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We thank BGP Americas for a partial financial support of this work. The first
author is grateful to Huub Douma for inspiring discussions and for suggesting
the name ``seislet''. This publication is authorized by the Director, Bureau
of Economic Geology, The University of Texas at Austin.
Appendix
A
Review of plane-wave destruction
This appendix reviews the basic theory of plane-wave destruction
(Fomel, 2002).
Following the physical model of local plane waves, we
define the mathematical basis of plane-wave destruction filters via
the local plane differential equation (Claerbout, 1992)
![\begin{displaymath}
{\frac{\partial P}{\partial x}} +
{\sigma\,\frac{\partial P}{\partial t}} = 0\;,
\end{displaymath}](img69.png) |
(19) |
where
is the wave field, and
is the local slope, which may
also depend on
and
. In the case of a constant slope,
equation A-1 has the simple general solution
![\begin{displaymath}
P(t,x) = f(t - \sigma x)\;,
\end{displaymath}](img73.png) |
(20) |
where
is an arbitrary waveform. Equation A-2 is
nothing more than a mathematical description of a plane wave.
If we assume that the slope
does not depend on
, we can
transform equation A-1 to the frequency domain, where it
takes the form of the ordinary differential equation
![\begin{displaymath}
{\frac{d \hat{P}}{d x}} +
i \omega\,\sigma\, \hat{P} = 0
\end{displaymath}](img75.png) |
(21) |
and has the general solution
![\begin{displaymath}
\hat{P} (x) = \hat{P} (0)\,e^{i \omega\,\sigma x}\;,
\end{displaymath}](img76.png) |
(22) |
where
is the Fourier transform of
. The complex
exponential term in equation A-4 simply represents a shift
of a
-trace according to the slope
and the trace separation
.
In the frequency domain, the operator for transforming the trace
to the neighboring trace
is a multiplication by
. In other words, a plane wave can be perfectly
predicted by a two-term prediction-error filter in the
-
domain:
![\begin{displaymath}
a_0 \, \hat{P} (x) + a_1\, \hat{P} (x-1) = 0\;,
\end{displaymath}](img81.png) |
(23) |
where
and
. The goal of
predicting several plane waves can be accomplished by cascading
several two-term filters. In fact, any
-
prediction-error
filter represented in the
-transform notation as
![\begin{displaymath}
A(Z_x) = 1 + a_1 Z_x + a_2 Z_x^2 + \cdots + a_N Z_x^N
\end{displaymath}](img84.png) |
(24) |
can be factored into a product of two-term filters:
![\begin{displaymath}
A(Z_x) = \left(1 - \frac{Z_x}{Z_1}\right)\left(1 - \frac{Z_x}{Z_2}\right)
\cdots\left(1 - \frac{Z_x}{Z_N}\right)\;,
\end{displaymath}](img85.png) |
(25) |
where
are the zeroes of
polynomial A-6. According to equation A-5,
the phase of each zero corresponds to the slope of a local plane wave
multiplied by the frequency. Zeroes that are not on the unit circle
carry an additional amplitude gain not included in
equation A-3.
In order to incorporate time-varying slopes, we need to return to
the time domain and look for an appropriate analog of the phase-shift
operator A-4 and the plane-prediction
filter A-5. An important property of plane-wave
propagation across different traces is that the total energy of the
propagating wave stays invariant throughout the process: the energy of
the wave at one trace is completely transmitted to the next trace.
This property
is assured in the frequency-domain solution A-4 by the fact
that the spectrum of the complex exponential
is
equal to one. In the time domain, we can reach an equivalent effect
by using an all-pass digital filter. In the
-transform notation,
convolution with an all-pass filter takes the form
![\begin{displaymath}
\hat{P}_{x+1}(Z_t) = \hat{P}_{x} (Z_t) \frac{B(Z_t)}{B(1/Z_t)}\;,
\end{displaymath}](img87.png) |
(26) |
where
denotes the
-transform of the corresponding
trace, and the ratio
is an all-pass digital filter
approximating the time-shift operator
. In
finite-difference terms, equation A-8 represents an
implicit finite-difference scheme for solving equation A-1
with the initial conditions at a constant
. The coefficients of
filter
can be determined, for example, by fitting the filter
frequency response at low frequencies to the response of the
phase-shift operator. This leads to a version of Thiran's
maximally-flat all-pass fractional-delay filters (Välimäki and Laakso, 2001; Thiran, 1971).
Taking both dimensions into consideration,
equation A-8 transforms to the prediction equation
analogous to A-5 with the 2-D prediction filter
![\begin{displaymath}
A(Z_t,Z_x) = 1 - Z_x \frac{B(Z_t)}{B(1/Z_t)}\;.
\end{displaymath}](img92.png) |
(27) |
In order to characterize several plane waves, we can cascade several
filters of the form A-9 in a manner similar to that of
equation A-7. A modified version of the filter
, namely the filter
![\begin{displaymath}
C(Z_t,Z_x) = A(Z_t,Z_x) B(1/Z_t) = B(1/Z_t) - Z_x B(Z_t)\;,
\end{displaymath}](img94.png) |
(28) |
avoids the need for polynomial division. In case of a
3-point filter
, the 2-D filter A-10 has exactly
six coefficients. It consists of two columns, each column having three
coefficients and the second column being a reversed copy of the first
one.
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2013-03-02