Equivalence of the spectral and integral migration algorithms

Expression (15.224) represents Stolt s (1978) Fourier-based migration formula. 15.4-5 Equivalence of the spectral and integral migration algorithms In the previous sections we analyzed independently two different approaches to wave-field migration 1) based on the Kirchhoff integral formula, and 2) Stolt s Fourier-based migration. It is important to understand that these two approaches are equivalent. 

in the spectral approach, wavefield extrapolation from one horizontal level to another is implemented by multiplying its spectrum by the complex conjugate of the vertical derivative of the spatial spectrum of the Green s function for the Helmholtz equation. 

Let us prove the equality (15.225). It can be shown that the function g kx,ky, z to) satisfies the 1-D Helmholtz equation 

g is the 1-D Green s function for the medium with the constant velocity cq = 1 (see formula (13.79))  

Substituting equation (15.225) into (15.223), we arrive at the integral formula 

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