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Radar Clutter ModelingMaria S. GRECODipartimento di Ingegneria dell’InformazioneUniversity of PisaVia G. Caruso 16, I-56122, Pisa, Italym.greco@iet.unipi.it

Clutter modeling2What is the clutter?Clutter refers to radio frequency (RF) echoes returned fromtargets which are uninteresting to the radar operators andinterfere with the observation of useful signals.Such targets include natural objects such as ground, sea,precipitations (rain, snow or hail), sand storms, animals (especiallybirds), atmospheric turbulence, and other atmospheric effects,such as ionosphere reflections and meteor trails.Clutter may also be returned from man-made objects such asbuildings and, intentionally, by radar countermeasures such aschaff.2/157

Clutter reflectivity: General Concepts3We will focus particularly on sea and land(or ground) clutterRadar cross section (RCS)RCS per unit illuminated area, σo (m2/m2)RCS per unit illuminated volume, η (m2/m3)Radar equation and pattern-propagation factor FSea clutter RCS and spikesLand clutter RCSSea and land clutter statisticsThe compound-Gaussian modelClutter spectral models3/157

Clutter reflectivity4A perfectly smooth and flat conducting surface acts as a mirror,producing a coherent forward reflection, with the angle of incidenceequal to the angle of reflection. If the surface has some roughness,the forward scatter component is reduced by diffuse, non-coherentscattering in other directions.radarFor monostatic radar,clutter is the diffusebackscatter in thedirection towards theradarθiθrspecular reflectionθi ough surface4/157

Radar Cross Section (RCS)5The IEEE Standard for RCS in square meters isPsσ 4πΩpiwherePs power (watts) scattered in a specified directionfrom the target having RCS σΩ solid angle (steradians) over which Ps is scatteredpi power density (watts/m2) of plane wave at target5/157

Radar Cross Section (RCS)6Range of RCS Values (dBm2)Echo power directlyproportional to RCSFactors that influenceRCS:size, shape, materialcomposition, moisturecontent, surface coatingand roughness,orientation, polarization,wavelength, multipathRCS of common objects6/157

Normalized RCS σ07The normalized clutter reflectivity, σ0, is defined as the total RCS, σ,of the scatterers in the illuminated patch, normalized by the area, Ac,of the patch and it is measured in units of dBm2/m2.ψElevationRhRadar pulseρθelRφgrR θazAzimuthσ 0 σ / AcantennafootprintθazClutter patchρ secφgrThe factor α accounts for the actual compressed pulseshape and the azimuth beamshaperange resolutionAc αρ Rθ az sec(φ gr )local grazing angle7/157

Volume reflectivity η8The volume clutter reflectivity, η, is defined as the total RCS, σ, ofthe scatterers in the illuminated volume, normalized by the volumeitself, Vc, and it is measured in units of dBm2/m3.R θazPencil beamRθazR θelθelη σ / VcρRadar pulseVc αρ R 2θ azθelone-way 3dB elevation beamwidth8/157

Radar Equation and propagation factor F9For monostatic radar, received power Pr from a target withRCS σ is2 2PGλ4tPr σF3 44π( )RPt transmit powerG antenna gainR distance of target from antennaF the pattern-propagation factor, the ratio of fieldstrength at a point to that which would be present if freespace propagation had occurredA clutter measurement provides either σF4 or σoF4.Even so, normally the data are reported as being σ or σo.9/157

Discrete and distributed clutter10DISCRETERCS depends on aspect angle, multipath environment,frequency, and polarizationRCS values up to 30 dBm2 are commonRCS above 40 dBm2 rare, except in built-up areasNominal RCS values:60 dBm2very large ship or building250/40 dBmlarge building or shipsmall building/house30/20 dBm2trucks/automobiles20/10 dBm2DISTRIBUTEDAverage RCS so times A, where A is illuminated surfacearea (footprint) of a range-azimuth cell10/157

Sea clutter: Dependence on grazing angle11At near vertical incidence, the backscatter is quasi-specular andvaries inversely with surface roughness with a maximum at verticalincidence for a perfectly smooth surface.At medium grazing angles the reflectivity shows a lowerdependence on grazing angle (plateau region).0 dB10 log σOBelow some critical angle( 10º, depending on theroughness) the reflectivityreduces rapidly with smallergrazing angles (interferenceregion, where propagation isstrongly affected bymultipath scattering andshadowing).0oθcGRAZING ANGLE θθo90o11/157

Empirical model for sea clutter σ012Nathanson tables [Nat69]The "standard" beginning 1969, updated 1990HH and VV POLs; 0.1 , 0.3 ,1 , 3 , 10 , 30 , 60 grazingMany data sources, 60 different experimentsUHF to millimeter wavelengthsReported by sea state up to state 6Averaged without separating by wind or wave directionGreatest uncertainties at lower frequencies and 3 Reported RCS generally larger than typical because(a) experimenters tend to report strongest clutter and(b) over-water ducting enhances apparent RCS[Nat69] F.E. Nathanson, Radar Design Principles, McGraw Hill, 196912/157

Empirical model for sea clutter σ013GIT model [Hor78]Variables: radar wavelength, grazing angle, wind speed, winddirection from antenna boresight, wave heightEmploys separate equations: for HH and VV polarization, andfor 1 to 10 GHz and 10 to 100 GHzThe 1-10 GHz model based on data available for grazing anglesof 0.1 to 10o and average wave heights up to 4 m (correspondsto significant wave heights of 6.3 m)Few liable σoF4 data available at 3o grazing and below,and for dependencies on wind and wave directionsGraphs from the model appear to give “best guesses” of σoF4versus grazing angles less than 10o[Hor78] M.M. Horst, F.B. Dyer, M.T. Tuley, “Radar Sea Clutter Model”, IEEE InternationalConf. Antennas and Propagation, Nov. 1978, pp 6-10.13/157

Empirical model for sea clutter σ014GIT model: Wind speed dependenceHH POL, 10 GHzCross-wave direction,2 m signif. wave height,winds 3, 5, 10, 20 m/sσoF4 increases with windspeed,Critical angle unchanged,because wave heightassumed fixed.14/157

Empirical model for sea clutter σ015GIT model: Dependence on significant waveheight h1/3HH POL, 10 GHzCross-wave direction,10 m/s wind speed,h1/3 0.5, 2, and 6 mIn plateau region, σoF4 isindependent of h1/3 (forfixed wind speed)σoF4 increases with h1/3(multipath reduces criticalangle) at angles 1o15/157

Empirical model for sea clutter σ016GIT model: Comparison between HH and VVPOL, 10 GHzWind speed/wave height inequilibriumσoF4 increases with windspeed and h1/3HH/VV ratio increases withincreased surface roughnessand reduced grazing angleHH VV at small angles underrough conditions at 1.25 and10 GHz16/157

Sea clutter spikes17Bishop (1970) classified sea clutter at X-band into four types:Noise-like-clutter which appears similar to thermal noisewith no apparent sign of periodicity;Clumped clutter which appears as a mixture of noise-likereturns and discrete clumped clutter returns that oftenfades rapidly;Spiky clutter which consists largely of a collection ofclumped returns with short persistence, e.g., 1 to 2s;Correlated spiky clutter consisting of persistent clumpedreturns. At any one time, an A-scope trace looks like a combwith semi-randomly spaced teeth. The whole pattern movesat speeds up to 40 knots. Individual spikes persisted for 10to 40 s and the clutter level between spikes was virtuallyzero.17/157

Empirical model for land clutter σ018σo vs incidence angle for rough ground [Ula89]4.25 GHz data, high moisturecontentσo is insensitive to surfaceroughness at 10o (80o grazing)Same insensitivity toroughness observed at 80ograzing in 1.1 GHz and 7.25GHz dataSame shapes but lower σo fordry conditions[Ula89] Ulaby, F.T. and Dobson, M.C, Handbook of Radar Scattering Statistics for Terrain,Artech House, Norwood MA, 198918/157

Land clutter σ0 at low grazing angles37 Rural Sites [Bil02]190-20-30-40-50-60RANGE POLRES. (M)1-80 atial averages of σoF4 for θ 8o grazingLarger spreads in σoF4 atlower frequenciesResolution 150 & 15/36 mHH and VV polarizationsAt each frequency, themedian spatial average isroughly –30 dBNote: σoF4 can be larger atthe lower frequenciesMEAN OF σO F4 (dB)-10X-BANDFREQUENCY (MHz)[Bil02] J.B. Billingsley, Low-angle radar land clutter – Measurements and empirical models,William Andrew Publishing ,Norwich, NY, 2002.19/157

Clutter statistics: effect of spatial resolution20The scattered clutter can be written as the vector sum fromN random scatterersz σ i exp [ jφi ]NRCS of a single scattererphase termWith low resolution radars, N is deterministic and very high in eachilluminated cell. Through the application of the central limittheorem (CLT) the clutter returns z can be considered as Gaussiandistributed, the amplitude r z is Rayleigh distributed and themost important characteristic is the radar cross section. r2 p( r ) 2 exp 2 u ( r )σ σ 2r20/157

Clutter statistics: effect of spatial resolution21This is not true with high resolution systems. With reduced cellsize, the number of scatterers cannot be longer consideredconstant but random, then improved resolution reduces the averageRCS per spatial resolution cell, but it increases the standarddeviation of clutter amplitude versus range and cross-range and, inthe case of sea clutter, versus time as well.Jakeman and Pusey [Jak78] showed that a modification of the CLTto include random fluctuations of the number N of scatterers cangive rise to the K distribution (for amplitude PDF):1 N Z R Z aie jϕi N N i 12-D random walkK distributed if N is a negativebinomial r.v. (Gaussian distributed if Nis deterministic, Poisson, or binomial)N E{N},{ai}i.i.d.,{ϕ i}i.i.d.21/157

The compound-Gaussian model22In general, taking into account the variability of the local power τ,that becomes itself a random variable, we obtain the so-calledcompound-Gaussian model, then r2 p(r τ ) exp u (r )τ τ 2r p (r ) p( r τ ) p(τ ) dτ ; 0 r 0According to the CG model:z(n) τ (n)x(n)x(n) xI (n) jxQ (n)Texture: non negative randomprocess, takes into account thelocal mean powerSpeckle: complex Gaussianprocess, takes into account thelocal backscattering22/157

The compound-Gaussian model23Particular cases of CG model (amplitude PDF):νK (Gamma texture)GK (GeneralizedGamma texture)LNT (log-normaltexture)W, Weibull 4ν 4ν µ 4ν pR (r ) ν 1r Kν 1 r u (r ) 2 Γ (ν ) µ µ νb 2br ν pR ( r ) Γ (ν ) µ b2 rνν b 2 0 τ exp τ µ τ dτ r22 1exp 2 ln (τ δ ) dτpR ( r ) 2 τ 22πσ 0 τ 2σ r2c r pR ( r ) b b c 1exp ( r b) c u ( r )23/157

The K model24The K model is a special case ofthe compound-Gaussian model:N negative binomial r.v.τ (local clutter power) Gamma distributedAmplitude R K distributedK-PDF (amplitude PDF)1ν 10.8Gamma-PDF (texture PDF)0.6ν 4.530.42.50.200ν 0.5ν 0.52123r/E{R}456p τ (τ )pR (r) E{R}ν 1.5ν 301.5ν 1ν 1.5ν 101The order parameter ν is ameasure of clutter spikinessThe clutter becomes spikier as ν decreasesGaussian clutter:ν ν 20.5000.511.522.53texture (τ)24/157

The multidimensional compound-Gaussian model25 In practice, radars process M pulses at time, thus, to determine theoptimal radar processor we need the M-dimensional joint PDF Since radar clutter is generally highly correlated, the joint PDF cannotbe derived by simply taking the product of the marginal PDFs The appropriate multidimensional non-Gaussian model for use in radardetection studies must incorporate the following features:1) it must account for the measured first-order statistics (i.e., theAPDF should fit the experimental data)2) it must incorporate pulse-to-pulse correlation between data samples3) it must be chosen according to some criterion that clearlydistinguishes it from the multitude of multidimensional non-Gaussianmodels, satisfying 1) and 2)25/157

The multidimensional compound-Gaussian model26 If the Time-on-Target (ToT) is short, we can consider the texture asconstant for the entire ToT, then the compound-Gaussian model degeneratesinto the spherically invariant random process (SIRP) proposed by Conte andLongo [Con87] for modeling the radar sea clutter. By sampling a SIRP we obtaina spherically invariant random vector (SIRV) whose PDF is given by pZ ( z ) 10 ( πτ )M z H M 1 z exp pτ (τ ) dττM where z [z1 z2 . . . zM]T is the M-dimensional complex vector representing theobserved data. A random process that gives rise to such a multidimensional PDF can bephysically interpreted in terms of a locally Gaussian process whose power levelτ is random. The PDF of the local power τ is determined by the fluctuation model of thenumber N of scatterers.26/157

Properties of SIRVs27 The PDF of a SIRV is a function of a non negative quadratic form:q(z) (z mz ) H M 1 (z mz ) A SIRV is a random vector whose PDF is uniquely determined by thespecification of a mean vector mz, a covariance matrix M, and acharacteristic first-order PDF pτ(τ):pZ ( z ) 1(π ) MMhM ( q ( z ) ) hM(q) must be positive andmonotonically decreasing First-order amplitude PDF: q hM ( q ) τ M exp pτ (τ ) dτ τ 0{ } r2 2pR (r ) 2 h1 2 , σ 2 E { R 2 } E zσ σ r A SIRV is invariant under a linear transformation: if z is a SIRV withcharacteristic PDF pτ(τ), then y Az b is a SIRV with the same characteristicPDF pτ(τ).27/157

Sea and land clutter APDFs28First order statistics:The homogeneity of sea allows to consider its spatial distribution equivalentto the temporal distribution. The same is not true for land clutter. The mostcommon distributions for the sea clutter are the K, GK and Weibull.Because of the large spatial variability of land clutter the statistics in spaceand time are different. The most common are in the table below.1st orderTimeRayleigh, Rice, Weibull,Log-normalSpaceWeibull, Log-normal28/157

Power Spectral Density (PSD) models29 The question is: How to specify the clutter covariance matrix and thepower spectral density? Correct spectral shape impacts clutter cancellation and target detectionperformance. The clutter spectrum is not concentrated at zero Doppler only,but spreads at higher frequencies. There are several reasons for the clutter spreading:Wind-induced variations of the clutter reflectivity (sea waves,windblown vegetations, etc. ).Amplitude modulation by the mechanically scanning antenna beam.Pulse-to-pulse instabilities of the radar system components.Transmitted frequency drift. The pulse-to-pulse fluctuation is generally referred to as internalclutter motion (ICM).29/157

PSD models30The PSD is often modeled as having a Gaussian shape:( f mfSG ( f ) S0G exp 2σ 2f )2 This is usually a mathematical convenience rather than any attempt atrealism. Often the Doppler spectrum will be strongly asymmetric andthe mean Doppler shift, mf, may not be zero. Clearly for land cluttermf is usually zero, but for rain and sea clutter in general mf 0 andwill be dependent on the wind speed and direction.From velocity/Doppler relationship v fDλ/2,standard dev. of scatterer velocity is σv σfλ/2Wind changes bandwidths, but typical σv areRain/chaff σv 1 to 2 m/sSea σv 1 m/sLand σv 0 to 0.5 m/s30/157

PSD models: windblown ground clutter31The Gaussian PSD model was proposed by Barlow [Bar49] for windblownclutter spectra,spectra for noncoherent radar systems and over limited spectraldynamic ranges (up to a level 20 dB below the peak level and to a maximumDoppler velocity of 0.67 m/s)Essentially all modern measurements of ground clutter spectra, with increasedsensitivity compared to those of Barlow, without exception show spectralshapes wider than Barlow’s Gaussian in their tailsIt had become theoretically well understood from 1965-67 on, that branchmotion in windblown vegetation generates spectra wider than GaussianIn a much referenced later report, Fishbein et al. [Fis67] introduced thepower-law clutter spectral shape:break-point Doppler frequency where theshape function is 3 dB below its peak zeroDoppler levelPac ( f ) nsin( π n)12πf c 1 ( f f c )nn is the shapeparameter31/157

PSD models: windblown ground clutter32Common values of power-law exponent n used in PSD modeling are usually on theorder of 3 or 4, but sometimes greaterThe evidence that clutter spectra have power-law shapes over spectral dynamicranges reaching 30 to 40 dB below zero-Doppler peaks is essentially empirical,not theoretical.However, there is no simple physical model or fundamental underlying reasonrequiring clutter spectral shapes to be power law.Recently, Billingsley [Bil91] showed that measurements at MIT-LL of windblownground clutter power spectra to levels substantially lower than most earliermeasurements (i.e., 60 to 80 dB below zero-Doppler peaks) indicate spectralshapes that fall off much more rapidly than constant power-law at the lowerlevels, at rates of decay approaching exponential:exponentialPac ( f ) λβe λβ exp e f 2 4λ is the radar transmissionwavelength and βe is theexponential shape parameter32/157

PSD models: windblown ground clutter33Then, recent studies have demonstrated that the groundclutter spectrum of windblown trees consists of threecomponents: coherent component slow diffuse component fast-diffuse componentThe coherent component was the results of radarreturns from steady objects such as buildings, highwaysand from movable objects at rest.The coherent component is at zero Doppler.33/157

Slow-diffuse & fast-diffuse components34The slow-diffuse component is the consequence of motions ofobjects with moderate inertia (tree branches).The slow-diffuse component occupies a relative narrow regionaround zero Doppler.The spectrum is approximately symmetrical and its spectraldensity in dB scale decreases linearly with increasing absolutevalues of Doppler frequency.The fast-diffuse component is the result of movements in lightobjects such as a tree leaves. This component has a spectraldensity similar to a band-limited noise. Its magnitude is usuallycompared to other components.The spectral extent is of the same order as the Doppler shiftsthat corresponds to the wind speed.34/157

PSD models: windblown ground clutter3535/157

PSD models: windblown ground clutter36Spectral shapes having equal AC (Fluctuating) Power- Source: Billingsley (1996)Each spectrum for windspeed of about 20 mphThe Gaussian shapereported by Barlow (1949)The power-law shape fromFishbein et al. (1967)The exponential shapefrom Billingsley andLarrabee ( 1987)36/157

Ground clutter spectra: X-band3737/157

Ground clutter spectra: S-band3838/157

Variation of the spectral slope diffuse components39S-band39/157

L-Band forest PSD vs wind speed40Approximate lineardependence of power densityin dB versus velocity, for allwind speedsFor VHF through X band,measured spectral shapesversus Doppler velocityfound to be essentially thesameSource: Billingsley (1996).40/157

Sea clutter PSD41The relative motion of the sea surface with respect to theradar causes an intrinsic Doppler shift of the return fromindividual scatterers.Because the motion of the scattering elements havevarying directions and speeds the total echo contains aspectrum of Doppler frequencies.Two effects are of interest: the spectral shape and width the mean Doppler shift of the entire spectrum.41/157

Sea clutter PSD42The spectrum of sea clutter is sometimes assumed to haveGaussian shape. An approximate relationship between the-3dB bandwidth f of the spectrum and sea state S (Douglasscale) has been derived by Nathanson: f 3.6 f 0(GHz ) SThe standard deviation of the Gaussian spectrum is related to fby the expression:σ f 0.42 fRecently more complex and realistic models have been proposedfor sea clutter PSD. We are going to analyze them later on.42/157

43Radar Clutter:Live recorded data43/157

IPIX Radar Description44Transmitter frequency agility (16 frequencies, X-band) H and V polarizations, switchable pulse-to-pulse pulse width 20 ns to 5000 ns PRF 0 to 20 KHzReceiver coherent receiver 2 linear receivers; H or V on each receiver quantization: 8 to 10 bits sample rate: 0 to 50 MHz BW 5.5 MHzAntenna parabolic dish (2.4 m) pencil beam (beamwidth 1.1 ) grazing angle 1 , fixed or scanningSource: Defense Research Establishment Ottawa.44/157

Sea clutter temporal behaviour (30 m)45The spikes have different behaviour in the two like-polarizations (HH and VV)The vertically polarized returns appearto be a bit broader but less spikyThe dominant spikes on the HHrecord persist for about 1-3 s.45/157

46Sea clutterLong waves or swellsWaves and whitecaps46/157

Sea clutter temporal behaviour (30 m)47Waves 80 cm highIPIX radarWaves 2.4 m high47/157

Data Description48Dataset19980204 22375319980204 22084919980204 22322019980204 22402419980204 223506Date and 22:32:2002/04/199822:40:2402/04/199822:35:06# Range cells2828282827Start range3201 m3201 m3201 m3201 m3201 mRange res.60 m30 m15 m9m3mPulse width400 ns200 ns100 ns60 ns20 ns# Sweep6000060000600006000060000Sample per cell6000060000600006000060000PRF1 KHz1 KHz1 KHz1 KHz1 KHzRF-freq.9.39 GHz9.39 GHz9.39 GHz9.39 GHz9.39 GHzSSSSSRadar andwave geometry48/157

Statistical Analysis: Amplitude ModelsLN, log-normalW, WeibullpR ( r ) 49 1 exp 2 ln ( r δ ) 2 u ( r ) 2σ r 2πσ 21c r pR ( r ) b b c 1exp ( r b) c u ( r )νK (Gamma texture)GK (Generalized Gammatexture)LNT (log-normaltexture)pR ( r ) 4ν 4ν µ 4ν rKr u (r ) ν 1 ν 12 Γ (ν ) µ µ νb 2br ν pR ( r ) Γ (ν ) µ b2 rνν b 2 0 τ exp τ µ τ dτ r22 1 pR ( r ) exp lnτδ() dτ2 2 τ 22πσ 0 τ 2σ r249/157

Histogram and moments50 A histogram is a graphical representation used to plot densityof data, and often for density estimation. A histogram consists of tabular frequencies, shown asadjacent rectangles, erected over discrete intervals (bins),with an area equal to the frequency of the observations in theinterval. The height of a rectangle is also equal to thefrequency density of the interval, i.e., the frequency dividedby the width of the interval. The total area of the histogram isequal to the number of data. A histogram may also be normalized displaying relativefrequencies. In that case the total area is 1. The bins must beadjacent, and often are chosen to be of the same size.50/157

Statistical Analysis: Results - 15 m-2100HistoWLNKLNTGK10-310PDFAveraged Clutter Power10511-410VVHHVH0.1VV data-5100481216Range Cell20240.012800.050.1Amplitude (V)0.150.2 With resolution of 60 m, 30 m,and 15 m a very good fittingwith the GK-PDF. Negligible differences amongpolarizationsNormalized Moments1000100DataWLNKLNTGK101123456Order51/157

Statistical Analysis: Results - 3 m52100HistoWLNKLNTGK101PDF With resolution of 9 m and 3 mhistograms with very long tails Not big differences amongpolarizations, but generally HHdata spikier than VV data0.10.01HH data0.00101000.10.260.30.40.5Amplitude (V)0.60.70.810HistoWLNKLNTGKPDF110Normalized Moments1050.1410DataWLNKLNTGK3102100.0110VV data0.00100.10.20.30.40.5Amplitude (V)0.60.70.81123456Order52/157

53Average spectral models53/157

Sea clutter average centimetres or less. Generated by turbulent gusts of nearsurface wind; their restoring force is the surface tension.Longer gravity waves (sea or swell) with wavelengths rangingfrom a few hundred meters to less than a meter. Swells areproduced by stable winds and their restoring force is theforce of gravity.54/157

Sea clutter average spectra55In the literature, it has been often assumed that the seaclutter has Lorentzian spectrum (i.e., autoregressive oforder 1).Autoregressive (AR) models with the order P rangingfrom 2 to 5 have also been proposed for modelling radarclutter.For the sea surfaces some experimental analysis at smallgrazing angle, C and X-bands, indicate that the sea Dopplerspectrum cannot be expressed by the Bragg mechanismonly, but also by wave bunching (super-events).55/157

Sea clutter average spectra56Lee et al. showed that the spectral lineshapes can bedecomposed into three basis functions which areLorentzian, Gaussian, and Voigtian (convolution of theGaussian and Lorentzian):S( f ) Γ 2π 2( f f L ) ( Γ 2π )22S( f ) peak of the Lorentzian functionaπexp ( x 2 ) f f VfVe 2 x a dxΓ 1 characteristic scatterer lifetimea Γ 2 πf Ve shape parameterfV centre of the Voigt function56/157

How to estimate the clutter PSD57The PSD can be estimated parametrically (or model-based) ornon-parametrically, without any hypothesis on the model.Non parametrically, we used the periodogram defined as:Z( f )P( f ) M2where Z ( f ) is the Fourier Transform of the data and M thenumber of samples.There are many variants of the periodogram (for instance,method of Welch, Blackman and Tukey) [see e.g. Stoica andMoses book on Power Spectral Analysis].57/157

Sea clutter average spectra-35 10-34 1058The spectrum is the sum of two basisfunctions among: the Gaussian, theLorentzian, and the Voigtian, withdifferent Doppler peaks.HH polarizationnon-paramV Gnon-par: periodogramGauss and Voigtian basis functions-3PSD3 10-35 10VV polarization-32 10-34 10-3non-paramV G1 10-30-1000 -800 -600 -400 -2000200 400 600 800 1000Frequency (Hz)High peak (Voigtian): 450 HzLow peak (Gaussian): 320 HzHigh peak (Voigtian): 410 HzLow peak: (Gaussian): 250 HzPSD3 10-32 10-31 100-1000 -800 -600 -400 -2000200 400 600 800 1000Frequency (Hz)58/157

AR modelling59An Autoregressive process of order P, AR(P), is characterized by the difference equation:PZ ( n) aP ,k Z ( n k ) W ( n)k 1where the coefficientsaP ,kare the process parameters, and W(n) is white noiseFor estimating the AR(P) parameters, we use the Yule-Walker equationsRz (1)Rz ( P 1) aP ,1 Rz (1) Rz (0) R ( 1) R (0) a R (2) z z P ,2 z Rz (1) a R(1 P) R( 1)R(0)R(P)zz z P,P z Ra rIn our case, we don’t know the “true” correlation of the clutter, so we estimate it as1Rˆ Z ( m ) NN 1 m z ( n) z ( n m)k 059/157

AR modelling60We replace the estimated correlation to the true one and we solve the linearsystemˆ 1rˆaˆ RNormalized PSDso obtaining an estimate of the characteristic parameters of the PSD.10-110-210-310-410-510-6PeriodogramAR (3)We tried AR(P) with P 1 up to16.AR(3) model shows good fittingwith data and seems to capturephysical phenomena.-400-2000200Frequency (Hz)400Goodcompromisebetweenmodel complexity and fittingaccuracy.Example of periodogram calculated on 60,000 HH polarized data.60/157

61Ground clutter data61/157

Data recorded at Wolseley site with MIT-LL Phase One radar 62PHASE ONE radar parametersSource: MIT-LL, courtesy Mr. J. B. Billingsley Frequency Band (MHz)VHFUHFL-BandS-Band16543512303240 PRF500 Hz Polarization (TX/RX)VV or HH Range Resolution150, 36, 15 m Azimuth Resolution13 5 3 X-Band92001 Peak Power10 KW (50 KW at X-Band) Antenna ControlStep or Scan through Azimuth Sector Tower Height60 or 100 10 Km Sensitivity-60 dB Amount of Data25 Tapes/Site Acquisition Time2 Weeks/Site1 62/157

X-band ground clutter data in open agricultural terrain63703azimuth2D clutter map703(360 )The illuminated area was covered byagricultural crops (83%), deciduoustrees (11%), lakes (4%), and ruralfarm buildings (2%).11316azimuthsamplesrange3D clutter mapVV polarizationWhite areas: low reflectivity(field surfaces)1(270 )Black areas: high reflectivity(buildings, fencelines, trees, bushesaligned along roads)316(5.7 Km)rangesamples1(1 Km)63/157

Ground clutter data analysis64310210VV polarization4th range intervalhistogramGaussPDF101The analysis, performed on each rangeinterval has shown that I and Q PDFsdeviate considerably from Gaussian:the clutter amplitude is not Rayleighdistributed-1100.4-2100.35histogramVV polarization4th range intervaluniform-3100.30.25PDF-410-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05I component0.20.15The phase is uniformlydistributed(it may be not for DC offset,quantization effects, etc.)0.10.050-4-3-2-101Phase (radians)23464/157

Ground clutter data analysis 65121010810610estimateKLNWeib.HH polarization4th range interval310HH polarization4th range interval410210210110010234Moment order56histogramKLNRayleighWeibullPDFNormalized moments101st and 2nd range intervals:the Weibull distributionprovides the best fitting0103rd and 4th range intervals:the data show a behaviour thatis intermediate betweenWeibull and log-normal-110-21000.050.1Amplitude0.150.265/157

Ground clutter data analysis: windblown vegetation 66Analyzed clutter data:- recorded at Katahdin Hill site by Lincoln Laboratory.- Phase One X-band stationary antenna.- HH-polarization, PRF 500 Hz, 76 range gates.3D power map210histogramRayleighWindblown treesPDF101Number ofpulserepetitiontime intervals176-110Range cells307201Data set courtesy of Barrie Billingsleyof MIT – Lincoln Laboratory-21000.010.020.03 0.04 0.05Amplitude0.060.070.08These data are Gaussian.66/157

Spectral model on windblown vegetation671WPExpGaussPL2PL3-110-210PSD-3PSD, 35th range cell.10-410-510-610-710-250 -200 -150 -100 -50050 100Frequency (Hz)Cell #35ExpGaussPL2PL3β/σ/fc/fc5.95 (Hz m)-123.63 Hz1.02 Hz6.33 HzNon-Linear Least Squares(NLLS) method is usedfor parameter estimation:150200250θ Log NLLS arg min Log10 Pac ( f i ,θ) Log10S( fi )θ2i67/157

Radar cross section (RCS) RCS per unit illuminated area, σo (m 2/m 2) RCS per unit illuminated volume, η (m 2/m 3) Radar equation and pattern-propagation factor F Sea clutter RCS and spikes Land clutter RCS Sea and land clutter statistics The compound-Gaussian model Clutter spectral models

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