Log-stretch DMO correction
The frequency-wavenumber DMO correction  described in this section is computationally intensive. Specifically, for each output frequency ω0, one has to apply the phase-shift exp(−iω0tnA), 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   . 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 τ0 and tn of equation (12b):
where, for convenience, a constant scalar with its unit in time is omitted. Hence, the inverse relationships are given by
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
and the expression for the midpoint variable y0 in the log-stretch domain is given by
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.
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).
To circumvent the logarithmic computation, a variation of the phase-shift term in equation (19) is given by Notfors and Godfrey . 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:
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 Pn(yn, h, tn) to common-offset sections Pn(yn, tn; h).
- 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.
- 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 Pn(ky, Ω0; h), and obtain the dip-moveout-corrected data P0(ky, Ω0; h) in the log-stretch Fourier transform domain.
- Perform 2-D inverse Fourier transform to obtain the dip-moveout corrected common-offset section P0(y0, T0; 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(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.
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- Black et al. (1993), Black, J., Schleicher, K. L. and Zhang, L., 1993, True-amplitude imaging and dip moveout: Geophysics, 58, 47–66.
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- 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.
- Prestack partial migration
- Frequency-wavenumber DMO correction
- Integral DMO correction
- Velocity errors
- Variable velocity
- Turning-wave migration