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 equation, equation and so on. Although the meanings of , 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
A simple way of deriving the equation is by a
second-order Taylor series expansion of the extrapolation
wavenumber function of the variables and : ,
. |_0 +
. |_0 +
+ 12 . |_0 ^2 + . |_0 + 12 . |_0 ^2 .
Introducing oneway.3d.b into taylor.3d, we obtain
an equivalent form for the 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 (),