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In anisotropic media, when the reflector is dipping with
respect to the normal to the isotropic axis of symmetry
(horizontal direction for VTI) the incident and reflected
aperture angle differ.
This difference is caused by the fact that,
although the phase slowness is function of the propagation angle,
Snell law requires that the
components parallel to the reflector
of the incident and reflected slowness vectors
must match at the interface.
However, we can still define an ``average'' aperture angle and ``average'' dip angle using the following relationships:
| |
(8) |

where the and are the
phase angles of the downgoing and upgoing plane waves,
respectively.

**cig-aniso-v2
**

Figure 1
Sketch representing the reflection of a plane wave in an anisotropic medium.

Figure shows the geometric
interpretation of these angles.
Notice that the average dip angle
is different from the
true geological dip angle ,and that the average aperture angle is
obviously different from the true
aperture angles and .However, the five angles are related
and, if needed, the true angles can be derived
from the average angles
Rosales and Biondi (2005).

The transformation to the angle domain transforms
the prestack image from the migrated subsurface offset domain ,to the angle domain by a slant stack transform.
The transformation axis is thus the physical dip
of the image along the subsurface offset;
that is,
.The dip angles can be similarly related to the midpoint dips
in the image;
that is,
.Following
the derivation of acoustic isotropic ADCIGs
by Sava and Fomel (2003)
and of converted-waves ADCIGs by
Rosales and Rickett (2001),
we can write the following relationships between
the propagation angles and the derivative
measured from the wavefield:

| |
(9) |

| (10) |

| (11) |

where and are
the phase slownesses for the source and receiver
wavefields, respectively.
We obtain the expression for the offset dip
by taking the ratio of equation 11
with equation 9,
and similarly for the midpoint dips
by taking the ratio of equation 10
with equation 9,
and after some algebraic manipulations,
we obtain the following expressions:
| |
(12) |

| (13) |

In contrast with the equivalent relationships valid
for isotropic media,
these relationships depend on both the aperture angle and the dip angle .The expression for the offset dip
(equation 9)
simplifies into the known relationship valid in isotropic media
when either the difference between the phase slownesses is zero,
or the dip angle is zero.
In VTI media this happens for flat geological dips.
In a general TI medium this condition is fulfilled when the geological
dip is normal to the axis of symmetry.
Solving for and we obtain the following:

| |
(14) |

| (15) |

where for convenience I substituted the symbol for the ``normalized slowness difference''
.
Substituting
equation 15
in equation 14,
and
equation 14
into equation 15,
we get the following two quadratic
expressions that can be solved to estimate
the angles as a function
of the dips measured from the image:

| |
(16) |

| (17) |

These are two independent quadratic equations in
and that can be solved independently.
If the ``normalized slowness difference''
between the slowness
along the propagation directions of the source and receiver wavefields
are known,
we can directly compute and ,and then the true and .One important case in this category is when we image converted waves.
For anisotropic velocities, the slownesses depend on the propagation
angles, and thus the normalized difference depends on
the unknown and .In practice, these equations can be solved by
a simple iterative process that starts by
assuming the ``normalized difference'' to be equal to zero.
In all numerical test I conducted
this iterative process converges to the correct solution in
only a few iterations, and thus is not computationally demanding.

The dependency of equations 16 and 17
from the slowness function is also an impediment to the
use of efficient Fourier-domain methods to perform
the transformation to angle domain,
because the slowness function cannot be assumed to be constant.
Fortunately,
the numerical examples shown below indicate
that for practical values of the anisotropy parameters
the dependency of the estimate from the dip angles
can be safely ignored for small dips,
and it is unlikely to constitute a problem for steep dips.

** Next:** Kinematic analysis of ADCIGs
** Up:** Biondi: Anistropic ADCIGs
** Previous:** Anisotropic parameters used for
Stanford Exploration Project

5/3/2005