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

Next: Taking weak lateral variations Up: Sripanich & Fomel: Time Previous: Introduction

# Theory

The time-domain coordinates used in time migration are related to the Cartesian depth coordinates through the knowledge of image rays (Figure 2), which have orthogonal slowness vector to the surface (Hubral, 1977). For each subsurface location , an image ray travels through the medium and emerges at with traveltime . The forward maps and can be obtained with the knowledge of the interval velocity . We can also define the inverse maps and for the time-to-depth conversion process. Similar description of coordinates relation also holds in 3D.

imageray
Figure 2.
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 .

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):

 (2)

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:

 (3)

 (4)

Combining equations 3 and 4 gives

 (5)

To solve for the interval velocity, two additional equations are needed (Cameron et al., 2007; Li and Fomel, 2015):
 (6) (7)

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.

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

Next: Taking weak lateral variations Up: Sripanich & Fomel: Time Previous: Introduction

2018-11-16