K-distribution

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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[edit]

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[edit]

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[edit]

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

Differential equation[edit]

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[edit]

  1. ^ a b c d Redding (1999)

Sources[edit]

  • 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

Further reading[edit]

  • Jakeman, E. (1980) "On the statistics of K-distributed noise", Journal of Physics A: Mathematics and General, 13, 31–48
  • Ward, K.D.; Tough, Robert J.A; Watts, Simon (2006) Sea Clutter: Scattering, the K Distribution and Radar Performance, Institution of Engineering and Technology. ISBN 0-86341-503-2