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The irregular geometries of land data acquired in rugged terrain make
a datuming algorithm based on the Kirchhoff integral convenient.
Berryhill 1979 gives the following expression
for the upward continuation of a scalar wave field:
| |
(1) |
where U(0,z,t) is one trace of the upward continued wave field at datum
elevation z and lateral position x=0. The function is a time delayed
filtered version of the input traces on the original datum.
The assumptions that go into the derivation of this equation are
locally valid for non-planar datums and two dimensional wave fields.
For a wave field transformation from one datum to another,
the discretized form of equation (1) is
a summation where each input trace Ui is weighted and time shifted.
Each output trace
is calculated by
performing a sum of the form:
| |
(2) |
where Ui(t-ti) is a filtered input trace recorded at location i
along the lower datum
and delayed by travel time ti.
The effect of the filtering operation is to compensate for the Hankel tail.
Ai is an amplitude term incorporating spherical divergence and
geometric terms, is the angle between
the normal to the input surface and the straight line
travel time path connecting input location i and output location j,
ti is the time shift corresponding to the travel time
between the input and output locations.
Figure 1 illustrates the impulse response of upward continuation
and downward continuation for three diffractors at three different depths.
The energy from each diffractor is distributed along a hyperbolic trajectory.
The trajectory is the same for all depths.
It is natural and convenient to do the calculation in the frequency domain
because each input trace is shifted by a time invariant amount.
The algorithm is cast in domain
so that each individual output trace is calculated by
performing a sum of the form:
| |
(3) |
Because the time shift is performed in the frequency domain, there is no need
to interpolate. Anti-aliasing is easily handled by filtering the input
traces as necessary. Another advantage of performing the computation in the
frequency domain is that a half-order differential operator is easily applied
to compensate for the Hankel tail.
updnimp1
Figure 1 Impulse response of three diffractors at three different depths for
upward continuation and downward continuation. If you have the electronic
version of this document, you can press the button to see a movie
of the impulse response for diffractors at different depths.
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| |
Kirchhoff datuming is typically implemented in a serial fashion
by looping over the input traces as indicated by equations (2)
and (3).
The algorithm can be applied to zero-offset data or to shot and receiver
gathers.
Pre-stack datuming is done by extrapolating the shots and the geophones
separately using equation (3) Berryhill (1984).
The extension to three dimensions is straight forward and has the same
algorithmic form.
Equations (1) through (3) are for upward continuation.
For downward continuation
the conjugate transpose process is used.
Next: Data Parallel Implementation
Up: IMPLEMENTATION
Previous: IMPLEMENTATION
Stanford Exploration Project
11/17/1997