Angle-domain common-image gathers in generalized coordinates

Eliminating Spatial Dependency

This section presents a method for eliminating the spatial dependency of generalized ADCIG transforms through a judicious stretching of the subsurface offset axis. Revisiting how one can introduce a stretch using equation 11, one obvious restriction to make is maintaining uniform Cartesian spatial sampling. I accomplish this by enforcing $\frac{\partial h_{x_1}}{\partial {x_1}}=1$ such that

$${\rm tan}\gamma = - \frac{\partial \xi_3}{\partial h_{\xi_1}}\lef... ...partial x_3}{\partial \xi_3}\right/ \frac{\partial x_1}{\partial \xi_1}\right].$$ (24)

The next step is to specify the relationship between $h_{\xi_1}$ and $\xi_1$ that enables us to calculate $\frac{\partial h_{\xi_1}}{\partial {\xi_1}}$. One useful ansatz solution is

$$\frac{\partial h_{\xi_1}}{\partial {\xi_1}}= \left. \frac{\partial x_1}{\partial \xi_1}\right/ \frac{\partial x_3}{\partial \xi_3}.$$ (25)

Substituting equation 25 into equation 24 generates the following ADCIG

$${\rm tan}\gamma = - \frac{\partial \xi_3}{\partial h_{\xi_1}}.$$ (26)

Equation 25 implies that if we can define an appropriate coordinate system stretch for each $\boldsymbol{\xi}$ location, we may still recover the reflection opening angle using Fourier-based techniques.

Angle-domain common-image gathers in generalized coordinates