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Quantitative measurements

The radar has been found to have broader utility and its utility has now matured for observing weather phenomena. In effect, the “clutter background” that atmospheric phenomena represent for a primary aircraft surveillance radar application becomes the “signal” interpreted in meteorological applications of radar. The meteorological radar relies on meteorological targets distributed in space and occupies a large fraction of the spatial resolution cells observed by the radar.

Moreover, it is necessary to make quantitative measurements of the received signal’s characteristics in order to estimate such parameters as precipitation rate, precipitation type, air motion, turbulence, and wind shear. In addition, because so many radar resolution cells contain useful information, meteorological radars require high-data rate recording systems and effective means for real-time display Understanding the nature of sea clutter is crucial to the successful modeling of sea clutter as well as to facilitate target detection within sea clutter.

To this end, an important question to ask is whether sea clutter is stochastic or deterministic. Since the complicated sea clutter signals are functions of complex (sometimes turbulent) wave motions on the sea surface, while wave motions on the sea surface clearly have their own dynamical features that are not readily described by simple statistical features, it is thus very appealing to understand sea clutter by considering some of their dynamical features.

In the past decade, Haykin have carried out analysis of some sea clutter data using chaos theory, and concluded that sea clutter was generated by an underlying chaotic process. Recently, their conclusion has been questioned by a number of researchers. In particular, Unsworth have demonstrated that the two main invariants used by Haykin, namely the “maximum likelihood of the correlation dimension estimate” and the “false nearest neighbors” are problematic in the analysis of measured sea clutter data, since both invariants may interpret stochastic processes as chaos.

They have also tried an improved method, which is based on the correlation integral of Grassberger and Procaccia and has been found effective in distinguishing stochastic processes from chaos. Still, no evidence of determinism or chaos has been found in sea clutter data. To reconcile ever growing evidence of stochasticity in sea clutter with their chaos hypothesis, recently, Haykin et al. have suggested that the non-chaotic feature of sea clutter could be due to many types of noise sources in the data.

To test this possibility, McDonald and Damini have tried a series of low-pass filters to remove noise; but again they have failed to find any chaotic features. Furthermore, they have found that the commonly used chaotic invariant measures of correlation dimension and Lyapunov exponent, computed by conventional ways, produce similar results for measured sea clutter returns and simulated stochastic processes, while a nonlinear predictor shows little improvement over linear prediction. While these recent studies highly suggest that sea clutter is unlikely to be truly chaotic, a number of fundamental questions are still unknown.

For example, most of these studies are conducted by comparing measured sea clutter data with simulated stochastic processes. We can ask: can the non-chaotic nature of sea clutter be directly demonstrated without resorting to simulated stochastic processes? Recognizing that simple low-pass filtering does not correspond to any definite scales in phase space, can we design a more effective method to separate scales in phase space and to test whether sea clutter can be decomposed as signals plus noise?

Finally, will studies along this line be of any help for target detection within sea clutter? In this paper, we employ the direct dynamical test for deterministic chaos developed by Gao and Zheng to analyze 280 sea clutter data measured under various sea and weather conditions. The method offers a more stringent criterion for detecting low-dimensional chaos, and can simultaneously monitor motions in phase space at different scales. However, no chaotic feature is observed from any of these scales.

But very interestingly, we find that sea clutter can be conveniently characterized by the new concept of power-law sensitivity to initial conditions (PSIC), which generalizes the defining property for chaotic dynamics, the exponential sensitivity to initial conditions (ESIC). There is currently a debate in the radar community as to whether sea clutter is stochastic or deterministic in nature. Conventionally, high-resolution radar sea clutter has been modelled by a stochastic compound k-distribution [I]. Haykin et al.

[2, 31 performed an analysis on sea clutter data sets using nonlinear techniqucs and concluded that sea clutter was generated by an underlying chaotic process. This analysis relied on two main chaotic invariants, the ‘maximum likelihood estimation of the correlation dimension’ (DML value) [4] and ‘false nearest neighhours’ (FNN) [5]. The Lyapunov exponents were also measured, where the number of exponents to be measured was determined from the FNN calculation. In [3] the results of surrogate data tests were also reported and it was concluded that sea clutter was a nonlinear process.

From this finding a nonlinear predictor function was then applied to predict the sea clutter which was claimed to improve the prediction performance of maritime surveillance radar. However, the surrogate test [6] performed in [3] was based on the Fourier transfonn (FT). Several problems have been identified with the FT surrogate test (Le. spurious detection of nonlinearity for random systems with strong periodic components [7, 81). Recently, the surrogate statistic in [3] has been shown to imply chaotic behaviour for white and correlated noise signals [9] and replaced by a more robust statistic.

More recently, a new surrogate test was developed specifically to measure the [lo]. The nonlinear prediction network used in [3] was not compared with the performance of a linear prediction network. Recently this comparison has been performed in [I I ] where it was shown that the clutter prediction functions were well approximated by a linear function and that nonlinear predictor network functions provided little or no further improvement in performance. In [3] it was reported that sea clutter had fractional DML values in the range 4. 1-4. 5 and FNN global dimension in the range 5-6.

It could be inferred from these results that the system is low-dimensional and fractal which is symptomatic of chaos. From the FNN result reported in [3], 5-6 Lyapunov exponents were calculated, resulting is positive and negative values, indicative of chaos. More recently, in [I21 the Kolmogorov-Sinai entropy was derived from the values of the Lyapunov exponents and found to be positive which is also a signature of chaos. In this paper it is shown that two of the invariants that the nonlinear analysis of [2] and [3] relied upon cannot be employed to determine the nature of an unknown system.

Each invariant is taken in turn and its robustness tested. It is shown that each invariant can lead to the false detection of chaos for known stochastic time series. An alternative method is then proposed to test the ature of sea clutter for stochastic or deterministic behaviour. The examination of the behaviour of the correlation dimension d, drawn from the correlation integral C(r) of Grassherger et al. For increasing embedding dimension dE is robust at distinguishing white and correlated stochastic times series from deterministic ones. Therefore it is suggested that such a method is used first to determine the nature of an unknown system.

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