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## Mixed-domain solutions

Mixed-domain solutions to the one-way wave equation usually consist of terms computed in the Fourier domain for a reference of the extrapolation medium, followed by a finite-differences correction applied in the space-domain. For equation (9), a generic mixed-domain solution has the form:
 (13)
where a0 and b0 are reference values for the medium characterized by the parameters a and b, and the coefficients , and take different forms according to the type of approximation. As for usual Cartesian coordinates, is applied in the Fourier domain, and the other two terms are applied in the space domain. If we limit the space-domain correction to the thin lens term, , we obtain the equivalent of split-step Fourier (SSF) method Stoffa et al. (1990) in Riemannian coordinates.

Appendix A details the derivations for two types of expansions: pseudo-screen Huang et al. (1999), and Fourier finite-differences Biondi (2002); Ristow and Ruhl (1994).

• Pseudo-screen:

The coefficients for the pseudo-screen solution to equation (13) are    (14)
where a0 and b0 are reference values for the medium characterized by parameters a and b. In the special case of Cartesian coordinates, a=s and b=1, equation (13) with coefficients equation (14) takes the familiar form  (15)
where the coefficients c1 and b take different values for different orders of the finite-differences term: c15=(c1,b)=(0.50,0.00) and c45=(c1,b)=(0.50,0.25). When (c1,b)=(0.00,0.00) we obtain the usual split-step Fourier equation.

• Fourier finite-differences:

The coefficients for the Fourier finite-differences solution to equation (13) are    (16)
where  (17)
a0 and b0 are reference values for the medium characterized by the parameters a and b. In the special case of Cartesian coordinates, a=s and b=1, equation (13) with coefficients equation (16) takes the familiar form:  (18)
where the coefficients c1 and b take different values for different orders of the finite-differences term: c1=0.5, b=0.0 for , or c1=0.5, b=0.25 for .When c1=b=0.0 we obtain the usual split-step Fourier equation.

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10/23/2004