Angle-domain common-image gathers in generalized coordinates

Tilted Cartesian coordinates

Tilted Cartesian coordinates are a useful generalized migration coordinate system (see Figure 3a). () use this mesh in a plane-wave migration scheme where the coordinate system is oriented toward the plane-wave take-off angle to improve large-angle propagation accuracy. A tilted Cartesian mesh is defined by

$$\left[ \begin{array}{c} x_1\\ x_3 \end{array} \right] = \left[ \... ...\end{array}\right] \left[ \begin{array}{c} \xi_1 \\ \xi_3 \end{array} \right],$$ (14)

where $\theta$ is the tilt angle. The partial derivative transform matrix is

$$\left[ \begin{array}{cc} \frac{\partial x_1}{\partial \xi_1}& \fr... ...in} \theta \\ {\rm sin} \theta & {\rm cos} \theta \end{array}\right],$$ (15)

which leads to the following ADCIG equation:

$$- \left. \frac{\partial \xi_3}{\partial h_{\xi_1}}\right\vert _{\... ...lpha + {\rm sin} \theta {\rm sin} \alpha \right) } = {\rm tan} \gamma.$$ (16)

Thus, calculating ADCIGs in tilted Cartesian coordinates directly recovers the correct reflection opening angle. Note that setting $\theta=0^\circ$ recovers the Cartesian expression in equation 8.

Angle-domain common-image gathers in generalized coordinates