On the measurement operator for scattering in layered media

Peter C. Gibson

doi:10.3934/ipi.2017005

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2017, Volume 11, Issue 1: 87-97. Doi: 10.3934/ipi.2017005

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On the measurement operator for scattering in layered media

Peter C. Gibson

Dept. of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J1P3, Canada

Received: January 31, 2016

Revised: August 31, 2016

Published: March 2018

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Abstract

We describe new mathematical structures associated with the scattering of plane waves in piecewise constant layered media, a basic model for acoustic imaging of laminated structures and in geophysics. Using explicit formulas for the reflection Green's function it is shown that the measurement operator satisfies a system of quasilinear PDE with smooth coefficients, and that the sum of the amplitude data has a simple expression in terms of inverse hyperbolic tangent of the reflection coefficients. In addition we derive a simple geometric description of the measured data, which, in the generic case, yields a natural factorization of the inverse problem.

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Mathematics Subject Classification: Primary: 35R05, 74J20; Secondary: 86A22.

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