next up previous [pdf]

Next: Bibliography Up: Fomel and Liu: Seislet Previous: Conclusions


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)

{\frac{\partial P}{\partial x}} +
{\sigma\,\frac{\partial P}{\partial t}} = 0\;,
\end{displaymath} (19)

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 A-1 has the simple general solution
P(t,x) = f(t - \sigma x)\;,
\end{displaymath} (20)

where $f(t)$ is an arbitrary waveform. Equation A-2 is nothing more than a mathematical description of a plane wave.

If we assume that the slope $\sigma$ does not depend on $t$, we can transform equation A-1 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} (21)

and has the general solution
\hat{P} (x) = \hat{P} (0)\,e^{i \omega\,\sigma x}\;,
\end{displaymath} (22)

where $\hat{P}$ is the Fourier transform of $P$. The complex exponential term in equation A-4 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 $x-1$ to the neighboring trace $x$ 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} (23)

where $a_0 = 1$ 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} (24)

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} (25)

where $Z_1,Z_2,\ldots,Z_N$ 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 $e^{i
\omega\,\sigma}$ 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} (26)

where $\hat{P}_x (Z_t)$ denotes the $Z$-transform of the corresponding trace, and the ratio $B(Z_t)/B(1/Z_t)$ is an all-pass digital filter approximating the time-shift operator  $e^{i \omega \sigma}$. 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 $x$. The coefficients of filter $B(Z_t)$ 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

A(Z_t,Z_x) = 1 - Z_x \frac{B(Z_t)}{B(1/Z_t)}\;.
\end{displaymath} (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 $A(Z_t,Z_x)$, namely the filter
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} (28)

avoids the need for polynomial division. In case of a 3-point filter $B(Z_t)$, 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.

next up previous [pdf]

Next: Bibliography Up: Fomel and Liu: Seislet Previous: Conclusions