Angle-domain common-image gathers in generalized coordinates

Cartesian coordinate ADCIGs

For constant velocity media in conventional Cartesian geometry, 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 these variables.

Mathematically, these relationships are

$$\left[ \begin{array}{c} \frac{\partial t}{\partial h_{x_1}} \fr... ...begin{array}{c} {\rm sin} \gamma {\rm cos} \gamma \end{array} \right],$$ (7)

where $s$ is slowness, $\alpha$ is reflector dip, and $\gamma$ is the reflection opening angle. The right-hand-side of equations 7 are derived by (). Equations 7 can be rewritten as

$$- \left. \frac{\partial x_3}{\partial h_{x_1}}\right\vert _{x_1,t... ...}{\partial h_{x_1}}\right/ \frac{\partial t}{\partial x_3}= {\rm tan} \gamma,$$ (8)

where the negative sign derives from use of the implicit functions theory (, ). () note that Cartesian ADCIGs become pathogenically degenerate in situations where $\frac{\partial t}{\partial x_3}\rightarrow 0$ (i.e. for steeply dipping structures where $\alpha\rightarrow 90^\circ$ in Figure 1). However, vertically oriented structures are, generally, not well imaged in Cartesian coordinates because of limited steep-angle propagation in downward extrapolation.

Finally, because equation 7 has no explicit geometric-dependence, Fourier-based methods can calculate the reflection opening angle directly in the wavenumber domain

$${\rm tan}\gamma = - \frac{k_{h_{x_1}}}{k_{x_3}},$$ (9)

where $k_{h_{x_1}}$ and $k_{x_3}$ are the wavenumbers in the $h_{x_1}$ and ${x_3}$ directions, respectively.

Angle-domain common-image gathers in generalized coordinates