One-way wavefield extrapolation

The expression for $k_\xi_3$ forms the basis for a one-way extrapolation operator that can be used to propagate wavefields on generalized coordinate meshes. This requires that a wavefield at step $\xi_3$ (i.e., $\mathcal{U}(\xi_3,k_\xi_1,k_\xi_2;\omega)$) be propagated to the next step $\xi_3+\Delta \xi_3$ (i.e., $\mathcal{U}(\xi_3+\Delta \xi_3,k_\xi_1,k_\xi_2,\omega)$) according to  

$$\mathcal{U}(\xi_3+\Delta \xi_3,k_\xi_1,k_\xi_2,\omega) = \ma... ...xi_1,k_\xi_2;\omega) \; {\rm e}^{-{\rm i} k_\xi_3\Delta \xi_3}.$$ (32)

The back-projection of residuals required for waveform inversion can be implemented easily according to the adjoint process of equation 32,  

$$\mathcal{U}(\xi_3+\Delta \xi_3,k_\xi_1,k_\xi_2,\omega) = \ma... ...xi_1,k_\xi_2;\omega) \; {\rm e}^{+{\rm i} k_\xi_3\Delta \xi_3}.$$ (33)

Note that the coefficients above are spatially variant which requires employing a typical approach (e.g. split-step Fourier, FFD or phase-screens) for developing a mixed $\omega-{\bf x}$ domain exponential operators. This study uses the split-step Fourier approach detailed in Shragge (2006a) using the extrapolation wavenumber $k_\xi_3$ defined by equation 31.