# 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 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, 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),  Gelfand and Levitan (1955), which was extended by Marchenko.  The Gelfand-Levitan-Marchenko equation is of current interest for imaging with internally scattered waves.

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 .