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## One-dimensional earth and impulsive source

Let's define u0 as the primary wavefield and u1 the surface-related, first-order multiple wavefield recorded at the surface. If the earth varies only as a function of depth, then u will not depend on both s and g, but only on the offset, h=g-s. In this one-dimensional case, equation (10) becomes
 (11) (12)
where h'=g'-s. Equation (12) clearly represents a convolution, so can be computed by multiplication in the Fourier domain such that

 U1(kh) = U0(kh)2, (13)

where Ui(kh) is the Fourier transform of ui(h) defined by
 (14)

Equation (13) can be extended to deal with a smoothly varying earth by considering common shot-gathers (or common midpoint gathers) independently, and assuming the earth is locally one-dimensional in the vicinity of the shot e.g., Rickett and Guitton (2000):

 U1(kh,s) = U0(kh,s)2. (15)

Next: Two-dimensional earth Up: Surface-related multiple prediction theory Previous: Surface-related multiple prediction theory
Stanford Exploration Project
4/29/2001