K-distribution
Encyclopedia
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 distribution
s, 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:
having another gamma distribution, this time with mean μ and shape parameter ν. The result is that X has the following density function (x > 0):
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
: 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 μ and shape parameter ν.
This distribution derives from a paper by Jakeman and Pusey (1978).
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
that arises as the consequence of a statistical or probabilistic model used in Synthetic Aperture Radar
Synthetic aperture radar
Synthetic-aperture radar is a form of radar whose defining characteristic is its use of relative motion between an antenna and its target region to provide distinctive long-term coherent-signal variations that are exploited to obtain finer spatial resolution than is possible with conventional...
(SAR) imagery. The K distribution is formed by compounding
Compound probability distribution
In probability theory, a compound probability distribution is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution F with an unknown parameter θ that is distributed according to some other distribution G, and then...
two separate probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
s, 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 X has a gamma distribution with mean σ and shape parameter L, with σ being treated as a random variableRandom variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
having another gamma distribution, this time with mean μ and shape parameter ν. The result is that X has the following density function (x > 0):
where K is a modified Bessel function of the second kind. In this derivation, the K-distribution is a compound probability distribution
Compound probability distribution
In probability theory, a compound probability distribution is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution F with an unknown parameter θ that is distributed according to some other distribution G, and then...
. It is also a product distribution
Product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions...
: 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 μ and shape parameter ν.
This distribution derives from a paper by Jakeman and Pusey (1978).
Moments
The mean and variance are given bySources
- Redding, Nicholas J. (1999) Estimating the Parameters of the K Distribution in the Intensity Domain http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA368069. 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
Further reading
- 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