Introduction:
Returns from the sea surface, i.e., sea clutters have a very adverse impact on the performance of the maritime surveillance radars. The radar system that is being put into service for examining the current maritime scenario receives returned signals that are actually a cumulative contribution of returned signals scattered by sea surface as well as the target object. This contribution of these scattered signals causing sea clutter may result partial or complete unintelligibility of the target object thereby reducing its identification or detection considerably. The recovery to these losses of detection signals can be recovered through the application of signal processing techniques being selected after the identification of the characteristic features of the clutter. The features associated with the clutter are being exploited for the above mentioned recovery process. Identification of clutter requires proper description of the feature. Modeling of clutter is way to identify the current type. The clutter models present a realistic and tractable set of information. Characterization of microwaves back from the scattering from the sea surface through the use of Maxwell’s equation based description of electromagnetic field has not been found to be very useful. Parameterized statistical models are more suitable in this context, specifying the probability of the clutter and its contribution to the back scatter. The ideal clutter models are supposed to be realistic; easily tractable incorporating a pseudo philosophy making it in direct correlation with clear understanding of radar operation and clutter and target returns.
Gaussian clutter models
The complex Gaussian process can provide a model for sea clutter phenomena. It has been the basis of the conventional signal processing and clutter modeling. A radar return is being termed as a complex signal in phase and with quadrature components El and Eq. When the ocean is being illuminated by a low resolution radar system the ocean, the returning signal has random contribution from the many independent scattering structures. These scattered rays will add to give the resultant signal that corresponds to the radar on return. The simple discrete scatterer model for scattering gives rise to the standard Gaussian or thermal noise. The discrete scatterers contribute to compose the field.
N
E = ? an ————— (1)
n=1
For an arbitrary N, the detailed information of individual contribution and its type i.e., a would provide answer to this question. As central limit theorem says, with increase in value of N, electric field corresponding to I and Q components and its distribution would get normalized. Other known facts relate to contributors have zero means and they have same variance but are uncorrelated. The central limit theorem as states is the sum of a large number of normally distributed independent random variables with well behaved distributions of the constituent random variables.
Non-Gaussian clutter
As we have seen the complex Gaussian process provides a dependable model for sea clutter in a low resolution radar system. In case of high resolution radar, within a range cell the numbers of independent scattering structures are limited. These radars are capable of resolving some of the larger scale structure and hence the arguments behind Gaussian model falls apart. The Gaussian model includes the contribution of all independent scatters including the larger ones. As cluttering observed with the high resolution radar system is subjected to sharp fluctuations than the one in case of Gaussian clutter, then achieved improvisation in signal to noise ratio and potential detection can get offset since the cluttering is now more target-like exceeding the thresholds based on a Gaussian model of the clutter regularly. Regaining this loss in performance will require the development of non-Gaussian clutter models. With a focus of quantities the non- Gaussian clutter model is modeled with the normalized moments of intensity data including the ‘spiky’ non-Gaussian behavior in probability distribution function:
————— (2)
As we have already looked into the structure of sea surface, the whole structured is being characterized by length scales making it a complex mega unit with the scales ranging from 1cm or less i.e., foams and ripples, to swell structures that are many meters in size. The time-scales characteristic of sea surface motions also range from a sub-second unit to that of many seconds. So as a whole, we can see the presence of many effective independent small-scale structures within a high- resolution range cell creating speckle-like clutter. This cluttering actually decorrelates over a short period of time with small scale structures move through a distance of the order of half a radar wavelength. Modulation of these small-scale structures is achieved through the inclusion of slowly changing large scale structures which actually results the changing of local power of the Rayleigh-like returns from the small scale structure. The non-Gaussian clutter decomposes into a rapidly de-correlating, locally located Rayleigh process, with local power modulation through a much more slowly varying process, forms the basis the K distribution and other compound models for clutter.
The gamma distribution of local power and the K distribution
The local speckle process actually de-correlates in a time characteristic of the motion of small scale structure on the sea surface through a distance of half a radar wavelength; typically this is of the order of several milliseconds. Immediate decorrelation can be effected by the use of a frequency agile waveform; frequency agility does not, however, decorrelate the more slowly varying background modulation due to large scale structure. Thus we are able to obtain many independent samples of the local power in a time in which it does not change appreciably. These give an estimate of the local power x. By analysing a sufficiently large quantity of data a large set of independent measurements of x can be built up. These can be used to identify a good model for Pc (x). It has been found that the gamma distribution provides the best fit to most of the available data.
————— (3)
Other choices for Pc (x) could be made, and would provide other potentially useful clutter models that would retain many of the attractive features of the K distribution. However, it is unlikely that they would result in distributions that are as well characterised in terms of tabulated functions, nor that they would have the property of infinite divisibility possessed by the K distribution.
When (3) is substituted by full probability distribution function of full envelope of the signal i.e.
————— (4)
we find that the pdf of the clutter envelope is given by
————— (5)
We see that this integral can be evaluated in terms of the modified Bessel or K function that gives its name to the model. Fortunately, no knowledge of the properties of these Bessel functions is required if we are to evaluate quantities of interest, such as probabilities of false alarm or intensity moments. In each case we merely take the Gaussian result and integrate it over the gamma distribution of x. In this way we find that
————— (6)
————— (7)
Modeling the power spectrum of K distributed clutter
We have seen how the compound model, and in particular the K distribution, is able to describe the single point statistics of a non-Gaussian process. This compound K representation can be extended to the modeling of power spectra. The first requirement of the model is that the total power in the spectrum is gamma distributed. Thus we expect
————— (8)
to be gamma distributed i.e. that the pdf of x takes the form (3). As an illustration we take the simple Gaussian spectrum with unit power
————— (9)
as our fundamental building block. We then form the clutter power spectrum as
————— (10)
This will now satisfy (equation)) automatically. By letting the parameters ? and ?? depend on x, either deterministically or stochastically, we can model the statistics of the returns in given Doppler bins (i.e. the value taken by the power spectrum for given values of the frequency) in a variety of ways. Thus we can write the nth moment of the power at a given frequency as
————— (11)
The conditional probability P(?, ?? |x) accommodates a wide variety of models. Rather than consider the general case further we will consider several special cases, discussing the extent to which they make contact with experimental data and models. Much of the published analysis of Doppler spectra of clutter presents the data in terms of an effective n parameter that is in general frequency dependent and is defined by
————- (12)
is commonly observed to decrease (i.e. the frequency component in the power spectrum becomes more ‘spiky’ or non-Gaussian) as the Doppler frequency increases. It is this feature that we particularly wish to reproduce in our model.
The simplest model merely represents the power spectrum as the product of a gamma variate and with ? and ?? taking fixed values. While this model is straightforward to analyse and simulate it necessarily implies that
————- (13)
and so cannot reproduce the frequency dependence of the effective shape parameter.
Introduction of a dependence of the spectral width on the local power x of the clutter is physically reasonable (a breaking wave feature way produces a larger cross section and a greater spread of velocities/Doppler frequencies). The simplest way to incorporate this into the model is through a deterministic dependence
i.e. ? = ? (x) or P(? |x) = ?(? – ? (x)) so that
————- (14)
We now make the choice ?(x) = , primarily to facilitate the analysis; nonetheless the monotonic growth of ? with x captured by this model is sensible. Thus we have so
————- (15)
that the effective shape parameter can be calculated from ————- (16)
as usual K is the modified Bessel function. The following plot of the effective shape parameter as a function of frequency shows that this simple model is able to reproduce qualitatively the behaviour seen in coherent clutter data.
The application of the compound representation to the modelling of coherent clutter is much less thoroughly developed than is its application to single point statistics and is the subject of an on-going research effort.
