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Generating the Azimuth moveout operator in VTI media

The AMO operator in anisotropic v(z) media is constructed by cascading a forward and an inverse 3-D v(z) DMO operators. An angular transformation, equal to the desired azimuth correction, is applied to the inverse operator. Therefore, to build the AMO operator we must first build the 3-D anisotropic v(z) operator. Artley et al. (1993) suggested an approach to build a kinematically exact 3-D DMO operator in v(z) media. Alkhalifah (1997c) used the same approach to build 3-D DMO operators in VTI v(z) media. Following their approach, we construct the 3-D DMO operator by solving a system of six nonlinear equations to obtain six unknowns that include, among other things, the zero-offset time and surface position of the specular reflection point. The traveltimes are calculated and tabulated using a VTI v(z) ray tracing. Because velocity varies only vertically, each ray propagating in the subsurface is contained in a vertical plane; therefore, 2-D raytracing is sufficient to calculate the traveltimes. The total traveltime is:

tsg = ts+tg,

and therefore the gradient vector,

{\bold \bigtriangledown} t_{sg} = {\bf \bigtriangledown} t_s+{\bf \bigtriangledown} t_g = {\bf p_s}+ {\bf p_g}\end{displaymath}

has a direction that is normal to reflector dip. Because the zero-offset slowness vector ${\bf p_0}$ is also in the direction that is normal to reflector dip, then ${\bf p_0}$ is a scaled sum of the slownesses of the rays from the source ${\bf p_s}$ and receiver ${\bf p_g}$ to the specular point reflection. Therefore,  
{\bf p_0}= \lambda ({\bf p_s}+{\bf p_g}).\end{displaymath} (4)
Considering the z-component gives

p_{0z} = \lambda (p_{sz}+p_{gz}),\end{displaymath}


\lambda = \frac{p_{0z}}{p_{sz}+p_{gz}}.\end{displaymath}


p_{0z}=\cos[\theta(p_0,t_0)] s(p_0,t_0),\end{displaymath}

p_{sz}=\cos[\theta(p_s,t_s)] s(p_s,t_s),\end{displaymath}


p_{gz}=\cos[\theta(p_g,t_g)] s(p_g,t_g),\end{displaymath}

where s is the slowness and $\theta$ is the phase angle, both of which are calculated using ray tracing and tabulated as a function of rayparameter p and the traveltime t. Then  
\lambda = \frac{\cos[\theta(p_0,t_0)] s(p_0,t_0)}{\cos[\theta(p_s,t_s)] s(p_s,t_s)+\cos[\theta(p_s,t_s) s(p_s,t_s)}.\end{displaymath} (5)

Substituting equation (5) into the x-and y-components of equation (4) provides two of the six nonlinear equations needed to be solved. The other four equations are given by
0 &=& \xi (p_g,t_g)\cos\phi_g - \xi (p_s,t_s)\cos\phi_s + 2h
 ...0,t_0) - \tau (p_s,t_s)
\\  0 &=& \tau (p_0,t_0) - \tau (p_g,t_g),\end{eqnarray} (6)
Equation (6) is the requirement that the surface distances, $\xi$, along the inline component from both the source and receiver to the specular reflection point (SRP) add to equal the source-receiver offset, 2h. Equation (7) is the requirement that the distances along the crossline component to the SRP are equal for the source and receiver. Equations (8) and  (9) imply that the vertical times, $\tau$ from the source, the receiver, and the zero-offset surface positions to the SRP are equal. Both $\xi$ and $\tau$ are calculated using ray tracing and then stored in a table as a function of ray parameter p and the traveltime t.

The inverse operator is calculated in the same way as the forward operator, but now we must calculate tn or the total traveltime tsg instead of t0, which is known. Subsequently, x0 and y0 are calculated in the same way as the forward approach.

To build the AMO operator, the output of the forward 3-DMO operator t0(tn,px,py), x0(tn,px,py), and y0(tn,px,py) are inserted into the inverse 3-D DMO operator (another reference, Alkhalifah and Biondo, ). Prior to applying the inverse operator the axes are rotated with an angle given by the desired azimuth correction. The result is an AMO operator given by





where x0, and y0, correspond to the adjoint (inverse) operator in the rotated domain. The rotation angle is the azimuth correction angle. This is basically the same approach used by Alkhalifah and Biondi (1998) for isotropic v(z) media. For anisotropic media, I use anisotropic ray tracing.

Figure 3
AMO operators for VTI homogeneous media for an input and output offset of 2 km and correction to only the azimuth. The azimuth corrections from left to right are 15, 30, and 45 degrees, respectively. The velocity is 2 km/s, $\eta$=0.1, and the input NMO corrected time is 2 s.

Figure 3 shows three AMO operators that correspond to three different azimuth correction angles in a VTI homogeneous medium. From left to right, the azimuth correction angles are 15, 30, and 45 degrees, respectively. The input and output offset are the same and equal to 2 km. Though these operators have a general saddle shape, they are different from their isotropic counterparts (Figure 1). They are considerably stretched and thus do not have the vertical size that the isotropic operators have. Additional differences will be apparent later when we take a closer look at the anisotropic operators. Though the general shape of the AMO operator is practically the same between the three azimuth corrections, the size is very much dependent on the amount of azimuth correction; the larger the azimuth correction the larger the AMO operator. Clearly, for zero azimuth correction the operator reduces to a point. The size dependence of the operator on azimuth holds regardless of the medium. The shape of the operator, however, is very much independent of azimuth correction. This phenomenon holds for homogeneous as well as v(z) isotropic media Alkhalifah and Biondi (1998). As a result, we will, as Alkhalifah and Biondi (1998) did, use a single azimuth correction for most of the examples shown in this paper, that is a 30 degrees azimuth correction.

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