Fast time-to-depth conversion and interval velocity estimation in the case of weak lateral variations |

imageray
The relationship between time-domain coordinates and the Cartesian depth coordinates. An example image ray with slowness vector normal to the surface travels from the source
into the subsurface. Every point along this ray is mapped to the same distance location
in the time coordinates with different corresponding traveltime
.
Figure 2. |
---|

In time domain, one operates with the time-migration velocity estimated from migration velocity analysis (Yilmaz, 2001; Fomel, 2003a,b). In a laterally homogeneous medium, corresponds theoretically to the RMS velocity:

(1) |

where we denote throughout the text. The inverse process to recover interval velocity can be done through the Dix inversion (Dix, 1955):

where the subscript is used to denote the Dix-inverted parameter. A simple conversion from to reduces then to a straightforward integration over time to obtain a map.

On the other hand, in the case of laterally heterogeneous media, Cameron et al. (2007) proved that the Dix-inverted velocity can be related to the true interval velocity by the geometrical spreading of the image rays traced telescopically from the surface as follows:

where the geometrical spreading satisfies,

Combining equations 3 and 4 gives

To solve for the interval velocity, two additional equations are needed (Cameron et al., 2007; Li and Fomel, 2015):

Equation 6 indicates that is constant along each image ray, and equation 7 denotes the eikonal equation of image ray propagation. Equations 5-7 amount to a system of PDEs that can be solved for the interval velocity as well as the maps and needed for the time-to-depth conversion process.

Fast time-to-depth conversion and interval velocity estimation in the case of weak lateral variations |

2018-11-16