Fourier Bases-Expansion Contraction Integral Equation for Inversion Highly Nonlinear Inverse Scattering Problem

2020 ◽  
Vol 68 (6) ◽  
pp. 2206-2214
Author(s):  
Kuiwen Xu ◽  
Lu Zhang ◽  
Zhun Wei
Keyword(s):  
Integral Equation ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Highly Nonlinear

AN INVERSE SCATTERING PROBLEM FOR NONLOCAL POTENTIALS I: THE FAMILY OF PHASE EQUIVALENT WAVEFUNCTIONS

1986 ◽  
Vol 01 (07) ◽  
pp. 449-454 ◽  
Author(s):  
V.M. MUZAFAROV
Keyword(s):  
Integral Equation ◽  
Schrödinger Equation ◽  
Inverse Scattering ◽  
Schrodinger Equation ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattering Data ◽  
The Family ◽  
Consistent Approach ◽  
Nonlocal Potentials

We develop a consistent approach to an inverse scattering problem for the Schrodinger equation with nonlocal potentials. The main result presented in this paper is that for the two-body scattering data, given the problem of reconstructing both the family of phase equivalent two-body wavefunctions and the corresponding family of phase equivalent half-off-shell t-matrices, is reduced to solving a regular integral equation. This equation may be regarded as a generalization of the Gel’fand-Levitan equation.

Direct nonlinear Q-compensation of seismic primaries reflecting from a stratified, two-parameter absorptive medium

Geophysics ◽  
2010 ◽  
Vol 75 (2) ◽  
pp. V13-V23 ◽  
Author(s):  
Kristopher A. Innanen ◽  
Jose E. Lira
Keyword(s):  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Text Structure ◽  
Nonlinear Inversion ◽  
Reflection Data ◽  
Born Series ◽  
Highly Nonlinear ◽  
Reference Medium ◽  
Absorptive Medium

[Formula: see text]-compensation of seismic primaries that have reflected from a stratified, absorptive-dispersive medium may be posed as a direct, nonlinear inverse scattering problem. If the reference medium is chosen to be nonattenuating and homogeneous, an inverse-scattering [Formula: see text]-compensation procedure may be derived that is highly nonlinear in the data, but which operates in the absence of prior knowledge of the properties of the subsurface, including its [Formula: see text] structure. It is arrived at by (1) performing an order-by-order inversion of a subset of the Born series, (2) isolating and extracting a component of the resulting nonlinear inversion equations argued to enact [Formula: see text]- compensation, and (3) mapping the result back to data space. Once derived, the procedure can be understood in terms of nonlinear interaction of the input primary reflection data: the attenuation of deeper primaries is corrected by an operator built (automatically) using the angle- and frequency variations of all shallower primaries. A simple synthetic example demonstrates the viability of the procedure in the presence of densely sampled, broadband reflection data.

scholarly journals A Regularised Total Least Squares Approach for 1D Inverse Scattering

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 216
Author(s):  
Andreas Tataris ◽  
Tristan van Leeuwen
Keyword(s):  
Integral Equation ◽  
Least Squares ◽  
Inverse Scattering ◽  
Least Squares Method ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Stability Estimate ◽  
Total Least Squares ◽  
Scattering Data ◽  
Least Squares Approach

We study the inverse scattering problem for a Schrödinger operator related to a static wave operator with variable velocity, using the GLM (Gelfand–Levitan–Marchenko) integral equation. We assume to have noisy scattering data, and we derive a stability estimate for the error of the solution of the GLM integral equation by showing the invertibility of the GLM operator between suitable function spaces. To regularise the problem, we formulate a variational total least squares problem, and we show that, under certain regularity assumptions, the optimisation problem admits minimisers. Finally, we compute numerically the regularised solution of the GLM equation using the total least squares method in a discrete sense.

scholarly journals Imaging of bi-anisotropic periodic structures from electromagnetic near-field data

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Dinh-Liem Nguyen ◽  
Trung Truong
Keyword(s):  
Inverse Scattering ◽  
Maxwell Equations ◽  
Field Data ◽  
Periodic Structures ◽  
Near Field ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Factorization Method ◽  
Unique Determination

AbstractThis paper is concerned with the inverse scattering problem for the three-dimensional Maxwell equations in bi-anisotropic periodic structures. The inverse scattering problem aims to determine the shape of bi-anisotropic periodic scatterers from electromagnetic near-field data at a fixed frequency. The factorization method is studied as an analytical and numerical tool for solving the inverse problem. We provide a rigorous justification of the factorization method which results in the unique determination and a fast imaging algorithm for the periodic scatterer. Numerical examples for imaging three-dimensional periodic structures are presented to examine the efficiency of the method.

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