#### Location

San Diego, CA

#### Start Date

1-1-1985 12:00 AM

#### Description

Probabilistic inversion methodology (e.g., finding the most probable inhomogeneity given the measurements) has been applied to many inverse scattering problems with either Gaussian or non-Gaussian statistical models of possible scatterers. All of our past investigations have dealt either with weak inhomogeneities, for which the Born approximation is valid, or with highly reflective scatterers, for which the Kirchhoff approximation is presumed to be adequate. In this paper we consider an approach to the general probabilistic inversion problem involving strong inhomogeneities by the application of the Pontryagin maximum principle generalized to 3D space. The key feature of the present approach to inverse scattering is the treatment of the wave equation, relating each scattered field to each incident field and the state of the inhomogeneity, as a continuous set of side conditions. This set of side conditions is handled by the Lagrange multiplier method, Our procedure, in analogy with that of Pontryagin, is to first determine the most probable state of the inhomogeneity given both the scattered wave fields and the conjugate Lagrange multiplier fields. The final stationarization with respect to both fields yields finally a set of nonlinear coupled integral equations defined on the localization domain.

#### Book Title

Review of Progress in Quantitative Nondestructive Evaluation

#### Volume

4A

#### Chapter

Chapter 3: Theoretical and Applied Inverse Methods

#### Section

Theory

#### Pages

525-534

#### DOI

10.1007/978-1-4615-9421-5_59

#### Copyright Owner

Springer-Verlag US

#### Copyright Date

January 1985

#### Language

en

#### File Format

Application/pdf

Exact Solution of Probabilistic Inverse Problem Pertaining to the Scattering of Elastic Waves from General Inhomogeneities

San Diego, CA

Probabilistic inversion methodology (e.g., finding the most probable inhomogeneity given the measurements) has been applied to many inverse scattering problems with either Gaussian or non-Gaussian statistical models of possible scatterers. All of our past investigations have dealt either with weak inhomogeneities, for which the Born approximation is valid, or with highly reflective scatterers, for which the Kirchhoff approximation is presumed to be adequate. In this paper we consider an approach to the general probabilistic inversion problem involving strong inhomogeneities by the application of the Pontryagin maximum principle generalized to 3D space. The key feature of the present approach to inverse scattering is the treatment of the wave equation, relating each scattered field to each incident field and the state of the inhomogeneity, as a continuous set of side conditions. This set of side conditions is handled by the Lagrange multiplier method, Our procedure, in analogy with that of Pontryagin, is to first determine the most probable state of the inhomogeneity given both the scattered wave fields and the conjugate Lagrange multiplier fields. The final stationarization with respect to both fields yields finally a set of nonlinear coupled integral equations defined on the localization domain.