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Following the physical model of local plane waves, we can define the
mathematical basis of the planewave destructor filters as the local
plane differential equation
 
(98) 
where P(t,x) is the wave field, and is the local slope, which may
also depend on t and x. In the case of a constant slope,
equation () has the simple general solution
 
(99) 
where f(t) is an arbitrary waveform. Equation () is
nothing more than a mathematical description of a plane wave.
If the slope does not depend on t, we can transform
equation () to the frequency domain, where it takes the
form of the ordinary differential equation
 
(100) 
and has the general solution
 
(101) 
where is the Fourier transform of P. The complex
exponential term in equation () simply represents a shift
of a ttrace according to the slope and the trace separation
x.
In the frequency domain, the operator for transforming the trace
at position x1 to the neighboring trace at position x is a
multiplication by . In other words, a plane wave
can be perfectly predicted by a twoterm predictionerror filter in
the FX domain:
 
(102) 
where a_{0} = 1 and . The goal of
predicting several plane waves can be accomplished by cascading
several twoterm filters. In fact, any FX predictionerror
filter represented in the Ztransform notation as
 
(103) 
can be factored into a product of twoterm filters:
 
(104) 
where are the zeroes of
polynomial (). According to equation (),
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 ().
In order to incorporate timevarying slopes, we need to return to
the time domain and look for an appropriate analog of the phaseshift
operator () and the planeprediction
filter (). An important property of planewave
propagation across different traces is that the total energy of the
transmitted wave stays invariant throughout the process. This property
is assured in the frequencydomain solution () 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 allpass digital filter. In the Ztransform notation,
convolution with an allpass filter takes the form
 
(105) 
where denotes the Ztransform of the corresponding
trace, and the ratio B(Z_{t})/B(1/Z_{t}) is an allpass digital filter
approximating the timeshift operator (). In
finitedifference terms, equation () represents an
implicit finitedifference scheme for solving equation ()
with the initial conditions at a constant x. The coefficients of
filter B(Z_{t}) can be determined, for example, by fitting the filter
frequency response at small frequencies to the response of the
phaseshift operator. The Taylor series technique (equating the
coefficients of the Taylor series expansion around zero frequency)
yields the expression
 
(106) 
for a threepoint centered filter B_{3}(Z_{t}) and the expression
 

 
 (107) 
for a fivepoint centered filter B_{5}(Z_{t}). It is easy to generalize
these expressions to longer filters.
Figure shows the
phase of the allpass filters B_{3}(Z_{t})/B_{3}(1/Z_{t}) and
B_{5}(Z_{t})/B_{5}(1/Z_{t}) for two values of the slope in
comparison with the exact linear function of equation ().
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 9 Phase of the implicit
finitedifference shift operators in comparison with the exact
solution. The left plot corresponds to , the right plot
to .
In two dimensions, equation () transforms to the
prediction equation analogous to () with the 2D
prediction filter
 
(108) 
In order to characterize several plane waves, we can cascade several
filters of the form () in a manner similar to that of
equation (). In the examples of this chapter, I use a
modified version of the filter A(Z_{t},Z_{x}), namely the filter
 
(109) 
which avoids the need for polynomial division. In case of the 3point
filter (), the 2D filter () has exactly
six coefficients, with the second t column being a reversed copy of
the first column. When filter () is used in data
regularization problems, it can occasionally cause undesired
highfrequency oscillations in the solution, resulting from the
nearNyquist zeroes of the polynomial B(Z_{t}). The oscillations are
easily removed in practice with appropriate lowpass filtering.
In the next subsection, I address the problem of estimating the local
slope with filters having form (). Estimating
the slope is a necessary step for applying the finitedifference
planewave filters on real data.
Next: Slope estimation
Up: Regularizing local plane waves
Previous: Regularizing local plane waves
Stanford Exploration Project
12/28/2000