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DSR for anisotropic media

The 3-D prestack DSR operator in frequency and wave number domain is Biondi and Palacharla (1996); Claerbout (1985)
DSR(K_{m},K_{h},z,\omega)=\sqrt{\frac{w^2}{v_{(mS,z)}^2} +
 ...v_{(mG,z)}^2} +
where Kmx is the CDP in-line wavenumber, Kmy is the CDP cross-line wavenumber, Khx is the offset in-line wavenumber, Khy is the offset cross-line wavenumber, and v(mG,z) and v(mS,z) are the velocities expressed as a function of the survey geometry.

Alkhalifah (1997) derives the dispersion equation for transversely anisotropic media. He assumes that the vertical S-wave velocity is equal to zero (VS0=0) and that the dispersion relation for the 3-D prestack DSR can be rewritten in a more general equation as follows:
&&DSR(K_{m},K_{h},z,\omega)=K_{z}= \nonumber \\  &&
where vv(mS,v) and vv(mG,v) are the vertical velocity as a function of the shot and receiver (survey geometry). Notice that if ${\delta=0}$ the vertical velocity (Vv) is equal to the migration velocity, there is no problem plotting the results in the depth coordinate.

Equation (2) is expressed in pseudo-depth by the following equation:  
K_{z}=K_{\tau}\frac{1}{v_{v}}.\end{displaymath} (3)

Rewriting the DSR equation equation (2) in pseudo-depth ($\tau$) for the 2-D case, we have :  
K_{\tau}= \sqrt{w^2
 ...v^2(K_{mx}+K_{hx})^2}{1-2\frac{v^2}{w^2}\eta(K_{mx}+K_{hx})^2}}\end{displaymath} (4)

Applying the split-step approximation to equation (2), gives the DSR used in a 3-D prestack depth migration Malcotti and Biondi (1998) as follows:
&&DSR(k_{m},k_{h},z,\omega) \cong \nonumber \\ &&\frac{v_{(mS,z...
 ...ft(\frac{\omega}{v_{(mG,z)}} -

where vSref(z) and vGref(z) are the reference velocities defined by the geometry of the survey and depth. In the case of 3-D prestack migration algorithm, this DSR is rewritten using the common-azimuth approximations Biondi and Palacharla (1996).

Equation (5) shows that in anisotropic media, the split-step approximation is obtained by assuming that ${\eta}$ is constant for every depth step. In the seismic synthetic examples that I show in the results section, I use two different approaches to migrate the anisotropic Marmousi data set. In the first approach, I use a constant ${\eta}$. In the second approach I define a number of reference ${\eta}$'s, in order to obtain a better migrated image of the dipping reflectors. This is an approach that gives impressive results but it should be generalized to handle lateral ${\eta}$ variations, where ${\eta}$ variations are the velocity variation.

Other possible solution to include lateral ${\eta}$ variation is using different ${\eta}$'s defined in the same fashion that reference velocities are defined but migrate with just one reference velocity (Alkhalifah 1998, personal communication).

Alkhalifah and Tsvankin (1994), assuming that the vertical S-wave velocity is equal to zero vS0=0, introduce an anisotropic parameter called ${\eta}$. This anisotropic parameter ${\eta}$ can be written as a function of Thomsen's parameters (${\delta}$ and ${\epsilon}$) as follows:  
\eta= \frac{\epsilon - \delta}{1 + 2\delta},\end{displaymath} (6)

In this paper I set ${\delta=0}$ to avoid working with the ratio of the vertical velocity and migration velocity in equation (5).

The variable ${\eta}$ is expressed as a function of the NMO velocity (VNMO) as follows:  
\eta= \frac{1}{2} \frac{v_{h}^2}{V_{NMO}^2} -1,\end{displaymath} (7)
where the vh is the horizontal velocity. For isotropic medium, ${\eta=0}$ and VNMO=vv, where VNMO is short spread NMO velocity. This parameter ${\eta}$ contains the information about the ratio between the horizontal velocity vh and vertical velocity.

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Next: Results Up: Malcotti: Anisotropic depth migration Previous: Introduction
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