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,
has a direction that is normal to reflector dip. Because the zero-offset slowness vector is also in the direction that is normal to reflector dip, then is a scaled sum of the slownesses of the rays from the source and receiver to the specular point reflection. Therefore,
where s is the slowness and 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
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
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
yAMO(tn,px,py)=y0(tn,px,py)-y0,(t0,px,py),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 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.