Angle-domain common-image gathers in generalized coordinates

Defining subsurface stretch

One question arising from the geometric factors in equation 7 is what does the term $\frac{\partial h_{x_1}}{\partial h_{\xi_1}}$ represent? One way to evaluate this quantity is use a partial derivative expansion

$$\frac{\partial h_{x_1}}{\partial h_{\xi_1}}= \frac{\partial h_{x_... ...partial x_1}{\partial \xi_1}\right/\frac{\partial h_{\xi_1}}{\partial {\xi_1}},$$ (8)

to isolate three separate terms. I interpret each contribution in the following way:

• $\frac{\partial h_{x_1}}{\partial {x_1}}$ - a scale factor of the transformation between $h_{x_1}$ and $x_1$ usually given by $h_{x_1}=x_1$ such that $\frac{\partial h_{x_1}}{\partial {x_1}}=1$;
• $\frac{\partial x_1}{\partial \xi_1}$ - the partial derivative mapping between the two coordinate systems derivable from equations 4; and
• $\frac{\partial h_{\xi_1}}{\partial {\xi_1}}$ a scale factor of the transformation between $h_{\xi_1}$ and $\xi_1$ normally defined as $h_{\xi_1} = \xi_1$ such that $\frac{\partial h_{\xi_1}}{\partial {\xi_1}}=1$, but permitted to have a parametric form $h_{\xi_1} = A(\xi_1)$.

Using the above partial derivative expansion allows us to write a general ADCIG relationship

$${\rm tan}\gamma = - \frac{\partial \xi_3}{\partial h_{\xi_1}} \le... ...al h_{x_1}}{\partial {x_1}} \frac{\partial x_1}{\partial \xi_1}\right) \right].$$ (9)

In the examples below, unless stated otherwise, I assume regular sampling with linear wavefield shifting such that the following hold,

$$\left[\begin{array}{c} h_{x_1} \\ h_{\xi_1} \end{array}\right] = ... ...partial h_{x_1}}{\partial {x_1}}=\frac{\partial h_{\xi_1}}{\partial {\xi_1}}=1,$$ (10)

which reduces the complexity of the general coordinate ADCIG expressions to

$${\rm tan}\gamma =- \frac{\partial \xi_3}{\partial h_{\xi_1}} \lef... ...partial x_3}{\partial \xi_3}\right/ \frac{\partial x_1}{\partial \xi_1}\right].$$ (11)

Angle-domain common-image gathers in generalized coordinates