# K-distribution

The K-distribution is a probability distribution that arises as the consequence of a statistical or probabilistic model used in Synthetic Aperture Radar (SAR) imagery. The K-distribution is formed by compounding two separate probability distributions, one representing the radar cross-section, and the other representing speckle that is a characteristic of coherent imaging.

The model used to represent the observed intensity $X$ involves compounding two gamma distributions. In each case a reparameterisation of the usual form of the family of gamma distributions is used, such that the parameters are:

• the mean of the distribution, and
• the usual shape parameter.

## Density

The model is that random variable $X$ has a gamma distribution with mean $\sigma$ and shape parameter $L$, with $\sigma$ being treated as a random variable having another gamma distribution, this time with mean $\mu$ and shape parameter $\nu$. The result is that $X$ has the following probability density function (pdf) for $x>0$:[1]

$f_X(x;\mu,\nu,L)= \frac{2}{x} \left( \frac{L \nu x}{\mu} \right)^\frac{L+\nu}{2} \frac{1}{\Gamma(L)\Gamma(\nu)} K_{\nu-L} \left( 2 \sqrt{\frac{L \nu x}{\mu} } \right)$,

where $K$ is a modified Bessel function of the second kind. In this derivation, the K-distribution is a compound probability distribution. It is also a product distribution:[1] it is the distribution of the product of two independent random variables, one having a gamma distribution with mean 1 and shape parameter $L$, the second having a gamma distribution with mean $\mu$ and shape parameter $\nu$.

This distribution derives from a paper by Jakeman and Pusey (1978).

## Moments

The mean and variance are given[1] by

$\operatorname{E}(X)= \mu$
$\operatorname{var}(X)= \mu^2 \frac{ \nu+L+1}{L \nu} .$

## Other properties

All the properties of the distribution are symmetric in $L$ and $\nu$.[1]

### Differential equation

The pdf of the K-distribution is a solution of the following differential equation:

$\left\{\begin{array}{l} \mu x^2 f''(x)-\mu x (L+\nu -3) f'(x)+f(x) (\mu (L-1) (\nu -1)-L \nu x)=0, \\ f(1)=\frac{2 \left(\frac{L \nu }{\mu }\right)^{\frac{L}{2}+\frac{\nu}{2}} K_{\nu -L}\left(2 \sqrt{\frac{L \nu }{\mu }}\right)}{\Gamma (L) \Gamma (\nu )}, \\ f'(1)=\frac{2 \left(\frac{L \nu}{\mu}\right)^{\frac{L+\nu}{2}} \left((L-1) K_{L-\nu} \left(2 \sqrt{\frac{L \nu}{\mu}}\right)-\sqrt{\frac{L \nu}{\mu}} K_{L-\nu +1}\left(2 \sqrt{\frac{L \nu}{\mu}}\right)\right)}{\Gamma (L) \Gamma (\nu)} \end{array}\right\}$

## Notes

1. ^ a b c d Redding (1999)

## Sources

• Redding, Nicholas J. (1999) Estimating the Parameters of the K Distribution in the Intensity Domain [1]. Report DSTO-TR-0839, DSTO Electronics and Surveillance Laboratory, South Australia. p. 60
• Jakeman, E. and Pusey, P. N. (1978) "Significance of K-Distributions in Scattering Experiments", Physical Review Letters, 40, 546–550 doi:10.1103/PhysRevLett.40.546