# Integral 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

In migration principles, we reviewed the migration process based on Kirchhoff summation. Dip-moveout correction also can be formulated as an integration process [1][2][3]. The integral DMO correction is particularly the preferred method for data with irregular spatial sampling and 3-D data with large variations in source-receiver azimuths (processing of 3-D seismic data).

The traveltime trajectory associated with the dip-moveout correction operator is given by the following equation (Section E.3):

 ${\displaystyle {\frac {y_{0}^{2}}{h^{2}}}+{\frac {\tau _{0}^{2}}{t_{n}^{2}}}=1.}$ (21)

Equation (21) describes an ellipse with the following properties (Figure 5.1-12):

1. Semi-major axis in midpoint y0 direction: a = h.
2. Semi-minor axis in time τ0 direction: b = tn.

The ellipse of equation (21) in the y0τ0 plane describes the impulse response of a dip-moveout operator applied to nonzero-offset data with offset 2h. In Figure 5.1-12, the coordinate of midpoint M is yn and y0 coordinate is referenced to yn. Note also that the maximum lateral extent of the ellipse — the aperture of the DMO operator, is equal to offset 2h. Figures 5.1-6b and 5.1-11 show the DMO ellipses associated with the impulse responses of the frequency-wavenumber and log-stretch DMO operators.

Analogous to the semicircle superposition technique for migration (migration principles), DMO correction can be viewed as mapping of an amplitude A0 at time tn on a normal-moveout-corrected trace at midpoint yn to an amplitude A1 at time τ0 on a trace at midpoint y0. The vertical excursion ΔtDMO and the horizontal excursion ΔyDMO denoted in Figure 5.1-12 are given by equations (9a, 9b). If trace spacing in the midpoint direction is Δy, then the lateral excursion is ΔyDMOy traces, the maximum excursion being hy traces.

 ${\displaystyle \Delta t_{DMO}=t_{n}-\tau _{0},}$ (9a)

 ${\displaystyle \Delta t_{DMO}=t_{n}\left(1-{\frac {1}{A}}\right).}$ (9b)

While the kinematics of the DMO correction operator is given by equation (21), the question remains as to the amplitude and the phase of the operator. Although the method was first described by Deregowski and Rocca [1], a formal derivation of integral DMO correction with amplitude preserving characteristics is given by Black [3]. Mathematical treatise of the problem is quite involved, and we shall only refer to the results of the analysis.

Rewrite equation (21) explicitly in terms of the normal-moveout-corrected time variable tn as

 ${\displaystyle t_{n}=\alpha \tau _{0},}$ (22)

where

 ${\displaystyle \alpha ={\frac {1}{\sqrt {1-{\frac {y_{0}^{2}}{h^{2}}}}}},}$ (23)

and y0h. Given the output sample time τ0 on the DMO ellipse, equation (22) gives the input sample time tn (Figure 5.1-12). The output sample value Pout(y0, τ0; h) is computed by summing over the input sample values Pin(yn, tn; h) over the DMO operator aperture

 ${\displaystyle P_{out}={\frac {\Delta y}{2\pi }}\sum \limits _{y}\left[{\frac {2\alpha ^{2}-1}{h}}{\sqrt {\tau _{0}}}\ \rho (t_{n})\ast P_{in}\right],}$ (24)

where Δy is the trace spacing in midpoint direction.

Equation (23) is adapted from Black [3] and is similar to equation (4-5) which describes the Kirchhoff summation. For the 2-D application of DMO correction, the ρ(tn) filter has an amplitude spectrum of the form ${\displaystyle {\sqrt {\omega _{n}}},}$ with ωn being the temporal frequency associated with the input time variable tn, and a phase spectrum equal to π/4.

In the integral implementation of DMO correction by Deregowski and Rocca [1], and Deregowski [2], the term α is set to unity in equation (24). Moreover, in the Liner [4] and Bleistein [5] implementation of integral DMO correction, the term 2α2 − 1 in equation (24) is replaced with α2(2α2 − 1). Nevertheless, within the context of a conventional processing sequence which includes geometric spreading correction prior to DMO correction, the amplitude scaling (2α2 − 1)/h described here preserves relative amplitudes. Other DMO amplitude scaling strategies include those suggested by Sorin and Ronen [6] and Gardner and Forel [7].

In practical implementations of the integral DMO correction, a user-defined aperture commonly is imposed on the operator to avoid aliasing along the steep flanks of the DMO ellipse, especially at late times. When the DMO ellipse is truncated before it reaches its fullest lateral extent, amplitude distribution along the elliptic trajectory is adjusted accordingly, such that the amplitude is tapered to zero at the truncation point on the ellipse. Figure 5.1-13 shows the impulse responses of an integral DMO operator based on equation (24) for 1000-m, 2000-m, and 3000-m offsets. Compare with the impulse responses in Figure 5.1-11 and note the truncation at steep flanks of the DMO ellipses. The problem of spatial aliasing due to undersampling and the adverse effect of irregular sampling on DMO correction are particularly relevant for 3-D data; as such, these issues will be dealt with in processing of 3-D seismic data.

## References

1. Deregowski and Rocca (1981), Deregowski, S. M. and Rocca, F., 1981, Geometrical optics and wave theory for constant-offset sections in layered media: Geophys. Prosp., 29, 374–406.
2. Deregowski, 1987, Deregowski, S. M., 1987, An integral implementation of dip-moveout: Geophys. Trans. of Geophys. Inst. of Hungary, 33, 11–22.
3. Black et al. (1993), Black, J., Schleicher, K. L. and Zhang, L., 1993, True-amplitude imaging and dip moveout: Geophysics, 58, 47–66.
4. Liner, 1990, Liner, C. L., 1990, General theory and comparative anatomy of dip-moveout: Geophysics, 55, 595–607.
5. Bleistein (1990), Bleistein, N., 1990, Born DMO revisited: 60th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1366–1369.
6. Sorin and Ronen (1989), Sorin, V. and Ronen, J., 1989, Ray-geometrical analysis of dip moveout amplitude distribution: Geophysics, 54, 1333–1335.
7. Gardner and Forel (1990), Gardner, G. F. H. and Forel, D., 1990, Amplitude preservation equations for DMO: Geophysics, 55, 485–487.