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High-order plane-wave destructors

Following the physical model of local plane waves, we can define the mathematical basis of the plane-wave destructor filters as the local plane differential equation  
 \frac{\partial P}{\partial x} +
 \sigma\,\frac{\partial P}{\partial t} = 0\;,\end{displaymath} (98)
where P(t,x) is the wave field, and $\sigma$ 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  
 P(t,x) = f(t - \sigma x)\;,\end{displaymath} (99)
where f(t) is an arbitrary waveform. Equation ([*]) is nothing more than a mathematical description of a plane wave.

If the slope $\sigma$ does not depend on t, we can transform equation ([*]) to the frequency domain, where it takes the form of the ordinary differential equation  
 \frac{d \hat{P}}{d x} +
 i \omega\,\sigma\, \hat{P} = 0\end{displaymath} (100)
and has the general solution  
 \hat{P} (x) = \hat{P} (0)\,e^{i \omega\,\sigma x}\;,\end{displaymath} (101)
where $\hat{P}$ is the Fourier transform of P. The complex exponential term in equation ([*]) simply represents a shift of a t-trace according to the slope $\sigma$ and the trace separation x.

In the frequency domain, the operator for transforming the trace at position x-1 to the neighboring trace at position x is a multiplication by $e^{i \omega\,\sigma}$. In other words, a plane wave can be perfectly predicted by a two-term prediction-error filter in the F-X domain:  
 a_0 \, \hat{P} (x) + a_1\, \hat{P} (x-1) = 0\;,\end{displaymath} (102)
where a0 = 1 and $a_1 = - e^{-i \omega\,\sigma}$. The goal of predicting several plane waves can be accomplished by cascading several two-term filters. In fact, any F-X prediction-error filter represented in the Z-transform notation as  
 A(Z_x) = 1 + a_1 Z_x + a_2 Z_x^2 + \cdots + a_N Z_x^N\end{displaymath} (103)
can be factored into a product of two-term filters:  
 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} (104)
where $Z_1,Z_2,\ldots,Z_N$ 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 time-varying slopes, we need to return to the time domain and look for an appropriate analog of the phase-shift operator ([*]) and the plane-prediction filter ([*]). An important property of plane-wave propagation across different traces is that the total energy of the transmitted wave stays invariant throughout the process. This property is assured in the frequency-domain solution ([*]) by the fact that the spectrum of the complex exponential $e^{i \omega\,\sigma}$ is equal to one. In the time domain, we can reach an equivalent effect by using an all-pass digital filter. In the Z-transform notation, convolution with an all-pass filter takes the form  
\hat{P}_{x+1}(Z_t) = \hat{P}_{x} (Z_t) \frac{B(Z_t)}{B(1/Z_t)}\;,\end{displaymath} (105)
where $\hat{P}_x (Z_t)$ denotes the Z-transform of the corresponding trace, and the ratio B(Zt)/B(1/Zt) is an all-pass digital filter approximating the time-shift operator ([*]). In finite-difference terms, equation ([*]) represents an implicit finite-difference scheme for solving equation ([*]) with the initial conditions at a constant x. The coefficients of filter B(Zt) can be determined, for example, by fitting the filter frequency response at small 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  
 B_3(Z_t) = 
 \frac{(1-\sigma)(2-\sigma)}{12}\,Z_t^{-1} + 
 ...2+\sigma)(2-\sigma)}{6} +
 \frac{(1+\sigma)(2+\sigma)}{12}\,Z_t\end{displaymath} (106)
for a three-point centered filter B3(Zt) and the expression
B_5(Z_t) & = & 
 ..._t +
for a five-point centered filter B5(Zt). It is easy to generalize these expressions to longer filters. Figure [*] shows the phase of the all-pass filters B3(Zt)/B3(1/Zt) and B5(Zt)/B5(1/Zt) for two values of the slope $\sigma$ 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.

Figure 9
Phase of the implicit finite-difference shift operators in comparison with the exact solution. The left plot corresponds to $\sigma=0.5$, the right plot to $\sigma=0.8$.
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In two dimensions, equation ([*]) transforms to the prediction equation analogous to ([*]) with the 2-D prediction filter  
 A(Z_t,Z_x) = 1 - Z_x \frac{B(1/Z_t)}{B(Z_t)}\;.\end{displaymath} (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(Zt,Zx), namely the filter  
 C(Z_t,Z_x) = A(Z_t,Z_x) B(Z_t) = B(Z_t) - Z_x B(1/Z_t)\;,\end{displaymath} (109)
which avoids the need for polynomial division. In case of the 3-point filter ([*]), the 2-D 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 high-frequency oscillations in the solution, resulting from the near-Nyquist zeroes of the polynomial B(Zt). The oscillations are easily removed in practice with appropriate low-pass filtering.

In the next subsection, I address the problem of estimating the local slope $\sigma$ with filters having form ([*]). Estimating the slope is a necessary step for applying the finite-difference plane-wave filters on real data.

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