3D shot-profile migration in ellipsoidal coordinates

Integral Elliptic Representations

The integral elliptic representations of the ellipsoidal coordinate system are given by the following three equations (, ):

$$\beta = c \int_{b}^{\xi_1} \frac{{\rm d} \xi_1}{(c^2-\xi_1^2)(\xi_1^2-c^2)} = F\left[ \sqrt{1-\frac{b^2}{c^2}},{\rm sin}^{-1} \left( \sqrt{ \frac{1-\frac{b^2}{\xi_1^2}}{1-\frac{b^2}{c^2}} } \right) \right],$$

$$\gamma = c \int_{0}^{\xi_1} \frac{{\rm d} \xi_1}{(b^2-\xi_1^2)(c^2-\xi_1^2)} = F\left[ \frac{b}{c}, {\rm sin}^{-1}\left(\frac{\xi_2}{b} \right) \right],$$ (A-1)

$$\alpha = c \int_{c}^{\xi_2} \frac{{\rm d} \xi_2}{(\xi_2^2-b^2)(\xi_2^2-c^2)} = F\left[ \frac{b}{c},\frac{\pi}{2} \right] - F\left[ \frac{b}{c},{\rm sin}^{-1}\left(\frac{c}{\xi_3}\right) \right] ,$$

where $F\left[ \cdot \right]$ are elliptic integrals of the first kind defined by

$$u = F\left[\phi,k\right] = \int_{0}^{\phi} \frac{{\rm d}\theta}{\sqrt{1 - k^2 {\rm sin}^2 \theta}},$$ (A-2)

where the elliptic modulus $k$ satisfies $0 < k^2 <1$ and $\phi$ is the Jacobi amplitude. Solutions to equation A-2 are calculated using the method of arithmetic-geometric mean and descending Landen Transformation described in ().

The integral transforms are invertible and can be represented in terms of Jacobi elliptic functions ${\rm dc}(u,k), {\rm nd}(u,k)$ and ${\rm sn}(u,k)$ (, ):

$$\xi_1 = b\,{\rm nd} \left[ \beta,\sqrt{1-\frac{b^2}{c^2}} \right],$$

$$\xi_2 = b\,{\rm sn} \left[\gamma, \frac{b}{c} \right],$$ (A-3)

$$\xi_3 = c\,{\rm dc} \left[ \alpha,\frac{b}{c} \right],$$

where $k$ is again elliptic modulus and $u$ is defined by equation A-2. The Jacobi elliptic functions are calculated using the method of the arithmetic-geometric mean described in ().

3D shot-profile migration in ellipsoidal coordinates