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Next: Slope estimation
Up: Fomel: Plane-wave destructors
Previous: Introduction
Following the physical model of local plane waves, we can define the
mathematical basis of the plane-wave destruction filters as the local
plane differential equation
![\begin{displaymath}
\frac{\partial P}{\partial x} +
\sigma\,\frac{\partial P}{\partial t} = 0\;,
\end{displaymath}](img7.png) |
(1) |
where
is the wave field, and
is the local slope, which may
also depend on
and
. In the case of a constant slope,
equation (1) has the simple general solution
![\begin{displaymath}
P(t,x) = f(t - \sigma x)\;,
\end{displaymath}](img12.png) |
(2) |
where
is an arbitrary waveform. Equation (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 (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}](img14.png) |
(3) |
and has the general solution
![\begin{displaymath}
\hat{P} (x) = \hat{P} (0)\,e^{i \omega\,\sigma x}\;,
\end{displaymath}](img15.png) |
(4) |
where
is the Fourier transform of
. The complex
exponential term in equation (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 at
position
to the neighboring trace
and at position
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}](img20.png) |
(5) |
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}](img23.png) |
(6) |
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}](img24.png) |
(7) |
where
are the zeroes of
polynomial (6). According to equation (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 (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 (4) and the plane-prediction
filter (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 (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}](img26.png) |
(8) |
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 (8) represents an
implicit finite-difference scheme for solving equation (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. The Taylor series technique (equating the
coefficients of the Taylor series expansion around zero frequency)
yields the expression
![\begin{displaymath}
B_3(Z_t) =
\frac{(1-\sigma)(2-\sigma)}{12}\,Z_t^{-1} +
...
...sigma)(2-\sigma)}{6} +
\frac{(1+\sigma)(2+\sigma)}{12}\,Z_t
\end{displaymath}](img31.png) |
(9) |
for a three-point centered filter
and the expression
for a five-point centered filter
. The derivation of
equations (9-10) is detailed in the appendix. It
is easy to generalize these equations to longer filters.
Figure 1 shows the phase of the all-pass filters
and
for two values of the
slope
in comparison with the exact linear function of
equation (4). As expected, the phases match the exact line
at low frequencies, and the accuracy of the approximation increases
with the length of the filter.
|
---|
phase
Figure 1. Phase of the implicit
finite-difference shift operators in comparison with the exact
solution. The left plot corresponds to the slope of
, the right plot
to .
|
---|
|
---|
Taking both dimensions into consideration,
equation (8) transforms to the prediction equation
analogous to (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}](img41.png) |
(11) |
In order to characterize several plane waves, we can cascade several
filters of the form (11) in a manner similar to that of
equation (7). In the examples of this paper, I use 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}](img43.png) |
(12) |
which avoids the need for polynomial division. In case of the 3-point
filter (9), the 2-D filter (12) 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. When filter (12) is used in data regularization
problems, it can occasionally cause undesired high-frequency
oscillations in the solution, resulting from the near-Nyquist zeroes
of the polynomial
. The oscillations are easily removed in
practice with appropriate low-pass filtering.
In the next section, I address the problem of estimating the local
slope
with filters of form (12). Estimating
the slope is a necessary step for applying the finite-difference
plane-wave filters on real data.
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Next: Slope estimation
Up: Fomel: Plane-wave destructors
Previous: Introduction
2014-03-29