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 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.
The model is that random variable has a gamma distribution with mean and shape parameter , with being treated as a random variable having another gamma distribution, this time with mean and shape parameter . The result is that has the following probability density function (pdf) for :
where 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: it is the distribution of the product of two independent random variables, one having a gamma distribution with mean 1 and shape parameter , the second having a gamma distribution with mean and shape parameter .
This distribution derives from a paper by Jakeman and Pusey (1978).
The mean and variance are given by
All the properties of the distribution are symmetric in and .
The pdf of the K-distribution is a solution of the following differential equation:
- Redding (1999)
- Redding, Nicholas J. (1999) Estimating the Parameters of the K Distribution in the Intensity Domain . 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
- 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