recorded data | = | ![]() |
Most generally, the operator is elastic two-way wave equation
modeling, although making the acoustic assumption simplifies matters. In both
cases, the modeling operator is non-linearly dependent on the earth model to be
estimated. For many reasons (including, but not limited to: computational
expense, model non-uniqueness/nullspace, and sensitivity to starting model),
full-waveform techniques are rarely applied successfully in today's conventional
seismic processing environment. The LSJIMP method can be abstracted in a similar
qualitative fashion, using the notation of the previous sections:
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velocity, reflection coefficient, crosstalk model |
---|---|---|
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= | ![]() |
Quantities like imaging velocity, the measured reflection coefficient of the
multiple generators, and the crosstalk model are assumed to be fixed. Some
LSJIMP implementations might depend only implicitly on velocity or reflection
coefficient. For the sake of argument, however, let us assume that the LSJIMP
operator, , is a non-linear function of these parameters, which the basic
LSJIMP inversion makes no attempt to optimize. A multiple-free estimate of the
primaries obviously enhances our ability to estimate imaging velocity,
regardless of the method, and also permits us to model crosstalk noise from
primaries that are below the onset of the seabed pure multiple. Thus the
simplest nonlinear iteration of the LSJIMP method would proceed as follows,
where the superscript k denotes that an operator or model vector is attached
to the
nonlinear iteration:
iterate { | ||
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velocity, reflection coefficients, crosstalk model |
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= | ![]() |
---|---|---|
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updated velocity, crosstalk model |
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Nonlinear updating of the reflection coefficients of the multiple generators is
in general a more difficult, and potentially more valuable, problem. If a
multiple generator's pure multiple is obscured by other events, the reflection
coefficient estimation scheme outlined in section may produce
inaccurate estimates, which the spatial regularization may not account for. For
example, in the 2-D field data example shown in Chapter
,
the pure multiple for one multiple generator happens to be overlapped over
almost the entire line by a prominent pegleg from another multiple generator.
I propose a nonlinear reflection coefficient updating scheme which obtains
perturbations by fitting unmodeled events in the LSJIMP data residual,
. If the initial reflection coefficient is perfect, then after
convergence,
will contain only uncorrelated noise. If it is
imperfect, then we also expect to see correlated events left over in the
residual. Because LSJIMP separates each multiple mode indpendently in the model
space (
), we can simply apply the forward model for that mode
(
) to obtain an estimate of the particular multiple in data
space,
.
The main idea of my updating scheme is to compute a scalar update to the
reflection coefficient of the multiple generator,
, such that
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(13) |
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(14) |
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(15) |