# Log-stretch DMO correction

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 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.

 ${\displaystyle P_{0}\left(k_{y},\omega _{0};h\right)=\int {\frac {2A^{2}-1}{A^{3}}}\times P_{n}\left(k_{y},t_{n};h\right){\rm {exp}}\left(-i\omega _{0}t_{n}A\right)dt_{n}.}$ (14a)

 ${\displaystyle \tau _{0}={\frac {t_{n}}{A}}}$ (12b)

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

 ${\displaystyle T_{0}={\rm {ln}}\ \tau _{0},}$ (15a)

and

 ${\displaystyle T_{n}={\rm {ln}}\ t_{n},}$ (15b)

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

 ${\displaystyle \tau _{0}=e^{T_{0}},}$ (16a)

and

 ${\displaystyle t_{n}=e^{T_{n}}.}$ (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

 ${\displaystyle T_{0}=T_{n}-{\rm {ln}}\ A_{e},}$ (17a)

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

 ${\displaystyle y_{0}=y_{n}-{\frac {h^{2}k_{y}}{A_{e}\Omega _{0}}},}$ (17b)

where

 ${\displaystyle A_{e}={\sqrt {1+{\frac {h^{2}k_{y}^{2}}{\Omega _{0}^{2}}}}}.}$ (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.

 ${\displaystyle y_{0}=y_{n}-{\frac {h^{2}k_{y}}{t_{n}A\omega _{0}}},}$ (12a)

 ${\displaystyle A={\sqrt {1+{\frac {h^{2}k_{y}^{2}}{t_{n}^{2}\omega _{0}^{2}}}}}.}$ (13)

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

 ${\displaystyle P_{0}\left(k_{y},\Omega _{0};h\right)={\rm {exp}}\left(-i{\frac {h^{2}k_{y}^{2}}{A_{e}\Omega _{0}}}+i\Omega _{0}{\rm {ln}}\ A_{e}\right)P_{n}\left(k_{y},\Omega _{0};h\right).}$ (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:

 ${\displaystyle P_{0}\left(k_{y},\Omega _{0};h\right)={\rm {exp}}\left[i\Omega _{0}\left({\sqrt {1+{\frac {h^{2}k_{y}^{2}}{\Omega _{0}^{2}}}}}-1\right)\right]P_{n}\left(k_{y},\Omega _{0};h\right).}$ (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

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