Multiscale modeling of sea clutter radar returns · Multiscale modeling of sea clutter radar...

Multiscale modeling of sea clutter radar returns

Jianbo Gao

PMB InTelliGence, LLC, West Lafayette, IN 47906

Mechanical and Materials Engineering, Wright State University

[email protected]://www.gao.ece.ufl.edu/

Gao, Jianbo (PMB InTelliGence) Target detection within sea clutter July 2013 1 / 41

Outline

Background and challenges

Part I: Fitting nonstationary sea clutter data

Part II: Correlation and fractal behavior in sea clutter data

Part III: Understanding the nonlinear dynamics of sea clutter usingscale-dependent Lyapunov exponent

Part IV: Cascade multifractal modeling of sea clutter

Conclusions


Sea clutter

Backscattered returns from a patch of the sea surface illuminated bya transmitted radar pulse —turbulent wave motions + multipathpropagation


Source of sea clutter data

14 sea clutter measurements from Prof. Simon Haykin; eachmeasurement contains 14 range bins, a few bins hit a small target

Each measurement was made under certain weather and seaconditions (wave height varied from 0.8 m to 3.8 m; wind conditionsvaried from still to 60 km/hr)

B1 B

2 ⋅⋅⋅ B

i B

14

h φ

⋅⋅⋅ (Primary)

Bi−1

(Secondary)

Bi+1

(Secondary)

Ri

h : Antenna height φ : Grazing angle R

i : Range (distance from the radar)

B1 ~ B

14 : Range bins

o

Target


Significance and Challenges of sea clutter modeling

Sea clutter analysis is an important theoretical problem

Target detection within sea clutter is important to coastal andnational security, to navigation safety, and to environmentalmonitoring

New models for sea clutter may shed new light on wirelesscommunication channel characterization and signal detection

Over a thousand papers have been published. Numerous methods andnew concepts including chaos and fractal theory have been tried.

By now, the nature of sea clutter is still unclear

Simple and reliable methods for detecting low observable targetswithin sea clutter have yet to be developed


Why sea clutter modeling is difficult? — Nonstationary!

(i) Data viewed at different time and scales appear very different.

(ii) Subplot (e): signals cannot be characterized as ideal random fractals or

autoregressive (AR) processes.

46 46.02 46.04 46.06 46.08 46.10

0.51

1.52

2.5

Time (sec)

Am

plitu

de

80 80.02 80.04 80.06 80.08 80.1012345

Time (sec)

Am

plitu

de

6 6.5 7 7.5 8012345

Time (sec)

Am

plitu

de

0 20 40 60 80 100 1200

5

10

Time (sec)

Am

plitu

de

0 4 8 12 16−2

−1.5

−1

−0.5

0

0.5

log2 m

log 2 [m

2 Var

(X(m

) )]

(a) (b)

(c)

(d)

(e)


Failure of direct distributional analysis of sea clutter

(i) Distr. tried: Weibull, log-normal, K, compound Gaussian, log-Weibull

(ii) K distr. f (x) =√

2ν√µΓ(ν)2ν−1

(√2νµ x)ν

Kν−1

(√2νµ x), x ≥ 0

is among the best; but the fitting can be poor— Can’t help with target detection — Culprit: data isnonstationary!

0 0.5 1 1.5 20.1

0.2

0.3

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0.8

PD

F

x

0 0.5 1 1.5 20.1

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x

PD

F

DataK distribution fit


(a) (b)


Part I: Our approach to fit nonstationary sea clutter data

Denote the sea clutter amplitude data by y(n), n = 1, 2, · · ·Denote the differenced data of sea clutter by,x(n) = y(n + 1)− y(n), n = 1, 2, · · ·Fit x(n), n = 1, 2, · · · using Tsallis distribution.

Why such a strategy works?I Consider white Gaussian noise, u(i), i = 1, 2, · · · . It is stationary!I Standard Brownian motion (or random walk): v(n) =

∑ni=1 u(i) is

nonstationary, because the variance of v(n) is proportional to (time) nI K-distr. can be derived by assuming a random walk model for

scatterers!


Tsallis distribution

Obtained by maximizing the Tsallis entropy under 2 constraints.

The distr: when 1 < q < 3,

p(x) =1

Zq[1 + β(q − 1)x2]1/(1−q),

where Zq is a normalization constant.

When q = 1 & 2, it reduces to the normal & Cauchy distr.

When 5/3 < q < 3, the distribution is heavy-tailed

Significance: provides foundation for the heavy-tailed and α-stabledistr.


Heavy-tailed distribution

Pareto distr: P[X ≥ x ] =(

bx

)α, x ≥ b > 0, α > 0

where α and b are the shape & the location parameters.

In the discrete time case, we have Zipf distr.

Heavy-tailed distr: P[X ≥ x ] ∼ x−α, x →∞When α < 2, the variance and all higher than 2nd-order moments donot exist.when α ≤ 1, the mean also diverges.

Cauchy distr (also called Lorentzian distr) with PDF f (x) = lπ(l2+x2)

is an example with α = 1


Stable laws and Levy motions

Paul Levy (teacher of Mandelbrot, the Father of fractal geometry)posed such a question: When will the distribution for the sum of therandom variables and those being summed have the same functionalform?

Stable laws are the unique class of distributions that have such aproperty.

Stable laws include Gaussian distr as a special case; in thenon-Gaussian case, the distributions are heavy-tailed

Levy motions: random walk processes whose increments arecharacterized by stable laws


The meaning of stable laws and Levy motions

Normal distr & central limit theorem describe daily, mundane life— Many lucky people live through such a life happily.

Occasionally one has to take on an unplanned journey, during whichmany unexpected and exciting (or terrible) things happen.

Such a journey could be related to hate, love, patriotism, and so on,as illustrated by numerous classic poems, fictions and movies.

Kolmogorov was pondering: Stable laws with infinite varianceshould be observed more often than the normal distr. In reality ...?

Abundant examples of heavy-tailed distributions have been found:Amount of Internet traffic, topology of networks (eg, power-lawnetworks), distr. of the size of the power outages, ...

Fundamental question: How do stable laws arise?


Deriving Tsallis distr by maximizing Tsallis entropy

Tsallis entropy aims to characterize a type of motion whosecomplexity is neither regular nor fully chaotic/random, by employing aparameter q, that best describes the motion.

It’s defined by HTq = 1

q−1

(1−

∑mi=1 pi

q)

.

In the continuous case, it is HTq = 1

q−1

(1−

∫∞−∞ d( x

σ )[σp(x)]q)

.

It reduces to the Shannon entropy when q → 1.

Tsallis distr can be derived by maximizing Tsallis entropy under 2constraints,

I Total prob. is 1:∫∞−∞ p(x)dx = 1.

I Second normalized moment is known:∫∞−∞[x2 − σ2][p(x)]qdx = 0.


Generalized Tsallis distribution

We may generalize the Tsallis distr by replacing the 2nd constraint by∫∞−∞[xα − σα][p(x)]qdx = 0. Then the distr becomes

p(x) =1

Zq[1 + β(q − 1)xα]1/(1−q)

This is our starting point for modeling sea clutter.

To model turbulent motions, Christian Beck (2000) obtained thesame distr. through a different approach, which is considerably morecomplicated than our approach.


Modeling of sea clutter

(Symbol: data; curve: Tsallis fit)

−2 −1 0 1 210

−4

10−2

100

x

PD

F

−2 −1 0 1 210

−4

10−2

100

x

PD

F

−2 −1 0 1 210

−4

10−2

100

x

PD

F

−2 −1 0 1 210

−4

10−2

100

x

PD

F

DataTsallis fit

(a) (b)

(c) (d)

Data is ready. So whatis the challenge here?

Data is highlynonstationary. It’s notvery meaningful toperform distributionalanalysis on original data.

How about thedifferenced data? Itworks!

Parameters are helpfulfor target detection.


Diffusive models for sea clutter

Tough and Ward (TW: 2002): the electric field returned from Ncoherently illuminated scatterers is written as a two-dimensionalvector sum of the contributions ak from the individual scatterers

E =N∑

k=1

ak

It can be thought of as the resultant of a random walk of N steps.

TW assume that ak are independent, isotropically distributed andstatistically identical.

Based on this model, TW have derived elegant stochastic differentialequations and associated Fokker-Planck equations, and re-casted theK distribution as a Gaussian speckle process with its local powermodulated by a gamma distribution.


Significance of the diffusive models

The reformulation interprets sea clutter as Gaussian speckle overshort periods, with de-correlation time on the order 0.1 s,corresponding to transit of small scale sea surface features throughthe microwave wavelength

The modulation represented by the gamma variate is considered to beassociated with the large scale structure of the sea surface, partiallyresolved by the high resolution radar; its de-correlation time istypically of the order of seconds or more.

In summary, two time scales have been identified: 0.1 s and ∼ 1 s

Question: Between and beyond these two time scales, is thereanything interesting going on within sea clutter?


Failure of direct distributional analysis of sea clutter

(i) Distr. tried: Weibull, log-normal, K, compound Gaussian, log-Weibull

(ii) K disrt. f (x) =√

2ν√µΓ(ν)2ν−1

(√2νµ x)ν

Kν−1

(√2νµ x), x ≥ 0

is among the best; but the fitting can be poor— Can’t help with target detection — Culprit: data isnonstationary!

0 0.5 1 1.5 20.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

PD

F

x

0 0.5 1 1.5 20.1

0.2

0.3

0.4

0.5

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0.8

x

PD

F



(a) (b)


Part II: How to better model sea clutter? – Correlationsand fractal behavior

We need multiscale analyses methods

One of the simplest multiscale analyses is the structure-functionbased multifractal formulation

Introduction to fractal & multifractalI A part is (exactly or statistically) similar to another part, or to the

whole.I Clouds; mountains; trees; etc.I Power-law relation

— a straight line in a log-log plot (scaling)I Many (or possibly infinitely many)

power-law relations—Multifractal.


Random walks and their analysis

Remove the mean values from {x(i)} process, denote it as {u(i)}Random walk: y(n) =

∑ni=1 u(i)

Independent u(i)’s (a drunk)–no correlation:E [y(m)2] = m · E [u(i)2] ∼ m

Fluctuation analysis (FA):F (2)(m) =

⟨|y(n + m)− y(n)|2

⟩∼ mζ(2)

Hurst parameter H = H(2) = ζ(2)/2—H = 1/2: no or short-range correlation; 0 < H < 1/2:anti-persistent correlation; 1/2 < H < 1: persistent correlation

Power-spectral density: Sy (f ) ∼ f −(2H+1), Sx(f ) ∼ f −(2H−1)


Multifractal analysis:structure function technique

F (q)(m) = 〈|y(i + m)− y(i)|q〉 ∼ mζ(q) ?q < 0: emphasizes small absolute increments of y(i);q > 0: emphasizes large absolute increments of y(i).

H(q) = ζ(q)/q

Monofractal: ζ(q) linear in q (ζ(0) = 0);H(q) constant.

Multifractal: ζ(q) nonlinear in q;H(q) varies with q.


Variation of H(q) for 14 range bins within ameasurement

Sea clutter data with target have larger H(q) and are multifractals

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q

H(q

)


Target detection within sea clutter

H(2) is much larger when the range bins hit a target

Sea clutter data are multifractals, and that other q values can alsorobustly detect targets within sea clutter

0 5 10 151

1.1

1.2

1.3

1.4

Bin Number

H(2

)

Primary Target Bin

Bins With Targets


Accuracy of target detection across measurements

The pdf’s for H of data with and without target completely separate— 100% accuracy!

0.6 0.8 1 1.2 1.4 1.60

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H(2)

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cyno targetsprimary targets


Part III: Exploring the nonlinear dynamics of sea clutter

Understanding the nature of sea clutter is crucial to the successfulmodeling of sea clutter as well as to facilitate target detection withinsea clutter

An important question to ask is whether sea clutter is stochastic ordeterministic

Can sea clutter be fully characterized by statistical means? — No!I Sea clutter signals are functions of complex (sometimes turbulent)

wave motions on the sea surfaceI Wave motions on the sea surface clearly have their own dynamical

features that are not readily described by simple statistical features

Sea clutter modeling using dynamical systems theory, especially chaostheory, is very appealing


State-of-the-art of the field

Haykin et al. (1997, 1999, 2001): Sea clutter data are chaotic

Recently, many researchers (Noga, Davies, Cowper, Unsworth et al,Gao &Yao, · · · ) have challenged their finding

I Gao & Yao: directly showed that sea clutter data are not chaotic in arigorous mathematical sense

I Others: (indirectly) showed that sea clutter data may not bedistinguished from simulated stochastic processes having similarstatistical features as sea clutter

Haykin et al. (2001): non-chaotic features of sea clutter are due tonoise

McDonald & Damini (2004): no chaos in sea clutter after low-passfiltering to remove noise


Problem formulation

This is a clamorous field

Consensus: sea clutter data are not chaotic

Have we gained much by this type of research? —No!

Fundamental questions:

I The wave-turbulence interactions on the sea surface clearly have theirdistinct dynamics. Can we properly and directly characterize them bygoing beyond the standard chaos theory and conventional statisticaldescriptions?

I Will studies along this line be of any help for target detection withinsea clutter? 11


Our approach

Recently, we have developed a new multiscale complexity measure,the scale-dependent Lyapunov exponent (SDLE)

The SDLE is able to classify various types of complex data, includinglow-dimensional chaos, stochastic oscillations, random 1/f processes,random Levy processes, and complex motions with multiple scalingbehaviors

It is able to unambiguously distinguish chaos from noise

We shall apply the SDLE to analyze almost 400 sea clutter datasetsavailable to us— In contrast, most published works only considered 1 or 2datasets


Scale dependent Lyapunov exponent (SDLE): Definition(Gao et al. Phys. Rev. E, 2006)

Consider an ensemble of trajectories in phase space

Denote the initial separation between two nearby trajectories by ε0,and their average separation at time t and t + ∆t by εt and εt+∆t ,respectively

Being defined in an average sense, εt and εt+∆t can be readilycomputed from any processes, even if they are non-differentiable

When ∆t → 0, we have

εt+∆t = εteλ(εt)∆t or λ(εt) =

ln εt+∆t − ln εt∆t

ordεtdt

= λ(εt)εt

Given a time series data, the smallest ∆t = sampling time


SDLE: Computation

Basic idea: form proper ensemble averages

Phase space reconstruction: Vi = [x(i), x(i + L), ..., x(i + (m − 1)L],x(i): given time series, m: embedding dimension, L: delay time

Introducing a sequence of shells

εk ≤ ‖Vi − Vj‖ ≤ εk + ∆εk , k = 1, 2, · · ·

where εk & ∆εk are arbitrarily chosen small distances

λ(εt) =

⟨ln ‖Vi+t+∆t − Vj+t+∆t‖ − ln ‖Vi+t − Vj+t‖

⟩∆t

Ensemble average is taken within a shell— add condition |i − j | > (m − 1)L for chaotic systems

Similar algorithm used in computing time-dependent exponent curves

(Gao & Zheng, Phys. Lett. A 1993; Europhys. Lett. 1994; Phys. Rev. E, 1994)


Classification of complex motions

Chaotic motions

I Low-dimensional deterministic chaos: λ(ε) ∼ constI Noisy chaos: λ(ε) ∼ −γ ln εI Noise-induced chaos — without noise, motion is regular

(Gao et al. Phys. Rev. Lett. 1999, 2002; Hwang & Gao Phys. Rev. E 2000):λ(ε) ∼ −γ ln ε

1/f 2H+1 processes: λ(ε) ∼ Hε−1/H

α-stable Levy processes: λ(ε) ∼ 1αε−α

Stochastic oscillations: both λ(ε) ∼ −γ ln ε and λ(ε) ∼ Hε−1/H

Complex motions with multiple scaling behaviors


Example: Stochastic Lorenz system

dx/dt = −10(x − y) + Dη1(t),dy/dt = −xz + 28x − y + Dη2(t),dz/dt = xy − 8

3z + Dη3(t),

−1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2

0

0.5

1

1.5

2

ε

λ(ε)

D=0D=1D=2D=3D=4

log10


A classic issue: distinguishing chaos from noise

Often adopted assumption:finite positive Lyapunov exponent and non-integral fractal dimension

Chaos in brain, chaos in heart, chaos in sea clutter— Chaos is everywhere!

Counter-example: a 1/f process with Hurst parameter H has a fractaldimension 1/H and finite positive Lyapunov exponent (or entropy).

Running on a wild beach, hunting for chaotic beast: “Here is afootprint”, “Here is another”...Turning around, astonishingly foundthese are but their own footprints!

Distinguishing chaos from noise is no longer a problem!


Multiscale analysis of sea clutter

Small ε, λ(ε) = −γ ln ε;Large ε, λ(ε) ∼ ε−1/H (not shown here, see Hu, Gao et al. 2005, 2006)

10−0.6

10−0.1

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λ(ε)

ε0 5 10 15

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1.2

Bin number

γ

Without targetWith target

(a) (b)

Primary target bin


Target detection within sea clutter

∼ 400 datasets, measured with wave height 0.8− 3.8 m, winds 0− 90km/hr, were obtained from Prof. Simon Haykin of McMaster Univ

Target: spherical block of styrofoam of size 1 m, wrapped with wire mesh

A threshold value of 0.9 for γ yields an almost perfect classification:

I Hypothesis H0: sea clutter without target, γ < 0.9I Hypothesis H1: sea clutter with target, γ > 0.9


Significance of the parameter γ

Error doubling time: the average time for an initial error ε0 todouble

An error grows faster when the doubling time is shorter

A faster error growth means a more rapid loss of initial information379,0-1

Using the definition for SDLE, we have

ln εt = ln ε0 +

∫ t

0λ(εt)dt

Let εTdb= 2ε0, then Tdb is given by

ln 2 =

∫ Tdb

0λ(εt)dt


Significance of the parameter γ (Cont’d)

First approximation: Tdb ∝ 1/λ(ε)

More rigorously, we can prove that − ln ε0 � 1 decays to a saturationor limiting value of 0 exponentially,

− ln εt = − ln ε0e−γt

A zero magnitude of ln εt amounts to a unit error.

It is proper to call γ the relaxation parameter of information loss


Cascade multifractal modeling of sea clutter

Construction of multiplicative multifractal processes


Cascade multifractal modeling of sea clutter

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plitu

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itude

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(a) Non target (b) Primary target

(d) Primary target (c) Non target


Target detection using Dq spectrum

Scaling for cascade model: Mq(ε) =∑

i wqi ∼ ετ(q)

Dq = τ(q)/(q − 1)

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Primary Secondary

Non target

Primary Secondary

Non target


Conclusions

We have shown that sea clutter data are highly nonstationary andmultiscaled

We have developed a new distributional analysis approach to describesea clutter

We have developed a highly accurate (close to 100%) fractal-basedmethod for detecting low observable targets within sea clutter

I The accuracy of target detection within sea clutter critically dependson the time scale range of 30 msec to about 1 sec we identified

I On time scales smaller than 30 msec, sea clutter data are actually quitesmooth

I On time scales longer than 1 sec, sea clutter data are very irregular

Analysis of sea clutter using SDLE revealed distinctive noisy dynamicsthat can also be used for target detection

Cascade multifractal can simulate sea clutter data very well, and canalso be used for target detection


Multiscale modeling of sea clutter radar returns · Multiscale modeling of sea clutter radar...

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