylor series expansion

Therefore, the wavefield perturbation at depth caused by a
slowness perturbation at depth under the influence of the
background wavefield at the same depth (forward scattering
operator 8) can be written as

Similarly, the slowness perturbation at depth caused by a
wavefield perturbation at depth under the influence of the
background wavefield at the same depth (adjoint scattering
operator 11) can be written as

The mixed-domain implementation of the forward and adjoint scattering
operators A-3 and A-4, is summarized on
the following tables:

Numeric implementation of wave-equation migration velocity analysis operators |

2013-08-29