# Chi-square target models

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

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:

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

$sigma_\left\{av\right\}$ 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. $m$ is the number of degrees of freedom divided by 2. 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.

werling 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:

werling 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

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

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

werling II

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

werling 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

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

werling IV

Similar to Swerling III, but the RCS varies from pulse to pulse rather than from scan to scan.

werling V (Also known as Swerling 0)

Constant RCS (m $o$ infinity).

References

* Skolnik, M. Introduction to Radar Systems: Third Edition. McGraw-Hill, New York, 2001.

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