# Chi-squared 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 radar cross-section (RCS) of a given object using a distribution in the location-scale family of the chi-squared distribution.

$p(\sigma) = \frac{m}{\Gamma(m) \sigma_{av}} \left ( \frac{m\sigma}{\sigma_{av}} \right )^{m - 1} e^{-\frac{m\sigma}{\sigma_{av}}} I_{[0,\infty)}(\sigma)$

where $\sigma_{av}$ 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 degree of freedom divided by 2. The degree of freedom used in the chi-squared probability density function is a positive number related to the target model. Values of $m$ between 0.3 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-squared distribution is equal to $m$−1/2, larger values of $m$ will result in smaller fluctuations. If $m$ equals infinity, the target's RCS is non-fluctuating.

## Swerling Target Models

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

### Swerling I

A model where the RCS varies according to a Chi-squared 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.

### Swerling II

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

### Swerling III

A model where the RCS varies according to a Chi-squared 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

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

### Swerling IV

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

### Swerling V (Also known as Swerling 0)

Constant RCS ($m\to\infty$). also known as infinite degree of freedom

## References

• Skolnik, M. Introduction to Radar Systems: Third Edition. McGraw-Hill, New York, 2001.
• Swerling, P. Probability of Detection for Fluctuating Targets. ASTIA Document Number AD 80638. March 17, 1954.