Angle-domain common-image gathers in generalized coordinates

Cartesian Coordinates

For constant velocity media in Cartesian coordinates, a straightforward link exists between differential changes in the travel time, $t$, of rays connecting the source-reflector and reflector-receiver paths to changes in the subsurface offset, $h_{x_1}$, and depth, $x_3$, coordinates. Figure 1a shows the geometry of the variables described above.

rays
Figure 1. Zoomed in view illustrating reflection opening angle geometry. a) Cartesian coordinates. b) Generalized coordinates. Adapted from Sava and Fomel (2003). [NR]

Mathematically, these relationships are expressed in the following two equations

$$\left. \left[ \begin{array}{c} \frac{\partial t}{\partial h_{x_1}... ...eft[ \begin{array}{c} {\rm sin} \gamma \\ {\rm cos} \gamma \end{array} \right],$$ (1)

where $s$ is slowness, $\alpha$ is reflector dip, and $\gamma$ is the reflection opening angle. Using the implicit functions theory, equations 1 can be rewritten as

$$\left. \frac{\partial x_3}{\partial h_{x_1}}\right\vert _{t,x_1}=... ...t}{\partial h_{x_1}}\right/ \frac{\partial t}{\partial x_3}= - {\rm tan}\gamma.$$ (2)

I introduce a negative sign in the right-hand-side of the equation above to be consistent with the notation of Sava and Fomel (2003). The left-hand side of equation 2 can be calculated in the frequency-wavenumber domain

$${\rm tan}\gamma = - \frac{k_{h_{x_1}}}{k_{x_3}},$$ (3)

where $k_{h_{x_1}}$ and $k_{x_3}$ are the wavenumbers in the $h_{x_1}$ and ${x_3}$ directions, respectively. Note that because equation 2 does not depend explicitly on $\boldsymbol{x}$, we may use Fourier-based methods to calculate the opening angle, $\gamma$, directly.

Angle-domain common-image gathers in generalized coordinates