DSR for anisotropic media

The 3-D prestack DSR operator in frequency and wave number domain is (, )

$$\begin{eqnarray} DSR(K_{m},K_{h},z,\omega)=\sqrt{\frac{w^2}{v_{(mS,z)}^2} + \fra... ...v_{(mG,z)}^2} + \frac{1}{4}[(K_{mx}+K_{hx})^2+(K_{my}+K_{hy})^2]},\end{eqnarray}$$ (34)

where K_mx is the CDP in-line wavenumber, K_my is the CDP cross-line wavenumber, K_hx is the offset in-line wavenumber, K_hy 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 (V_S0=0) and that the dispersion relation for the 3-D prestack DSR can be rewritten in a more general equation as follows:

$$\begin{eqnarray} &&DSR(K_{m},K_{h},z,\omega)=K_{z}= \nonumber \\ && \frac{v_{(m... ... (\frac{v_{(mG,z)}}{w})^2[(K_{mx}+K_{hx})^2+(K_{my}+K_{hy})^2]})},\end{eqnarray}$$ (35)

where v_v(mS,v) and v_v(mG,v) are the vertical velocity as a function of the shot and receiver (survey geometry). Notice that if ${\delta=0}$ the vertical velocity (V_v) is equal to the migration velocity, there is no problem plotting the results in the depth coordinate.

Equation () is expressed in pseudo-depth by the following equation:  

$$K_{z}=K_{\tau}\frac{1}{v_{v}}.$$ (36)

Rewriting the DSR equation equation () in pseudo-depth ($\tau$) for the 2-D case, we have :  

$$K_{\tau}= \sqrt{w^2 -\frac{v^2(K_{mx}-K_{hx})^2}{1-2\frac{v^... ...v^2(K_{mx}+K_{hx})^2}{1-2\frac{v^2}{w^2}\eta(K_{mx}+K_{hx})^2}}$$ (37)

Applying the split-step approximation to equation (), gives the DSR used in a 3-D prestack depth migration () as follows:

$$\begin{eqnarray} &&DSR(k_{m},k_{h},z,\omega) \cong \nonumber \\ &&\frac{v_{(mS,z... ...ft(\frac{\omega}{v_{(mG,z)}} - \frac{\omega}{v_{Gref(z)}}\right)].\end{eqnarray}$$ (38)

where v_Sref(z) and v_Gref(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 ().

Equation () 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).

(), assuming that the vertical S-wave velocity is equal to zero v_S0=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:  

$$\eta= \frac{\epsilon - \delta}{1 + 2\delta},$$ (39)

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

The variable $\eta$ is expressed as a function of the NMO velocity (V_NMO) as follows:  

$$\eta= \frac{1}{2} \frac{v_{h}^2}{V_{NMO}^2} -1,$$ (40)

where the v_h is the horizontal velocity. For isotropic medium, ${\eta=0}$ and V_NMO=v_v, where V_NMO is short spread NMO velocity. This parameter $\eta$ contains the information about the ratio between the horizontal velocity v_h and vertical velocity.