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Let's define u_{0} as the primary wavefield and u_{1}
the surfacerelated, firstorder 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=gs.
In this onedimensional 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

U_{1}(k_{h}) = U_{0}(k_{h})^{2},

(13) 
where U_{i}(k_{h}) is the Fourier transform of u_{i}(h) defined by
 
(14) 
Equation (13) can be extended to deal with a
smoothly varying earth by considering common shotgathers (or common
midpoint gathers) independently, and assuming the earth is locally
onedimensional in the vicinity of the shot e.g., Rickett and Guitton (2000):

U_{1}(k_{h},s) = U_{0}(k_{h},s)^{2}.

(15) 
Next: Twodimensional earth
Up: Surfacerelated multiple prediction theory
Previous: Surfacerelated multiple prediction theory
Stanford Exploration Project
4/29/2001