Inverse scattering for a layered acoustic medium using the first‐order equations of motion

Geophysics ◽  
1983 ◽  
Vol 48 (2) ◽  
pp. 163-170 ◽  
Author(s):  
M. S. Howard
Keyword(s):  
Inverse Scattering ◽  
Equations Of Motion ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Normal Incidence ◽  
Acoustic Medium ◽  
Transmission Losses ◽  
First Order ◽  
Marchenko Equation

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.

Density and compressibility profiles of a layered acoustic medium from precritical incidence data

Geophysics ◽  
1981 ◽  
Vol 46 (9) ◽  
pp. 1244-1246 ◽  
Author(s):  
Shimon Coen
Keyword(s):  
Inverse Scattering ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Normal Incidence ◽  
Refractive Index Profile ◽  
Acoustic Medium ◽  
Index Profile ◽  
Incidence Data ◽  
Layered Fluid

The density and compressibility profiles of a layered fluid are obtained from the reflection coefficient due to plane waves at two precritical angles of incidence and all the frequencies. The inverse scattering problem for a layered fluid, at oblique incidence, is transformed to an equivalent inverse scattering problem for a layered refractive index profile, at normal incidence. The latter inverse scattering problem is transformed to an inverse scattering problem in quantum mechanics whose solution is obtained by the Gelfand‐Levitan theory.

TWO VERSIONS OF QUASI-NEWTON METHOD FOR MULTIDIMENSIONAL INVERSE SCATTERING PROBLEM

1993 ◽  
Vol 01 (02) ◽  
pp. 197-228 ◽  
Author(s):  
SEMION GUTMAN ◽  
MICHAEL KLIBANOV
Keyword(s):  
Newton Method ◽  
Inverse Scattering ◽  
Born Approximation ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Circular Cylinders ◽  
Iterative Sequence ◽  
Scattering Field ◽  
Quasi Newton

Suppose that a medium with slowly changing spatial properties is enclosed in a bounded 3-dimensional domain and is subjected to a scattering by plane waves of a fixed frequency. Let measurements of the wave scattering field induced by this medium be available in the region outside of this domain. We study how to extract the properties of the medium from the information contained in the measurements. We are concerned with the weak scattering case of the above inverse scattering problem (ISP), that is, the unknown. spatial variations of the medium are assumed to be close to a constant. Examples can be found in the studies of the wave propagation in oceans, in the atmosphere, and in some biological media. Since the problems are nonlinear, the methods for their linearization (the Born approximation) have been developed. However, such an approach often does not produce good results. In our method, the Born approximation is just the first iteration and further iterations improve the identification by an order of magnitude. The iterative sequence is defined in the framework of a Quasi-Newton method. Using the measurements of the scattering field from a carefully chosen set of directions we are able to recover (finitely many) Fourier coefficients of the sought parameters of the model. Numerical experiments for the scattering from coaxial circular cylinders as well as for simulated data are presented.

The Classical Connection

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Wave Equation ◽  
Inverse Scattering ◽  
Physical System ◽  
Equations Of Motion ◽  
Scattering Problem ◽  
Classical Mechanics ◽  
Inverse Scattering Problem ◽  
Alternative Approach ◽  
Classical Wave Equation ◽  
The Relationship

This chapter discusses the relationship between the elegance of the classical Lagrangian and Hamiltonian formulations of mechanics and optics. In physics, action is a mathematical functional which takes the trajectory, or path, of the system as its argument and has a real number as its result. Classical mechanics postulates that the path actually followed by a physical system is that for which the action is minimized, or, more generally, is stationary. The action is defined by an integral, and the classical equations of motion of a system can be derived by minimizing the value of that integral. The chapter first provides an overview of Lagrangians, action, and Hamiltonians in order to draw out an alternative approach to finding equations of motion. It then considers the classical wave equation and classical scattering and concludes with an analysis of the classical inverse scattering problem.

scholarly journals The Fixed Angle Scattering Problem with a First-Order Perturbation

2021 ◽  
Author(s):  
Cristóbal J. Meroño ◽  
Leyter Potenciano-Machado ◽  
Mikko Salo
Keyword(s):  
Order Term ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Carleman Estimates ◽  
Related Equation ◽  
Fixed Angle ◽  
First Order ◽  
Angle Scattering ◽  
First Order Perturbation

AbstractWe study the inverse scattering problem of determining a magnetic field and electric potential from scattering measurements corresponding to finitely many plane waves. The main result shows that the coefficients are uniquely determined by 2n measurements up to a natural gauge. We also show that one can recover the full first-order term for a related equation having no gauge invariance, and that it is possible to reduce the number of measurements if the coefficients have certain symmetries. This work extends the fixed angle scattering results of Rakesh and Salo (SIAM J Math Anal 52(6):5467–5499, 2020) and (Inverse Probl 36(3):035005, 2020) to Hamiltonians with first-order perturbations, and it is based on wave equation methods and Carleman estimates.

Numerical solution of the system of Marchenko integral equations

2021 ◽  
Vol 65 (3) ◽  
pp. 159-165
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Algebraic Equations ◽  
System Of Differential Equations ◽  
Scattering Problems ◽  
First Order ◽  
First Order Differential Equations

In this paper, inverse scattering problems for a system of differential equations of the first order are considered. The Marchenko approach is used to solve the inverse scattering problem. The system of Marchenko integral equations is reduced to a linear system of algebraic equations such that the solution of the resulting system yields to the unknown coefficients of the system of first-order differential equations. Illustrative examples are provided to demonstrate the preciseness and effectiveness of the proposed technique. The results are compared with the exact solution by using computer simulations.

Stable methods for solving the inverse scattering problem for a cylinder

1981 ◽  
Vol 89 (3-4) ◽  
pp. 181-188 ◽  
Author(s):  
David Colton ◽  
Andreas Kirsch
Keyword(s):  
Cross Section ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Initial Approximation ◽  
Constructive Method ◽  
Infinite Cylinder ◽  
Unknown Boundary ◽  
Closed Curves

SynopsisIt is shown that the inverse scattering problem for an infinite cylinder can be stabilized by assuming a priori that the unknown boundary of the cylindrical cross section lies in a compact family of continuously differentiable simple closed curves. A constructive method for determining the shape of this boundary is given under the assumption that an initial approximation is known and that the scattering cross section is known forn distinct incoming plane waves in the resonant region.

Far field patterns and inverse scattering problems for imperfectly conducting obstacles

1989 ◽  
Vol 106 (3) ◽  
pp. 553-569 ◽  
Author(s):  
T. S. Angell ◽  
David Colton ◽  
Rainer Kress
Keyword(s):  
Boundary Condition ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Impedance Boundary Condition ◽  
Far Field ◽  
Scattering Problems ◽  
Impedance Boundary ◽  
Field Patterns

AbstractWe first examine the class of far field patterns for the scalar Helmholtz equation in ℝ2 corresponding to incident time harmonic plane waves subject to an impedance boundary condition where the impedance is piecewise constant with respect to the incident direction and continuous with respect to x ε ∂ D where ∂ D is the scattering obstacle. We then examine the class of far field patterns for Maxwell's equations in subject to an impedance boundary condition with constant impedance. The results obtained are used to derive optimization algorithms for solving the inverse scattering problem.

Far field patterns and the inverse scattering problem for electromagnetic waves in an inhomogeneous medium

1988 ◽  
Vol 103 (3) ◽  
pp. 561-575 ◽  
Author(s):  
David Colton ◽  
Lassi Päivärinta
Keyword(s):  
Inhomogeneous Medium ◽  
Inverse Scattering ◽  
Integral Identity ◽  
Electromagnetic Waves ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Optimal Solution ◽  
Far Field ◽  
Field Patterns

AbstractWe consider the scattering of time harmonic electromagnetic waves by an inhomogeneous medium of compact support. It is first shown that the set of far field patterns of the electric fields corresponding to incident plane waves propagating in arbitrary directions is complete in the space of square-integrable tangential vector fields defined on the unit sphere. We then show that under certain conditions the electric far field patterns satisfy an integral identity involving the unique solution of a new class of boundary value problems for Maxwell's equations called the interior transmission problem for electromagnetic waves. Finally, it is indicated how this integral identity can be used to formulate an optimization scheme yielding an optimal solution of the inverse scattering problem for electromagnetic waves.

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