Split-Step Algorithm

This is a fast method for the solution of parabolic PDE’s, relying on the FFT implementation of the Fourier Transform to speed things up. Here it’s applied to an acoustic/seismic problem, following the development of Kuperman/Jackson in “Imaging of Complex Media with Acoustic and Seismic Waves”. Consider first the Helmholtz equation for a point source at r’,z’:
eq1_ssa
where G is the Green’s function, K the wavenumber (function of the frequency omega and sound speed c). Assuming azimuthal symmetry, G may be expressed as a product of two functions:
eq2_ssa
and similarly K, now a product of (constant) K0 and index of refraction n. Substituting into the Helmholtz equation gives two PDE’s:
eq3_ssa
The first PDE has Bessel functions as solutions; taking the assymptotic outgoing Hankel function solution and substituting it into the second, with the narrow angle approximation (second derivative of psi with respect to r much smaller than first derivative wrt r), one finds for the second (parabolic) PDE:
eq4_ssa
where chi is the fourier transform of psi. Assuming the variation of n is insignificant, in the wave-space domain the PDE and solution are:
eq5_ssa
Finally, the inverse FT gives the field as a function of depth (Delta r = r-r0, r0 the boundary value):
eq6_ssa

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