Forward Scattering Series for 2-Parameter Acoustic Media: Analysis and Implications to the Inverse Scattering Task Specific Subseries (Communications in Computational Physics, 2008)
We study the 2-parameter acoustic Born series for an actual medium withconstant velocity and a density distribution. Using a homogeneous background wedefine a perturbation, the difference between actual and reference medium (we usebackground and reference as synonyms), which exhibits an anisotropic behavior dueto 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 thesame speed as the waves in the background medium. Scattering theory decomposesthe actual wave field into an infinite series where each term contains the perturbationand the propagators in the background medium. Hence, in this formalism, all propagations occur in the background medium and the actual medium is included onlythrough the perturbations which scatter the propagating waves. The density-only perturbation has an isotropic and an anisotropic component. The anisotropic componentis 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 actsto 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 forwardseries 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 termis designed to accomplish. This is a remarkable behavior for a formalism that predictsthe 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 velocityis always constant. In the density-only perturbation, the forward series only seeks topredict the correct amplitudes. Finally, we extend the analysis to a wave propagatingin a medium where both density and velocity change. By selecting a convenient setof parameters, we find a conceptual framework for the multiparameter Born series.This framework provides an insightful analysis that can be mapped and applied to theconcepts and algorithms of the inverse scattering series.