Delayed-shot migration in TEC coordinates

Tilted elliptical-cylindrical coordinates

One question to be addressed is what coordinate system geometry optimally conforms to the impulse response of a conical wavefield? I assert that the best geometry is that of the TEC coordinate system shown in Figures 1 and 2. One advantage is that the breadth of the first extrapolation step at the surface allows multiple streamers of a single sail line to be positioned directly on a single mesh. Hence, this geometry is applicable to both narrow- and wide-azimuth acquisition. A second advantage is that one direction of large-angle propagation can be handled by coordinate system tilting, while the other is naturally handled by the ellipticality of the mesh. (Note that the geometry of another natural mesh - cylindrical polar coordinates - would not be a judicious choice for because the geometry permits migration of only single-streamer data and has singular points located on the surface at the first extrapolation step.)

I set up the migration geometry of the elliptical-cylindrical mesh as follows:

• $\xi_3 \in [0,\infty]$ is the extrapolation direction, where surfaces of constant $\xi _3$ form concentric elliptical cylinders, shown in Figure 1a.
• $\xi_2 \in [0,2\pi)$ is the crossline direction, where surfaces of constant $\xi _2$ are folded hyperbolic planes, shown in Figure 1b; and
• $\xi_1 \in [-\infty,\infty]$ is the inline direction, where surfaces of constant $\xi _1$ are 2D elliptical coordinate meshes, shown in Figure 1c;

TECgeom
Figure 1. Constant surfaces of the elliptical-cylindrical coordinate system (with zero inline tilt). Cartesian coordinate axes are given by the vector diagram. a) constant $\xi _3$ surfaces forming confocal elliptical-cylindrical shells that represent the direction of extrapolation direction. b) constant $\xi _2$ surfaces representing folded hyperbolic planes. c) constant $\xi _1$ surfaces representing 2D elliptical meshes. NR

The mapping relationship between the two coordinate systems, adapted from Arfken (1970), is

$$\left[\begin{array}{c} x_1 x_2 x_3 \end{array}\right] = \le... ...m {sinh} \xi_3 \rm {sin} \xi_2 {\rm cos} \theta \end{array}\right],$$ (13)

where $\theta$ is the inline tilt angle of the coordinate system and parameter $a$ controls the coordinate system breadth. Panels 2a and 2b show the TEC coordinate system at $0^\circ$ and $25^\circ$ tilt angles, respectively.

TEC
Figure 2. Four extrapolation steps in $\xi _3$ of an TEC coordinate system, where the $\xi _1$ and $\xi _2$ coordinate axes are oriented in the inline and crossline directions, respectively. a) 0$^\circ$ tilt angle. b) 25$^\circ$ tilt angle. NR

Delayed-shot migration in TEC coordinates