Applications of plane-wave destruction filters |

Following the physical model of local plane waves, we can define the
mathematical basis of the plane-wave destruction filters as the local
plane differential equation

where is an arbitrary waveform. Equation (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 (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 (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 at
position to the neighboring trace^{} and at position 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 (6). According to equation (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 (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 (4) and the plane-prediction
filter (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 (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

for a three-point centered filter and the expression

for a five-point centered filter . The derivation of equations (9-10) is detailed in the appendix. It is easy to generalize these equations to longer filters.

Figure 1 shows the phase of the all-pass filters and for two values of the slope in comparison with the exact linear function of equation (4). As expected, the phases match the exact line at low frequencies, and the accuracy of the approximation increases with the length of the filter.

phase
Phase of the implicit
finite-difference shift operators in comparison with the exact
solution. The left plot corresponds to the slope of
, the right plot
to .
Figure 1. |
---|

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

which avoids the need for polynomial division. In case of the 3-point filter (9), the 2-D filter (12) 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. When filter (12) 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 . The oscillations are easily removed in practice with appropriate low-pass filtering.

In the next section, I address the problem of estimating the local slope with filters of form (12). Estimating the slope is a necessary step for applying the finite-difference plane-wave filters on real data.

Applications of plane-wave destruction filters |

2014-03-29