Angle-domain common-image gathers in generalized coordinates

Elliptic coordinates

Elliptic coordinates (see Figure 3c) are a useful coordinate system for performing 2D shot-profile migration (, ). An elliptic mesh is defined by

$$\left[ \begin{array}{c} x_1\\ x_3 \end{array} \right] = \left[ \... ...os} \xi_1\\ a {\rm sinh} \xi_3 {\rm sin} \xi_1 \end{array} \right].$$ (20)

The partial derivative transformation matrix is

$$\left[ \begin{array}{cc} \frac{\partial x_1}{\partial \xi_1}& \fr... ...{\rm cos} \xi_1 & {\rm cosh} \xi_3 {\rm sin} \xi_1 \end{array}\right],$$ (21)

which leads to the following ADCIG equation:

$$- \left. \frac{\partial \xi_3}{\partial h_{\xi_1}}\right\vert _{\... ...xi_3 {\rm cos} \xi_1 {\rm sin} \alpha \right) } = {\rm tan} \gamma.$$ (22)

Thus, calculating ADCIGs in elliptic coordinates with Fourier-based methods will directly recover the true reflection opening angle.