  There exists a paradox in dip moveout (DMO) in seismic data processing. The
paradox is why Notfors and Godfrey's approximate time log-stretched DMO can
produce better impulse responses than the full log DMO, and why Hale's f-k DMO
is correct although it was based on two inaccurate assumptions for the midpoint
repositioning and the DMO time relationship? Based on the asymptotic analysis
of the DMO algorithms, we find that any form of correctly formulated DMO must
handle both space and time coordinates properly in order to deal with all dips
accurately. The surprising improvement of Notfors and Godfrey's log DMO on Bale
and Jakubowicz's full log DMO was due to the equivalent midpoint repositioning
by transforming the time-related phase shift to the space-related phase shift.
The explanation of why Hale's f-k DMO is correct although it was based on two
inaccurate assumptions is that the two approximations exactly cancel each other
in the f-k domain to give the correct final result.

Analysis of PDEs;Data Analysis, Statistics and Probability;Geophysics