Seislet transform and seislet frame

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# 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.

# 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)

 (19)

where is the wave field, and 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
 (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 does not depend on , we can transform equation A-1 to the frequency domain, where it takes the form of the ordinary differential equation

 (21)

and has the general solution
 (22)

where is the Fourier transform of . The complex exponential term in equation A-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 to the neighboring trace is a multiplication by . In other words, a plane wave can be perfectly predicted by a two-term prediction-error filter in the - domain:

 (23)

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
 (24)

can be factored into a product of two-term filters:
 (25)

where 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 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

 (26)

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 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

 (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
 (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.

 Seislet transform and seislet frame

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

2013-07-26