Finite-difference solutions to the one-way wave equation

Alternative solutions to the one-way wave-equation are obtained with pure finite-differencing methods in the $\ww-x$ domain, which can be implemented either as implicit (24), or as explicit methods (41). For the same stencil size, implicit methods are more accurate and robust than explicit methods, but harder to implement in 3D. However, explicit methods of comparable accuracy can be designed using larger stencils.

For implicit methods, various approximations to the square root in oneway.3d lead to approximate equations of different orders of accuracy. For downward continuation in Cartesian coordinates, those methods are known by their respective angular accuracy as the $15^\circ$ equation, $45^\circ$ equation and so on. Although the meanings of $15^\circ$, $45^\circ$ are undefined in ray coordinates where the extrapolation axis is time, we can still write approximations for the numerical finite-difference solutions using analogous approximations.

With the notation

$$\; k_o^2 = \frac{\lp\ww s\rp^2}{\czz} - \lp\frac{\cz}{2\czz}\rp^2 \;,$$ (18)

we can simplify the one-way wave oneway.3d as  

$$\kqz = i \frac{\cz}{2\czz} + \sqrt{ \; k_o^2 - \lb \frac{\cx... ...2 -i \frac{\cy}{\czz}\kqy \rb - \frac{\cxy}{\czz} \kqx\kqy }\;.$$ (19)

A simple way of deriving the $15^\circ$ equation is by a second-order Taylor series expansion of the extrapolation wavenumber $\kqz$ function of the variables $\kqx$ and $\kqy$: , =0,=0 + . |_0 + . |_0 +
+ 12 . |_0 ^2 + . |_0 + 12 . |_0 ^2 .

Introducing oneway.3d.b into taylor.3d, we obtain an equivalent form for the $15^\circ$ equation in a semi-orthogonal 3D Riemannian space:  i 2 + k_o + i2 k_o + 12 k_o 2 k_o^2- ^2
+ i2 k_o + 12 k_o 2 k_o^2 -^2
+ 12 k_o 2^2 k_o^2 -.

fifteen.3d specialized for the case of 2D coordinate systems obtained by ray tracing is further discussed in Appendix A.

For the particular case of Cartesian coordinates ($\cx=\cy=\cz=0, \cxx=\cyy=\czz=1, \cxy=0, \; k_o=\ww s$),

$$\kqz \approx \ww s - \frac{1}{2\ww s} \lp \kqx^2 + \kqy^2 \rp \;,$$ (20)

which is the usual form of the $15^\circ$ equation.