Angle-domain common-image gathers in generalized coordinates

Sheared Cartesian Coordinates

A sheared Cartesian coordinate system (see Figure 2) is an instructional, though impractical, generalized coordinate system for shot-profile migration. A sheared Cartesian mesh is defined by

$$\left[ \begin{array}{c} x_1\\ x_3 \end{array} \right] = \left[ \... ...\end{array}\right] \left[ \begin{array}{c} \xi_1 \\ \xi_3 \end{array} \right],$$ (16)

where $\theta$ is the shearing angle. The transformation matrix is

$$\left[ \begin{array}{cc} \frac{\partial x_1}{\partial \xi_1}& \fr... ...ay}{cc} 1 & {\rm sin} \, \theta \\ 0 & {\rm cos} \, \theta \end{array}\right],$$ (17)

which leads to the following differential travel-time equations

$$\left[ \begin{array}{c} \frac{\partial t}{\partial h_{\xi_1}}\\ ... ...} \, \gamma \\ {\rm cos} \, \theta \, {\rm cos} \, \gamma \end{array} \right].$$ (18)

The computation for ADCIGs in sheared Cartesian coordinates is

$${\rm tan}\, \gamma = - {\rm cos} \,\theta \,\frac{\partial \xi_3}{\partial h_{\xi_1}}.$$ (19)

From equation 19, we can see that the apparent dips must be filtered initially by ${\rm cos} \, \theta$ in order to recover the true reflection opening angle.

Sheared Figure 2. Example of a sheared Cartesian coordinate system with a shear angle of 25$^\circ$.[NR]

Angle-domain common-image gathers in generalized coordinates