Fourier finite-difference

In one of its most general forms Ristow and Ruhl (1994), we can write k_z as  

$$k_z= {k_z}_o+ \omega\left[1- \frac{s}{s_o} \left(\sum\limits... ...hoose n} \S^{2n} \delta_n \right)\right] \left(s - s_o\right),$$ (2)

where $${\frac{1}{2} \choose n}$$ are binomial coefficients for integer numbers n, s represents the spatially variable slowness function at depth z, s_o is a constant approximation to s, and $\delta_n$ is a sum of terms derived from s and s_o. The FFD equation is obtained using two Taylor series expansions of the SSR equations written for k_z and k_zo. I give a full derivation of Equation (2) in Appendix A.

Higher accuracy can be achieved by replacing the Taylor expansion in Equation (2) with Muir's continuous fraction expansion Claerbout (1985). The equivalent form of the general Fourier finite-difference propagator is:  

$$k_z= {k_z}_o+ \omega\left[1- \frac{s}{s_o} \left(\sum\limits... ...\frac{\S^2}{a_n+b_n \S^2} \right)\right] \left(s - s_o\right),$$ (3)

where a_n and b_n are coefficients that, in general, depend on the medium and the constant reference slownesses, s and s_o. The coefficients a_n and b_n are derived either by identification of terms between Equation (2) and Equation (3), after the approximation $\frac{1}{1-x^2}\approx 1+x^2$, or by an optimization problem as described by Ristow and Ruhl (1997).