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![]() | Seismic wave extrapolation using lowrank symbol approximation | ![]() |
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Let
be the seismic wavefield at
location
and time
. The wavefield at the next time
step
can be approximated by the following mixed-domain
operator (Wards et al., 2008)
Assuming small steps
in equation (1), one can
build successive approximations for the phase function
by
expanding it into a Taylor series. In particular, let us represent the
phase function as
When either the velocity gradient
or the time step
are small, the Taylor expansion (5) can be reduced to
only two terms, which in turn reduces equation (1) to the
familiar expression (Etgen and Brandsberg-Dahl, 2009)
In rough velocity models, where the gradient
does not
exist, one can attempt to solve the eikonal
equation 3 numerically or to apply approximations other than
the Taylor expansion (5). In the examples of this
paper, we used only the
term.
Note that the approximations that we use, starting from equation (1), are focused primarily on the phase of wave propagation. As such, they are appropriate for seismic migration but not necessarily for accurate seismic modeling, which may require taking account of amplitude effects caused by variable density and other elastic phenomena.
The computational cost for a straightforward application of
equation (1) is
, where
is the total size
of the three-dimensional
grid. Even for modest-size problems,
this cost is prohibitively expensive. In the next section, we describe
an algorithm that reduces the cost to
, where
is a small number.
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![]() | Seismic wave extrapolation using lowrank symbol approximation | ![]() |
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