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 ca n 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 th e midpoint repositioning and the DMO time relationship? Based on the asympt otic analysis of the DMO algorithms, we find that any form of correctly for mulated DMO must handle both space and time coordinates properly in order t o 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 equi valent 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 cor rect final result.