One-way Riemannian wavefield extrapolation

The acoustic wave-equation for wavefield, $\mathcal{U}$, in a generalized Riemannian space is  

$$\nabla^2\mathcal{U}= - \omega^2\ss^2 \left(\mathbf{x} \right)\mathcal{U},$$ (19)

where the $\omega$ is frequency, $\ss$ is the propagation slowness, and $\nabla^2$ is the Laplacian operator  

$$\Delta \mathcal{U}= \frac{1}{\sqrt{\vert\mathbf{g}\vert}}\,... ...t(\,m^{ij}\,\frac{\partial \mathcal{U}}{\partial \xi_j}\right).$$ (20)

Substituting equation 20 into 19 generates a Helmholtz equation appropriate for propagating waves through a 3D space  

$$\frac{1}{\sqrt{\left\vert \mathbf{g} \right\vert}} \frac{\pa... ...tial \W}{\partial \xi_j } \right)= - \omega^2\ss^2 \mathcal{U}.$$ (21)

Expanding the derivative terms and multiplying through by $\sqrt{\left\vert \mathbf{g} \right\vert}$yields  

$$\frac{\partial m^{\ii\jj} }{\partial \xi_i } \frac{\partial ... ...t{\left\vert \mathbf{g} \right\vert}\omega^2 \ss^2 \mathcal{U}.$$ (22)

Defining n_j as

$$n_j=\frac{\partial m^{\ii\jj} }{\partial \xi_i } = \frac{\p... ...}}{\partial \xi_2 } + \frac{\partial m^{3j}}{\partial \xi_3 } ,$$ (23)

leads to a more compact notation of equation 22  

$$n_j \frac{\partial \W}{\partial \xi_j } + m^{\ii\jj} \frac{\... ...t{\left\vert \mathbf{g} \right\vert}\omega^2 \ss^2 \mathcal{U}.$$ (24)

Developing a wave-equation dispersion relation is achieved by replacing the partial differential operators acting on wavefield $\mathcal{U}$ with their Fourier domain duals  

$$\left(m^{ij} k_\xi_i-i n_j \right)k_\xi_j= \sqrt{\left\vert \mathbf{g} \right\vert}\omega^2\ss^2,$$ (25)

where $k_\xi_i$ is the Fourier domain dual of differential operator $\frac{\partial}{\partial \xi_i}$. Equation 25 represents the dispersion relationship for wavefield propagation on a generalized 3-D Riemannian space.