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| Predictive painting of 3-D seismic volumes | |
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Plane-wave destruction originates from a local plane-wave model for
characterizing seismic data (Fomel, 2002). The
mathematical basis is the local plane differential equation
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(1) |
where is the wave field and is the local slope,
which may also depend on and (Claerbout, 1992). In the case of a
constant slope, equation 1 has the simple general solution
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(2) |
where is an arbitrary waveform. Equation 2 is
nothing more than a mathematical description of a plane wave. Assuming
that the slope varies in time and space, one can design
a local operator to propagate each trace to its neighbors.
Let represent a seismic section
as a collection of traces:
, where corresponds to
for A plane-wave destruction operator
(Fomel, 2002) effectively predicts each trace from its
neighbor and subtracts the prediction from the original trace. In the
linear operator notation, the plane-wave destruction operation can be
defined as
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(3) |
where is the destruction residual, and is the
destruction operator defined as
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(4) |
where stands for the identity operator, and
describes prediction of trace from trace
. Prediction of a trace consists of shifting the original trace
along dominant event slopes. The prediction operator is a numerical
solution of equation 1 for local plane wave propagation
in the direction. The dominant slopes are estimated by minimizing
the prediction residual using regularized least-squares
optimization. I employ shaping
regularization (Fomel, 2007a) for controlling the smoothness of the
estimated slope fields. In the 3-D case, a pair of inline and
crossline slopes,
and
, and a pair
of destruction operators, and , are
required to characterize the 3-D structure. Each prediction in 3-D occurs in
either inline or crossline direction and thus conforms to
equation 4. However, as explained below in the discussion
of Dijkstra's algorithm, it is possible to arrange all 3-D traces
in a sequence for further processing.
Prediction of a trace from a distant neighbor can be accomplished by
simple recursion. Predicting trace from trace is simply
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(5) |
If is a reference trace, then the prediction of
trace is
. I call
the recursive operator
predictive painting. Once the elementary prediction operators
in equation 4 are determined by plane-wave destruction,
predictive painting can spread information from a given trace to its
neighbors recursively by following the local structure of seismic
events. The next section illustrates the painting concept using 2-D
examples.
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| Predictive painting of 3-D seismic volumes | |
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Next: Predictive painting in 2-D
Up: Fomel: Predictive painting
Previous: Introduction
2013-07-26