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:

*t*_{sg} = *t*_{s}+*t*_{g},

(4) |

(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

(6) | ||

(7) | ||

(8) | ||

(9) |

The inverse operator is calculated in the same way as the forward operator,
but now we must calculate *t*_{n} or the total traveltime *t*_{sg} instead of *t _{0}*, which
is known.
Subsequently,

To build the AMO operator, the output of the forward
3-DMO operator *t _{0}*(

*t*_{AMO}[*t _{0}*(

*x*_{AMO}(*t*_{n},*p*_{x},*p*_{y})=*x _{0}*(

*y*_{AMO}(*t*_{n},*p*_{x},*p*_{y})=*y _{0}*(

Figure 3

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.

8/21/1998