# Inverse Problems and Data Assimilation Wiki and International Community

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scattering:inverse_scattering

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## Inverse Scattering

Inverse Scattering is concerned with the use of waves and wave fields to infer knowledge about unknown objects.

Inverse Scattering investigated for example in acoustics, electromagnetics and elasticity, but also in quantum mechanics or solid state physics, as well as in meteorology and medicine. Applications range from nondestructive testing to medical imaging. As in many other fields, you can approach the topic from an algorithmical point of view, from theory of uniqueness and stability or from particular application areas. Here, we provide links and material for a wide range of different views.

### Existence, Uniqueness and Stability

Existence is the question whether a given set of observation always corresponds to some source of the problem under consideration. For inverse scattering problems, we know that the measured fields usually are smooth. But errors are not as smooth as the “true” fields. Thus, for real measurements we do not have existence of a possible source. In general, we do not have existence for inverse scattering problems.

Uniqueness is the question whether a given set of measurements uniquely determines a quantity under investigation. For example, it is well-known that in general, a set of sources in space is not uniquely determined by its far field pattern. The same radiating field can be generated by a range of different source distributions.

Stability is concerned with the dependence of the reconstruction on errors in the meaured data. Many inverse problems are instable in the sense that small measurement errors in the data can lead to huge errors in the corresponding reconstructions.

Problems which are either not unique or not stable or do not satisfy the existence condition are called ill-posed according to Hadamard.

### Methods for Inverse Scattering Problems

#### Field Reconstructions

Field reconstruction methods deal with the calculation of wave fields from measured data. Often, scattered fields are measured on some surface, for example a microphone array. The task is then to calculate the scattered fields in a region which is often inaccessible to the user.

#### Inverse Obstacle Scattering

Inverse Obstacle Scattering is concerned with the reconstruction of the location, the shape or properties of objects which are in some inaccessible region of space. The objects can be bounded objects, for example some tumor in medical imaging, but they can also be surfaces, for example rough surfaces which are investigated by satellite based scatterometers in oceanography or meteorology.

#### Inverse Medium Scattering Methods

For important applications, waves penetrate a medium under consideration. Often, the medium is inhomogeneous or piecewise homogeneous. In this case we speak of inverse medium scattering. The methods for inverse medium scattering need to be different from inverse obstacle scattering, since we now search for a function in space, not only for a boundary function.