Mixed-domain solutions to the one-way wave-equation

We can use equation (12) to construct a numerical solution to the one-way wave equation in the mixed $\omega-k$, $\omega-x$ domain. The extrapolation wavenumber described in equation (12) is, in general, a function which depends on several quantities

$$k_\zeta= k_\zeta\left(\ss,\c_j\right)\;,$$ (14)

where $\ss \left(\zeta,\xi,\eta\right)$ is the space-variable slowness, and $\c_j\left(\zeta,\xi,\eta\right)=\left\{\c_{\xi},\c_{\eta},\c_{\zeta},\c_{\xi\xi},\c_{\eta\eta},\c_{\zeta\zeta},\c_{\xi\eta}\right\}$ are coefficients which are computed numerically from the definition of the coordinate system, as indicated by equations (8). For any given coordinate system, $\c_j$ can be regarded as known.

Next, we write the extrapolation wavenumber $k_\zeta$ as a first-order Taylor expansion relative to a reference medium:  

$$k_\zeta= {k_\zeta}_0 + \left . \frac{\partial k_\zeta}{\pa... ...right \vert _{ \ss_0,{\c_j}_0} \left(\c_j - {\c_j}_0 \right)\;,$$ (15)

where $\ss \left(\zeta,\xi,\eta\right)$ and $\c_j\left(\zeta,\xi,\eta\right)$represent the spatially variable slowness and coordinate system parameters, and $\ss_0$ and ${\c_j}_0$ are the constant reference values in every extrapolation ``slab'' Sava (2000).

As usual, the first part of equation (15), corresponding to the extrapolation wavenumber in the reference medium ${k_\zeta}_0$,is implemented in the Fourier domain, while the second part, corresponding to the spatially variable medium coefficients, is implemented in the space domain.

If we make the further simplifying assumptions that $k_\xi\approx 0$ and $k_\eta\approx 0$, we can write  

$$k_\zeta= {k_\zeta}_0 + \left . \frac{\partial k_\zeta}{\part... ...}}\right \vert _{0} \left(\c_{\zeta}- {\c_{\zeta}}_0 \right)\;,$$ (16)

where

$$\begin{eqnarray} \left . \frac{\partial k_\zeta}{\partial\ss}\right \vert _{0} &... ...\zeta\zeta}}_0\left(\omega\ss_0\right)^2 - {\c_{\zeta}}_0^2} } \;.\end{eqnarray}$$ (17)

Equation (16) is motivated by a wavefront normal propagation approximation. By ``0'', we denote the reference medium ($\ss_0,{\c_j}_0)$.We could also use many reference media, followed by interpolation, similarly to the technique of Gazdag and Sguazzero (1984).

For the particular case of Cartesian coordinates ($\c_{\zeta}=0, \c_{\zeta\zeta}=1$), equation (16) reduces to

$$k_\zeta= {k_\zeta}_0 + \omega\left(\ss - \ss_0 \right)\;,$$ (18)

which corresponds to the popular Split-Step Fourier (SSF) extrapolation method Stoffa et al. (1990).