DSR for anisotropic media

The 3-D prestack DSR operator in frequency and wave number domain is Biondi and Palacharla (1996); Claerbout (1985)

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

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:

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

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

$$K_{z}=K_{\tau}\frac{1}{v_{v}}.$$ (3)

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

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

$$\begin{eqnarray} &&DSR(k_{m},k_{h},z,\omega) \cong \nonumber \\ &&\frac{v_{(mS,z... ...ft(\frac{\omega}{v_{(mG,z)}} - \frac{\omega}{v_{Gref(z)}}\right)].\end{eqnarray}$$ (5)

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 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 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:  

$$\eta= \frac{\epsilon - \delta}{1 + 2\delta},$$ (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 (V_NMO) as follows:  

$$\eta= \frac{1}{2} \frac{v_{h}^2}{V_{NMO}^2} -1,$$ (7)

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.