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If an upward-propagating wave field is sampled uniformly at a given depth level,
it can be accurately continued upwards to an arbitrary point on the earth's
surface by a Kirchhoff summation operator. Let *P*(*t*,*x*_{i},*y*_{j},0) be the wave
field recorded by an array of unevenly spaced receivers on the surface.
We want to resample the wave field with an uniform sampling interval.
We consider the wave field recorded on the earth's surface to be the
upward continuation of a wave field *P*(*t*,*x*_{k},*y*_{l},*z*_{0}) that is recorded
by an array of uniformly spaced receivers at the depth *z*_{0}.
This relation is described mathematically by

| |
(1) |

where is a Kirchhoff upward continuation operator.
For a given *P*(*t*,*x*_{i},*y*_{j},0), we can estimate *P*(*t*,*x*_{k},*y*_{l},*z*_{0})
by minimizing the errors between the given input data and the data modeled
by upward continuation, which is as follows:

| |
(2) |

This is a linear least squares inverse problem. It can be efficiently solved
by using the conjugate gradient method.
Once *P*(*t*,*x*_{k},*y*_{l},*z*_{0}) is known, we can upward continue the wave field to
the surface and resample it, as follows:

| |
(3) |

where is another upward continuation operator.
The choice of *z*_{0} is determined from two aspects. The aperture of the
upward continuation operator for a given maximum time-dip of input data
is inversely proportional to *z*_{0}. Therefore, to achieve high efficiency,
we want to use small *z*_{0}. However, if the trace sampling intervals of
the original data are large, using small *z*_{0} causes the conjugate operator
of the upward continuation to be aliased.

An assumption made by this algorithm is that the velocities above
the depth level *z*_{0} are known, which is much weaker than the assumption
of the full knowledge of the velocity structure made
by the least squares migration (Cole, 1992; Ji, 1992).
The formulation presented above is well suited for resampling marine
data because the depth level *z*_{0} can be set within the water layer that has
a constant velocity and does not generate any useful events. It should also
work for land data recorded in an area of simple near-surface
velocity structure. For land data with complicated near surface structures,
a similar scheme can be formulated, in which the unevenly sampled traces
are modeled as the reverse time propagation of the wave field sampled
uniformly at a level above the earth's surface.

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** Up:** Zhang & Claerbout: Trace
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Stanford Exploration Project

11/17/1997