Computational and asymptotic aspects of velocity inversion

Abstract

We discuss computational and asymptotic aspects of the Born inversion method and show how asymptotic analysis is exploited to reduce the number of integrations in an f-k like solution formula for the velocity variation. The output of this alternative algorithm produces the reflectivity function of the surface. This is an array of singular functions—Dirac delta functions which peak on the reflecting surfaces—each scaled by the normal reflection strength at the surface. Thus, imaging of a reflector is achieved by construction of its singular function and estimation of the reflection strength is deduced from the peak value of that function. By asymptotic analysis of the application of the algorithm to the Kirchhoff representation of the backscattered field, we show that the peak value of the output estimates the reflection strength even when the condition of small variation in velocity (an assumption of the original derivation) is violated. Furthermore, this analysis demonstrates that the method provides a migration algorithm when the amplitude has not been preserved in the data. The design of the computer algorithm is discussed, including such aspects as constraints due to causality and spatial aliasing. We also provide O‐estimates of computer time. This algorithm has been successfully implemented on both synthetic data and common‐midpoint stacked field data.