# Log-stretch DMO correction

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 |

The frequency-wavenumber DMO correction ^{[1]}^{[2]} described in this section is computationally intensive. Specifically, for each output frequency *ω*_{0}, one has to apply the phase-shift exp(−*iω _{0}t_{n}A*), scale by (2

*A*

^{2}− 1)/

*A*

^{3}, and sum the resulting output over input time

*t*as described by equation (

_{n}**14a**). A computationally more efficient DMO correction can be formulated in the logarithmic time domain

^{[3]}

^{[4]}

^{[5]}

^{[6]}

^{[7]}. The log-stretch time variable enables linearization of the coordinate transform equation (

**12b**), and as a result, the DMO correction is achieved by a simple multiplication of the input data with a phase-shift operator in the Fourier transform domain.

**(**)

**(**)

Define the following logarithmic variables that correspond to the time variables *τ*_{0} and *t _{n}* of equation (

**12b**):

**(**)

and

**(**)

where, for convenience, a constant scalar with its unit in time is omitted. Hence, the inverse relationships are given by

**(**)

and

**(**)

Our goal is to derive equations for DMO correction in the log-stretch coordinates (*y*_{0}, *T*_{0}). The transform relation between the input log-stretch time variable *T _{n}* and the output log-stretch time variable

*T*

_{0}is given by

**(**)

and the expression for the midpoint variable *y*_{0} in the log-stretch domain is given by

**(**)

where

**(**)

The variable Ω_{0} is the Fourier transform dual of the variable *T*_{0} in the log-stretch domain. Equations (**17a**, **17b**) and (**18**) correspond to equations (**12a**, **12b**) and (**13**) in the log-stretch domain. Mathematical details of the derivation of equations (**17a**, **17b**) are left to Section E.3.

**(**)

**(**)

The log-stretch dip-moveout correction process is achieved by the following relationship (Section E.3):

**(**)

Note that the relationship of input *P _{n}*(

*k*, Ω

_{y}_{0};

*h*) to output

*P*

_{0}(

*k*, Ω

_{y}_{0};

*h*) given by equation (

**19**) computationally is much simpler than that of equation (

**14a**). The log-stretch domain implementation of DMO correction involves application of a phase-shift given by the exponential in equation (

**19**) to the input data; whereas, the frequency-wavenumber implementation involves an integral transform given by equation (

**14a**).

To circumvent the logarithmic computation, a variation of the phase-shift term in equation (**19**) is given by Notfors and Godfrey ^{[5]}. As in most log-stretch formulations of DMO correction, this reference assumes that under DMO correction the midpoint variable is invariant; hence, by way of equation (**17b**), the first term in the exponential of equation (**19**) drops out. A further approximation, In *A _{e}* =

*A*− 1, and use of the definition for

_{e}*A*given by equation (

_{e}**18**) then lead to the following expression for DMO correction:

**(**)

We now outline the steps in dip-moveout correction in the log-stretch domain:

- Start with prestack data in midpoint-offset
*y − h*coordinates,*P*(*y*,*h*,*t*) and apply normal moveout correction using a dip-independent velocity*v*. - Sort the data from moveout-corrected CMP gathers
*P*(_{n}*y*,_{n}*h*,*t*) to common-offset sections_{n}*P*(_{n}*y*,_{n}*t*;_{n}*h*). - Apply the logarithmic stretch in the time direction based on equation (
**15b**) so as to map each common-offset section*P*(_{n}*y*,_{n}*t*;_{n}*h*) in*y*coordinates to_{n}− t_{n}*P*(_{n}*y*,_{n}*T*;_{n}*h*) in*y*coordinates._{n}− T_{n} - Perform 2-D Fourier transform of each common-offset section in the log-stretch domain.
- Apply the phase-shift given by the exponential in equation (
**20**) to each common-offset section*P*(_{n}*k*, Ω_{y}_{0};*h*), and obtain the dip-moveout-corrected data*P*_{0}(*k*, Ω_{y}_{0};*h*) in the log-stretch Fourier transform domain. - Perform 2-D inverse Fourier transform to obtain the dip-moveout corrected common-offset section
*P*_{0}(*y*_{0},*T*_{0};*h*) in the log-stretch domain. - Undo the logarithmic stretch as in step (c) in the time direction based on equation (
**16a**) so as to obtain the dip-moveout-corrected data*P*(*y*_{0},*τ*_{0};*h*).

Figure 5.1-11 shows the impulse responses of a log-stretch DMO operator based on equation (**20**) for 1000-m, 2000-m and 3000-m offsets. The impulse responses greatly resemble those of the frequency-wave-number DMO correction described earlier (Figure 5.1-6b). Field data examples of DMO correction presented in this chapter mostly have been created using a log-stretch algorithm.

## References

- ↑ Hale (1984), Hale, D., 1984, Dip moveout by Fourier transform: Geophysics, 49, 741–757.
- ↑ Black et al. (1993), Black, J., Schleicher, K. L. and Zhang, L., 1993, True-amplitude imaging and dip moveout: Geophysics, 58, 47–66.
- ↑ Bolondi et al., 1982, Bolondi, G., Loinger, E. and Rocca, F., 1982, Offset continuation of seismic sections: Geophys. Prosp., 30, 813–828.
- ↑ Bale and Jacubowicz, 1987, Bale, R. and Jacubowitz, H., 1987, Poststack prestack migration: 57th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 714–717.
- ↑
^{5.0}^{5.1}Notfors and Godfrey, 1987, Marcoux, M. O., Godfrey, R. J., and Notfors, C. D., 1987, Migration for optimum velocity evaluation and stacking: Presented at the 49th Ann. Mtg. European Asn. Expl. Geophys. - ↑ Liner, 1990, Liner, C. L., 1990, General theory and comparative anatomy of dip-moveout: Geophysics, 55, 595–607.
- ↑ Zhou et al. (1996), Zhou, B, Mason, I. M., and Greenalgh, S.A., 1996, Accurate and efficient shot-gather dip-moveout processing in the log-stretch domain: Geophys. Prosp., 43, 963–978.

## See also

- Prestack partial migration
- Frequency-wavenumber DMO correction
- Integral DMO correction
- Velocity errors
- Variable velocity
- Turning-wave migration