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The time-domain coordinates $ (x_0,t_0)$ used in time migration are related to the Cartesian depth coordinates $ (x,z)$ through the knowledge of image rays (Figure 2), which have orthogonal slowness vector to the surface (Hubral, 1977). For each subsurface location $ (x,z)$ , an image ray travels through the medium and emerges at $ (x_0, 0)$ with traveltime $ t_0$ . The forward maps $ x_0(x,z)$ and $ t_0(x,z)$ can be obtained with the knowledge of the interval velocity $ v(x,z)$ . We can also define the inverse maps $ x(x_0,t_0)$ and $ z(x_0,t_0)$ for the time-to-depth conversion process. Similar description of coordinates relation also holds in 3D.

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 $ x_s$ into the subsurface. Every point along this ray is mapped to the same distance location $ x_s$ in the time coordinates with different corresponding traveltime $ t_s$ .
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In time domain, one operates with the time-migration velocity $ v_m(x_0,t_0)$ estimated from migration velocity analysis (Yilmaz, 2001; Fomel, 2003a,b). In a laterally homogeneous medium, $ v_m$ corresponds theoretically to the RMS velocity:

$\displaystyle w_m(t_0) = \frac{1}{t_0}\int_0^{t_0} w\big(z(t)\big) dt~,$ (1)

where we denote $ w=v^2$ throughout the text. The inverse process to recover interval velocity $ v(z)$ can be done through the Dix inversion (Dix, 1955):

$\displaystyle w_d(t_0) = \frac{d}{dt_0} \big(t_0 w_m(t_0)\big)~,$ (2)

where the subscript $ d$ is used to denote the Dix-inverted parameter. A simple conversion from $ w_d(t_0)$ to $ w(z)$ reduces then to a straightforward integration over time to obtain a $ z(t_0)$ 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 $ Q(x_0,t_0)$ of the image rays traced telescopically from the surface as follows:

$\displaystyle w_d(x_0,t_0) = \frac{w\big(x(x_0,t_0),z(x_0,t_0)\big)}{Q^2(x_0,t_0)}~,$ (3)

where the geometrical spreading $ Q$ satisfies,

$\displaystyle \nabla x_0 \cdot \nabla x_0 = \frac{1}{Q^2}~.$ (4)

Combining equations 3 and 4 gives

$\displaystyle \nabla x_0 (x,z) \cdot \nabla x_0 (x,z) = \frac{w_d(x_0(x,z),t_0(x,z))}{w(x,z)}$ (5)

To solve for the interval velocity, two additional equations are needed (Cameron et al., 2007; Li and Fomel, 2015):
$\displaystyle \nabla x_0 (x,z)\cdot \nabla t_0 (x,z)$ $\displaystyle =$ $\displaystyle 0~,$ (6)
$\displaystyle \nabla t_0 (x,z) \cdot \nabla t_0 (x,z)$ $\displaystyle =$ $\displaystyle \frac{1}{w(x,z)}~.$ (7)

Equation 6 indicates that $ x_0$ 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 $ v(x,z)$ as well as the maps $ x(x_0,t_0)$ and $ z(x_0,t_0)$ needed for the time-to-depth conversion process.

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