| Abstract: | An extension of the multidimensional Born inversion technique for acoustic waves is described. In earlier work, a perturbation in reference sound velocity was determined by assuming that the reference velocity was constant. In this extension, we allow the reference velocity to be a function of the depth variable z. The output of this method is a high-frequency bandlimited reflectivity function of the subsurface. The reflectivity function is an array of bandlimited singular functions scaled by the normal reflection strength. Each singular function is a Dirac delta function of a scalar argument which measures distance normal to a reflecting interface. Thus, the reflectivity function is an indicator map of subsurface reflectors equivalent to the map produced by migration. In addition to the assumption of small perturbation, the method requires that the reflection data reside in the high frequency regime in a well-defined sense. The method is based on the derivation of an integral equation for the perturbation in sound velocity from a known reference velocity. When the reference velocity is constant, the integral equation admits an analytic solution as a multifold integral of the reflection data. Further high frequency asymptotic analysis simplifies this integral considerably and leads to an extremely efficient numerical algorithm for computing the reflectivity function. The development of a computer code to implement this constant-reference-velocity solution is published elsewhere. For a reference velocity c(z) we can no longer invert the integral equation exactly. However, we can write down an asymptotic high-frequency approximation for the kernal of the integral equation and an asymptotic solution for the perturbation. The computer implementation of this result is designed along the same lines as the code for constant background velocity. In tests the total processing time for this algorithm with depth-dependent background velocity is usually considerably less than that required by a standard Kirchhoff migration algorithm. The method is implemented as a migration technique and compared with alternative migration algorithms on the flanks of the salt dome. |
