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![]() | Inversion of up and down going signal for ocean bottom data | ![]() |
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The derivation for decomposing over-under pressure waves into up-going and down-going signals is best done in the Fourier domain.
Denote
and
to be the Fourier transformed measurement of compressional waves at depth
(over) and
(under). Theoretically,
can be viewed as a sum of the up-going
and down-going
components. Likewise for
:
Down-going waves visit the over array () before visiting the under array (
). Therefore,
, when shifted forward in time, would match the signal
. Similarily, up-going waves visit the under array first. Therefore,
, when shifted forward in time would match the signal
. This relationship is equivalent to a phase-shift in the Fourier domain:
where
and
is the usual dispersion relation. Finally, substituting equation 4 into 3 yields the formula for the up-going and down-going waves at the receivers:
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Data acquisition using over-under arrangement is often used to elimate receiver ghosts and water reverberation. For a thorough review of this method, please see Sonneland et al. (1986). Although over-under arrays are rarely placed at the sea floor in real seismic surveying, this technique allows easy generation of up- and down-going data at the sea bottom in synthetic examples using the simpler acoustic wave equation.
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![]() | Inversion of up and down going signal for ocean bottom data | ![]() |
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