Riemannian wavefield extrapolation

Riemannian wavefield extrapolation Sava and Fomel (2003) generalizes solutions to the Helmholtz equation  

$$\Delta\mathcal{U}=-\omega^2 s^2 \mathcal{U}\;,$$ (1)

to coordinate systems that are different from simple Cartesian, where extrapolation is performed strictly in the downward direction. In equation (1), s is slowness, $\omega$ is temporal frequency, and $\mathcal{U}$ is a monochromatic wave.

The acoustic wave-equation in Riemannian coordinates can be written as:  

$$c_{\zeta\zeta}\dtwo{\mathcal{U}}{\zeta} + c_{\xi\xi}\dtwo{\m... ...\xi\partial \eta} = - \left (\omega s \right )^2 \mathcal{U}\;,$$ (2)

where coefficients c_ij are functions of the coordinate system and can be computed numerically for any given coordinate system Sava and Fomel (2003).

We can simplify the Riemannian wavefield extrapolation method by dropping the first-order terms in equation (2). According to the theory of characteristics for second-order hyperbolic equations Courant and Hilbert (1989), these terms affect only the amplitude of the propagating waves. To preserve the kinematics, it is sufficient to keep only the second order terms of equation (2):  

$$c_{\zeta\zeta}\dtwo{\mathcal{U}}{\zeta} + c_{\xi\xi}\dtwo{\m... ...\xi\partial \eta} = - \left (\omega s \right )^2 \mathcal{U}\;.$$ (3)

From equation (3) we can derive the following dispersion relation:  

$$- c_{\zeta\zeta}k_\zeta^2 - c_{\xi\xi}k_\xi^2 - c_{\eta\eta}... ...a^2 - c_{\xi\eta}k_\xi k_\eta= - \left (\omega s \right )^2 \;,$$ (4)

where $k_\zeta$, $k_\xi$ and $k_\eta$ are wavenumbers associated with the Riemannian coordinates $\zeta$, $\xi$ and $\eta$.

For one-way wavefield extrapolation, we need to solve the quadratic equation (4) for the wavenumber of the extrapolation direction $k_\zeta$,and select the solution with the appropriate sign to extrapolate waves in the desired direction:  

$$k_\zeta= \sqrt{ \frac{\left (\omega s\right )^2}{c_{\zeta\z... ...}k_\eta^2 - \frac{c_{\xi\eta}}{c_{\zeta\zeta}}k_\xi k_\eta }\;.$$ (5)

The 2D equivalent of equation (5) takes the form:  

$$k_\zeta= \sqrt{ \frac{\left (\omega s\right )^2}{c_{\zeta\zeta}} - \frac{c_{\xi\xi}}{c_{\zeta\zeta}}k_\xi^2 }\;.$$ (6)

In ray coordinates, defined by $\zeta\equiv \tau$ and $\xi\equiv \gamma$, we can re-write equation (6) as  

$$k_\tau= \sqrt{ \left (\omega s \aa\right )^2 - \left (\frac{\aa}{J} k_\gamma\right )^2 }\;,$$ (7)

where $\aa$ is velocity and J is geometrical spreading. We can simplify the computations by the notation

$$\left\{ \begin{array} {l} a = s \aa \;, \\ b = \frac{\aa}{J} \;,\end{array}\right.$$ (8)

therefore, equation (7) takes the form  

$$k_\tau= \sqrt{ \left (\omega a \right )^2 - \left (bk_\gamma\right )^2} \;.$$ (9)