Log-stretch DMO correction

Seismic Data Analysis

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 described in this section is computationally intensive. Specifically, for each output frequency ω₀, one has to apply the phase-shift exp(−iω₀t_nA), scale by (2A² − 1)/A³, and sum the resulting output over input time t_n 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.

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

(14a)

$\tau _{0}={\frac {t_{n}}{A}}$

(12b)

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

$T_{0}={\rm {ln}}\ \tau _{0},$

(15a)

and

$T_{n}={\rm {ln}}\ t_{n},$

(15b)

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

$\tau _{0}=e^{T_{0}},$

(16a)

and

$t_{n}=e^{T_{n}}.$

(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 T_n and the output log-stretch time variable T₀ is given by

$T_{0}=T_{n}-{\rm {ln}}\ A_{e},$

(17a)

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

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

(17b)

where

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

(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.

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

(12a)

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

(13)

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

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

(19)

Note that the relationship of input P_n(k_y, Ω₀; h) to output P₀(k_y, 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 . 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_e − 1, and use of the definition for A_e given by equation (18) then lead to the following expression for DMO correction:

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

(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 P_n(y_n, h, t_n) to common-offset sections P_n(y_n, t_n; h).
3. 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_n − t_n coordinates to P_n(y_n, T_n; h) in y_n − T_n 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 P_n(k_y, Ω₀; h), and obtain the dip-moveout-corrected data P₀(k_y, 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.

See also

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

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