Suppose the data set is a collection of seismograms
uniformly sampled in space.
In other words, the data is numbers in a (*t*,*x*)-plane.
For example, the following filter
destroys any wavefront
aligned along the direction of a line containing both the ``+1''
and the ``-1''.

(22) |

(23) |

(24) |

A two-dimensional filter filter ! two-dimensional that can be a dip-rejection filter like (22) or (23) is filter ! dip-rejection

(25) |

Fitting the filter to two neighboring traces
that are identical but for a time shift, we see that
the filter coefficients (*a*,*b*,*c*,*d*,*e*) should turn out to be
something like (-1,0,0,0,0) or
(0,0,-.5,-.5, 0),
depending on the dip (stepout) of the data.
But if the two channels are not fully coherent, we expect to see
something like
(-.9,0,0,0,0) or
(0,0,-.4,-.4,0).
To find filters such as (24),
we adjust coefficients to minimize the power out
of filter shapes, as in

(26) |

With 1-dimensional filters, we think mainly of power spectra, and with 2-dimensional filters we can think of temporal spectra and spatial spectra. What is new, however, is that in two dimensions we can think of dip spectra (which is when a 2-dimensional spectrum has a particularly common form, namely when energy organizes on radial lines in the -plane). As a short (three-term) 1-dimensional filter can devour a sinusoid, we have seen that simple 2-dimensional filters can devour a small number of dips.

4/27/2004