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This appendix develops a WEMVA scattering operator Sava and Biondi (2004)
for use in transmission wavefield waveform inversion. The
extrapolation operator, , is given by,
where kz is the depth wavenumber and is the
extrapolation depth step. The extrapolation wavenumber in depth is
where is temporal frequency and is the horizontal
The vertical wavenumber can be separated into two components, one
corresponding to the background medium, , and one
corresponding to a perturbation, , such that,
In a first-order approximation, we can relate these two extrapolation
wavenumbers by a Taylor-series expansion,
where s(x,z) is the slowness and corresponds to a
Within any depth slab we can extrapolate the wavefield from the top,
either in the perturbed or in the background medium. The wavefields
at the bottom of the slab, and , related by,
Equation 34 is a direct statement of the Rytov
approximation, because the wavefields at the bottom of the slab
correspond to different phase shifts related by a linear equation.
Thus, we obtain the wavefield perturbation at the bottom of the
slab by subtracting the background wavefield from the
perturbed wavefield :
For the Born approximation, we further assume that the wavefield
differences are small so that we linearize the exponential
function according to . With this approximation we write the
following downward continued scattered wavefield,
which, in operator form is
The Born operator may be implemented in the Fourier domain relative to
a constant reference slowness in any individual slab. In this case,
where is a damping parameter to avoid division by zero.
Figure shows the amplitude weighting demanded by
the filter in equation 38 for five different
frequencies for slowness 0.5 s/km.
Figure 1 Example of the Born amplitude weighting
function demanded by the WEMVA theory for a slowness of 0.5 s/km
and a damping factor of 0.001.