3D shot-profile migration in ellipsoidal coordinates

Integral Confocal Ellipsoidal Equations

A second definition for confocal ellipsoidal coordinates uses auxiliary variables defined through integral transforms. () defines the following Jacobi elliptic integral transforms for each coordinate system axis:

$$\beta = \int_{b}^{\xi_1} \frac{c \, {\rm d} \xi_1}{(c^2-\xi_1^2)(\xi^2-c^2)},$$

$$\gamma = \int_{0}^{\xi_2} \frac{c \, {\rm d} \xi_2}{(b^2-\xi_2^2)(c^2-\xi^2)},$$ (A-5)

$$\alpha = \int_{c}^{\xi_3} \frac{c \,{\rm d} \xi_3}{(\xi_2^2-b^2)(\xi^2-c^2)},$$

where axes $[\beta,\gamma,\alpha]$ are conformal to the $[\xi_1,\xi_2,\xi_3]$ axes, but are stretched by the integral transforms defined in equation 5. Axes $[\beta,\gamma,\alpha]$ are defined on the following ranges: $0 \le \xi_1 \le \infty$, $0 \le \xi_2 \le \infty$ and $0 \le \xi_3 \le \infty$. Additional information on the integral transforms can be found in Appendix A. Figure 3 illustrates the relative stretching for each axis for the wide-azimuth geometry case presented in figure 2.

IntegralTransform
Figure 3. Integral transform stretches for the $\boldsymbol {\xi }$ axes given by equation 5. Top panel: $\beta -\xi _1$ coordinate stretch. Middle panel: $\gamma -\xi _2$ coordinate stretch. Bottom panel: $\alpha -\xi _3$ coordinate stretch.[NR]

The integral ellipsoidal-coordinate Helmholtz equation is (, ):

$$\nabla^2 U = (\xi_2^2-\xi_3^2)\frac{\partial^2}{\partial \beta^2}... ...2} U+ (\xi_1^2-\xi_2^2)\frac{\partial^2}{\partial \alpha^2}U = -\omega^2 s^2 U.$$ (A-6)

Note that this definition effectively rescales the $\boldsymbol {\xi }$ coordinate axes to eliminate the first-order partial-differential terms in equation 2. This represents the most important theoretical result in this paper, as it removes the main implementation difficulty. In addition, integral ellipsoidal coordinates are defined globally, not just in octants, which eliminates the non-uniqueness noted above.

Obtaining a dispersion relationship from the expression in equation 6 is fairly straightforward. Replacing the partial differential terms with their Fourier-domain counterparts (i.e. $\frac{\partial}{\partial j} \leftrightarrow -{\rm i} k_j$, for $j=\alpha,\beta,\gamma$) and solving for the extrapolation direction wavenumber $k_\alpha$ yields

$$k_\alpha = \sqrt{ A^2 \omega^2 s^2 - B^2 k_\beta^2 - C^2 k_\gamma^2 },$$ (A-7)

where

$$A = \frac{1}{\sqrt{\xi_1^2-\xi_2^2}}, \quad B = \sqrt{\frac{\xi_2... ...2} }, \quad {\rm and} \quad C = \sqrt{\frac{\xi_1^2-\xi_3^2}{\xi_1^2-\xi_2^2}}.$$ (A-8)

In general, equation 7 is inexact because the $A, B$ and $C$ coefficients in equations 8 (and possibly slowness) vary spatially along the $\beta$ and $\gamma$ axes.

3D shot-profile migration in ellipsoidal coordinates