Seislet transform and seislet frame |

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.

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)

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

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:

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

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

Taking both dimensions into consideration,
equation A-8 transforms to the prediction equation
analogous to A-5 with the 2-D prediction filter

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 |

2013-07-26