# Difference between revisions of "The slant-stack transform"

(added page) |
(added links) |
||

Line 19: | Line 19: | ||

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. | 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 <ref name=ch06r41>Taner (1977), Taner, M. T., 1977, Simplan — plane-wave stacking: Presented at the 37th Ann. Eur. Assoc. Explor. Geophys. Mtg.</ref> 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]] <ref name=ch06r30>Ottolini, 1982, Ottolini, R., 1982, [[Migration]] of seismic data in angle-midpoint coordinates: Ph.D. thesis, Stanford University.</ref> and [[velocity analysis]] <ref name=ch06r38>Schultz and Claerbout, 1978, Schultz, P. S. and Claerbout, J. F., 1978, Velocity estimation by wavefront synthesis: Geophysics, 43, 691–712.</ref> <ref name=ch06r11>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.</ref> and <ref name=ch06r17>Gonzalez-Serrano, 1982, Gonzalez-Serrano, A., 1982, Wave-equation [[velocity analysis]]: Ph.D. thesis, Stanford University.</ref>. Alam and Lasocki <ref name=ch06r2>Alam and Lasocki (1981), Alam, A. and Lasocki, L., 1981, Slant stack and applications: Presented at the 41st Ann. Eur. Assoc. Expl. Geophys. Mtg.</ref> and Alam and Austin <ref name=ch06r1>Alam and Austin (1981), Alam, A. and Austin, J., 1981, [[Multiple attenuation]] using slant stacks: Tech. Rep., Western Geophysical Company.</ref> discussed possible applications in [[trace interpolation]] and [[multiple attenuation]], respectively. McMechan and Yedlin <ref name=ch06r28>McMechan and Yedlin (1981), McMechan, G. and Yedlin, M., 1981, Analysis of dispersive waves by wavefield transformation: Geophysics, 46, 869–879.</ref> devised a method to obtain phase velocity curves for dispersive waves using [[slant-stack transformation]] (Section F.1). Clayton and McMechan <ref name=ch06r9>Clayton and McMechan (1981), Clayton, R. W. and McMechan, G., 1981, Inversion of refraction data by wave-field extrapolation: Geophysics, 46, 860–868.</ref> 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. | + | 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 <ref name=ch06r41>Taner (1977), Taner, M. T., 1977, Simplan — plane-wave stacking: Presented at the 37th Ann. Eur. Assoc. Explor. Geophys. Mtg.</ref> 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]] <ref name=ch06r30>Ottolini, 1982, Ottolini, R., 1982, [[Migration]] of seismic data in angle-midpoint coordinates: Ph.D. thesis, Stanford University.</ref> and [[velocity analysis]] <ref name=ch06r38>Schultz and Claerbout, 1978, Schultz, P. S. and Claerbout, J. F., 1978, Velocity estimation by wavefront synthesis: Geophysics, 43, 691–712.</ref> <ref name=ch06r11>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.</ref> and <ref name=ch06r17>Gonzalez-Serrano, 1982, Gonzalez-Serrano, A., 1982, Wave-equation [[velocity analysis]]: Ph.D. thesis, Stanford University.</ref>. Alam and Lasocki <ref name=ch06r2>Alam and Lasocki (1981), Alam, A. and Lasocki, L., 1981, Slant stack and applications: Presented at the 41st Ann. Eur. Assoc. Expl. Geophys. Mtg.</ref> and Alam and Austin <ref name=ch06r1>Alam and Austin (1981), Alam, A. and Austin, J., 1981, [[Multiple attenuation]] using slant stacks: Tech. Rep., Western Geophysical Company.</ref> discussed possible applications in [[trace interpolation]] and [[multiple attenuation]], respectively. McMechan and Yedlin <ref name=ch06r28>McMechan and Yedlin (1981), McMechan, G. and Yedlin, M., 1981, Analysis of dispersive waves by wavefield transformation: Geophysics, 46, 869–879.</ref> devised a method to obtain phase velocity curves for dispersive waves using [[slant-stack transformation]] ([[Multichannel filtering techniques for noise and multiple attenuation#F.1 Analysis of guided waves|Section F.1]]). Clayton and McMechan <ref name=ch06r9>Clayton and McMechan (1981), Clayton, R. W. and McMechan, G., 1981, Inversion of refraction data by wave-field extrapolation: Geophysics, 46, 860–868.</ref> devised a method to invert a refracted wavefield, which involves [[downward continuation]] in the slant-stack domain ([[Multichannel filtering techniques for noise and multiple attenuation#F.2 Wavefield extrapolation in the τ − p domain|Section F.2]]). Based on [[downward continuation]] of a slant-stack gather, Schultz (1982) developed a technique to estimate interval velocities. |

==References== | ==References== |

## Latest revision as of 11:40, 26 September 2014

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