Log-stretch DMO correction

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Seismic Data Analysis
Seismic-data-analysis.jpg
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 ω0, one has to apply the phase-shift exp(−0tnA), scale by (2A2 − 1)/A3, and sum the resulting output over input time tn as described by equation (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.


(14a)


(12b)

Define the following logarithmic variables that correspond to the time variables τ0 and tn of equation (12b):


(15a)

and


(15b)

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


(16a)

and


(16b)

Our goal is to derive equations for DMO correction in the log-stretch coordinates (y0, T0). The transform relation between the input log-stretch time variable Tn and the output log-stretch time variable T0 is given by


(17a)

and the expression for the midpoint variable y0 in the log-stretch domain is given by


(17b)

where


(18)

The variable Ω0 is the Fourier transform dual of the variable T0 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.


(12a)


(13)

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


(19)

Note that the relationship of input Pn(ky, Ω0; h) to output P0(ky, Ω0; 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).

Figure 5.1-11  Impulse response of a log-stretch dip-moveout operator with source-receiver (S-G) offsets (a) 0 m, (b) 1000 m, (c) 2000 m, and (d) 3000 m.

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 Ae = Ae − 1, and use of the definition for Ae given by equation (18) then lead to the following expression for DMO correction:


(20)

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

  1. 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.
  2. Sort the data from moveout-corrected CMP gathers Pn(yn, h, tn) to common-offset sections Pn(yn, tn; h).
  3. Apply the logarithmic stretch in the time direction based on equation (15b) so as to map each common-offset section Pn(yn, tn; h) in yn − tn coordinates to Pn(yn, Tn; h) in yn − Tn coordinates.
  4. Perform 2-D Fourier transform of each common-offset section in the log-stretch domain.
  5. Apply the phase-shift given by the exponential in equation (20) to each common-offset section Pn(ky, Ω0; h), and obtain the dip-moveout-corrected data P0(ky, Ω0; h) in the log-stretch Fourier transform domain.
  6. Perform 2-D inverse Fourier transform to obtain the dip-moveout corrected common-offset section P0(y0, T0; h) in the log-stretch domain.
  7. 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(y0, τ0; 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

  1. Hale (1984), Hale, D., 1984, Dip moveout by Fourier transform: Geophysics, 49, 741–757.
  2. Black et al. (1993), Black, J., Schleicher, K. L. and Zhang, L., 1993, True-amplitude imaging and dip moveout: Geophysics, 58, 47–66.
  3. Bolondi et al., 1982, Bolondi, G., Loinger, E. and Rocca, F., 1982, Offset continuation of seismic sections: Geophys. Prosp., 30, 813–828.
  4. 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. 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.
  6. Liner, 1990, Liner, C. L., 1990, General theory and comparative anatomy of dip-moveout: Geophysics, 55, 595–607.
  7. 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

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Log-stretch DMO correction
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