Angle-domain common-image gathers in generalized coordinates

Generalized coordinate ADCIGs

Figure 1b illustrates a scenario similar to that in Figure 1a, but for a more general coordinate system. The reflection opening angle, $\gamma$, and the reflector dip, $\alpha$, obviously remain unchanged in the subsurface; however, the orientations of the $h_{\xi _1}$ and $\xi_3$ axes used to estimate $\gamma$ now differ. The key question is, which quantities in the ADCIG calculation are affected by this change of variables?

Answering this question requires properly formulating the derivative operators, $\frac{\partial}{\partial x_3}$ and $\frac{\partial}{\partial h_{x_1}}$, in equations 7 in the generalized coordinate system variables $\boldsymbol{\xi}=[\xi_1,\xi_3]$ and $\boldsymbol{h_{\xi}} = [h_{\xi_1},h_{\xi_3}]$. Appendix A shows how these derivatives can be specified using Jacobian change-of-variable arguments. Assuming that the subsurface-offset axes are formed by uniform wavefield shifts, such that the geometries of $\boldsymbol{\xi}$ and $\boldsymbol{h_{\xi}}$ are equivalent Appendix A derives the following expression for generalized coordinate ADCIGs:

$$- \left. \frac{\partial \xi_3}{\partial h_{\xi_1}}\right\vert _{\... ...os} \alpha + \frac{\partial x_1}{\partial \xi_3} {\rm sin} \alpha \right)}.$$ (10)

Note that if the $\boldsymbol{\xi}$-coordinate system satisfies the Cauchy-Riemann differentiability criteria (, )

$$\frac{\partial x_1}{\partial \xi_1}= \frac{\partial x_3}{\partial... ...uad \frac{\partial x_3}{\partial \xi_1}= - \frac{\partial x_1}{\partial \xi_3},$$ (11)

equation 10 then reduces to

$$-\left. \frac{\partial \xi_3}{\partial h_{\xi_1}}\right\vert _{\xi_1,t} = {\rm tan} \gamma.$$ (12)

This is the generalized coordinate equivalent of the Cartesian expression in equation 7. A physical meaning of the criteria in equations 11 is that the coordinate system must behave isotropically (i.e. dilatationally and rotationally) in the neighborhood of every grid point. Three canonical examples, two of which satisfy equations 11, are discussed in the following section.

Similar to Cartesian coordinates, elliptic coordinate ADCIGs become insensitive where structural dips cause $\frac{\partial t}{\partial \xi_3}\rightarrow 0$. However, this insensitivity can be minimized when using generalized coordinate systems, because structural dips appear at different angles in different translated elliptic meshes. Figures 2c-d illustrate this by showing a different coordinate shift for a different shot-location than that presented in panels 2a-b. Note the changes in structural dip in the right-hand-side of the elliptic coordinate panels. Thus, while ADCIGs calculated on one elliptic grid may be insensitive to certain structure locally, mesh translation ensures that ADCIGs are sensitive globally. Imaging steep dips in elliptic coordinates, though, is limited by the accuracy of wide-angle one-way wavefield extrapolation.

Finally, one may calculate reflection opening angles in the wavenumber domain for coordinate systems satisfying equations 11

$${\rm tan} \gamma = - \frac{k_{h_{\xi_1}}}{k_{\xi_3}},$$ (13)

where $k_{h_{\xi_1}}$ and $k_{\xi_3}$ are the wavenumbers in the $h_{\xi _1}$ and ${\xi_3}$ directions, respectively. While some non-orthogonal coordinate systems might satisfy equations 11, most practical applications will have orthogonal $k_{h_{\xi_1}}$ and $k_{\xi_3}$.

Angle-domain common-image gathers in generalized coordinates