Year:    2008

Communications in Computational Physics, Vol. 3 (2008), Iss. 1 : pp. 136–159

Abstract

We study the 2-parameter acoustic Born series for an actual medium with constant velocity and a density distribution. Using a homogeneous background we define a perturbation, the difference between actual and reference medium (we use background and reference as synonyms), which exhibits an anisotropic behavior due to the density distribution. For an actual medium with a constant velocity, the reference velocity can be selected so that the waves in the actual medium travel with the same speed as the waves in the background medium. Scattering theory decomposes the actual wave field into an infinite series where each term contains the perturbation and the propagators in the background medium. Hence, in this formalism, all propagations occur in the background medium and the actual medium is included only through the perturbations which scatter the propagating waves. The density-only perturbation has an isotropic and an anisotropic component. The anisotropic component is dependent on the incident direction of the propagating waves and behaves as a purposeful perturbation in the sense that it annihilates the part of the Born series that acts to correct the time to build the actual wave field, an unnecessary activity when the reference velocity is equal to the one in the actual medium. This means that the forward series is not attempting to correct for an issue that does not exist. We define the purposeful perturbation concept as the intrinsic knowledge of precisely what a given term is designed to accomplish. This is a remarkable behavior for a formalism that predicts the scattered wave field with an infinite series. At each order of approximation the output of the series is consistent with the fact that the time is correct because the velocity is always constant. In the density-only perturbation, the forward series only seeks to predict the correct amplitudes. Finally, we extend the analysis to a wave propagating in a medium where both density and velocity change. By selecting a convenient set of parameters, we find a conceptual framework for the multiparameter Born series. This framework provides an insightful analysis that can be mapped and applied to the concepts and algorithms of the inverse scattering series.