# Scattering theory

# Scattering Theory

When considering waves propagating in a complicated medium, it is often useful to consider the medium to be composed of a simpler *background* wavespeed profile, with perturbations from that background profile, known as *scatterers,* superimposed on the background. Correspondingly, it is possible to separate the wavefield in the absence of the scatterers as the *incident* field, and
to represent a *scattered field* as the *total field minus the incident field.* In disciplines other than wave propagation,
such an approach is called simply *perturbation theory.*

## Different types of scattering

There are numerous types models of scattering that are appropriate for particular physical problems, scattering length scales as compared with the characteristic wavelengths of the waves impinging on the scatterers. The phenomena of reflection and diffraction may be approached from the direction of scattering theory.

### Examples from physics

- Compton Scattering
- Mie Scattering
- Rayleigh Scattering
- Thomson Scattering
- Fresnel Diffraction
- Fraunhofer Diffraction.

### Examples seen in exploration seismic

These latter two are special cases of the Kirchhoff Diffraction equation, which is also known as the Rayleigh-Sommerfeld Diffraction formula, or as the Kirchhoff-approximate modeling formula.

Another important scattering result is the Lipmann-Schwinger equation,^{[1]} whose terms represent
primary and multiple scattering. An approximation for small perturbations, known as the Dictionary:Born approximation, yields
a result known as the Born-approximate modeling formula.

## Applications of scattering theory

Scattering accounts for a large collection of observed wave phenomena. Scattering formulas, therefore, are important
tools in the modeling of wave propagation. Scattering theory also provides the basis of the *inverse scattering problem*
wherein the size, shape, and material parameters for the scatterers are inferred from the observed waves.

Landmark contributions to inverse scattering theory
were made by Bojarski (1981), ^{[2]} Gelfand and Levitan (1955),^{[3]} which was extended by Marchenko. ^{[4]} ^{[5]}The Gelfand-Levitan-Marchenko equation is of current interest for imaging with internally scattered waves.^{[6]}

The seismic migration problem has been treated successfully as the solution of an inverse scattering problem by a number of investigators,
through the construction of approximate inverse operators, using the Born or Kirchhoff-approximate modeling formulas .
^{[7]} ^{[8]}^{[9]}
^{[10]}

## References

- ↑ Lippmann, B. A., & Schwinger, J. (1950). Variational principles for scattering processes. I. Physical Review, 79(3), 469.
- ↑ Bojarski, N. N. (1981). Exact inverse scattering theory. Radio Science, 16(6), 1025-1028.
- ↑ Gel’fand, I. M. & Levitan, B. M., "On the determination of a differential equation from its spectral function". American Mathematical Society Translations, (2)1:253–304, 1955.
- ↑ Marchenko A., "Sturm-Liouville Operators and Applications", Birkhäuser, Basel, 1986.
- ↑ Burridge, R. (1980) The Gelfand-Levitan, the Marchenko, and the Gopinath-Sondhi integral equations of inverse scattering theory, regarded in the context of inverse impulse-response problems. Wave motion, 2(4), 305-323.
- ↑ Wapenaar, K., Broggini, F., Slob, E., & Snieder, R. (2013). Three-dimensional single-sided Marchenko inverse scattering, data-driven focusing, Green’s function retrieval, and their mutual relations. Physical Review Letters, 110(8), 084301.
- ↑ Cohen, J., F. G. Hagin, and N. Bleistein (1986). ”Three‐dimensional Born inversion with an arbitrary reference.” GEOPHYSICS, 51(8), 1552-1558. doi: 10.1190/1.1442205
- ↑ Beylkin, G. (1985). Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform. Journal of Mathematical Physics, 26(1), 99-108.
- ↑ Miller, D., Oristaglio, M., & Beylkin, G. (1987). A new slant on seismic imaging: Migration and integral geometry. Geophysics, 52(7), 943-964.
- ↑ Bleistein, N., J. K. Cohen, & J. W. Stockwell Jr, (2001). Mathematics of multidimensional seismic imaging, migration, and inversion (Vol. 13). Springer Science & Business Media.