One-way Riemannian wavefield extrapolation

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One-way Riemannian wavefield extrapolation

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

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

(19)

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

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

(20)

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

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

(21)

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

$\begin{displaymath} \frac{\partial m^{\ii\jj} }{\partial \xi_i } \frac{\partial ... ...t{\left\vert \mathbf{g} \right\vert}\omega^2 \ss^2 \mathcal{U}.\end{displaymath}$

(22)

Defining n_j as

$\begin{displaymath} n_j=\frac{\partial m^{\ii\jj} }{\partial \xi_i } = \frac{\p... ...}}{\partial \xi_2 } + \frac{\partial m^{3j}}{\partial \xi_3 } ,\end{displaymath}$

(23)

leads to a more compact notation of equation 22

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

(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

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

(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.

Next: Solving for the extrapolation Up: REFERENCES Previous: Generalized Riemannian Geometry

Stanford Exploration Project
1/16/2007