Chi-square target models

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Chi-Square target models were introduced by Peter Swerling and are used to describe the statistical properties of the radar cross-section of complex objects.

1 General Target Model
2 Swerling Target Models
3 References

[edit] General Target Model

Swerling target models give the RCS of a given object based on the chi-square probability density function, which has the following form:

$p(\sigma) = \frac{m}{\Gamma(m) \sigma_{av}} \left ( \frac{m\sigma}{\sigma_{av}} \right )^{m - 1} e^{-\frac{m\sigma}{\sigma_{av}}}$

$σ a v$ refers to the mean value of sigma. This is not always easy to determine, as certain objects may be viewed the most frequently from a limited range of angles. For instance, a sea-based radar system is most likely to view a ship from the side, the front, and the back, but never the top or the bottom. $2 m$ is the number of Degrees of freedom (statistics). While the number of degrees of freedom used in the chi-square probability density function is an integer value in statistics, it can assume any positive real number in the target model. Values of m between .3 and 2 have been found to closely approximate certain simple shapes, such as cylinders or cylinders with fins.

Since the ratio of the standard deviation to the mean value of the chi-square pdf is equal to m^-1/2, larger values of m will result in less fluctuations. If m equals infinity, the target's RCS is non-fluctuating.

[edit] Swerling Target Models

Swerling target models are special cases of the Chi-Square target models with specific degrees of freedom. There are five different Swerling models, numbered I through V:

[edit] Swerling I

A model where the RCS varies according to a Chi-square probability density function with two degrees of freedom ( $m = 1$ ). This applies to a target that is made up of many independent scatterers of roughly equal areas. As little as half a dozen scattering surfaces can produce this distribution. Swerling I describes a target whose radar cross-section is constant throughout a single scan, but varies independently from scan to scan. In this case, the pdf reduces to

$p(\sigma) = \frac{1}{\sigma_{av}} e^{-\frac{\sigma}{\sigma_{av}}}$

Swerling I has been shown to be a good approximation when determining the RCS of objects in aviation.

[edit] Swerling II

Similar to Swerling I, except the RCS values returned are independent from pulse to pulse, instead of scan to scan.

[edit] Swerling III

A model where the RCS varies according to a Chi-square probability density function with four degrees of freedom ( $m = 2$ ). This PDF approximates an object with one large scattering surface with several other small scattering surfaces. The RCS is constant through a single scan just as in Swerling I. The pdf becomes

$p(\sigma) = \frac{4\sigma}{\sigma_{av}^2} e^{-\frac{2\sigma}{\sigma_{av}}}$