# Time-variant dip filtering

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

The slant-stack domain is convenient for implementing dip filtering. To illustrate this, consider the problem of attenuating the strong ground roll in the field record in Figure 6.3-20a, which was obtained from a walkaway noise test. For simplicity, ignore the backscattered Rayleigh waves, since removing them would mean computing negative p traces. Figure 6.3-20b shows the τ − p gather obtained from this field data set. Phase velocity is varied from 500 m/s to over 10 000 m/s. The ground roll generally has very low phase velocity and is mapped to the left of the solid line at 2500 m/s, labeled as E in Figure 6.3-20b.

Suppose that the τ − p gather traces that contain the ground roll are used to reconstruct the t − x gather with the original offsets. The reconstructed gather shown in Figure 6.3-20c contains only the dips that we want to remove from the original wavefield. If this reconstructed gather is subtracted from the original gather (Figure 6.3-20a), the result is the dip-filtered shot record shown in Figure 6.3-20d. An alternate way to compute the dip-filtered shot record is to use, for reconstruction, the slant-stack traces that are only within zone F to the right of the solid line in Figure 6.3-20b. With either approach, the amplitudes at the boundary between the pass and reject zones, the solid line at 2500 m/s in Figure 6.3-20b, must be tapered to reduce artifacts in the reconstructed t − x gather.

Dip filtering in the slant-stack domain should be nearly equivalent to the f − k dip filtering process described in frequency-wavenumber filtering. Figure 6.2-1b shows the 2-D amplitude spectrum of the original field record of Figure 6.2-1a, which is the same data set as in Figure 6.3-20a. The reject zone is defined by the fan in Figure 6.2-1c, which is equivalent to zone E to the left of the vertical line in Figure 6.3-20b.

When compared with the slant-stack output (Figure 6.3-20d), the result of f − k dip filtering of the field data set in Figure 6.3-20e suggests basically no difference in performance. However, with the slant-stack approach, dip filtering can be applied in a time-variant manner. This means that the boundary between the pass and reject zones need not be vertical as in Figure 6.3-20b. Also, with the slant-stack technique, we can work with data that are irregularly spaced along the offset axis. This is not the case for the f − k method of dip filtering, since the fast Fourier transform requires data with equal trace spacing. On occasion, dip filtering also is incorporated into multiple attenuation in the slant-stack domain to further eliminate multiples.

An application of time-variant dip filtering to reduce the cable truncation effects on slant-stack gathers (Figure 6.3-17) is provided by Kelamis and Mitchell [1]. Their procedure is outlined below:

1. Map a single trace in the offset domain to the slant-stack domain over a specified range of p-values (equations 4a, 4b).
2. Apply a time-varying filter by muting inner and outer portions of the slant-stack gather. The mute functions are prescribed using a velocity filter that depends on time and offset.
3. Repeat steps (a) and (b) for all offset traces in the input gather and sum the resulting slant-stack gathers.
4. Apply rho filtering to the summed slant-stack gather.
5. Following a specific process in the slant-stack domain, such as deconvolution, apply inverse linear moveout correction for a specified offset value, and sum over the p-range (equations 5a, 5b). Repeat for all offsets; the output is the slant-stack processed offset data.

 ${\displaystyle \tau =t-px,}$ (4a)

 ${\displaystyle S(p,\tau )=\sum _{x}P(x,\tau +px),}$ (4b)

 ${\displaystyle t=\tau +px.}$ (5a)

 ${\displaystyle P(x,t)=\sum _{p}S(p,t-px).}$ (5b)

## References

1. Kelamis and Mitchell (1989), Kelamis, P. G. and Mitchell, A. R., 1989, Slant-stack processing: First Break, 7, 43–54.