For the case of a plane layered Earth, for a particular slowness to be present in the cross-correlation seismograms, then there must be ambient energy incident with that slowness. Similarly, if the incident energy has one predominant slowness then this will overwhelm the rest. In this case, a filter should be applied so the resultant seismograms contain information with all slowness values.

In this paper, we have applied a filter to cross-correlation common shot gathers in the slant-stack (linear ) domain. The transformation to the slant-stack domain is easily invertible Thorson (1984). It also represents a decomposition into plane waves of given slowness. Since this is the energy which we would like to balance, it is a natural domain to work in both practically and theoretically.

The filter that was applied in this domain tapered the r.m.s.
values of each *p* trace by a Gaussian weighting function. The width of
the Gaussian was determined by the range of slowness values desired in
the output.

For a *v*(*z*) model illuminated by plane waves from below, each plane wave
will contribute to one slowness value, and so the slant-stack
decomposition well separates the energy from each incident angle.
The left panel of Figure 12 shows
a cross-correlation seismogram generated for
single horizontal layer illuminated by plane waves from all directions
in the subsurface. However some of the plane waves were stronger than others,
so the seismogram is dominated by energy with positive *p*-values. This can
be seen in the slant-stack domain shown in the right panel.

pw.plane.b4
Cross-correlation seismograms over a single horizontal layer
with dominant energy from one incident angle. Right panel: common shot gather.
Left panel: slant-stack.
Figure 12 |

pw.plane.after
Cross-correlation seismograms over a single horizontal layer
after slowness filtering. Right panel: common shot gather.
Left panel: slant-stack.
Figure 13 |

pw.block.b4
Cross-correlation seismograms over fault block model
with dominant energy from a single source in the subsurface.
Right panel: common shot gather. Left panel: slant-stack.
Figure 14 |

pw.block.after
Cross-correlation seismograms over fault block model
after slowness filtering.
Right panel: common shot gather. Left panel: slant-stack.
Figure 15 |

The filtered slant-stack of the same data is shown in
Figure 13, along with
the resultant seismogram after inversion back to the *t*-*x* domain. The
primary and first order multiple reflection events are now more coherent.

For a model with lateral velocity variations, we expect less successful
results. Plane waves in the subsurface are not necessarily plane waves
when they reach the surface, so the slant-stack transformation does not
separate the energy as cleanly.
Similarly if the sources of the ambient noise are with in the zone of
interest, the corresponding wavefronts will be diverging when they reach the
surface, contributing to a range of *p*-values. These problems can be
seen in Figures 14 and 15, where there is
an extra dipping event towards the bottom of the gather.

It should be noted that this method of filtering will boost under-represented slownesses even if they contain no signal, in which case noise will be boosted. It would be better if a filtering scheme was designed that only boosted slownesses containing useful information.

11/11/1997