# The parabolic Radon 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 |

An alternative to stretching in the time direction to attain the linear form of the Radon transform (equations **13a**, **13b**) is given by Hampson ^{[1]}. First, the input CMP gather is NMO corrected using the hyperbolic moveout equation

**(**)

where *t _{n}* is the time after NMO correction, and

*v*is the hyperbolic moveout correction velocity function. Resulting moveouts of the events, which were originally hyperbolic, are now

_{n}*approximately*parabolic:

**(**)

where *τ* is the two-way zero-offset time, and *q* is the parameter that defines the curvature of the parabola.

In the coordinates of the NMO-corrected gather *d*(*h, t _{n}*), equations (

**10a**,

**10b**) take the forms

**(**)

and

**(**)

The strategy for computing the hyperbolic Radon transform by performing least-squares minimization for each frequency component of the input data in the stretched coordinates also applies to the moveout-corrected data to compute the parabolic radon transform. Fourier transform the moveout-corrected CMP gather *d*(*h, t _{n}*) in the direction of the moveout correction time variable

*t*. Correspondingly, apply Fourier transform to equation (

_{n}**18b**) with respect to

*t*to obtain

_{n}

**(**)

where *ω _{n}* is the Fourier dual of

*t*.

_{n}For each *ω _{n}*, define

**d**′ :

*d′*(

*h, ω*) and

_{n}**u**:

*u*(

*q, ω*) as complex vectors in

_{n}*h*and

*q*, respectively. With the new definitions of

**d**′ and

**u**, equation (

**19**) can then be written in the matrix form of equation (

**15**), where

**L**now is a complex matrix given by equation (

**F-29**) of Section F.3. The solution that minimizes the least-squares error vector

**e**:

*e*(

*h, ω*), now defined in the

_{n}*ω*domain, accordingly, is given by equation (

_{n}**16**).

The complex vectors **d**′ and **u** have lengths *n _{h}*, the number of offsets, and

*n*, the number of constant

_{q}*q*values used in the transform defined by equation (

**19**), respectively. The complex matrix

**L**then has dimensions

*n*. For a typical field data set,

_{h}× n_{q}*n*= 60 and

_{h}*n*= 60; hence, the complex matrix

_{q}**L**may have dimensions of 60 × 60. As for equation (

**13b**), instead of solving one single problem using equation (

**18b**) in the moveout correction time

*t′*domain that involves a very large matrix, we solve

*n*problems, where

_{ω}*n*is the number of frequencies

_{ω}*ω*, in the Fourier transform domain using equation (

_{n}**19**) involving a small matrix

**L**of equation (

**F-29**).

We now outline the velocity-stack processing based on the discrete parabolic radon transform ^{[1]}.

- Start with a CMP gather,
*d*(*h, t*) and apply NMO correction,*d*(*h, t*)._{n} - Fourier transform in the
*t*direction,_{n}*d*(*h, ω*)._{n} - For each
*ω*, set up the_{n}**L**matrix (equation**F-29**) based on the geometry of the CMP gather and solve for**u**of equation (**16**) using the singular-value decomposition (Section F.3). - Inverse Fourier transform to get
*u*(*q, τ*), the parabolic radon transform. - Perform a desired operation, such as muting the zone of multiples, in the parabolic radon transform domain.
- Perform inverse mapping back to the offset domain to get the modeled NMO-corrected CMP gather
*d′*(*h, t*). During this inverse mapping, multiples, primaries, or all of the hyperbolic events can be modeled._{n} - Undo the moveout correction to get the modeled CMP gather
*d′*(*h, t*).

As events on the NMO-corrected CMP gather deviate from the ideal parabolic form, there can be degradation in the ability to map those events into the Radon-transform domain (*q, τ*) ^{[1]}. Moreover, stretch muting that is normally required after NMO correction can remove the far-offset data significantly. The *t*^{2}-stretching circumvents these shortcomings and replaces the moveout correction of the CMP gather. In practice, both hyperbolic and parabolic schemes in various forms are used to attenuate multiples.

## Other equations

**(**)

**(**)

**(**)

**(**)

**(**)

**(**)

## References

- ↑
^{1.0}^{1.1}^{1.2}Hampson (1986), Hampson, D., 1986, Inverse velocity stacking for multiple elimination: J. Can. Soc. Expl. Geophys., 22, 44–55.

## See also

- Velocity-stack transformation
- The discrete radon transform
- Practical considerations
- Impulse response of the velocity-stack operator
- Field data examples
- Radon-transform multiple attenuation