Angle-domain common-image gathers in generalized coordinates

Bibliography

Lee, M. W. and S. Y. Suh, 1985, Optimization of one-way wave-equations (short note): Geophysics, 50, 1634-1637.

Appendix A

ADCIG coordinate transform

This appendix addresses how to express operators $\frac{\partial}{\partial x_3}$ and $\frac{\partial}{\partial h_{x_1}}$ in generalized coordinate systems to derive equation 10. I first assume that generalized coordinate systems are related to the Cartesian variables through a bijection (i.e., one-to-one mapping)

$$x_1 = f(\xi_1,\xi_3) \quad {\rm and} \quad x_3 = g(\xi_1,\xi_3)$$ (26)

with a non-vanishing Jacobian of coordinate transformation, $J_{\boldsymbol{\xi}}$. The bijection between a generalized and Cartesian coordinate system allows us to rewrite the left-hand-sides of equations 7 as (, )

$$\frac{\partial t}{\partial {x_1}} = \frac{1}{J_{\boldsymbol{\xi}}... ...rac{1}{J_{\boldsymbol{\xi}}}\frac{\partial (x_1,t) }{ \partial (\xi_1,\xi_3) }.$$ (27)

Expanding the Jacobian notation leads to

$$\left[ \begin{array}{c} \frac{\partial t}{\partial \xi_1}\frac{\p... ...egin{array}{c} {\rm sin} \alpha \\ {\rm cos} \alpha \end{array} \right].$$ (28)

The right-hand-sides of equations A-3 are analogous to those derived by (). Cross-multiplying the expressions by factors $\frac{\partial x_1}{\partial \xi_3}$ and $\frac{\partial x_3}{\partial \xi_3}$

$$\left[ \begin{array}{c} \frac{\partial x_1}{\partial \xi_3} \left... ... \\ \frac{\partial x_3}{\partial \xi_3} {\rm cos} \alpha \end{array} \right]$$ (29)

and adding the two expressions results in

$$\frac{\partial t}{\partial \xi_3}\left( \frac{\partial x_3}{\part... ...in} \alpha + \frac{\partial x_3}{\partial \xi_3} {\rm cos} \alpha \right).$$ (30)

A similar argument can be used to construct the equations for the subsurface-offset axis. The bijection between the generalized coordinate and Cartesian subsurface-offset axes allows for the left-hand-side of equations 7 to be rewritten as

$$\frac{\partial t}{\partial h_{x_1} } = \frac{1}{J_{\boldsymbol{h}... ...\boldsymbol{h}}}\frac{\partial (h_{x_1},t) }{ \partial (h_{\xi_1},h_{\xi_3}) },$$ (31)

where $J_{\boldsymbol{h}}$ is the subsurface-offset Jacobian of transformation. Expanding the Jacobian notation leads to

$$\left[ \begin{array}{c} \frac{\partial t}{\partial h_{\xi_1}}\fra... ...egin{array}{c} {\rm cos} \alpha \\ {\rm sin} \alpha \end{array} \right].$$ (32)

The right-hand-side of equations A-7 are again analogous to those given by (). Cross-multiplying the expressions by factors $\frac{\partial h_{x_1}}{\partial h_{\xi_1}}$ and $\frac{\partial h_{x_3}}{\partial h_{\xi_1}}$

$$\left[ \begin{array}{c} \frac{\partial h_{x_1}}{\partial h_{\xi_1... ...{\partial h_{x_3}}{\partial h_{\xi_1}} {\rm sin} \alpha \end{array} \right],$$ (33)

and subtracting the two expressions above yields

$$\frac{\partial t}{\partial h_{\xi_1}}\left( \frac{\partial h_{x_1... ... \alpha - \frac{\partial h_{x_3}}{\partial h_{\xi_1}} {\rm sin} \alpha \right).$$ (34)

An expression for ADCIGs can be obtained by dividing equation A-9 by equation A-5

$$\frac{\frac{\partial t}{\partial h_{\xi_1}}}{\frac{\partial t}{\p... ...os} \alpha + \frac{\partial x_1}{\partial \xi_3} {\rm sin} \alpha \right)}.$$ (35)

One question arising from the geometric factors in equation A-10 is what do the terms $\frac{\partial h_{x_1}}{\partial h_{\xi_1}}$, $\frac{\partial h_{x_3}}{\partial h_{\xi_1}}$, $\frac{\partial h_{x_1}}{\partial h_{\xi_3}}$ and $\frac{\partial h_{x_3}}{\partial h_{\xi_3}}$ represent? I assume that the subsurface offset axes are generated by uniform wavefield shifting such that the following equations are valid:

$$\left[\begin{array}{c} h_{x_1} \\ h_{x_3} \\ h_{\xi_1}\\ h_{\x... ...x_1}{\partial \xi_3}\\ \frac{\partial x_3}{\partial \xi_3} \end{array}\right].$$ (36)

If the subsurface offset axes were generated by anything other than uniform shifting (e.g. $h_{x_1} = x_1^2$), then the assumptions behind equations A-11 would not be honored.

Using these identities in equation A-5 reduces equation A-10 to

$$- \left. \frac{\partial \xi_3}{\partial h_{\xi_1}}\right\vert _{\... ...os} \alpha + \frac{\partial x_1}{\partial \xi_3} {\rm sin} \alpha \right)},$$ (37)

where the two Jacobian transformations are equivalent (i.e. $J_{\boldsymbol{\xi}}=J_{\boldsymbol{h}}$). This completes the derivation of equation 10.

Angle-domain common-image gathers in generalized coordinates