Acoustic wave-equation in 3D generalized Riemannian spaces

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

$$\Delta\mathcal{U}= - \omega^2\ss^2 \left(\mathbf{x} \right)\mathcal{U},$$ (5)

where the $\omega$ is frequency, $\ss$ is the propagation slowness, and $\Delta$ is the Laplacian operator,  

$$\Delta \mathcal{U}= \frac{1}{\sqrt{\vert\mathbf{g}\vert}}\,... ...t(\,m^{ij}\,\frac{\partial \mathcal{U}}{\partial \xi_j}\right).$$ (6)

Substituting equation 6 into 5 generates a Helmholtz equation appropriate for propagating waves through a 3D space,  

$$\frac{1}{\sqrt{\left\vert \mathbf{g} \right\vert}} \frac{\pa... ...tial \W}{\partial \xi_j } \right)= - \omega^2\ss^2 \mathcal{U}.$$ (7)

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

$$\frac{\partial m^{\ii\jj} }{\partial \xi_i } \frac{\partial ... ...t{\left\vert \mathbf{g} \right\vert}\omega^2 \ss^2 \mathcal{U}.$$ (8)

Defining n_j as,

$$n_j=\frac{\partial m^{\ii\jj} }{\partial \xi_i } = \frac{\p... ...}}{\partial \xi_2 } + \frac{\partial m^{3j}}{\partial \xi_3 } ,$$ (9)

leads to a more compact notation of equation 8,  

$$n_j \frac{\partial \W}{\partial \xi_j } + m^{\ii\jj} \frac{\... ...t{\left\vert \mathbf{g} \right\vert}\omega^2 \ss^2 \mathcal{U}.$$ (10)

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

$$\left(m^{ij} k_\xi_i-i n_j \right)k_\xi_j= \sqrt{\left\vert \mathbf{g} \right\vert}\omega^2\ss^2,$$ (11)

where $k_\xi_i$ is the Fourier domain dual of differential operator $\frac{\partial}{\partial \xi_i}$. Equation 11 represents the dispersion relationship for wavefield propagation on a generalized 3-D Riemannian space.
 

• One-way wavefield extrapolation
• Split-Step Fourier Approximation