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## Differential equations and Fourier transforms

The chain rule for partial differentiation gives
 (66)
In our usual notation the Fourier representation of the time derivative is .Likewise, and the spatial derivatives are associated with kg , ks , kz ). Using these Fourier variables in the vectors of (66) and differentiating (58), (59), (60), and (61) to find the indicated elements in the matrix of (66), we get
 (67)

Let S be the sine of the takeoff angle at the source and let G be the sine of the emergent angle at the geophone. If the velocity v is known, then these angles will be directly measurable as stepouts on common-geophone gathers and common-shot gathers. Likewise, on a constant-offset section or a slant stack, observed stepouts relate to an apparent dip Y, and on a linearly moved-out common-midpoint gather, stepouts measure the apparent stepout H'. The precise definitions are
 (68) (69)
With these definitions the second and third rows of (67) become
 (70) (71)
The familiar offset stepout angle H is related to the LMO residual stepout angle H' by H' = H -pv. Setting H' equal to zero means setting kh' equal to zero, thereby indicating integration over h' , which in turn indicates slant stacking data with slant angle p. Small values of H' /v or refer to stepouts near to p.

Next: Processing possibilities Up: INTERVAL VELOCITY BY LINEAR Previous: Common-midpoint Snell coordinates
Stanford Exploration Project
10/31/1997