# 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.

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

where ${\displaystyle \sigma _{av}}$ refers to the mean value of ${\displaystyle \sigma }$. 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. ${\displaystyle m}$ 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 ${\displaystyle m}$ 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 ${\displaystyle m}$−1/2, larger values of ${\displaystyle m}$ will result in smaller fluctuations. If ${\displaystyle m}$ 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 (${\displaystyle m=1}$). This applies to a target that is made up of many independent scatterers of roughly equal areas. As few 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

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

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 (${\displaystyle m=2}$). 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

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

### Swerling IV

Similar to Swerling III, but the RCS varies from pulse to pulse rather than from scan to scan. Examples include some helicopters and propeller driven aircraft.

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

Constant RCS, corresponding to infinite degrees of freedom (${\displaystyle m\to \infty }$).

## 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.