Multiscale Characterization of Sea Clutter by Scale-Dependent Lyapunov Exponent

and Applied Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 ISRN Applied Mathematics Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 International Journal of Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Journal of Function Spaces and Applications International Journal of Mathematics and Mathematical Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2013


Introduction
Sea clutter, the backscatter of the transmitted radar signal by elements of the sea surface such as ocean sprays, capillary waves, wind waves, and swells, is highly complicated.The complexity of the signals comes from two sources, the rough sea surface, sometimes oscillatory, sometimes turbulent, and the multipath propagation of radar backscatter.From Figure 1, it is clear that the signal is not purely random, since the waveform can be fairly smooth on short time scales (Figure 1(a)).However, the signal is highly nonstationary, since the frequency of the signal (Figures 1(a) and 1(b)) and the randomness of the signal (Figures 1(c) and 1(d)) change over time drastically.Therefore, Fourier analysis or deterministic chaotic analysis of sea clutter cannot be effective.From Figure 1(e), where  ()  = ( −+1 + ⋅ ⋅ ⋅ +   )/, it can be concluded that neither autoregressive (AR) models nor textbook fractal models can describe the data.This is because AR modeling requires exponentially decaying autocorrelation (which amounts to Var( ()   ) ∼  −1 ), while fractal modeling requires the variation between Var( ()   ) and  to follow a power-law [1].(To see better the variation of Var( ()  ) with , log 2 [ 2 Var( ()  )] is plotted against log 2  in the figure.)However, neither AR nor true fractal behavior is observed in Figure 1(e).
Understanding of sea clutter will depend not only on finding suitable models to describe the surface scattering but also on knowledge of the complex behavior of the sea [2].As radars become more complex, to fully utilize their power, it has become increasingly important to accurately model their interaction with environment with ever greater fidelity.Therefore, the study of sea clutter continues to be an important aspect of the development of radar systems.
The most salient feature of multiscale signals is that the signals behave differently, depending upon the scale at which the data are examined [1].To understand why multiscale analysis of sea clutter is essential, it is important to highlight a few empirical observations that point to the multiscale nature of sea clutter: (1) real sea clutter data are highly nonstationary, on both short and long time scales.This has motivated us to fit the differentiated data of sea clutter by Tsallis distribution [13].(2) In the time scale range of 0.01 to 4 s, structure-function based multifractal characterization detects targets within sea clutter with very high accuracy, although the power-law fractal scaling is not very well defined [1,[23][24][25]; (3) when analyzed by multiplicative cascade multifractal [35], sea clutter again shows a fractal scaling break on time scales shorter than about 0.01 s.When the envelope of the sea clutter data is extracted by finding all the local maxima and subsequently analyzed, the cascade multifractal characterization becomes excellent; furthermore, log-normal distribution fits the envelope data of sea clutter excellently, but not the original sea clutter data [35].These observations motivate us to carry out a multiscale analysis of sea clutter by using scale-dependent Lyapunov exponent (SDLE) [1,45], to shed new light on the nature of sea clutter.
The remainder of the paper is organized as follows.In Section 2, we briefly describe the SDLE, focusing on its properties that are most relevant to sea clutter analysis.In Section 3, we use the SDLE to analyze sea clutter, consider whether such characterization can help detect targets within sea clutter, and compare SDLE with the conventional largest positive Lyapunov exponent.In Section 4, we make a few concluding remarks.

SDLE as a Multiscale Complexity Measure
2.1.SDLE: Definition and Computation.SDLE is a recently developed multiscale complexity measure [1,45].It has been used to characterize brainwave EEG data [46,47], heart rate variability (HRV) [48][49][50][51], financial time series [52], Earth's geodynamo [53], among others.SDLE is defined in a phase space through consideration of an ensemble of trajectories [1,45].In the case of a scalar time series (1), (2), . . ., (), a suitable phase space may be obtained by using time delay embedding [54] to construct vectors of the following form: where  and  are called the embedding dimension and the delay time, respectively.For chaotic systems,  and  have to be chosen according to certain optimization criterion [1].For a stochastic process, which is infinite-dimensional, the embedding procedure transforms a self-affine stochastic process to a self-similar process in a phase space, and often  = 2 is sufficient (this issue will be further explained later) [1,45].
We will now be more concrete.Denote the initial distance between two nearby trajectories by  0 , and their average distances at time  and  + Δ, respectively, by   and  +Δ , where Δ is small.The SDLE (  ) is defined by [1,45] Or equivalently by To compute SDLE, we can start from an arbitrary number of shells as follows: where   and   are reconstructed vectors,   (the radius of the shell) and Δ  (the width of the shell) are arbitrarily chosen small distances (Δ  is not necessarily a constant).Then, we monitor the evolution of all pairs of points (  ,   ) within a shell and take average.Equation ( 2) can now be written as where  and Δ are integers in unit of the sampling time, and the angle brackets denote average within a shell.Note that this computational procedure is similar to that for computing the so-called time-dependent exponent curves [55].Note that the initial set of shells serve as initial values of the scales; through evolution of the dynamics, they will automatically converge to the range of inherent scales.This is emphasized by the subscript  in   , when the scales become inherent,  can then be dropped.Also, note that when analyzing chaotic time series, the following condition: needs to be imposed when finding pairs of vectors within a shell, to eliminate the effects of tangential motions [1].This condition is often also sufficient for an initial scale to converge to the inherent scales [1].
To better understand SDLE, it is instructive to point out a relation between SDLE and the largest positive Lyapunov exponent  1 for true chaotic signals.It is given by [1] where  * is a scale parameter (e.g., used for renormalization when using Wolf et al. 's algorithm [56]), () is the probability density function for the scale  given by where  is a normalization constant satisfying ∫  * 0 () = 1 and () is the well-known Grassberger-Procaccia's correlation integral [57].
SDLE has distinctive scaling laws for different types of time series.Those most relevant to sea clutter analysis are listed here.
(1) For clean chaos on small scales, and noisy chaos with weak noise on intermediate scales, As an operational definition for chaos, we define chaos to be observing scaling of (9) on a scale range of (, ), where  > 1 is a coefficient [1,45].When low-dimensional chaos is concerned, one may require  ≥ 2.
(2) For clean chaos on large scales where memory has been lost and for noisy chaos (including noiseinduced chaos [58][59][60]) on small scales, where  > 0 is a parameter.
(3) For random 1/ 2+1 processes, where 0 <  < 1 is called the Hurst parameter which characterizes the correlation structure of the process: depending on whether  is smaller than, equal to, or larger than 1/2, the process is said to have antipersistent, short-range, or persistent long-range correlations [1,23] as follows: Note that  = 1/2 for the standard Brownian motion and  = 1/3 for turbulence.Note also that this scaling is quite independent of , except that the scale range defining (11) shrinks when  increases.Because of this,  = 2 again is preferred.To better appreciate the above properties, it is helpful to examine the chaotic Lorenz system with stochastic forcing as an example as follows: where   (),  = 1, 2, 3 are independent Gaussian noise forcing terms with mean 0 and variance  2 .Note that the system with a different parameter set was studied in [1,45].
The system is solved using the scheme of Exact propagator [61], where the exact solution of the Lorenz system is solved using a 4th order Runge-Kutta method with a time-step of ℎ = 0.002, and then a term  √ ℎ, where  is a Gaussian noise of mean 0 and variance 1, is added to the corresponding equations to take into account the noise.Figure 2 shows five curves, for the cases of  = 0, 1, 2, 3, 4. The computations are done using  = 4,  = 2, and 10000 points of the  component of the Lorenz system sampled at a time interval of 0.06.For the clean system, we observe two scaling laws.One is (9), () ≈ 0.9, for small , the other is (10), for large  where memory has been lost.For the noisy system, the scale region where the scaling law of (9) shrinks when the stochastic forcing is increased.Interestingly, although the part of the curve with () ∼ − ln  shifts to the right when noise is increased, the parameter  appears to not depend on the noise strength.(1) Shift a 1/  ,  = 2+1 process downward or upward at randomly chosen points in time by an arbitrary amount.For convenience, we call this procedure type 1 nonstationarity and the processes obtained broken-1/  processes.

Coping with
(2) At randomly chosen time intervals, concatenate randomly broken-1/  processes and oscillatory components or superimpose oscillatory components on broken-1/  processes.This procedure causes a different type of nonstationarity, which for convenience we will call type 2 nonstationarity.
We call the resulting random processes perturbed 1/  processes.Three examples of the () curves for such processes, where the frequency of the perturbations are, on average, 1% of the simulated data, are shown in Figure 3.We observe that (11) still holds very well when () > 0.02.Therefore, SDLE can readily characterize 1/ processes perturbed by either of the nonstationarities considered.
To understand why the SDLE can deal with type 1 nonstationarity, it suffices to note that type 1 nonstationarity causes shifts of the trajectory in phase space; the greater the nonstationarity, the larger the shifts.The SDLE, however, cannot be affected much by shifts, especially large ones, since it is based on the coevolution of pairs of vectors within chosen small shells.In fact, the effect of shifts is to exclude a few pairs of vectors that were originally counted in the ensemble  average.Therefore, so long as the shifts are not too frequent, the effect of shifts can be neglected, since ensemble average within a shell involves a large number of pairs of vectors.
Let us now turn to type 2 nonstationarity, which involves oscillatory components.Being regular, oscillatory components can only affect () where it is close to 0. Therefore, type 2 nonstationarity cannot affect the positive portion of () either.
Note that similar types of perturbations have been carried out to clean and noisy chaotic data, and the major features of ( 9) and ( 10) are also robust, as expected.
Finally, we examine whether SDLE can detect chaos from an intermittent time series with a long laminar phase during which neighboring trajectories do not diverge, and a rapid divergence over a small part of the state space.Such a case is most relevant to the transitions from a quiet sea surface to a turbulent sea disturbed by a sudden gust of wind, and vice versa.For this purpose, we study the logistic map as follows: Figure 4(a) shows an intermittent time series.We note that the chaotic part is very short compared with the periodic part.Figure 4(b) shows the SDLE curve for the time series.We clearly observe a plateau defining a chaotic motion.Also, note that the ln  scaling on very small scales indicates the transitions from periodic to chaotic motions, and vice versa (it is unstable, but not chaotic in the rigorous sense).

Multiscale Analysis of Sea Clutter
3.1.Sea Clutter Data.Fourteen sea clutter datasets were obtained from a website maintained by Professor Simon Haykin: http://soma.ece.mcmaster.ca/ipix/dartmouth/datasets.html.The measurement was made using the McMaster IPIX radar at Dartmouth, Nova Scotia, Canada.The radar was mounted in a fixed position on land 25-30 m above sea level, with an operating (carrier) frequency of 9.39 GHz (and hence a wavelength of about 3 cm).It was operated at low grazing angles, with the antenna dwelling in a fixed direction, illuminating a patch of ocean surface.The measurements were performed with the wave height in the ocean varying from 0.8 to 3.8 m (with peak heights up to 5.5 m) and the wind conditions varying from still to 60 km/hr (with gusts up to 90 km/hr).For each measurement, 14 areas, called antenna footprints or range bins, were scanned.Their centers are depicted as  1 ,  2 , . . .,  14 in Figure 5.The distance between two adjacent range bins was 15 m.One or a few range bins (say,  −1 ,   and  +1 ) hit a target, which was a spherical block of styrofoam of diameter 1 m wrapped with wire mesh.The locations of the three targets were specified by their azimuthal angle and distance to the radar.They were (128 0 , 2660 m), (130 0 , 5525 m), and (170 0 , 2655 m), respectively.The range bin where the target is strongest is labeled as the primary target bin.Due to the drift of the target, bins adjacent to the primary target bin may also have hit the target.They are called secondary target bins.For each range bin, there were 2 17 complex numbers, sampled with a frequency of 1000 Hz.

Sea Clutter Analysis.
The SDLE curves can be readily computed from sea clutter data.Figure 6(a) shows a typical example.We do not observe the chaotic scaling of (9) on any significant scale ranges.Therefore, sea clutter is not chaotic.While this result is consistent with the results of [29,30,[33][34][35][36][37], it merits noting that the SDLE method is a direct method and is more rigorous.
More interestingly, all the sea clutter data, whether it is with target or without target, yields the scaling of (10).Therefore, (10) describes a prevailing feature of sea clutter.To understand which feature that is as well as to understand the physical significance of , let us consider the problem of predictability with sea clutter.Specifically, we consider the concept of the error doubling time, defined as the average time for an initial error  0 to double: an error grows faster when the doubling time is shorter.In other words, a faster error growth means a more rapid loss of initial information.
What is the error doubling time for sea clutter data?To find this, we use (3) to get the following: Letting    = 2 0 , we find the error doubling time   given by As the first approximation, we may consider 1/() to be proportional to the doubling time.When () is given by (10), then the error doubling time is inversely proportional to .Therefore,  characterizes the speed of information loss.More rigorously, we can prove that − ln  0 ≫ 1 decays to a saturation or limiting value of 0 exponentially as follows: A zero magnitude of ln   amounts to a unit error.Therefore, it is proper to call  the relaxation parameter of information loss.The fact that  is larger for sea clutter with target implies that information loss in sea clutter with target is faster.Physically, this may be attributed to the generation of turbulence near the target.
Can the  parameter be used to help detect targets within sea clutter?The answer is positive, as shown in Figure 6(b).Clearly,  is very effective in distinguishing sea clutter with and without targets.For automatic detection purposes, we have analyzed almost 400 datasets available to us and found that a threshold value of 0.9 for  yields an almost perfect classification as follows.
When a signal is truly chaotic, SDLE is related to the conventional positive Lyapunov exponent by (7).Now that we have found sea clutter not to be chaotic, it is interesting to find out how different SDLE is from  1 .We have computed  1 for all the sea clutter data using Wolf et al. 's algorithm [56].Wolf et al. 's algorithm works as follows: one selects a reference trajectory and follows the divergence of its neighboring trajectory from it.Denote the reference and the neighboring trajectories by   = [(), ( + ), . . ., ( + ( − 1))],   = [(), ( + ), . . ., ( + ( − 1))],  = 1, 2, . .., and  = ,  + 1, . .., respectively.At the start of the time (which corresponds to  = 1),   is usually taken as the nearest neighbor of  1 .That is,  =  minimizes the distance between   and  1 .When time evolves, the distance between   and   also changes.Let the spacing between the two trajectories at time   and  +1 be    and  +1 , respectively.The variations of the Lyapunov exponent  estimated by conventional methods versus the 14 range bins for two HH measurements.Open circles denote the range bins with target, while asterisks denote the bins without target.The primary target bin is explicitly indicated by an arrow.For convenience, the largest   is taken as 1 unit.
To ensure that the separation between the two trajectories is always small, when  +1 exceeds certain threshold value, it has to be renormalized; a new point in the direction of the vector of  +1 is picked up so that   +1 is very small compared to the size of the attractor.After  repetitions of stretching and renormalizing the spacing, one obtains the following formula: Figures 7(a) and 7(b) show representative  1 for two measurements, where the largest  1 among the 14 bins of a measurement is taken as 1 unit.Clearly,  1 does not have any power of detecting targets within sea clutter.This suggests that  1 only characterizes noise in sea clutter, whether they hit targets or not.Therefore, the effectiveness of using (10) to detect targets within sea clutter suggests that SDLE is fundamentally different from the conventional Lyapunov exponent.

Concluding Remarks
To better characterize the nonstationary and multiscale nature of sea clutter, in this study, we have employed SDLE to analyze low-grazing angle sea clutter.We are able to directly show that sea clutter is not chaotic and reveal a new scaling law suggesting noisy dynamics for sea clutter.The new scaling law has an interesting interpretation in terms of intrinsic predictability of sea clutter.The fact that it can be used to accurately detect targets within sea clutter implies that it is a fundamental feature of sea clutter.

Figure 2 :
Figure 2: () curves for the clean and the noisy Lorenz system.

Figure 4 :
Figure 4: (a) An intermittent time series generated by the logistic map.(b) The SDLE curve for the time series.

Figure 5 :
Figure 5: A schematic showing how the sea clutter data were collected.

Figure 7 :
Figure 7: ((a) and (b)) The variations of the Lyapunov exponent  estimated by conventional methods versus the 14 range bins for two HH measurements.Open circles denote the range bins with target, while asterisks denote the bins without target.The primary target bin is explicitly indicated by an arrow.For convenience, the largest   is taken as 1 unit.