Angle-domain common-image gathers in generalized coordinates

Generalized Coordinate Extension

Figure 1b illustrates a scenario similar to that illustrated in panel a, but for generalized coordinates. The reflection opening angle, $\gamma$, and the reflector dip, $\alpha$, obviously remain unchanged; however, the orientations of the $h_{\xi_1}$ and $\xi_3$ axes used to estimate $\gamma$ now differ. The key question is which quantities in the ADCIG calculation are affected by this change of variables?

To answer these questions, I first assume that generalized coordinate systems are related to the Cartesian variables through a bijection (i.e., one-to-one mapping)

$$x_1 = f(\xi_1,\xi_3) \quad {\rm and} \quad x_3 = g(\xi_1,\xi_3).$$ (4)

I also assert that the subsurface offset axes can be defined such that the implicit functions theory is valid

• $h_{x_1}$ - a horizontal, but not necessarily linear, shift along the $x_1$ axis; and
• $h_{\xi_1}$ - a shift in the direction of the $\xi_1$ axis commonly, though not always, sharing the same geometric qualities as the $\xi_1$ axis.

The bijection between the general and Cartesian coordinate systems allows us to rewrite equations 1 as

$$\left. \left[ \begin{array}{c} \frac{\partial t}{\partial h_{\xi_... ...ft[ \begin{array}{c} {\rm sin} \gamma \\ {\rm cos} \gamma \end{array} \right].$$ (5)

Moving partial derivatives from the left side of the equation 5 to the right yields

$$\left. \left[ \begin{array}{c} \frac{\partial t}{\partial h_{... ...rtial x_3}{\partial \xi_3} {\rm cos} \gamma \end{array}\right].$$ (6)

The generalized coordinate ADCIG is given by the division by the two expressions in equation 6

$${\rm tan}\gamma = -\left. \frac{\partial \xi_3}{\partial h_{\xi_1... ..._3}{\partial \xi_3} \right/ \frac{\partial h_{x_1}}{\partial h_{\xi_1}}\right].$$ (7)

The bracketed terms generally introduce a geometric dependence of the ADCIGs on coordinates $\boldsymbol{\xi}$, which can preclude the use of Fourier-based methods for calculating ADCIGs. The expression $\left. \, \right\vert _{\xi_1,t}$ will be implicitly assumed for the remainder of the paper.

Angle-domain common-image gathers in generalized coordinates