# Anisotropic migration

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

A convenient way to understand the effect of anisotropy on migration is to refer to the zero-offset dispersion relation of the one-way scalar wave equation that describes wave propagation associated with exploding reflectors (migration principles), using the present notation,

 ${\displaystyle k_{z}={\frac {2\omega }{\alpha _{0}}}{\sqrt {1-\left({\frac {\alpha _{0}k_{y}}{2\omega }}\right)^{2}}},}$ (104)

where kz and ky are the wavenumbers associated with the depth z and midpoint y axes, ω is the temporal frequency and α is the medium velocity used to migrate a zero-offset compressional wavefield. The dispersion relation given by equation (104) is the basis from which all zero-offset finite-difference and frequency-wavenumber migration algorithms are developed (Sections D.1, D.4, and D.7). Hence, it makes good sense to examine what form this dispersion relation takes in the case of an anisotropic medium.

Accompanying the dispersion relation is the equation for wave extrapolation used in finite-difference and frequency-wavenumber algorithms (migration principles). Given a zero-offset upcoming wavefield P(y, z = 0, t) at the surface z = 0, for which a stacked section is a good approximation, the objective is to extrapolate it downward at depth steps of Δz using the extrapolation equation

 ${\displaystyle P(k_{y},z+\Delta z,\omega )=P(k_{y},z,\omega )\exp(-ik_{z}\Delta z).}$ (105)

This is then followed by invoking the imaging principle which states that the migrated section P(x, z, t = 0) at each depth z corresponds to the summation over the frequency components of the extrapolated wavefield, or equivalently collecting the extrapolated wavefield at time t = 0.

To account for the effect of anisotropy in migration, the medium velocity α0 in the dispersion relation given by equation (104) needs to be replaced with the P-wave phase velocity given by equation (84). This means that velocity used in anisotropic migration is dependent on the phase angle of the propagating wavefront (Figure 11.7-1). We shall first rewrite the dispersion relation of equation (104) explicitly in terms of the velocity

 ${\displaystyle \alpha (\theta )=\alpha _{0}(1+\delta \sin ^{2}\theta \cos ^{2}\theta +\varepsilon \sin ^{4}\theta ).}$ (84)

 ${\displaystyle {\frac {4\omega ^{2}}{k_{y}^{2}+k_{z}^{2}}}=\alpha _{0}^{2},}$ (106)

and replace α0 with the phase velocity of equation (84) specified for the reflector dip ϕ

 ${\displaystyle {\frac {4\omega ^{2}}{k_{y}^{2}+k_{z}^{2}}}=\alpha _{0}^{2}(1+\delta \sin ^{2}\phi \cos ^{2}\phi +\varepsilon \sin ^{4}\phi )^{2}.}$ (107)

Next, we shall use the relations (Section D.1)

 ${\displaystyle \sin \phi ={\frac {k_{y}}{\sqrt {k_{y}^{2}+k_{z}^{2}}}}}$ (108a)

and

 ${\displaystyle \cos \phi ={\frac {k_{z}}{\sqrt {k_{y}^{2}+k_{z}^{2}}}},}$ (108b)

for substitution into equation (106b) to get

 ${\displaystyle {\frac {4\omega ^{2}}{k_{y}^{2}+k_{z}^{2}}}=\alpha _{0}^{2}\left[1+{\frac {k_{y}^{2}(\delta k_{z}^{2}+\varepsilon k_{y}^{2})}{(k_{y}^{2}+k_{z}^{2})^{2}}}\right]^{2}.}$ (109)

By setting the anisotropy parameters ε = δ = 0, note that the anisotropic dispersion relation given by equation (109) reduces to the isotropic case given by equation (106).

A straightforward practical implementation of the anisotropic dispersion relation of equation (109) is within the framework of a frequency-wavenumber algorithm such as Stolt or phase-shift migration. By using an anisotropic dispersion relation defined for time migration, Anderson [1] have modified Fowler DMO and prestack time migration (prestack time migration) to account for anisotropy. Nevertheless, various practical forms of the dispersion relation can also be used to include the effect of anisotropy in frequency-space finite-difference explicit or implicit schemes [2] [3]. Although it is only a theoretically interesting conjecture, elliptical anisotropy may be accounted for by a vertical stretching in depth that follows an isotropic time migration [4].

The effect of anisotropy on an expanding wavefront is seen in Figure 11.7-28. Depending on the values of the Thomsen parameters, the wavefront can take various shapes, sometimes quite warped. These wavefronts can be considered as the kinematic aspect of an anisotropic migration impulse response. Note that, at certain dips, more migration takes place if anisotropy is accounted for as compared to isotropic migration. On the other hand, at some other dips, less migration may take place, depending on the anisotropy parameters.

A widely recognized case of anisotropy is from offshore West Africa. Figure 11.7-29 shows a popular example of anisotropic processing [6]. The data were processed by applying both isotropic and anisotropic DMO correction and migration. Note the better imaging of the steep fault planes by accounting for anisotropy in processing.

Note from the anisotropic dispersion relation given by equation (109) that it is important to supply an anisotropic migration algorithm with accurate estimates of the anisotropy parameters ε and δ. It is most likely safer to do isotropic migration than anisotropic migration with incorrect values for ε and δ. Alkhalifah and Larner [9] conducted some numerical studies for migration error in transversely isotropic media and concluded that, fortuitously, isotropic migration in the presence of anisotropy yields fairly accurate results for dips up to 50 degrees. For steeper dips, the isotropic assumption no longer is valid. Nevertheless, Alkhalifah and Larner [9] also caution the practitioner of the undesirable consequences of anisotropic migration using poorly estimated ε and δ parameters.

## References

1. Anderson et al. (1996), Anderson, J. E., Alkhalifah, T., and Tsvankin, I., 1996, Fowler DMO and time migration for transversely isotropic media: Geophysics, 61, 835–845.
2. Kitchenside, 1993, Kitchenside, P. W., 1993, 2-D anisotropic migration in the space-frequency domain: J. of Seis. Expl., 2, 7–22.
3. Ristow and Ruehl, 1997, Ristow, D. and Ruehl, T., 1997, Migration in transversely isotropic media using implicit operators: 67th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1699–1702.
4. Verwest, 1989, Verwest, B. J., 1989, Seismic migration in elliptically anisotropic media: Geophys. Prosp., 37, 149–166.
5. Thomsen, 1986, Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954–1966.
6. Alkhalifah et al., 1996, Alkhalifah, T., 1996, Transformation to zero offset in transversely isotropic media: Geophysics, 61, 947–963.
7. Uren et al., 1990a, Uren, N. F., Gardner, G. H. F. and McDonald, J. A., 1990a, The migrator’s equation for anisotropic media: Geophysics, 55, 1429–1434.
8. Sheriff, 1991, Sheriff, R. E., 1991, Encyclopedic dictionary of exploration geophysics: Soc. Expl. Geophys.
9. Alkhalifah and Larner (1994), Alkhalifah, T. and Larner, K., 1994, Migration error in transversely isotropic media: Geophysics, 59, 1405–1418.