Divergent and asymptotic nature of the time-offset Taner-Koehler series in reflection seismics

摘要

Power series (1) has been known in essence for more than four decades and has long been used to represent traveltime in reflection seismics. We show the series is degenerate, divergent, and an asymptotic expansion about the zero offset. The accuracy of the truncated series, though remarkable for small offsets, is usually poor for large offsets; equation (7) is a better alternative. The analytical results imply the formal invalidity of the proof in Al-Chalabi (1973) that rms velocity is less than or equal to stacking velocity. The asymptotic nature controls the accuracy of the series approximation, implying initial convergence and subsequent divergence. The use of the series must be limited to the range of convergence, which ends at the term with the smallest magnitude. The optimal strategy is to include not as many terms as possible but only the first several terms before the smallest, the ensuing error being of the order of the magnitude of the first discarded term. The asymptotic nature explains several of the convergence characteristics of the partial sum and the latter's oscillation about the true traveltime. The nonanalytic nature of traveltime also implies (a) the increasing order of an optimal polynomial does not necessarily improve accuracy while representing traveltime in terms of offset and (b) an optimal polynomial in terms of offset is an inadequate model of travel-time at large offsets. Another consequence is the feasibility of an asymptotic expansion about a nonzero offset, which eventuall leads to equation (7), apt for represeting traveltime in the far-offset range. The use of equations (6) and (7) and a regression analysis of the observed traveltimes over all offsets yield estimates of the coefficients of series (1), which are useful for moveout correction and have interpretation in terms of velocity models of the earth.