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| Carl
Tape and Qinya Liu
have employed adjoint methods in a 2D model of wave propagation
to produce sensitivity kernels. Our aim is to apply these techniques
to velocity structure models in 3D, both at a regional scale
and at the scale of the globe.
We
have constructed finite-frequency sensitivity kernels using
a 2D spectral-element code for elastic wave propagation. The
method we are using involves the interaction between a “regular”
wavefield, propagating from the source to the receiver, with
an “adjoint” wavefield, which propagates from
the receiver to the source. The source of the adjoint field
is the time-reversed regular field recorded at the receiver.
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fig
1 |
| Model
setup for computing the elastic wavefield in 2D. The elastic
wavefield can be separated into the two decoupled fields,
SH and P-SV. All three components of the wavefield are computed,
but the propagation is limited to a depth cross-section (hence
2D). Also shown are the possible body waves for each wavefield.
The receiver and source are placed at 40 km depth simply to
illustrate some of the simpler examples. The model is homogeneous
with absorbing boundaries plus a free surface on top to mimic
a half space (fig
1). |
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SH
wavefield, where we have time-reversed the S arrival to use
as the adjoint wavefield source. The snapshots for (a) and (b)
show the wavefields at t = 16.00s, where the source initiates
at t = 8.00s. The interaction field (c) represents the instantaneous
contribution by the interaction of the regular field and the
adjoint field toward the construction of the sensitivity kernel
for S-velocity perturbations (d). The kernel that forms is a
“banana-doughnut kernel”, as illustrated extensively
by the Princeton group (fig
2). (In this case it is not banana-shaped because
of the uniform velocity field, and there is no doughnut cross-section
because these are 2D, not 3D, kernels.)
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P-SV wavefield, where we have reversed the PS+SP
arrival, and the kernel
we are showing is for P-velocity perturbations. For the
instant in time
in the figures, we see that the interaction (labeled in
(c)) between the
regular wavefield and the adjoint wavefield is "painting"
the P portion
of the PS phase. The final shape of the PS+SP alpha-kernel
will look
like two truncated cigars. The outermost fringes in (d)
represent PS
and SP scatterers with comparable traveltimes to the PS
and SP phases
reflecting at the surface (fig 3).
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