Asymptotic analysis of the paradox in log-stretch dip moveout

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.