Time-to-depth conversion and seismic velocity estimation using time-migration velocity |

The seismic velocity and the Dix
velocity are connected through the quantity
, the
geometrical spreading of image rays.
is a scalar in 2-D
and a
matrix in 3-D. The simplest way to introduce
is the following. Trace an image ray
.
is the
starting surface point,
is the traveltime. Call this ray
*central*. Consider a small tube of rays around it. All these
rays start from a small neighborhood
of the point
perpendicular to the earth surface. Thus, they
represent a fragment of a plane wave propagating downward. Consider
the fragment of the wave front defined by this ray tube at time
.
Let
be the fragment of the tangent to the front at the
point
reached by the central ray at
time
, bounded by the ray tube (Figure 1). Then, in
2-D,
is the derivative
. In 3-D,
is the matrix of the derivatives
,
, where
derivatives are taken along certain mutually orthogonal directions
,
(Popov, 2002; Popov and Pšencik, 1978; Cervený, 2001).

Qdef
Illustration for the definition of geometrical spreading.
Figure 1. |
---|

The time evolution of the matrices and is given by

where it the velocity at the central ray at time , , and is the identity matrix. The absolute value of has a simple meaning: it is the geometrical spreading of the image rays (Popov, 2002; Popov and Pšencik, 1978; Cervený, 2001). The matrix , introduced in the previous section, relates to and as . Hence, .

In (Cameron et al., 2007), we have proven that

in 2-D, where is the time-migration velocity, and

in 3-D, is defined by equation 6 and can be determined from equation 7.

Time-to-depth conversion and seismic velocity estimation using time-migration velocity |

2013-03-02