A transversely isotropic medium with a tilted symmetry axis normal to the reflector |

If we restrict the observation to the immediate vicinity of the
reflection point, which means that we consider the moveout surface in
a small range of lags, we can approximate the typical irregular
wavefront in complex media by a plane, although the shapes of
wavefronts are arbitrary in heterogeneous media. Following the
derivation of Yang and Sava (2009) and using the geometry shown
in Figure 2, the source and receiver plane waves are described
by:

where and are the unit direction vectors of the source and receiver plane waves, respectively, is the unit vector orthogonal to the reflector at the image point, and vector indicates the image point position. is defined as the phase velocity in the locally homogeneous medium around the reflection point, and thus it is identical for both wavefields. is half the scattering angle (reflection angle).

We can also obtain the shifted source and receiver plane waves by
introducing the space- and time-lags

Solving the system of equations 10-11 leads to the expression

which characterizes the moveout function (surface) of space- and time-lags at a common-image point.

Furthermore, we have the following relations for the reflection
geometry:

where and are unit vectors normal and parallel to the reflection plane, respectively, and is the reflection angle. Vector characterizes the line representing the intersection of the reflection and the reflector planes. Combining equations 12-14, we obtain the moveout function for plane waves:

The quantity is defined as

(16) |

and represents the depth of the reflection corresponding to the chosen 2common-image gather (CIG) location. This quantity is invariant for different plane waves, thus assumed constant here. The vector is along the Earth's surface given by .

When incorrect velocity is used for imaging, and thus, an inaccurate reflection angle is assumed, based on the analysis in the preceding section, we can obtain the moveout function

where is the focusing depth of the corresponding reflection point, is the migration velocity, is the focusing error, and are vectors normal and parallel to the migrated reflector, respectively. Equation 17 describes the extended images moveout for a single seismic experiment and it is essentially identical to the similar formula obtained by Yang and Sava (2009) for isotropic media, but for using the phase velocity instead of the isotropic velocity.

A transversely isotropic medium with a tilted symmetry axis normal to the reflector |

2013-04-02