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We will now describe how to image either the
source distribution or the reflectivity distribution from passive
seismic data in a *v*(*x*,*y*,*z*) medium.
The sources are assumed to be distributed anywhere in space, and the
time histories of each source are assumed to be uncorrelated.
Neither the source location or time history are known.
Without loss of generality we conveniently assume one source and one
scatterer, but the resulting migration formula are also applicable to
multiple sources and scatterers.
For an earth model with a free-surface, a smoothly varying velocity
distribution and a single point scatterer at *x*_{o}, the wavefield,
, recorded at
due a source at is given by

where is the WKBJ Green's function for
an impulsive source at and a geophone at , and
is the complex source spectrum.
The first term on the RHS represents the direct arrival (see top
figure in Figure 1),
the second term represents the scattered field excited by the direct
arrival,
and the third term represents the scattered arrival that is generated
by a specular free-surface reflection (see middle figure in
Figure 1).
Here, *R* is the scattering coefficient, is a solution to
the eikonal equation for a source at *g* and a receiver at *g*', and
the geometrical-spreading factors have been harmlessly dropped.
The specular-reflection point on the free surface is denoted by *g*''
and the location of the recording geophone is denoted by *g*'.

To eliminate the unknown phase of the source, we crosscorrelate the
wavefield, recorded at
with recorded at to get

| |
(1) |

If we then assume source is a white source spectrum such
that , the source term can be dropped
entirely, leaving
where the first expression on the RHS corresponds to the correlation
of the conjugate direct wave recorded at geophone *g*_{o}
with the direct wave at *g*';
and the second term corresponds to
the correlation of the conjugate direct wave at *g*_{o}
with the ghost reflection recorded at *g*'.
The *other terms* correspond to the other correlations
which will not be needed for
imaging, and are presumed to cancel upon migration.
This last assumption about cancellation is similar to the
standard assumption in migration, i.e., all migrated arrivals
incoherently superimpose except those that are tuned to
the specified imaging condition.
For the above equation, we will tune the imaging conditions
so that either the terms are migrated
to image the source distribution, or the
terms are migrated to
image the reflectivity distribution.

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** Up:** Schuster & Rickett: Daylight
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Stanford Exploration Project

9/5/2000