A hybrid BIE + Born series modeling scheme: Generalized Born series

A new hybrid modeling method which incorporates boundary integral equations (BIE) and a Born series in solving multibody scattering problems is presented. This method, designated the GBS (generalized Born series), is based on perturbing the surface boundary integral equation (SBIE) matrix into a part to be easily inverted and another part that often need not be inverted at all. Under appropriate conditions the perturbed SBIEs can be efficiently solved by a Born series. Hence GBS avoids the inversion of the relatively much larger SBIE matrix of a pure BIE method and can require several orders of magnitude fewer computations. Even greater efficiency can be achieved if the analytic Green’s function of a scatterer is known [G.C. Gaunaurd and H. Uberall, J. Acoust. Soc. Am. 63, 1699–1712 (1978)]. In this case the BIEs are used to compute the Green’s operators of the irregular scatterers and GBS efficiently couples them to the analytic Green’s operator. The key advantage of GBS lies in interactive modeling. Once the Green’s operators of individual scatterers are computed then the responses to iterative changes in the location or orientation of a scatterer can be quickly computed. No more matrix inversions are necessary and only a few matrix-vector operations per frequency are required. Computer experiments suggest that many multibody problems can be modeled at each frequency by as few as four terms in the GBS. We have also devised a GBS formulation for layers with arbitrary geometry. Terms in the GBS are analagous to terms in the Bremmer series for plane layers. Hence, the GBS is somewhat akin to a generalization of the Bremmer series for interfaces of arbitrary geometry. Similar to the Bremmer series, the GBS can be used as a powerful interpretation tool by the geophysicist. Each term in the GBS has a clear physical meaning so that careful examination of a synthetic seismogram after each iteration provides clues to the genesis of ambiguous events.