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Published in IET Radar, Sonar and NavigationReceived on 27th July 2010Revised on 21st December 2010doi: 10.1049/iet-rsn.2010.0234

ISSN 1751-8784

Signal spectral modelling for airborne radar in thepresence of windshear phenomenaC. Moscardini1 F. Berizzi1 M. Martorella1 A. Capria2

1Department of Information Engineering, University of Pisa, Via Caruso 16, 56122 Pisa, Italy2RaSS National Laboratory, CNIT (National Inter-University Consortium for Telecommunications), Galleria Gerace 1,56124 Pisa, ItalyE-mail: [email protected]

Abstract: A number of significant airplane accidents have resulted from windshear encounters during takeoff or landing. In thesesituations the radar signal received may be affected by a strong ground clutter that may make the conventional windsheardetection algorithms implemented on ground-based systems unusable. Typically, the solution to this problem is to employ aclutter rejection filter and then process the filter output to derive weather information. In this study, a parametric bimodalspectral model of the raw radar signal from an airborne Doppler weather radar is proposed. The bimodal shape of the modelhas been defined as a superposition of clutter and windshear. The model can be used to define a windshear detectionalgorithm that can also directly estimate the physical weather parameters by using the estimated model parameters withoutusing a clutter rejection pre-filter. Owing to the impossibility of testing a windshear detection system in a realisticenvironment, a simulated windshear data base has been developed by National Aeronautics and Space Administration(NASA) and Federal Aviation Administration (FAA) during their windshear research programme. The purpose of thisstudy is to define a parametric bimodal spectral model and demonstrate its validity on the NASA-FAA windshear certificationdata set.

1 Introduction

In the field of aviation science, the term windshear generallyrefers to a wind speed or direction change experienced by anairplane at a particular distance and over a given length oftime. A large number of meteorological and physicalphenomena are known to be responsible for windshear:convective turbulences, gust fronts and terrain-influencedflows. Mostly, windshear is not strong enough to constitutea danger to an airplane flying. However, some windshearmay have a critical impact on flight safety at low altitudeand low speed, especially during takeoff and landingphases. In these flight phases the plane is close to theground and it has short time or room to manoeuvre. Aparticular weather condition known as ‘microburst’ cangenerate a hazardous low altitude windshear. This term wasused for the first time by Fujita’s in 1970 to indicate adowndraft of air that impacts on the ground and divergesinto a horizontal outflow [1, 2]. The outflow may expand inall directions within a 4 km radius. The duration of amicroburst is usually less than 10 min.

A research promoted by the International Civil AviationOrganisation (ICAO) in 2005, showed that windshear haverepresented a major contributing factor in at least 20–30civil aviation accidents that occurred during the period from1964 to 1985 in the USA [1]. In response to the windshearproblem, the Federal Aviation Administration (FAA) andthe National Aeronautics and Space Administration (NASA)

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signed a memorandum of agreement in July 1986 for acooperative research effort aimed at examining technicalfactors related to severe windshear detections [3, 4].

The scope of a windshear detection system is that ofalerting the pilot before the aircraft enters the area occupiedby the phenomenon [3, 4]. The general approach to define adetection algorithm is to characterise the phenomenon interms of spectral analysis and wind field estimation.Moreover, the windshear threat for an airplane is evaluatedand quantified by using this characterisation [5]. Theprincipal problem in low-altitude airborne scenario isthe presence of a strong clutter return that can modify theestimates of the weather phenomenon physical parameters.The main approach to resolve this problem is that ofimplementing a clutter rejection filter before applyinggeneral spectral estimation algorithms (e.g. Pulse Pairalgorithm) [6–8]. Another possible approach is to estimatewind fields (average wind speed with respect to the range)from the radar returns by using spectrum modelling coupledwith a pattern classification technique [6, 9].

In addition, during the FAA-NASA program, cited above,the procedures for testing and certifying airborne forwardlooking windshear detection systems have been defined.These regulations are collected in document of RadioTechnical Commission for Aeronautics (RTCA) [3]. Theprincipal and considerable problem of the certificationprocess is the impossibility of testing the proposed systemsin a realistic environment. Safety and operational factors

IET Radar Sonar Navig., 2011, Vol. 5, Iss. 7, pp. 796–805doi: 10.1049/iet-rsn.2010.0234

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forbid flight tests through hazardous windshear at lowaltitudes and slow aircraft speeds; for this reason, acombination of flight tests and simulations is used. Inparticular, ground clutter data collected by systemmanufactures superposed to simulated microburst datagenerated by the Terminal Area Simulation System (TASS)have being considered. For example, the performance of theNASA airborne forward-looking windshear radar systemevaluated through the mentioned certification procedurescan be found in [10].

In order to determine system performance under realisticconditions and to merge real clutter data with simulatedwindshear data, a comprehensive simulator of pulseDoppler airborne radar was developed by the NASALangley Research Center. This simulation program, calledAirborne Windshear Doppler Radar Simulation (ADWRS)has been extensively used in the NASA program toevaluate signal and data processing techniques proposed toreliably detect windshear in the presence of ground clutter[11]. Therefore a simulation programme of the signal returnmay be developed starting from the ADWRS, as suggestedin [3]. In this paper a customised version of ADWRS isused to generate the airborne weather radar signal.

The purpose of this paper is to define a spectral model for thesignal received by an airborne forward-looking radar in thepresence of windshear. The proposed model will be tested byapplying a procedure defined in [3] and summarised inSection 2. An accurate spectral analysis of the signal receivedin each range cell has been performed to define a range-dependent bimodal spectral model. Specifically, the twoidentified modes are relative to ground clutter and windshearsignal components. Therefore the model definition is jointlybased on the typical spectrum of clutter and windshearreturns. The validity of the model has been demonstrated andvalidated over various windshear certification cases. It isworth mentioning that the proposed model can be used todirectly estimate the characteristic parameters of windshear(e.g. mean wind velocity and dispersion of wind velocity asfunction of the range) even without performing clutterrejection. Knowing that the majority of windshear detectionalgorithms are based on estimating such parameters [6], awindshear detection algorithm may be implemented that isbased on the use of the proposed model. Although in anapproximated way, the performance of a windshear detectionalgorithm or a clutter rejection filter may be immediatelyevaluated by using the signal return generated with theproposed bimodal spectral model.

The remainder of this paper is organised as follows. Section2 introduces the received signal model and the systemgeometry and illustrates the windshear certification database.In Section 3, the preliminary spectral analysis of the receivedsignal is detailed. The definition of the bimodal spectralmodel and the estimation of the model parameters are dealtwith in Section 4. Simulation results are finally presented inSection 5 to prove the effectiveness of the proposed model.

2 System geometry and received signalmodel

The certification testing procedure of an airborne forward-looking radar windshear detection system, defined byNASA and FAA, is composed of three steps [3]:

1. Collect radar ground clutter data from selected airport areausing an airborne radar flown on specified flight routes;


2. Simulate the radar signal in an environment resulting fromthe combination of measured environmental ground clutterand a simulated windshear environment;3. Evaluate the performance of the proposed detectionsystem on a dataset simulated in accordance with thewindshear certification data base.

The second step (i.e. simulation) is necessary because ofthe impossibility of testing a windshear detection system ina realistic environment, and therefore a comprehensivewindshear database has been developed by NASA LangleyResearch Center to support the industry initiative ofproducing and certifying forward-looking windsheardetection systems. The certification testing regulationsuggests using an adapted version of the NASA LangleyResearch Center’s simulation programme (ADWRS) tocombine measured airport clutter radar signal data withsimulated weather radar signal compliant with thewindshear database. In this paper a customised version ofADWRS [11] and the NASA windshear database [3, 12]have been used to generate the radar received signal, whilethe clutter data have been simulated through the parametricclutter model directly implemented on ADWRS mainlyowing to the lack of real clutter data received by airborneradars. Regarding the third step of the certification process,the windshear database and the standard certificationscenarios have been used to demonstrate the validity ofthe proposed spectral model. In future developments theperformances of the detection algorithm, based on theanalysed spectral model, will be evaluated.

2.1 Received signal generation

The range-Doppler echo signal is simulated for a defined setof system parameters considering the geometry of a coherentairborne weather radar. This signal may be represented interm of its complex envelope as follows

s(tf , ts) = sw(tf , ts) + sc(tf , ts) + n(tf , ts) (1)

where tf represents the fast time and ts the slow time, sw(tf , ts)is the received signal relative to the weather echo, sc(tf , ts) andn(tf , ts) represent the contribute of ground clutter echo andreceiver noise, respectively. The simulation input valuesinclude the radar system parameters, the kinematiccharacteristics of the airborne platform, the antennaparameters and the scanning angle strategy. Other inputsspecify the characteristic of the weather phenomenon andthe ground clutter. From both the initial position of theaircraft and the initial scan direction of the radar antenna,the simulation consists of the generation of theinstantaneous received signal defined in (1). For each rangebin, the amplitude and phase of the received signal can beseen as the coherent sum of a number of contributions thatcame from both volumetric and superficial scatteringmechanism, the first being related to the weatherphenomenon and the latter being related to ground clutter.To implement such a concept, each resolution cell isdivided into a number of sub-cells. Each sub-cell isconsidered to be much smaller than the radar resolutioncell. The amplitude of the signal return from a genericscatterer in a sub-cell is calculated using the radar equationwith:

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† the radar cross section obtained from either the groundclutter model or the weather data, depending on the positionof the sub-cell (either volumetric or on the ground)† the antenna gain in the direction of the sub-cell.

The phase term is composed of a random phase term and adeterministic term representing the relative motion betweenthe scatterer and the aircraft. This procedure continues untila predefined number of pulses in the slow time areobtained. The random signal relative to each scatterer isconstant for all pulses (scan to scan fluctuation). A detaileddescription of the simulation programme can be found in[11, 13].

2.1.1 Ground clutter simulation: The ground cluttercharacteristics are described in an input file that accounts forthe type of ground surface. The available ground surfacetypes are: soil, grass, tall grass, trees, urban, wet snow anddry snow. Each type of surface automatically implies theusage of a set of parameters in the parametric groundecho reflectivity model. This analytical-empirical modelis based on fitting simulated clutter data to ground cluttermeasurements [11]. The model provides the ground areareflectivity s(qm, p) for the mth ground scatterer based on thedepression angle qm and p is the vector of the input parameters.

2.1.2 Windshear database description: The windsheardatabase is generated by the simulation programme TASSdeveloped at NASA Langley Research Center. TASS issubstantially a general-purpose multi-dimensional numericalweather model for studying convective phenomena such asmicrobursts, convective rain storms, gust fronts andhailstorms. The meteorological variables available in thedatabase are: the wind velocity components, temperature,radar reflectivity factor, density of water vapour, rainwatercontent, hailwater content and liquid cloud droplet water.The variables used to generate the radar signal are the windvelocity components and the radar reflectivity. Eachvariable is represented by an array of data in a three-dimensional space. The data base contains a large numberof different windshear cases representative of a wide rangeof events: small- to large-scale events, low- to high-reflectivity events, symmetrical and asymmetrical events,weak to hazardous windshear. Several of the numericallymodelled events included in the database represent realaccident or incident windshear cases where real observed

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data were available. The database is composed of sevenmain meteorological events sequentially numbered fromone to seven as follows:

1. Dallas-Ft. Worth (DFW): accident case with singlemicroburst2. Orlando, Florida: wet microburst3. Denver; Colorado: multiple microbursts4. Denver, Colorado: warm microburst5. Denver, Colorado: very dry microburst6. Florida: highly asymmetric microburst7. Montana: gust front

Each data set contains the output variables of TASSsimulation at a particular time. Each of the data sets reportsthe relative simulation time in the second part of the dataset name. Two of these seven cases, specifically case 3 andcase 5, have been produced at two different times in TASSsimulation. For this reason, there are nine certification datasets: 1_11, 2_37, 3_49, 3_51, 4_36, 5_40, 5_45, 6_14,7_27. A complete and detailed description of the databasecan be found in [12].

2.2 Windshear certification database

The certification scenarios have been jointly defined byNASA and FAA [3]. The certification test procedure hasbeen carefully planned to deal with a range of windshear-alert situations. The simulation plane route is specified bythe direction of takeoff or landing, the position respect tothe simulated weather phenomenon and flight conditions.The main predefined flight path scenarios are: aligned fortakeoff (near or far from microburst), and straight approach,curved approach or straight approach with drift. In thispaper takeoff or landing are considered:

1. aligned for takeoff, near microburst2. aligned for takeoff, far microburst3. straight approach (38 glide slope)

In Fig. 1, the geometry of the generic certification flightpath scenario is presented. The X–Y plane represents theground, in the nomenclature of NASA/FAA the Y-axis isnamed ‘North’ and X-axis ‘East’. The point indicated with(Xa, Ya) represents the initial position in the horizontal planeof the aircraft that it is moving along the runway with

Fig. 1 Path scenario definition

a Horizontal planeb Vertical plane


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velocity v. In the figure the runway is represented with adouble line arrow and its location is identified with theangle a. The point indicated with (X0,Y0) represents theorigin of windshear spatial domain that extends from thispoint to (X0 + DX,Y0 + DY). In the vertical plane along therunway the aircraft height above the ground is indicatedwith h and DZ represents the extension of windsheardomain above the ground.

The combination of these scenarios with the simulatedwindshear data sets forms the certification data set. Thecertification data set is summarised in Table 1. It should bereminded that the first two parts of the case number indicatethe windshear case while the third part of the case numberindicates the type of the scenario. The aircraft velocityvector is direct along the runway in the XY-plane and itsabsolute value is 150 knots for each certification scenario.

2.3 Simulated airborne radar system

The simulated radar system is a typical airbornemeteorological radar. It operates at a carrier frequency ofabout 10 GHz, the transmitted signal is characterised by apulse duration of 1 ms, and the pulse repetition frequency is6.5 KHz. The simulated antenna system is based on atypical modern X-band airborne weather radar parabolicantenna. In the following, only one antenna pointingdirection is considered during simulations.

3 Preliminary data analysis: spectral analysis

This section deals with the preliminary analysis of thecertification case data, as described in the previous section.Results are provided in terms of spectral distribution ofreceived radar signal as a function of range and Dopplerfrequency. In Fig. 2, six range-Doppler maps relative tosome representative windshear certification cases are shown.These results have been obtained by calculating the meanspectrum averaging on 300 realisations for each scenario.The spectrum relative to each range cell is evaluated as thesquare value of the Fourier Transform of the received signal.

Owing to the acquisition geometry during landing andtakeoff, the received signal is characterised by a strongclutter return. Under the hypothesis of an exact aircraftmotion compensation the spectral clutter contribution ismainly located around the zero Doppler for each range cell.The spectral term of the weather return is clearly visible inthe range-Doppler map and it shows an appreciableextension in both range and Doppler. The phenomenonpresents both negative and positive Doppler frequencies andit extends in range like typical weather phenomena such asrain. For example, in the first case shown in Fig. 2, the


range-Doppler map exhibits a shape known in the literatureas ‘S-form’. In order to define a range-dependent spectrummodel for each component, (weather and clutter), thespectrum of the received signal is analysed for each rangecell. Specifically, the characteristics of each range cellspectrum have been studied for all the windshear simulationcases in the range-Doppler domain. Some range sectionsrelative to two windshear simulation cases, respectively, thecase 1_11_3 and 3_49_3a are shown in Figs. 3 and 4. Thespectrum of the two selected cases can be consideredrepresentative of most of the windshear phenomena.

The presence of three typical spectrum regions is clearlyvisible: one dominated by the clutter, another dominated bythe weather return, and a last region characterised by thesuperposition of both the components. Examples with aclutter-only component are visible in Fig. 3 at distances of4.5 and 9.75 km, and in Fig. 4, at a distance of 5.85 km.The windshear-only spectral component can be seen inFig. 3 at a distance of 6.6 km. All the other cases, shown inFig. 3 and in Fig. 4 present the combination of both thespectral components.

The analysis conducted on the simulated clutter model, asdescribed in the previous section, allows defining a clutterspectral model for every range cell. Among a number ofpossible models, the spectral model chosen for fitting theclutter data is the power-law model (second order), whichwas presented in [14].

SPL(2)(f ) = Ac

1

pfc

1

1 + (f /fc)2 (2)

where fc is the cut-off Doppler frequency (where the shapefunctions is 3 dB below its peak zero-Doppler level) and Ac

represents the mean power value of the single range cell.A typical spectral model of the mean received power

spectral density used for representing meteorological targetsis the Gaussian model [2], which is analytically representedin (3)

SGauss(f ) = Awe(−(f −�f )2)/2s2

(3)

where the term Aw is related to the power of the receivedweather signal, �f and s2 are the spectral first- and second-order moments. It should be pointed out that theseparameters contain the necessary information to measuremeteorologically significant parameters. As all the particleswithin the sample volume move with some average radialvelocity, there is a mean frequency �f of the Dopplerspectrum that is shifted from the transmitted frequency. Themean Doppler frequency is related to the mean radial

Table 1 Certification data set

Case number Xa, km Ya, km H, m Alfa, deg Xo, km Yo, km DX, km DY, km DZ, km

1_11_3 27.4 0 405 90 24 24 8 8 2

2_37_3a 21.8 6.5 454 180 28.8 28.9 15 15 2

2_37_3b 210.2 1.1 454 90 28.8 28.9 15 15 2

3_49_3a 4.2 24.5 244 90 1.2 210.5 18 12 2

3_49_3b 8.5 26.5 454 0 1.2 210.5 18 12 2

3_51_3a 12.2 211 439 0 2 210.5 18 12 2

4_36_3 21.9 0 128 90 25 25 10 10 2

5_40_3 3.8 7.9 573.1 0 24.2 2.3 16 16 2

6_14_3a 14.6 23.3 449 0 8 23.5 10 10 2

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Fig. 3 Range sections relative to case 1_11_3

Fig. 2 Range Doppler maps for some representative windshear certification cases

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Fig. 4 Range sections relative to case 3_49_3a

velocity by the following relation

�f = 2�v

l(4)

where �v is the mean radial velocity and l is the transmittedwavelength.

Since the particles are in motion with respect to one other,there is also a Doppler spread s, often referred to as the widthof the Doppler spectrum. The Doppler spread is related to theparticles velocity dispersion sv around the mean value by thefollowing relation

s = 2sv

l(5)

This last parameter is an indicator of the turbulence associatedto the weather phenomenon. A general weather radar has tomake quantitative measures of these three parameters toevaluate the dynamic and the evolution of the weatherphenomenon, and to estimate parameters such asprecipitation rate, precipitation type, air motion andturbulence. In particular, for an airborne weather radar witha windshear detection system, the parameters of windvelocity are used to define and to quantify the windshearthreat for the airplane [5].

4 Bimodal spectral model definition

4.1 Model selection

The analysis conducted on the simulated signal model, asdescribed in the precedent section, allows defining abimodal spectral model for every range cell. The spectral


model of the total received signal, is obtained as

f(f ) = SGauss(f ) + SPL(2)(f ) + C (6)

where SPL(2)( f ) and SGauss( f ) are defined in (2) and (3),while the term C is a constant that contains the effects ofthe spectral additive white Gaussian noise and it has beenadded for improving the fitting between real weatherspectrum and Gaussian models. By substituting (2) and (3)into (6) the total spectral model becomes

f(f ) = Ac

1

pfc

1

1 + (f /fc)2 + Awe(−(f −�f )2)/2s2

+ C (7)

The bimodal spectral model f( f ) is dependent on the rangecell and it is defined for a discrete set of frequencies f (m),so (7) can be expressed as

f(xr, m) = Ac(r)1

pfc(r)

1

1 + (f /fc(r))2

+ Aw(r)e(−(f (m)−�f (r))2)/(2s2(r)) + C(r) (8)

where the model parameter vector for the range cell rth xr isdefined as

xr = [Ac(r), fc(r), AW(r), s2(r), �f (r), C(r)] (9)

As shown in (8) the bimodal parametric spectral model ischaracterised by six parameters are dependent on the range.In particular, two of these parameters are directly related tothe characteristic physical parameters of windshear: meanwind radial velocity and wind radial velocity dispersionaround the mean value.

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4.2 Model estimation

In the general theory of parametric spectral estimation, themodel estimation procedure suggests using the availabledata to estimate the parameters of the selected model,typically using some optimisation criterion [15]. Althoughthere are several criteria that can be used in this case,we concentrate on the least squares (LS) error criterion.This problem belongs to the general class ofunconstrained optimisation problem whose mathematicalformulation is

minx

f (x) (10)

where x [ Rn and f : Rn � R is a smooth function.The LS algorithms select values for the model parameters

that best match the model f(x, m) to the M observed dataym by minimising a particular special form of the objectivefunction

f (x) =∑M

m=1

r2m(x) (11)

where rm, namely the mth residual, is a smooth function fromRn to R and it is defined as follows

rm = f(x, m) − ym (12)

where f(x, m) is the defined model, which depends on theparameter vector x, evaluated at the M points where thedata ym are observed. In our problem, the observed data ym

are the values of the spectrum relative to the received signaldefined in (1), which can be obtained as follows

S(r, m) = |DFT (s(r, mTR))|2 m = 0, 1, . . . , M − 1;

r = 1, 2, . . . , N (13)

where N is the number of range cells and DFT indicates thediscrete Fourier transform and TR is the pulse repetitiontime. By using (10)–(13) the LS problem becomes

xr = minxr

∑M

m=1

r2m(xr) = min

xr

∑M−1

m=0

(f(xr, m) − S(r, m))2 (14)

where f(xr, m) is obtained from (8) as follows

f(xr, m) = f xr, f (m) = m

MTR

( )(15)

and xr indicates the parameter model vector estimated inevery range cell.

In the last forty years there has been a development of apowerful collection of algorithms for unconstrainedoptimisation of smooth functions. All algorithms require theuser to supply a starting point, which may be denoted byx0. The user, with the knowledge about the application andthe data set, may provide a reasonable estimate of thesolution for x0. We obtain the starting point x0(r) with aheuristic approach that automatically analyses a single rangespectrum S(r, m). The values fc(r) and s(r) are fixed byusing typical values assumed by the width of clutter andwindshear spectra. The value �f (r) is obtained by calculatingthe point of maximum of the spectrum S(r, m) outside a

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band around the zero Doppler. The parameters C(r) andAw(r) are set by calculating, respectively, the maximum andminimum of the spectrum S(r, m) outside the same bandaround the zero Doppler. The parameter Ac(r) are chosen byinverting (8) and by substituting with the chosen values ofthe parameters. By starting at a point x0, numericaloptimisation algorithms generate a sequence of iterationsthan terminate when either little progress can be made orwhen it seems that a solution point has been approximatedwith sufficient accuracy. In deciding how to move fromone iteration to the next, optimisation algorithms useinformation about the objective function at the current stepand possibly other information from earlier iterations.There are two fundamental strategies for moving from thecurrent point to a new iterate: line search strategy and trustregion strategy. In the line search strategy, the algorithmchooses a direction to move and searches along thisdirection a new iterate with a lower objective functionvalue. In the second strategy, at the current iterate k amodel mk of objective function is constructed and thenew iterate xk + p is found by solving the following sub-problem

minp

mk(xk + p) (16)

where xk + p lies inside the trust region, defined as a spherearound the iterate xk. If the candidate solution does notproduce a sufficient decrease of objective function the trustregion is made smallest and a new search is resolved.

Both the precedent methods generally use informationscontained in the gradient and the Hessian matrix of theobjective function. The special form of the objectivefunction in the LS problem permits expression of thegradient and the Hessian in term of the Jacobian matrix ofthe objective function, which is the matrix of first partialderivatives of the residual [15]. In many applications, thefirst partial derivatives of the residual and hence theJacobian matrix are relativity easy or inexpensive tocalculate. The most popular algorithms for resolving LSproblem fit into the line search and trust region frameworks;they are based on the Newton and quasi-Newtonapproaches with modifications that exploit the particularstructure of the objective function. A list of algorithms thatwe used in simulations follow: Gauss–Newton method,Levemberg–Marquardt method, large scale trust regionreflective. Details about this classic optimisation algorithmcan be found in [15]. Many practical application require theoptimisation of functions whose derivatives are notavailable. Problems of this kind can be solved byapproximating the gradient and possibly the Hessian usingfinite differences and using these approximate quantitieswithin the algorithms. The Gauss–Newton andLevemberg–Marquardt algorithms are tested both bycalculating Jacobian matrix and the approximated JacobianMatrix (with finite differences). In literature differentalgorithms have been developed that do not approximatethe gradient but they rather use the function values at a setof sample points to determinate a new iteration. Thesealgorithms are knows as derivative free optimisation (DFO)algorithms. Among the DFO-based algorithms we selectedthe Nelder–Mead algorithm, which is one of theoptimisation algorithms implemented in the Matlaboptimisation Toolbox. The number of iterations depends onthe stop criteria, which may include a lower bound on thesize of a step, a lower bound on the change in the value of


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the objective function during a step, maximum number ofiterations and function evaluations. The stop criteriarepresent an input to the optimisation algorithm and mayinfluence the performance of the algorithm. Several caseshave been analysed to test such optimisation algorithms.The results obtained by applying all the mentionedalgorithms with default stop criteria are presented in thenext section.

5 Simulation results

In this section the validity of the bimodal spectral model onthe certification data sets is proved. For this reason, someparameters are defined to evaluate the performance of themodel in Section 5.1. Section 5.2 contains the results interm of parameters defined for each certification data set.

5.1 Performance metrics definition

The accuracy of the model is evaluated in terms of the meanspectrum of the received signal. The mean spectrum of thereceived signal is obtained by averaging NR realisations asfollows

�S(r, m) =∑NR

i=1 Si(r, m)

NR

(17)

where Si(r, m) is the spectrum of the ith realisation of thereceived signal defined in (13). A value of NR ¼ 300 isused in the simulation.

To evaluate the model fitting performance with respect tothe dependence on range, some parameters are calculated.The first parameter that has been considered is the relativepercentage error, which is defined as

EP(r) WE{|�S(r, m) − f(xr, m)|2}

E{|�S(r, m)|2}100 (18)

where �S(r, m) and f(xr, m) are defined in (13) and (15), theoperator E{.} indicates the mean operation with respect tothe frequency. The second parameter that has beenconsidered is a normalised cross-correlation, which isdefined as

CR(r) WE{|�S(r, m)f(xr, m)|}

E{|�S(r, m)|2}(19)

This parameter equals one if the fitting is perfect.Since the performance metrics, defined in (18) and in (19),

do not present a strong variability when evaluated fordifferent optimisation algorithms and cases, they do notprovide a sensitive measure of model accuracy fitting. Forthis reason and since we are particularly interested in


evaluating the model fitting performance over the portion ofthe spectrum relative to the windshear, we define otherperformance metrics. In particular, we introduce theabsolute error of the mean and dispersion wind velocity.The absolute error of the mean radial velocity in function ofthe range is defined as

AE�v(r) = |�va(r) − �v(r)| (20)

where

�v(r) = l�f (r)

2(21)

and �va(r) is the actual mean radial velocity.The absolute error of the radial velocity dispersion as a

function of the range are defined as

AEs(r) = |sa(r) − sv(r)| (22)

where

sv(r) = ls(r)

2(23)

and sa(r) is the actual radial velocity dispersion.The actual values, �va and sa, are calculated in the same

scenario by simulating only the weather return and byapplying the pulse-pair algorithm to the spectrum of thereceived signal. Pulse-pair algorithm is an efficient and awidely used spectral moment estimation technique. Thespectral moment estimates are very accurate when a pureweather signal is immersed in white noise [8].

5.2 Results

The performance parameters defined in sub-section 5.1 arecalculated for each windshear simulation case. The meanvalues with respect to the range of the defined parameters,AE�v, AEs are calculated to eliminate the dependence on therange, in this way the results can be clearly shown. InTable 2, the maximum, minimum and mean value,calculated with respect to the different windshear case(presented in Table 1), of the parameters AE�v, AEs areshown for different optimisation algorithms.

Specifically for the Nelder–Mead algorithm the maximum,minimum and mean values, calculated with respect to range,of the parameters AE�v(r) and AEs(r) are shown in Table 3 fordifferent windshear cases (presented in Table 1).

The maximum, minimum and mean values, calculated withrespect to range, of the parameters EP(r) and CR(r) are shownin Table 4 for different windshear cases (presented in Table 1)and for the Nelder–Mead algorithm.

Table 2 Absolute error of the mean and dispersion wind velocity for different optimisation algorithms

Algorithms minc

AE�v maxc

AE�v meanc

AE�v minc

AEs maxc

AEs meanc

AEs

Gauss–Newton (Jacobian supplied) 0.748 35.874 17.069 0.662 3.470 2.008

Gauss–Newton (finite difference) 0.741 34.878 16.095 0.672 3.477 2.001

Levemberg–Marquardt (Jacobian supplied) 0.748 35.874 17.069 0.662 3.470 2.008

Levemberg–Marquardt (finite difference) 0.741 34.878 16.095 0.672 3.477 2.001

trust region reflective 0.349 0.940 0.575 0.390 0.816 0.566

Nelder–Mead 0.074 1.042 0.367 0.308 1.126 0.561

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Table 3 Absolute error of the mean and dispersion wind velocity

Case minr

AE�v (r ) maxr

AE�v (r ) meanr

AE�v (r ) minr

AEs(r ) maxr

AEs(r ) meanr

AEs(r )

1_11_3 0.000 0.830 0.176 0.032 0.555 0.308

2_37_3a 0.000 2.230 0.128 0.029 2.405 0.317

2_37_3b 0.015 1.514 0.291 0.020 1.394 0.359

3_49_3a 0.005 1.627 0.320 0.290 3.928 1.115

3_49_3b 0.001 0.676 0.074 0.145 1.138 0.396

3_51_3a 0.044 2.692 0.866 0.244 4.473 1.126

4_36_3 0.003 0.471 0.117 0.159 0.438 0.308

5_40_3 0.016 3.716 1.042 0.231 2.181 0.586

6_14_3a 0.001 3.657 0.293 0.067 2.852 0.533

Table 4 Relative percentage error and normalised cross-correlation parameters

Case minr

EP(r ) maxr

EP(r) meanr

EP(r) minr

CR(r) maxr

CR(r ) meanr

CR(r )

1_11_3 0.189 4.725 1.096 0.952 0.999 0.989

2_37_3a 0.143 4.566 0.514 0.954 0.999 0.995

2_37_3b 0.141 7.958 1.219 0.920 0.999 0.987

3_49_3a 0.168 3.855 1.043 0.957 0.999 0.989

3_49_3b 0.146 13.227 1.178 0.817 0.999 0.986

3_51_3a 0.451 4.821 1.785 0.952 0.996 0.983

4_36_3 0.291 1.231 0.573 0.989 0.998 0.994

5_40_3 0.363 5.985 2.138 0.940 0.996 0.978

6_14_3a 0.113 8.682 1.285 0.913 1.003 0.987

As an example, the results relative to the case 1_11_3 areshown in Fig. 5. In particular, the comparison between theactual spectrum (relative to the range cell 5.55 km) and theestimated model is shown in Fig. 5a. In this section, also

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the relative percentage error (as defined in (18)) is shownwith respect to the range. The comparison between theactual and estimated wind velocities and the relativeabsolute error, with respect to range, are shown in Fig. 5b.

Fig. 5 Results relative to case 1_11_3

a Comparison between actual spectrum and its estimateb Comparison between actual wind velocities and its estimatec Comparison between actual wind velocity dispersion and its estimate


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The comparison between the actual and estimated dispersionvelocities and the relative absolute error, with respect torange, are shown in Fig. 5c.

6 Conclusions

In this paper, a parametric bimodal spectral model of thereceived signal for airborne Doppler weather radar has beenproposed. The validity of the model has been demonstratedby using the NASA-FAA windshear certification data set.This approach may offer better windshear detectionperformance than the typical approach based onclutter filtering although this point is still to bedemonstrated. Moreover, this method can presentadvantages with respect to the modal analysis proposed in[7] because of the possibility to estimate the dispersion ofwind velocity as well as average wind velocity. Theprocedure of spectral model parameters estimation presentedhere may be used in the future for optimising windsheardetection algorithms.

7 References

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2 Serafin, R.J.: ‘Meteorological radar’, in Skolnik, M.I. (Ed.): ‘Radarhandbook’ (McGraw-Hill Press, 1990, 2nd edn.), pp. 23.1–23.33

3 RTCA SC-173 (Radio Technical Commission for Aeronautics – SpecialCommittee 173): ‘Minimum operational performance standards forairborne weather radar with forward-looking windshear capability’.RTCA, Technical report document no. RTCA/DO-220, 1993


4 The NASA Langley Research Center website. Available at http://www.nasa.gov/centers/langley/news/factsheets/Windshear.html, June 1992,accessed January 2008

5 Lewis, M.S.P., Robinson, A., Hinton, D.A., Bowles, R.L.: ‘Therelationship of an integral wind shear hazard to aircraft performancelimitations’. Technical report 109080, NASA Langley Research Center,1994

6 Baxa, E.G., Lai, Y.C., Kunkel, M.W.: ‘New signal processingdevelopment in the detection of low-altitude windshear with airborneDoppler radar’. Proc. Radar Conf. IEEE, Lynnfield, Massachusetts,April 1993, pp. 269–274

7 Byron, M.K.: ‘Adaptive clutter rejection filters for airborneDoppler weather radar applied to the detection of to low altitudewindshear’. Technical report 4457/DOT/FAA/RD-92/21, NASA, 1989

8 Passerelli, R.E., Siggia, A.D.: ‘The autocorrelation function and Dopplerspectral moments geometric and asymtotic interpretation’, J. Clim. Appl.Meteorol., 1983, 22, pp. 1776–1787

9 Kunkel, M.W.: ‘Spectrum modal analysis for the detection of lowaltitude windshear with airborne Doppler radar’. Technical report,NASA, 1992

10 Switzer, G.F., Britt, C.L.: ‘Performance of the NASA airborne radarwith the windshear database for forward-looking system’ (ResearchTriangle Institute NASA, TR, 1996)

11 Britt, C.L., Kelly, C.W.: ‘User’s guide for an airborne Doppler weatherradar simulation (ADWRS)’. Technical report 7473/029-05S NASA,Center for Aerospace Technology, 2002

12 Switzer, G.F., Proctor, F.H., Hinton, D.A.: ‘Windshear database forforward-looking systems certification’. Technical report RTI/4500/026-01F, NASA Langley Research Center, 1993

13 Moscardini, C., Berizzi, F., Martorella, M., Capria, A.: ‘Spectral modellingof airborne radar signal in presence of windshear phenomena’. Proc. Int.Conf. on Radar, Eurad 2009, Rome, Italy, pp. 533–536

14 Lombardo, P., Greco, M., Gini, F., Farina, A., Billingsley, J.B.: ‘Impactof clutter spectra on radar performance prediction’, IEEE Trans. Aerosp.Electron. Syst., 2001, 37, (3), pp. 1022–1038

15 Nocedal, J., Wright, S.J.: ‘Numerical optimization’ (Springer Press,2006, 2nd edn.)

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