The inverse scattering problem for a layered acoustic medium is considered from the first-order differential equations of motion, resulting in a vector formulation of the problem, and using a vector form of the Schrödinger inverse scattering methods. The result is a vector Marchenko equation. The differentiability constraints on the acoustic impedance are somewhat relaxed compared to the more standard approach of beginning with the wave equation. The solution for plane waves at normal incidence is given along with a good approximate solution which is easily obtainable and takes into account transmission losses not included in the normal WKBJ-Born approximation. A new solution for extracting separately the velocity and density of the medium using the reflection response for two different angles of incidence is given, which involves a nonlinear integral equation to relate the apparent traveltimes to depth.

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