# The slant-stack transform

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 |

In the 2-D Fourier transform, we learned that the 2-D Fourier transform is one way to decompose a wavefield into its plane-wave components, each with a unique frequency and each traveling at a unique angle from the vertical direction. In this section, we shall discuss the domain of ray parameter and present another way to decompose the wavefield into its plane-wave components.

Plane-wave decomposition of a wavefield, such as a common-shot gather, can be achieved by applying linear moveout and summing amplitudes over the offset axis. This procedure is called *slant stacking*. An underlying assumption of slant stacking is that of a horizontally layered earth model. Conventional processing is done primarily in midpoint-offset coordinates. Slant stacking replaces the offset axis with the ray parameter *p* axis. The ray parameter is the inverse of the horizontal phase velocity. A group of traces with a range of *p* values is called a slant-stack gather.

Several processing techniques have been devised, with varying degrees of success, in midpoint-ray-parameter coordinates. Examples include trace interpolation, multiple attenuation, time-variant dip filtering, refraction inversion, velocity analysis, migration before stack, and deconvolution. Taner ^{[1]} was the first to introduce the midpoint-ray-parameter coordinates. He discussed the interpretive use of plane-wave stacks, where several constant *p* sections are superpositioned over a restricted range of *p* values to enhance dipping events. Other processing methods were investigated later, such as migration ^{[2]} and velocity analysis ^{[3]} ^{[4]} and ^{[5]}. Alam and Lasocki ^{[6]} and Alam and Austin ^{[7]} discussed possible applications in trace interpolation and multiple attenuation, respectively. McMechan and Yedlin ^{[8]} devised a method to obtain phase velocity curves for dispersive waves using slant-stack transformation (Section F.1). Clayton and McMechan ^{[9]} devised a method to invert a refracted wavefield, which involves downward continuation in the slant-stack domain (Section F.2). Based on downward continuation of a slant-stack gather, Schultz (1982) developed a technique to estimate interval velocities.

## References

- ↑ Taner (1977), Taner, M. T., 1977, Simplan — plane-wave stacking: Presented at the 37th Ann. Eur. Assoc. Explor. Geophys. Mtg.
- ↑ Ottolini, 1982, Ottolini, R., 1982, Migration of seismic data in angle-midpoint coordinates: Ph.D. thesis, Stanford University.
- ↑ Schultz and Claerbout, 1978, Schultz, P. S. and Claerbout, J. F., 1978, Velocity estimation by wavefront synthesis: Geophysics, 43, 691–712.
- ↑ Diebold and Stoffa, 1981, Diebold, J. B. and Stoffa, P. L., 1981, The traveltime equation,
*τ − p*mapping and inversion of common midpoint data: Geophysics, 46, 238–254. - ↑ Gonzalez-Serrano, 1982, Gonzalez-Serrano, A., 1982, Wave-equation velocity analysis: Ph.D. thesis, Stanford University.
- ↑ Alam and Lasocki (1981), Alam, A. and Lasocki, L., 1981, Slant stack and applications: Presented at the 41st Ann. Eur. Assoc. Expl. Geophys. Mtg.
- ↑ Alam and Austin (1981), Alam, A. and Austin, J., 1981, Multiple attenuation using slant stacks: Tech. Rep., Western Geophysical Company.
- ↑ McMechan and Yedlin (1981), McMechan, G. and Yedlin, M., 1981, Analysis of dispersive waves by wavefield transformation: Geophysics, 46, 869–879.
- ↑ Clayton and McMechan (1981), Clayton, R. W. and McMechan, G., 1981, Inversion of refraction data by wave-field extrapolation: Geophysics, 46, 860–868.

## See also

- Introduction to noise and multiple attenuation
- Multiple attenuation in the CMP domain
- Frequency-wavenumber filtering
- The radon transform
- Linear uncorrelated noise attenuation
- Exercises
- Multichannel filtering techniques for noise and multiple attenuation