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![]() | Lowrank one-step wave extrapolation for reverse-time migration | ![]() |
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In this appendix, we prove the unconditional stability of one-step wave
extrapolation linear operator
in one-dimensional isotropic
media defined by:
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(48) |
Let us consider
where
corresponds to a
matrix with
entry given by
,
and
corresponds to a matrix with
entry given by
.
represents a matrix with
entry given by:
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(49) |
In order to bound the
norm of
we
estimate the
entry of
. For
,
we have
. For
:
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(50) |
For sufficiently small
,
satisfies
Let us define
. From equation 52, it is
clear that the map
is one to one. Substituting
into equation 53 gives
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(56) |
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(57) |
We now apply the above lemma to
. When
where
is the spatial dimension, we have
bounded. Therefore,
for sufficiently smooth
, we have
, for suffciently small
. Hence
and
. Since
and
as the Fourier transform
is an isometry, we have
.
When performing wave extrapolation, fix a final time
and propagate
steps, the operator is stable since
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(58) |
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![]() | Lowrank one-step wave extrapolation for reverse-time migration | ![]() |
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