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

Acknowledgments

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)

$\begin{displaymath} {\frac{\partial P}{\partial x}} + {\sigma\,\frac{\partial P}{\partial t}} = 0\;, \end{displaymath}$

(19)

where

is the wave field, and $\sigma$ 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

$\begin{displaymath} P(t,x) = f(t - \sigma x)\;, \end{displaymath}$

(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 $\sigma$ does not depend on , we can transform equation A-1 to the frequency domain, where it takes the form of the ordinary differential equation

$\begin{displaymath} {\frac{d \hat{P}}{d x}} + i \omega\,\sigma\, \hat{P} = 0 \end{displaymath}$

(21)

and has the general solution

$\begin{displaymath} \hat{P} (x) = \hat{P} (0)\,e^{i \omega\,\sigma x}\;, \end{displaymath}$

(22)

where $\hat{P}$ is the Fourier transform of

. The complex exponential term in equation A-4 simply represents a shift of a

-trace according to the slope $\sigma$ and the trace separation

In the frequency domain, the operator for transforming the trace to the neighboring trace 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 - domain:

$\begin{displaymath} a_0 \, \hat{P} (x) + a_1\, \hat{P} (x-1) = 0\;, \end{displaymath}$

(23)

where

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

prediction-error filter represented in the

-transform notation as

$\begin{displaymath} A(Z_x) = 1 + a_1 Z_x + a_2 Z_x^2 + \cdots + a_N Z_x^N \end{displaymath}$

(24)

can be factored into a product of two-term filters:

$\begin{displaymath} A(Z_x) = \left(1 - \frac{Z_x}{Z_1}\right)\left(1 - \frac{Z_x}{Z_2}\right) \cdots\left(1 - \frac{Z_x}{Z_N}\right)\;, \end{displaymath}$

(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 -transform notation, convolution with an all-pass filter takes the form

$\begin{displaymath} \hat{P}_{x+1}(Z_t) = \hat{P}_{x} (Z_t) \frac{B(Z_t)}{B(1/Z_t)}\;, \end{displaymath}$

(26)

where $\hat{P}_x (Z_t)$ denotes the

-transform of the corresponding trace, and the ratio

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

. 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

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

(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

$\begin{displaymath} C(Z_t,Z_x) = A(Z_t,Z_x) B(1/Z_t) = B(1/Z_t) - Z_x B(Z_t)\;, \end{displaymath}$

(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.

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

2013-03-02