An accurate formulation of log-stretch dip moveout in the frequency-wavenumber domain

摘要

In this paper, a new and accurate log-stretch F-K DMO formulation was presented. The result is identical to the one of Gardner (1991) except for a sign difference on the phase shift, which is caused by the definition of the 2-D Fourier transform. The derivation of the new DMO operator was based on the correct DMO relationships presented in Black et al. (1993), which take the repositioning of the midpoint into consideration. These correct DMO relationships, in common with Gardner's method, are parametric representations of the DMO process, but with better physical insight to the process than Gardner's method. No approximation was introduced during the derivation. Therefore, the new log-stretch DMO is accurate and can adequately handle steeply dipping reflectors. The final form of the new DMO is also identical to that of Liner (1990) with respect to phase shift, while the amplitude factor is different. The numerical impulse responses show that the difference in results between the new DMO and Liner's DMO is subtle. However, our new DMO operator should produce slightly stronger amplitudes for steep reflectors because Liner's DMO amplitude factor is always less than ours and decreases as the time dip (k/Ω) increases. The philosophy of our basic derivation is the same as Bale and Jakubowicz (1987). But the starting point is different: Bale and Jakubowicz start from the conventional NMO/DMO equation as did Hale (1984) while we start from the new DMO relationships from Black et al. (1993) who corrected the subtle flaw in Hale's DMO formulation, i.e., the repositioning of the midpoint. Therefore, we conclude that the subtle flaw of Hale's derivation is largely responsible for the inaccurate DMO impulse responses of Bale's DMO. We do not have a satisfactory explanation to why Hale's F-K DMO operator is correct although it was based on two inaccurate assumptions: First, the midpoint was not repositioned, i.e. x_0 = x_n. Second, the time relationship before and after DMO is not strictly correct: t_0 = t_nA. It might be that these two approximations (errors) somehow counteract each other in the F-K domain to make the final result correct. But they do not cancel each other in the log-stretch F-K domain. They do not even describe the DMO elliptical responses in the x-t domain in the first place.