# Practical aspects of slant stacking

Series | Investigations in Geophysics |
---|---|

Author | Öz Yilmaz |

DOI | http://dx.doi.org/10.1190/1.9781560801580 |

ISBN | ISBN 978-1-56080-094-1 |

Store | SEG Online Store |

First, we examine the interrelations between various domains used in seismic data processing. Consider a band-limited dipping event in the *t − x* domain as shown in Figure 6.3-13. The offset range is from 250 to 5000 m with a trace spacing of 50 m. This event is mapped along a radial line *AA′* in the *f − k* domain.

The slope of the radial line, *ω*/*k _{x}* is related to the horizontal phase velocity

*v*/sin

*θ*by the relationship

**(**)

Substitute for *p* = sin *θ*/*v* to find the relationship between the variables in the transform domain given by

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Figure 6.3-13 also shows the mapping of the dipping event to the *τ − p* domain. Note that a linear event in the *t − x* domain maps onto a point in the *τ − p* domain. Converse also is true — a linear event in the *τ − p* domain maps onto a point in the *t − x* domain.

A 1-D Fourier transform of the slant-stack traces in the time direction gives the amplitude spectrum in the *ω − p* domain, which also is shown in Figure 6.3-13. Actually, the *ω − p* plane describes the frequency dependency of horizontal phase velocity and is used in analyzing guided waves (Section F.1). The energy along the radial direction *AA′* in the *ω − k _{x}* domain is equivalent to that along the vertical direction

*BB′*in the

*ω − p*domain.

Figure 6.3-14 shows a spatially aliased dipping event. Again, as in Figure 6.3-13, the offset range is from 250 to 5000 m with a trace spacing of 50 m. The wraparound observed in the *ω − k _{x}* plane results from the inadequate spatial sampling of the event. Note that both the unaliased component, segment 1, and the aliased component, segment 2, map onto a single

*p*trace. We expect the spatially aliased part to map onto a number of negative

*p*traces. However, if this were the case, then the aliased frequency range (21 to 42 Hz) would be absent from the

*ω − p*plane in which only the positive

*p*values were included.

**Figure 6.3-16**Slant stack can be used for trace interpolation: (a) a*t − x*gather is transformed to a*τ − p*gather (c), and is reconstructed using a finer trace spacing (d). The corresponding*f − k*spectra show spatial aliasing in the original gather (b), which was eliminated after reconstruction (e).

We now outline the steps involved in slant-stack processing that includes forward and inverse *τ − p* transforms.

- Start with the offset data, apply linear moveout correction for a specified value of
*p*, and sum over offset (equations**4a**,**4b**). Repeat for a range of*p*values, the output is the slant-stack gather (Figure 6.3-8). - Apply a particular process in the slant-stack domain, such as dip filtering or deconvolution.
- Apply
*rho filtering*to the processed slant-stack gather. - Then, 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.

**(**)

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We illustrate the forward and inverse *τ − p* transforms using the synthetic CMP gather shown in Figure 6.3-15. This figure also shows the slant-stack and reconstructed CMP gather without any process applied, except the rho filter. The linear streaks labeled as *CT* on the slant-stack gather in Figure 6.3-15 are caused by the finite cable length. To minimize the streaks, for each trace in the *t − x* domain, Kelamis and Mitchell ^{[1]} limit the mapping from the *t − x* domain to the *τ − p* domain to a time-variant zone in the *τ − p* domain. Specifically, only one trace at a time from the *t − x* domain is mapped onto all *p* traces. The resulting *τ − p* gather is muted on both the low and high end of the *p*-axis in a time-varying manner. The mute functions are based on a representative primary velocity function.

During reconstruction of the *t − x* gather, we do not have to use the same trace spacing that was used for the original *t − x* gather. Consider the synthetic CMP gather in Figure 6.3-16a. The 2-D amplitude spectrum shows that frequencies above 48 Hz are spatially aliased (Figure 6.3-16b). This gather can be mapped to the slant-stack domain (Figure 6.3-16c) and reconstructed using a finer trace spacing (Figure 6.3-16d). The original trace spacing is 25 m; the reconstructed gather has a trace spacing of 12.5 m. The 2-D amplitude spectrum of the trace-interpolated gather shows that no frequencies are spatially aliased (Figure 6.3-16e). Nevertheless, note the missing high-frequency energy beyond 60 Hz. This energy mainly is along the steep direct arrival path in the input gather (Figure 6.3-16a) and is absent in the output gather (Figure 6.3-16d). We see that reconstruction can be successful even for spatially aliased data, provided dips do not have a wide range of variation.

## References

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

## See also

- Physical aspects of slant stacking
- Slant-stack transformation
- Slant-stack parameters
- Time-variant dip filtering
- Slant-stack multiple attenuation