One-way wave-equation in 3-D Riemannian spaces

Equation (6) can be used to describe two-way propagation of acoustic waves in a semi-orthogonal Riemannian space. For one-way wavefield extrapolation, we need to modify the acoustic wave equation (6) by selecting a single direction of propagation.

In order to simplify the computations, we introduce the following notation:

$$\begin{eqnarray} \c_{\zeta\zeta}&=& \frac{1 }{\AA^2} \;, \nonumber \\ \c_{\xi\xi... ...{\partial } {\partial \xi } \left(F\frac{\AA}{J}\right) \right]\;.\end{eqnarray}$$ (8)

All quantities in equations (8) can be computed by finite-differences for any choice of a Riemannian coordinate system which fulfills the orthogonality condition indicated earlier. In particular, we can use ray coordinates to compute those coefficients. With these notations, the acoustic wave-equation can be written as:  

$$\c_{\zeta\zeta}\frac{\partial^2 \mathcal{U}}{\partial \zeta^... ...rtial \xi\partial \eta} = - \frac{\omega^2}{v^2} \mathcal{U}\;.$$ (9)

For the particular case of Cartesian coordinates ($\c_{\xi}=\c_{\eta}=\c_{\zeta}=0, \c_{\xi\xi}=\c_{\eta\eta}=\c_{\zeta\zeta}=1, \c_{\xi\eta}=0$), the Helmholtz equation (9) takes the familiar form

$$\frac{\partial^2 \mathcal{U}}{\partial \zeta^2} + \frac{\pa... ...al{U}}{\partial \eta^2} = -\frac{\omega^2}{v^2} \mathcal{U}\;.$$ (10)

From equation (9), we can directly deduce the modified form of the dispersion relation for the wave-equation in a semi-orthogonal 3-D Riemannian space:  

$$- \c_{\zeta\zeta}k_\zeta^2 - \c_{\xi\xi}k_\xi^2 - \c_{\eta\e... ...\c_{\eta}k_\eta - \c_{\xi\eta}k_\xi k_\eta= - \omega^2\ss^2 \;.$$ (11)

For one-way wavefield extrapolation, we need to solve the second order equation (11) for the wavenumber of the extrapolation direction $k_\zeta$,and select the solution with the appropriate sign to extrapolate waves in the desired direction:  

$$k_\zeta= i \frac{\c_{\zeta}}{2\c_{\zeta\zeta}} \pm \sqrt{ \f... ...right] - \frac{\c_{\xi\eta}}{\c_{\zeta\zeta}} k_\xi k_\eta }\;.$$ (12)

The solution with the positive sign in equation (12) corresponds to propagation in the positive direction of the extrapolation axis $\zeta$.

For the particular case of Cartesian coordinates ($\c_{\xi}=\c_{\eta}=\c_{\zeta}=0, \c_{\xi\xi}=\c_{\eta\eta}=\c_{\zeta\zeta}=1, \c_{\xi\eta}=0$), the one-way wavefield extrapolation equation takes the familiar form

$$k_\zeta= \pm\sqrt{\left(\omega\ss\right)^2 -k_\xi^2 - k_\eta^2}\;.$$ (13)