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

We can use oneway.3d to construct a numerical solution to the one-way wave equation in the mixed $(\ww,k)$, $(\ww,x)$ domain. The extrapolation wavenumber described in oneway.3d is, in general, a function dependent on several quantities

$$\kqz = \kqz \lp s,c_j\rp \;,$$ (14)

where $s \ofq$ is slowness, and $c_j\ofq=\lc \cx,\cy,\cz,\cxx,\cyy,\czz,\cxy \rc$ are coefficients computed numerically from the definition of the coordinate system, as indicated by coefs.3d. Specifying a coordinate system, implicitly defines all coefficients c_j.

We can write the extrapolation wavenumber $\kqz$ as a first-order Taylor expansion relative to a reference medium:  

$$\kqz = {\kqz}_0 + \left. \frac{\partial \kqz}{\partial s }... ...rtial c_j}\right\vert _{ s_0,{c_j}_0} \lp c_j - {c_j}_0 \rp \;,$$ (15)

where $s \ofq$ and $c_j\ofq$represent the spatially variable slowness and coordinate system parameters, and s₀ and c_j0 are the constant reference values at every extrapolation step.

The first part of mixed.3d, corresponding to the extrapolation wavenumber in the reference medium ${\kqz}_0$,is implemented in the Fourier ($\ww-k$) domain, while the second part, corresponding to the spatially variable medium coefficients, is implemented in the space ($\ww-x$) domain.

If we make the further simplifying assumptions that $\kqx \approx 0$ and $\kqy \approx 0$, we can write  

$$\kqz = {\kqz}_0 + \left. \frac{\partial\kqz}{\partial s }\ri... ...l\kqz}{\partial\cz }\right\vert _{0} \lp \cz - { \cz}_0 \rp \;,$$ (16)

where

. s|_0 &=& 2s_0 4_0s_0^2 - _0^2 ,
. |_0 &=& - i_0 2_0^2 + _0^2-2_0 s_0^2 2_0^2 4_0s_0^2 - _0^2 ,
. |_0 &=& i 2_0 - _0 2_0 4_0s_0^2 - _0^2 .

By ``0'', I denote the reference medium (s₀,c_j0). In principle, we could also use many reference media, followed by interpolation, similarly to the phase-shift plus interpolation (PSPI) technique of (38).

For the particular case of Cartesian coordinates ($\cz=0, \czz=1$), mixed.3d.explicit reduces to

$$\kqz = {\kqz}_0 + \ww \lp s - s_0 \rp \;,$$ (17)

which corresponds to the popular split-step Fourier (SSF) extrapolation method (100).