Inverse Acoustic and Electromagnetic Scattering Theory: A Comprehensive Overview

Inverse scattering theory is a branch of applied mathematics that deals with the problem of reconstructing an object from measurements of the scattering of waves from that object. This theory has applications in a wide variety of fields, including radar, sonar, medical imaging, and geophysics.

In inverse acoustic scattering, the object is illuminated with sound waves and the scattered waves are measured. The goal is to use these measurements to reconstruct the shape and material properties of the object. In inverse electromagnetic scattering, the object is illuminated with electromagnetic waves and the scattered waves are measured. The goal is to use these measurements to reconstruct the shape and electrical properties of the object.

Inverse Acoustic and Electromagnetic Scattering Theory (Applied Mathematical Sciences Book 93)

by David Colton

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Language	:	English
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Inverse scattering theory is a challenging problem, as there is no unique solution to the problem. In other words, there are many different objects that could produce the same scattered waves. However, there are a number of techniques that can be used to reduce the number of possible solutions and to improve the accuracy of the reconstruction.

Mathematical Formulation

The mathematical formulation of inverse scattering theory is based on the wave equation. The wave equation is a partial differential equation that describes the propagation of waves. In the case of acoustic waves, the wave equation is given by:

\frac{\partial^2 p}{\partial t^2}- c^2 \nabla^2 p = 0

where:

* $p$ is the acoustic pressure * $t$ is time * $c$ is the speed of sound

In the case of electromagnetic waves, the wave equation is given by:

\nabla^2 E - \mu \epsilon \frac{\partial^2 E}{\partial t^2}= 0

where:

* $E$ is the electric field * $\mu$ is the permeability * $\epsilon$ is the permittivity

The wave equation can be solved using a variety of techniques, including the finite element method, the boundary element method, and the method of moments. Once the wave equation has been solved, the scattered waves can be calculated.

Inverse Scattering Algorithms

Once the scattered waves have been calculated, a variety of algorithms can be used to reconstruct the object. One common algorithm is the distorted Born iterative method. This algorithm starts with an initial guess for the object and then iteratively updates the guess until it converges to a solution.

Another common algorithm is the singular value decomposition (SVD) method. This algorithm uses the SVD to decompose the scattered waves into a set of orthogonal modes. The object can then be reconstructed by combining the modes in a way that minimizes the error between the calculated scattered waves and the measured scattered waves.

Applications

Inverse scattering theory has a wide variety of applications, including:

* Radar: Inverse scattering theory can be used to reconstruct the shape and material properties of targets from radar measurements. This information can be used to improve the performance of radar systems. * Sonar: Inverse scattering theory can be used to reconstruct the shape and material properties of objects from sonar measurements. This information can be used to improve the performance of sonar systems. * Medical imaging: Inverse scattering theory can be used to reconstruct the shape and material properties of organs and tissues from medical imaging measurements. This information can be used to diagnose and treat diseases. * Geophysics: Inverse scattering theory can be used to reconstruct the shape and material properties of geological structures from seismic measurements. This information can be used to explore for oil and gas and to study the Earth's interior.

Inverse scattering theory is a powerful tool that can be used to solve a wide variety of problems in a variety of fields. This theory is still under development, but it is already making a significant impact in many areas.

Additional Resources

* [Inverse Scattering Theory](https://en.wikipedia.org/wiki/Inverse_scattering_theory) * [Inverse Acoustic Scattering](https://www.sciencedirect.com/topics/engineering/inverse-acoustic-scattering) * [Inverse Electromagnetic Scattering](https://www.sciencedirect.com/topics/engineering/inverse-electromagnetic-scattering)

Inverse Acoustic and Electromagnetic Scattering Theory (Applied Mathematical Sciences Book 93)

by David Colton

4.3 out of 5

Language	:	English
File size	:	8356 KB
Screen Reader	:	Supported
Print length	:	535 pages

FREE