Parameter estimation for Pareto and K distributed clutter with noise

Abstract

The form of the z log z estimator is derived for both Pareto and K distributed clutter plus noise. When noise is included, numerical zero finding is required to obtain the shape parameter from the estimator, but it still provides a robust and accurate method that is relatively quick to compute. It is compared with two other methods. The method of moments is the simplest and fastest to compute, but less accurate than other methods if the clutter shape parameter is small. A constrained maximumlikelihood (ML) estimator is constructed by maximising the log likelihood function in one dimension to find the shape parameter, while holding the mean power and clutter to noise ratio constant. This estimator is robust and accurate, but relatively slow because numerical integration is required to calculate the likelihood function, along with numerical optimisation to find the maximum. If the noise power is unknown, it can be obtained using the first two intensity moments in combination with either the constrained ML or z log z estimator. These combinations provide more robust and accurate estimates than the third intensity moment.

Analysis of X-Band SAR Sea-Clutter Distributions at Different Grazing Angles

“…However, if the noise power is unknown, then the third-order moment is required, and the shape and noise power can be estimated by [46] a = 3 + 6t(6t 2 + s)…”

Analysis of polarimetric bistatic sea clutter using the NetRAD radar system

“…The formula is reported in equation (3). Another approach is based on the estimates of the mean of the data and of the mean of the logarithm of the data, as discussed in chapter 13 of [15] and in [29][30][31], and reported in equation (4) where N is the number of non-coherently integrated pulses.…”

“…The estimator based on the mean of the logarithm of the data in equation (4) has been extended to take into account the effect of thermal noise, either proposing suitable numerical methods to obtain a value of the shape parameter [31], or developing a closed-form of the estimator when more than a single pulse are non-coherently integrated together [29]. Moment matching approach has been also used, for instance estimating the second, fourth, and sixth moment as in [23], or exploiting the knowledge of the noise power PN as in equation (6), where the first and second moment of the intensity of the clutter plus noise data are used, as reported in chapter 5 and 13 of [15] and in [22,31].…”

K-clutter plus noise parameter estimation using fractional positive and negative moments

“…A good analysis of the effect of additive noise on some estimation methods is well presented and discussed in [2]. This estimation problem has been discussed and some interesting estimation procedures with different degrees of accuracies are proposed in the literature [3][4][5][6][7][8][9][10]. In this context, Watts et al [3][4][5] presented a method based on the first three integer intensity moments.…”

“…However, regarding its heavy computational load, this method can only be used as a benchmark to parameters estimation methods for compound Gaussian clutter with additive thermal noise. Recently, Bocquet [8] derived the [zlog(z)] estimator in terms of the generalized exponential integral function and the first two intensity moments. Because this method cannot lead to a closed form expression of the estimate of the shape parameter, one has to resort to numerical computations.…”