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

Theory

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.

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 $x_s$ 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 $t_s$ .

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:

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

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

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

where the geometrical spreading $Q$ satisfies,

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

Combining equations 3 and 4 gives

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

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

$$\nabla x_0 (x,z)\cdot \nabla t_0 (x,z) = 0~,$$ (6)

$$\nabla t_0 (x,z) \cdot \nabla t_0 (x,z) = \frac{1}{w(x,z)}~.$$ (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.

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