OCTOBER 1994 FREHLICH AND YADLOWSKY 1217Performance of Mean-Frequency Estimators for Doppler Radar and Lidar R. G. FREHLICH AND M. J. YADLOWSKY *Cooperative Institute for Research in the Environmental Sciences (CIRES), University of Colorado, Boulder, Colorado(Manuscript received 20 May 1993, in final form 14 January 1994) ABSTRACT The performance of mean-frequency estimators for Doppler radar and lidar measurements of winds is presentedin terms of two basic parameters: cI,, the ratio of the average signal energy per estimate to the spectral noiselevel; and ft, which is proportional to the number of independent samples per estimate. For fixed - and ~2, theCramer-Rao bound (CRB) (theoretical best performance) for unbiased estimators of mean frequency (normalizedby the spectral width of the signal), signal power, and spectral width are essentially independent of the numberof data samples M. For large (I,, the estimators of mean frequency are unbiased and the performance is independentof M. The spectral domain estimators and covariance based estimators are bounded by the approximate periodogram CRB. The standard deviation of the maximum-likelihood estimator approaches the exact CRB, whichcan be more than a factor of 2 better than the performance of the spectral domain estimators or covariancebased estimators for typical ~2. For small ~, the estimators are biased due to the effects of the uncorrelated noise(white noise ), which results in uniformly distributed "bad" estimates. The fraction of bad estimates is a functionof - and M with weak dependence on the parameter ft. Simple empirical models describe the standard deviationof the good estimates and the fraction of bad estimates. For Doppler lidar and for large q,, better performanceis obtained by using many low-energy pulses instead of one pulse with the same total energy. For small ~, theconverse is true.1. Introduction Measurement of wind fields using Doppler radar hasbeen routinely performed for many years. The designand operation of Doppler radar is reviewed by Doviakand Zrni6 ( 1993 ). The optical counterpart to Dopplerradar, coherent Doppler lidar, has become more important with the development of efficient COt (Menzieset al. 1984; Menzies 1986; Bilbro et al. 1986; Petheramet al. 1989; Post and Cupp 1990; Pearson and Rye1992; Gal-chen et al. 1992 ) and solid state lidars (Kavaya et al. 1989; Henderson et al. 1991, 1993). Thescatterers for Doppler radar are refractive-index fluctuations, hydrometeors, and insects. The scatterers forDoppler lidar are atmospheric aerosol particles. Doppler radar data are generated using a sequence of manypulses, usually separated by 0.5-5 ms. Doppler lidardata are produced with a single pulse that permits manymore estimates in a given time. The statistical description of the data is the same (ignoring ground clutterfor radar) because both signals are produced by thesuperposition of many randomly phased scattered * Current affiliation: BEIP, National Institute of Health, Bethesda,Maryland. Corresponding author address: R. G. Frehlich, Cooperative Institutefor Research in the Environmental Sciences (CIRES), University ofColorado, Boulder, CO 80309.fields. Both of these instruments estimate the radialvelocity of the scatterers from the Doppler frequencyor mean frequency of the signal by using various estimators. Most mean-frequency estimators are either spectraldomain or covariance estimators (Levin 1965; Zrnifi1979; Mahapatra and Zrnifi 1983; Miller and Rochwarger 1972; Doviak and Zrni~ 1993; May and Strauch1989; May et al. 1989; Rye and Hardesty 1993a,b,c;Sirmans and Bumgarner 1975); that is, they estimatethe mean frequency of the weather signal using estimates of the spectrum or estimates of the covariance(see also Kay and Marple 1981; Marple 1987). The ideal performance of an unbiased estimator isgiven by the Cramer-Rao bound (CRB) (Helstrom1968; Van Trees 1968; Scharf 1991; Frehlich 1994a).If an unbiased estimator approaches the CRB, it is amaximum-likelihood (ML) estimator. Approximatecalculations of the CRB have been discussed by Whittle(1953), Levin (1965), Zrnifi (1979), and Rye andHardesty (1993a,b). These approximations are basedon the spectrum of the signal, and therefore require asufficiently long data sequence for the spectrum to bewell defined [see Marple (1987) for a discussion of theeffects of finite data and discrete sampling]. The calculation of the exact CRB has been discussed by Frehlich (1994a), who introduced an approximate CRBbased on the average periodogram instead of the spectrum. We will compare the performance of variousmean-frequency estimators to the exact CRB and thec 1994 American Meteorological Society1218 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 11periodogram CRB. The CRB and the performance ofmean-frequency estimators depend on the parametersof the data. The system parameters are chosen to emphasize the most important physical mechanisms ofthe process. This reduces the parameter space to twobasic variables: q,, the ~atio of the average signal energyper estimate to the spectral level of the noise; and r,which is proportional to the number of independentsamples per estimate.. (For Doppler lidar q~.is also thenumber of effective photoelectrons coherently detectedper estimate.) For fixed - and r, the number of'datapoints M per estimate has little effect on performance,especially for large ~. This simplifies the design andanalysis of Doppler radar and lidar Systems.. The performance of mean-frequency estimators hasbeendescribed by its standard deviation (Zrni6 1979;Sirmans and Bumgarner 1975; Mahapatra and Zrni61983;' Doviakand Zrni6 1993; May and Strauch 1989;May et al, 1989; Rye and Hardesty 1993a,b) and thenumber of estimates that fall within a given region(Anderson 1991; Rye and Hardesty 1993a). We describe the performance of mean-frequency estimatorsusing an empirical model for the probability densityfunction (PDF) of the estimates. This will permit ameaningful qomparis0n t- the CRB when the estimatesare biased. For Doppler lidar, Rye and Hardesty (1993a) considered the question: Are many pulses of lowenergybetter than one pulse of the same total energy? Thisquestion will be addressed for the unbiased and biasedregimes.2. Basic system parameters Doppler radars typically employ a complex receiver( mixers that convert the radio signals to complex data)to measure positive 'and negative velocities (Doviakand Zrni6 1993). We consider complex data throughout this paper with 'specific references to the case ofreal data. Doppler radar or Doppler lidar data is wellapproximated as a zero-mean complex Gaussian rand0m vectorz with elements zk, which 'satisfy= 0, where angle: brackets denote ensemble average(Doviak and Zrni6 1993~ 'Helstrom 1968).. The'PDFof the data is jointly Gaussian and defined by thevariance matrix lt with elements Rkl = ~ ZkZ~ ). (1)We assume the signal model (Zrni6 1979) zk = sk exp(2~r&fTs) + n~, (2)where f(Hz) is the mean frequency; Ts (s) is the sampling interval; the random signal s~ is independent ofthe Uncorrelated noise n~, wheren~nj) -- O, nkn~) = Nb~_j; (3)N = ~ I n 2) is the average noise power; I n I denotesthe modulus of the complex variable n; and ~ is theKronecker delta symbol. For Doppler radar, the mean frequency fis definedas the first moment of'the signal spectrum 'that is afunction of the wind field and sensing volume of thepulse(Doviak and Zrni6 !993 ). This assumes that thesignal is stationary. For Doppler lidar, the mean frequency f is defined by the radial component 'of thevelocity of the scatterers in the center of the sensingvolume of a given range gate (Frehlich 1994b). Themean frequency is Well defined for a single shot andfor any random velocity field and random collectionof scatterers and also for nonstationary data. For bothDoppler rfidar and lidar, the signal model of Eq.' (2) isvalid. An estimate of the mean frequency is producedwith' M data' samples, '.which defines the total observation time per estimate as MTs (s). The average'.noise power is set to unity ~to simplifythe results. The data are assumed stationary; that is,Rkt = R~-l, and the co//ariance' reduces to R~ = SNRp~ exp(2~rificTs) + b~, (4)where SNR --' SIN is the signal-to-noise ratio, $= ( I s [ 2 ) is the average signal power, and 0k is the normalized covariance of the signal. The performance ofmean-frequency estimators will be presented for theGaussian covariance model o~ Zrni6 (1979): p~, = exp[-2~r~( wkTs)~], (5)where w (Hz) is the spectral Width of the signal --iththis model, the data are fully 'characterized by the system parameters (f, $, w) as well as the' experimentalparameters M and Ts. For Doppler lidar, this signalmodel is exact (Frel~lich 1994b) if there are no windfluctuations over the sensing Volume and a Gaussianlaser pulse with temporal powe~ profile Pr(t) (W) givenby P~(t) = exp -' (6)is transmitted through a telescope aperture, where a(s) is G = (81/27rW)-1. (7)The full width at half-maximum (FWHM) at (s) ofthe transmitter pulse is At = 2(ln2)1/2o'. (~) The 1/e full width /Xr (s) of the signal covariance Rk defines the signal correlation time (see Fig. l ) as 2 ~/2 Ar =--=4a. (9) ~rw For a Doppler lidar operating under ideal conditions [the detector noise is dominated by the local oscillatorOCTOBER 1994 FREHLICH AND YADLOWSKY 1219 0 200 400 1.0 ,I~~ 0.8e~ ~ 0.$~: 0.4-~ol~ 0.2~ 0.0 ~-' ~ ~ ' ~' ~ - ~ I ~ I ~ I ~ I8 [ -~ -4,~ -$ -8 -10 0 -40ffi 'o -50 io~ -70'~ -80'=m -00n -100 0 Range [m]600 800 1000 1200 1400Ap ~,/t ~ I~-~ I ~ I ~ I I I ' I ' I ' I ' . -~ I ~ I I I I I I I I I I I , I , I , 1 2 3 4 5 6 7 8 9 10 Time [microseconds] ~ I ~ I~ 10 20 30 40 50 Frequency [MHz] FIG. 1. Simulation of Doppler lidar data for a 2-tzm lidar (Henderson et al. 1993) with a Gaussian pulse [Eq. (6)] and SNR = 10,f= 5 MHz, w = 0.2 MHz, Ts = 20 ns. The range weighting functionPL(t - 2R/c) is shown for the signal at 2 and 6 #s. The FWHM zXr[Eq. (12)] of the pulse-sensing volume, the distance Ap [Eq. (14)]that the pulse moves in 4 t~s, the signal correlation time Ar [Eq.(9)], and the periodogram [Eq. ( 15)] are also shown.shot noise, the shot noise is determined by the Poissonstatistics of the detected photons, and there is no atmospheric refractive turbulence (Frehlich and Kavaya1991 )], the SNR is given by Pz~( t ) SNR(t) = he---~- */H(t), (10)where Po (t) (W) is the effective direct detection powermeasured by the detector, ~H(t) is the heterodyne efficiency, h = 6.626 x 10-34 J s is Planck's constant, e(Hz) is the optical frequency of the laser, and B (Hz)is the detector bandwidth. For ideal operation, r/e o~ x [K(R)]2t~(R)C(R)dR, (11)where no (electrons per photon) is the detector quantum efficiency, c (m s-I) is the speed of light in a homogeneous atmosphere, K(R) is the dimensionlessone-way irradiance extinction of the atmosphere atrange R (m), ~(R) (m-~ sr-~ ) is the atmospheric aerosol backscatter coefficient, PL(t) is the pulse powerprofile of the laser, and C(R) is the coherent responsivity of the Doppler lidar (lidar system dependence).The term PL(t - 2R/c)[K(R)]21~(R)C(R) describesthe range weighting for the SNR. For many cases ofinterest, the terms [K(R) ] 2~(R) C(R) are constant overthe range gate and the range weighting function is givenby the term PL(t - 2R/c) (see Fig. 1 ). The FWHMAr (m) of the range weighting function for the Gaussianpulse is then cat Ar = 2 ' (12)A simulation of Doppler lidar data (appendix) and therange weighting function P~(t - 2R/c) are shown inFig. 1 for typical parameters of a 2-#m lidar. At time2 tas the center of the pulse is located at range 300 mwith a width Ar = 140.5 m. At a time 4 #s later, thepulse is situated at a range of 900 m. The range resolution ZkR (m) for an observation time MTs is definedas ~R = At+ Ap, (13)where Ap (m) is the distance the range weighting function travels during the observation time; that is, MTsc Ap 2 (14)The temporal scale of the modulation of the signalcorresponds to the time required for the pulse to travela distance Ar, after which time a new collection ofindependent atmospheric scatterers are illuminated bythe pulse (see Fig. 1 ). The periodogram of the data is defined as (Doviakand Zrnifi 1993; Marple 1987) ~Ts M-I exp(2~r~--~km)2 /5(m) = ~--o z~ (15)and is shown in Fig. 1. This is an estimate for the spectrum at frequencies f = mAf, where Af = 1/MTs isthe frequency resolution of the periodogram. Themaximum frequency that can be observed withoutaliasing is the Nyquist frequency FN = MAf = 1/Ts(for real data FN = 1/2Ts). The frequency domainestimation of the mean frequency f involves extracting the location of the spectral feature of the randomperiodogram coefficients (or any spectral domain estimate) in the presence of random noise. It is reasonableto expect that the performance of a frequency domainestimator depends on the statistics of the signal andnoise around the spectral feature. If one would increasethe sampling interval Ts by a factor of 2 and also reducethe noise bandwidth by a factor of 2 to match the new1220 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUIVlE 11Nyquist frequency, the performance of the spectral domain mean-frequency estimator would be unchangedbecause the statistics of the periodogram around thespectral feature is unchanged. However, the frequencyrange that can be searched has been reduced by a factorof 2 and the SNR has increased by a factor of 2. Therefore, the SNR is not a convenient parameter to characterize the signal strength. A more useful parameter to describe the signal powerfor mean-frequency estimation is the ratio of the average signal energy per estimate E = SMTs to the spectral noise level No = NTs; that is, E - = -- = MSNR. (16) No(Note that for real data, No = 2Nrs and tI, = SNR M/2.) This is the basic parameter for matched filter estimation of known signals in uncorrelated noise that iscommonly used in communication and radar (Helstrom 1968; Van Trees 1968 ). For Doppler lidar withquantum limited detection (the additive noise is dominated by the local oscillator shot noise), this parameterhas a convenient physical interpretation. It is commonpractice to set the sampling rate 1 / rs equal to the noisebandwidth B. The SNR Eq. (10) can then be writtenas SNR(t) = PR(t)nu(t)rs, (17)where PR(t) = Po(t)/hv (photoelectrons per second)is the average detection rate of photoelectrons. Therefore, - = rl~qP~MTs. (18)~Since P~Mrs is the average number of photoelectronsgenerated per observation time Mrs, rI, is the effectivenumber of photoelectrons coherently detected per observation time or the number of "coherent photoelectrons'' (coined by Kavaya). For Doppler radar, thisinterpretation is not valid because quantum-limiteddetection is not achieved. However, the interpretationof Eq. (16) is meaningful for Doppler radar. The average number of photoelectrons per estimate has beenused by Menzies ( 1986 ) as a measure of performance. SNR has also been defined in terms ofa narrowbandSNR (Huffaker et al. 1984), which is the ratio of thepeak of the signal spectrum to the constant spectrallevel of the white noise. This narrowband SNR dependson the spectral width of the signal. The parameter (I,can be expressed in terms of a narrowband SNR byassuming all the spectral signal power is collected inone spectral bin of width/x f= 1/Mrs. Then, - is theratio of the average of this spectral signal to the averageof the spectral noise that is contained in the same spectral bin. The spectral width w is commonly normalized bythe Nyquist frequency FN = 1 / rs. For fixed range resolution (fixed MTs), this parameter changes with thesampling interval. A more convenient parameter is. theratio of the spectral width w to the frequency resolutiona f,where w 2~ = -- = wMrs = M,, (19) Mrs M~ = (20) Aris defined as the number of independent samples ofthe signal in the observation time Mrs. If the windfield is constant over the sensing volume of the pulse,then [ln(2)/2] 1/2 Ap ~2 , (21) w Arand the parameter 12 is proportional to Ap/Ar, thenumber of pulse widths Ar contained in the distance/Xp that the pulse travels during the observation time.Using the parameters of the simulated data presentedin Fig. 1 we have M~ = 4.44, which is approximatelythe number of modulated events or fades displayed inthe lidar signal. For Doppler lidar observations of uniform wind fields using a Gaussian pulse, the choice ofone independent sample per observation interval (M~= 1) produces ft = 0.450 and Ap/Ar = 2.40. ForDoppler lidar measurements of the atmosphericboundary layer, f~ < I. For the WSR-88D Dopplerradar, 0.25 < t2 < 5 (Doviak and Zrni6 1993). Forclear-air wind profilers, the signals are weak and themuch longer observation time per estimate produces~ > 100 (May and Strauch 1989). Zrni6 (1979) investigated the number of independent samples per estimate in terms of the reduction of the variance of theestimates for signal power. Zrni6's effective number ofindependent samples is also proportional to fh Sincethe choice of a definition for the number of independent samples per estimate is arbitrary, the parameter~ will be used to represent this basic physical quantity. The covariance of the data [Eqs. (4) and (5)] canbe written in terms of the new variables cI, and ~; thatis, Rk = e~cp M2 + 2~ri f 12k + ~k. (:>.2) wThe best performance of any unbiased estimator of thesignal parameters is given by the CRB. For fixed - andft, the CRB a/(Frehlich 1994a) for estimation of themean frequency normalized by the spectral width ware shown in Fig. 2 as a function of M. [The normalizedvariable af/w is suggested by the functional form ofEq. (22).] For sufficiently large M, the CRB for 'thenormalized mean frequency is independent of M forall the parameters 12. For larger ft, the CRB becomesindependent of M when the spectral width w is lessOCTOBER I994 FREHLICH AND YADLOWSKY 1221 1.1 1.0~-0.9 0.8 0.7 0.9 11.8 0.7 0.6 1.0 0.9 0.7 0.6I I I I I I II 10 100 M FIG. 2. The normalized CRBs affor estimation of mean frequencyf, as for estimating signal power S, and a~ for estimating spectralwidth w as a function of M, the number of samples per estimate, for- = l0 and ~2 = 0.2 (solid), 0.5 (dotted), 1.0 (dashed), 2.0 (dotdash), and 5.0 (dash-double dot).than the Nyquist frequency F~v~that is, when w < F~v/6 or, equivalently, when M > 6[2. This condition issatisfied for any realistic problem. The invariance of the CRB for fixed [2 and - produceuseful relationships for many practical problems. Sinceaf/w is independent of M, the CRB for mean frequencyaf is independent of M for fixed w. This implies thatthe total observation time per estimate MTs and rangeresolution is also fixed because [2 = wMTs is fixed. TheCRB for mean frequency af is independent of M forfixed observation time MTs (fixed range resolution)and fixed w. This result was also obtained by Rye andHardesty (1993a) using approximate empirical modelsfor Levin's approximate CRB. The exact CRB verifiesthe accuracy of this statement. The CRB for estimationof the average signal power S and the CRB for theestimation of the spectral width w are also shown inFig. 2. For both cases, the CRB is independent of M,provided that the spectral width is less than the Nyquistfrequency. Similar results are produced for other valuesof ~. This result is excellent motivation for the choiceof - and fi as the basic system parameters. Accuratecalculations of the CRB's of the signal parameters forvery large M can be made using the more efficient calculation at smaller M. For fixed MTs (fixed range resolution), the frequency resolution Affor the periodogram is also fixed. For fixed ~, [2, and MTs, the statisticsof the periodogram around the spectral peak will beunchanged because the shape of the signal peak compared to the frequency resolution Af is unchanged.This provides an intuitive explanation for the invariance of the CRB's for the estimation of the normalizedsignal parameters. The low SNR analytic approximation to the CRB for mean frequency can be written as(y)/W2 = 4~rl/2~/(~2 [Zrni6 1979, Eq. (A29)], whichis independent of M. However, the high SNR analyticapproximation to the CRB becomes a~/w2 = 12122/M3 [Zrni6 1979, Eq. (A28)], which is not independentof M. It has been shown that the high SNR approximation to the CRB is not valid (Frehlich 1994a). The radial component v (m s-l) of the velocity ofthe atmospheric scatterers is related to the mean frequency f by v = Xf/2 and the velocity estimationerror a~ (m s-l) = Xaf/2, where X (m) is the operatingwavelength. The velocity error is proportional to X ifaf is independent of X--that is, if - and [2 are independent of X and MTs (range resolution) is fixed. ForDoppler lidar, - is independent of X for fixed laserpower PL if the backscatter coefficient t3 oc X-t oc v[Eqs. ( 11 ) and ( 16)] and C(R) is independent of X,which is the normal case for comparison. Here, fi isindependent of X if w is independent of 3,; that is, thereare negligible velocity fluctuations over the sensingvolume of the pulse, and w is determined by the transmitted pulse shape (Frehlich 1994b). Velocity error isthen proportional to wavelength for fixed MTs (fixedrange resolution), fixed w, fixed power transmitted,and/~ oc 3,-t. This is a convenient benchmark for system design and for some atmospheric conditions/~ ccX-t (Srivastava et al. 1992). Rye and Hardesty (1993a)found that Levin's approximate CRB for mean-frequency estimation is proportional to 3`t/2 for fixedrange resolution, fixed signal power, fixed maximumvelocity, and fixed wTs (the spectral width is determined by the velocity fluctuations over the sensingvolume of the pulse).3. Mean-frequency estimators For Doppler lidar with short pulses, the spectralwidth of the signal w will be determined by the pulseprofile and therefore known a priori (Frehlich 1994b).If a long pulse is transmitted, the effects of wind shearand wind turbulence produce nonstationary data(Frehlich 1994b). For nonstationary data, spectral domain and covariance-based estimators are not rigorously defined and the maximum-likelihood estimatormay produce the best performance. For a typical experiment, many pulses will be transmitted and the SNRwill also be known if the atmosphere is stationary overthe total observation time. The mean-frequency estimators considered here will assume knowledge of theSNR and the spectral width w. For typical Dopplerradar operation, the spectral width is determined bythe wind fluctuations in the sensing volume, and mustalso be estimated. The pulse-pair algorithm does notrequire a priori knowledge. All the other algorithmsdiscussed in this section require knowledge of the SNRand spectral width w.1222 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 11a. Pulse pair (PP) A numerically efficient algorithm for mean-frequency estimation is the pulse-pair (PP) estimatorgiven by (Miller and Rochwarger 1972; Zrni~ 1979) 1 f=- 2--~ss arg(]~), (23)where/~k 1 M- l -k - Z zi+kz~ (24) M-kis an unbiased estimate for R~, the covariance at lagkTs with a fixed length of data and arg(z) denotes theargument of the complex variable z.b. Maximum-likelihood estimator (ML) The maximum-likelihood (ML) estimator is thevalue of the unknown parameters that maximizes thelog-likelihood function (Helstrom 1968; Van Trees1968), given byL(f, S, w) = --z*TIt-~(f, S, w)z -ln[llt(f,X, w)l] -Mln0r), (25)where I ItI denotes the determinant of the matrixFor the signal model considered here, I It(f, S,d w) lis independent off. The simplest ML estimator formean frequency f assumes knowledge of the signalpower $ and the spectral width w. For the covariancemodel of Eq. (4) (Zrnifi 1979),R~l(f, S, w) = D~(S, w) exp[2r/JTs(k - l)1, (26)where D(S, w) = It-~(f= 0, $, w): In this case, theML estimator for f is the value f that maximizes[ substitute Eq. (26) into the first term of Eq. (25) ] M-I L~ = - ~ dm cos(2~rTsmf), (27) m=0where M-m-IdmTM Z k=02~ 2k+mDk.k+m(S, W). (28)The ML estimator can be efficiently calculated usingthe fast Fourier transform (FFT) ~algorithm of lengthMs >~ M at discrete frequencies f = kfTs/Ms. Notethat the ML estimator is not given in terms of an estimate for the covariance.c. Periodogram maximum-likelihood estimator (PML) In the limit of large M, the FFT and periodogramcoefficients/5(m) become mutually uncorrelated andthe ML estimator for the mean frequency is the valueof fthat maximizes the log-likelihood function M-, L(f, S, w) = - ~ P(m,f, $, w) ' (29) m=Owhere (Doviak and Zrnifi 1993, chapter 5)P(m) = (iS(m)) = Ts ~ R~exp . (30) k=-(M-l)[Levin (1965) and Rye and Hardesty (1993a,b) usedthis ~gofithm with the spectrum substituted for P(m,f, S, w). For small ~ and large ~, this approximationresults in poorer perfomance; for example, for ~ = 0.5and - > 100 the velocity e~or is 15% larger than thePML estimator. ] The PML estimator is numericallymore efficient when computed in the time domain. Ifthe F~ coefficients are mutually unco~elated andstatistically independent, then the inverse of the covariance matrix R is Toeplitz and has the form of Eq.(26), where 1 ~-~ exp[2~im(k - l)/M] D~,(S,w)=~o P~~b7~ ' (31)Substituting this result in Eq. (28) produces the PMLestimator, which is also equivalent to implementingEq. (29) using the F~ algorithm. This estimator canbe written in te~s of estimates for the covafiance ~because the matrix R-~ is Toeplitz [see Eq. (28)]..d. Time series model estimators (AR, MV) Many spectral domain estimators have been proposed using models for the time series. These algorithms are discussed by Marple (1987) and computerprograms are provided. The autoregressive (AR) spectral estimator is given by Tspw PAR(P,f) = p , (32) I I + ~ a~ exp(-2*r/ficrs) l2 k=lwhere the coefficients a~ and pw are determined fi:omthe estimates of the covariance of the data. The orderp is an input parameter that must be selected. Another time series algorithm introduced by Capon(1969) has been advocated by Anderson (1991) forDoppler lidar parameter estimation. Marple (1987)calls this algorithm a minimum variance (MV) estimator, which can be written as 1 P 1 = ?l PAR(k,f) ' (33) P~v(P,f)For the spectral domain estimators, the mean frequency~ is determined by the maximum value of'theOCTOBER 1994 FREHLICH AND YADLOWSKY 1223spectral estimator. We have investigated the performance of the AR spectral estimator and the MV spectral estimator using the Yule-Walker and Burg methodfor estimating the autoregressive coefficients and usingthe order p that produced the best performance [ smallest standard deviation of the good estimates (see section4) ] for fixed (ae, ~2, M). The MV estimator with theYule-Walker method (MVYW) produced the bestoverall performance and is the only estimator of thisclass that will be considered here. For large ~, the bestorder for the MV estimators was p = 1, which corresponds to the PP estimator (Rye and Hardesty 1993b).For ~I, < 100, the optimal order increases, with thelargest increase for small f/. The optimal order shouldbe determined for a given choice of ~ and M.4. Performance of mean-frequency estimators The most useful description of the performance ofan estimator is its PDF. For fixed Ts and in the limitof large M, if the ML estimator is unbiased, it approaches the ideal performance of the CRB and thePDF approaches a Gaussian distribution. The periodogram (or any spectral estimator) of the random signalis random. For small enough - or SNR, there will berealizations where the spectral feature of the signal isburied in the noise. Then, the spectral domain estimators for mean frequency will choose the largest noisefeature that mimics the signal. This behavior was firstrecognized and analyzed by Shirakawa and Zrni6(1983) and Hardesty(1986). This qualitative description of a clump of localized good estimates around thetrue mean frequency sitting on a pedestal of uniformlydistributed "bad" estimates has been observed in manyother cases (Mahapatra and Zrni6 1983; May andStrauch 1989; Anderson 1991; Rye and Hardesty1993a). The PDF of the mean-frequency estimatorsf is therefore modeled as a Gaussian PDF centered onthe true mean frequency fand a uniformly distributedcomponent of bad estimates over the frequency range(0, F~v = 1/Ts), which is written as ~J l(1-b) [ _ (f~gf2)2]PDF(f) = brs + (~r~-iT~o exp/ , (34)where b is the fraction of bad estimates and g is thestandard deviation of the good estimates (see also Ryeand Hardesty 1993c). The PDF of the mean-frequencyestimators is estimated by its histogram with bin sizeAf. The parameters b and g of the model PDF aredetermined by minimizing the mean-square differencebetween the histogram of the estimates from the simulated data and the predicted histogram based on themodel PDF. An example of the histograms for 10 000estimates of the ML estimator and the PP estimator,the predicted histogram of the best-fit model parameters, and the model PDF are shown in Fig. 3. The ML1.0 =1 IIIIIIIIIll I li~l Illlll I I I I I Ill: 0.01 ~, ~ .,._,~ ~.,..,...~.~ ~ ~.,. ....... ~?77T~7:T~T~, - ] I ] I [ I i I [ I [ I ] I [ 0.1- +~+ + PP 0.01 . :cccccc,.-~a? xa~~*** ~ !"l' 4. 4' ~' 4- 4' '1' ~. ! I I [ I [ I I , I , I , I , 0 4 8 '~ 2 16 20 24 28 32 Mean Frequency Index k~ FIG. 3. The histoFams (crosses) from l0 000 ML and ?P estimatesof mean frequencyf= kf/Xf, where k/is the mean-frequency index,(I, = 10, ~ = 0.5, and M = 32. The best-fit' empirical model [Eq.(34)] for the PDF is indicated by the solid line and the histogramdetermined from the best-fit PDF is indicated by circles.estimator produces good agreement with the modelPDF. The PP estimator is a poorer fit to the modelPDF when - < 100. This is due to the nonlinear interaction of the signal and noise. The comparison of estimator performance with theCRB is complicated by the presence of the uniformdistribution of bad estimates. If th~ model PDF is agood approximation then it provides a complete statistical description of an estimator. For example, theensemble average of the mean-frequency estimator fis -]/Ts ~ ~ ~ b (f) = Jo fPDF(f)df= 5ys+ (1 - v)f, (35)which indicates a bias from the true mean frequencyf. The mean-square error of the estimator is defined as<(.f_f)2> = ae2 = Jo~/rs (f_f)2pDF(f)df b - 3r(1-3fTs+3f2r~)+(l-b)g2, (36)which has a minimum forfTs = 0.5. The CRB for abiased estimator is given by (Helstrom 1968) <(f_f)2) >~ a2~ = a~ = (l - b)2a~, (37)where a~ is the CRB for an unbiased estimate off. TheCRB for a biased estimate depends on the fraction of1224 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 11bad estimates b, which in turn depends on the estimatorf. The performance of estimators with the bias includedcan be compared to the CRB for unbiased estimatesby dividing Eqs.'(36) and (37),by ( 1 - b)2. From themodified Eq. (36), the adjusted .standard deviation ofthe estimator error becomes O'e ac= l-b' (38)which is to be compared with the CRB affor unbiasedestimators~ An empirical model for the fraction of bad estimatesb as a function Of - for fixed fi and 34 is ' b(,I,) 1 + [bo]] ' (39)In the previous section, it was shown that the CRB forestimation of the mean frequency fnormalized by thespectral width w was independent of the number 'ofObservation points M for fixed - and ft. This suggeststhe use 0fthe parameter gN -- g/w. An empirical functi6nal form for this parameter is = + ~. (40) gN(,I,) x ! \gol jThe simulation results shown in the following figureswere generated by averaging ten simulations of 2000 I IIIIIIII I I~1111111 I I III1~11 I IIIIII,I I I IIIIIIJ I IIIl!llJ I I IIIIII 1.4 - : , 32 ~ 2' ~ ~' ~=0.5 M= i. t ~ t _ 0~'~ I I I llllll I I IIIllll I I IIIIIII I I IIllll~ I I IIIIIII I I I llllll 'l II II~ - ~' *~P'"q ' ~"""1 , ~mini ' !'""~ ~ ~mml ' ~'""'1 , ~11.~ ' . ~ '" - . ~ 0.~ . .~: ~ 0. .~ ~ 0.4~- ' ' ~ 0.2- ,~ 0 ' ' I IIII1~11 I IIIIII ~11 I IIIII1~ I IIIII1~ ' I I IIIitll I IIIllll 0'10-1 100 101 .102 . 103. 104 10~ 106 ~ ' . FIG. 4. The standard deviation g of the"'good" ML estimmes formean frequency and the fraction b of "bad" estimates ~ a functionof - for fl = 0.5 and M = 32. The results' of the simulation areindicmed by circles and the best-fit empirical models [ Eqs. (39) and(40)] are indicated by 'the solid line. The exact CRB is denoted bythe dotted line and the empirical model c6~ected for bias by Eq.(38) is indicated by the dashed line. 1.4 , ,,,,,,,I , ~.~,,,,,I . .,.,,.I , ,,!,.q , ,-.,,,i , ~,,.!.,I., ,rll~ ~ 1.2 ~\ ~ = 0.5 'M=32 ~ ',.o 0.0 ,., . '~ %~ 0.6 .... ~J ~ - ~ ~_~ _ ,~ 0.4 "' 0.2 ..: 0.0 ........ r ........ ~ '"'"'ll ....... ~ ........ ~ ........ r" ~m.~ ~mmI ~'",1 ~'~ 0.8 026 ' J ' 0.4 0.2 , ....... , ..... ;,,, II I I IIII1~ I IIIIllll I IIIIIIll I I . ~0- 10~ 102 . 103 104 ~05 106 ~ FIG. 5. The standard deviation g of the "good" PML estimates formean frequency and the fraction .b of"bad" estimates as a functionof - for fl = 0.5 and M = 32. The results of the simulation areindicated by circles and the best-fit empirical models [ Eqs~ (39) and(40)] are indicated by the solid line. The exact CRB is denoted bythe dotted line and the pehodo~am CRB is indicated by t~e &~shedline ( Frehlich 1994a).realizations each for large (I,. As the number of goodestimates declined at small ~, the number of realizations was increased to maintain the number of goodestimates at approximately 2000, until the fraction ofbad estimates became larger than 0.9. 'The standarddeviation 0fthe ten estimates for b and g~v = g/w wereused to obtain the chi-squared best fit (press et al. 1986)foi' the parameters Of the empirical models Eqs. (39)and (40). The standard deviation of the good estimatesfor small - is difficult to estimate because a large number of realizations are required. The performance of the .ML estimator .is shown inFig. 4 for fl'= 0.5. The simulation results agree wellwith the best-fit models Eqs. (39) and (40). For largetI,, the bias of the estimator is small and' the stanclarddeviation of th~ good estimates g approaches the C'.RB~For small tI,, the comparison of the standard deviationac of thebiased estimate to the CRB. that includes theeffect of the .bias is poor. This is a typical .result for allthe estimators we investigated. The CRB with the biasincluded [Eq. (37).] is a poor indicator Of estim.atorperformance. The standard deviation 6i~ the good estimates g and the fraction of the bad estimates b providea complete, description of the performance of meanfrequency estimators whenthe model PDF defined byEq. (34) is valid. For small tI,, the standard deviationof the good estimates g is less than the CRB. This doesnot contradict ~he meaning of the CRB because in. thisOCTOBER 1994 FREHLICH AND YADLOWSKY 1225 0.82 ',, o.8 0.4 0.2 O.4 0.2 0,0 I I I Illllll I IIIl,lfl I IllllllI J Illllll ~0-~ 100 10~ 102 103 10~ 10~ 106 ~]G. 6. The standard deviation ~ of the "good" estimates for meanfrequency and the fraction b of "bad" estimates as a function offor ft = 0.5 and M = 32. The results of the simulation are given bythe best-fit empirical models [Eqs. (39) and (40)] for the ML (solid),PML (dotted), MVYW (dashed), and PP (dot-dash). The periodogram CRB is shown as the dash-double dot curve. The performance of the ML estimator for M = 32,64, 128 is shown in Fig. 8 for [2 = 0.5 and in Fig. 9 for[2 = 2. The standard deviation of the good estimateshas little dependence on the total number of data pointsM for $ > 10; that is, the mean-frequency error isindependent of the maximum observable frequencyF~v for fixed range resolution and fixed signal energyper estimate. For fixed ~, and increasing M, the fraction of bad estimates b increases because there are morenoise spikes that may be larger than the true meanfrequency signal peak of the log-likelihood function.5. Trade-off between pulse energy and number of pulses Rye and Hardesty (1993b) posed an importantquestion: Is it better to use many pulses with low energyor one pulse with the same total energy? For constantlaser power and for the case where accumulated datafrom many pulses approaches Levin's CRB, theyshowed that many low-energy pulses were better thanone high-energy pulse for narrow spectral width. Thisregime corresponds to large ~I, where there are no badestimates. The performance of mean-frequency estimators for K pulses per estimate can be compared tothe single pulse case for fixed ~. The data for eachpulse are assumed statistically independent. This is agood assumption for Doppler lidar because the signalregime the estimates are biased and are not boundedby the CRB for unbiased estimates. The performance of the PML estimator is shown inFig. 5 for [2 = 0.5. The simulation results agree wellwith the best-fit models Eqs. (39) and (40). When theestimates are unbiased (~ > 100), the performance isbounded by the periodogram CRB. The performance of the ML, PML, MVYW, andPP mean-frequency estimators are shown in Fig. 6 for[2 = 0.5 (typical Doppler lidar case) and in Fig. 7 for~ = 2.0 (typical Doppler radar case). The parametersof the empirical models Eqs. (39) and (40.) are givenin Table 1. All the spectral domain and covariancebased estimators are bounded by the periodogram CRBfor - > 100 (unbiased estimates). For the best estimators (ML, PML, AR, MV), the standard deviationof the good estimates g approaches w for small - andapproaches a fraction of w (~0.05 --~ 0.8) for large ~.This general feature produces a useful description ofmean-frequency accuracy in terms of the basic parameters ~, and [2. The dependence of b with rI, is similarfor the best estimators. The PDF of the PP estimatoris a poor fit to the empirical model Eq. (34) for $< 100 (see Fig. 3) and results in different performancecurves. For [2 = 0.5 and large ~I,, the ML estimator hasmore than a factor of 2 improvement in mean-frequency accuracy compared with the best spectral domain and covariance-based estimators. 1.2 , ...,..t . ,,...,..~ . .,,...i . ,.,.,.,I , ,,,,,.,I , ,,,,,,,I , ,,,,,, 1.0~*":'~:::..~ \~ P,=2 M=32 - 'e., ~ 0.8- ' \\ - ~.~ O.e- ~, ~ 0.2 ...... .~?.Z2~ ........ , ........ , ........ , ....... , ........ , ........ '~Ex~ ........ ~ ........ ~ ....... ~ ........ ~ ........ ~ ....... ~/ 0.8 ~ ~~X~~,, ,,,,,, ~~~,~ 0.6 ~ 0.4 0.2 0 0 , . ,.,.,d , , ,,..I . ,,..,,.I , , .,. '10q 100 10~ 102 103 104 10s 10~ FIG. 7. The s~n~d d~viat[o~ ~ of the "~ood" esfimttes fo~ m~tnfrtqutnc~ a~d tht fraction ~ of "bad" tsfimatts as a functio~ of -for ~ = Z0 and M = 3~. The ~tsu]ts of tht simulation art ~vcn bythe ~st-fit cmp~dc~ m~tis [~s. (39) and (40)] ~o~ ~ ~L (sofid),PML (dotttd), M~W (dashed), and PP (dot-dash). The pcdodo~am ~EB is shown as tht d~h~oub]t dot lint.1226 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 11 T~L- 1. Parameters of mean-frequency estimator performance. Parameters of the empirical models [Eqs. (39) and (40)] for performanceof mean-frequency estimators pulse pair (PP), maximum likelihood (ML), pefiodogram maximum likelihood (PML), and minimum variancewith Yule-Walker method (MVYW).Estimator M ~ go . e b x ~ bo a ~ML 32 5.0 6.6703 1.5707 0.4697 0.9277 0.1163 7.5556 1.1339 1.6094ML 32 2.0 2.8857 3.2457 0.1283 0.8970 0.0986 7.3809 1.2388 1.5998ML 32 1.0 2.2493 2.2808 0.1466 0.9033 0.1134 20.364 1.1699 4..4398ML 32 0.5 2.2087 1.4087 012020 0.9384 0.1486 9.7988 1.1992 2.0610ML 32 0.2 4.5910 0.1057 3.7896 11.858 0.2286 6.8265 1.1688 1.3177ML 64 5.0 3.9071 4.7631 0.0986 0.9022 0.0607 18.737 1.1971 2.9935ML 64 2.0 2.5652 2.7665 0.1437 0.9062 0.0925 9.6277 1.3505 1.7031ML 64 1.0 2.3752 2.2978 0.!452 0 869~ 0.1143 1&914 1.2968 3.3268ML 64 0.5 2.5108 1.4975 0.1806 0~8687 0.1433 10.497 1.2918 1.8985ML 64 0.2 0.3600 0.1758 1.7274 4.2255 0.2195 7.3158 1.2586 1.2036ML 128 5.0 3.8039 4.3628 ~.1062 0.9084 0.0590 24.519 1.3365 3.5752ML 128 2.0 2.81.34 2.6789 0.1498 0.8649 0.0944 1i.417 1.4833 1.7252ML 128 1.0 2.6969 2.1209 0.1558 0.8216 0.1115 14.981 1.4352 2.5919ML 128 0.5 3.0713 1.3'192 0.2043 0.8208 0.1410 10.286 1.4167 1.6258ML 128 0.2 1.7036 0.2663 1.0731 1.9396 0.2191 7.4160 1.3537 1.0564PP 32 5.0 11.897 0.5609 2.5659 2.4588 0.2214 2660.5 0.6675 62.257PP 32 2.0 6.8862 0.9061 1.5928 3.2176 0.2758 8.9822 0.8767 2.1189PP 32 1.0 5.3448 i.2442 1.0480 4.8220 0.3663 14 218 0.8727 939.91PP 32 0.5 5.3328 1.0789 1.2031 10.301 0.4884 150:96 0.8012 12.921PP' 32 0.2 7.1091 0.8191 1.7217 29.089 0.7103 6.8097 0.9680 1.3113PP 64 5.0 i~.453 0.8537 i.7115 3.0136 0.1793 18.7~2 0.8779 3.1407PP 64 2.0 15.016 0.7441 2.3529 7.8210 0.2644 5.6641 0.9440 1.3280PP 64 1.0 9.3629 1.1064 1.2673 10.691 0.3587 10489 0.8255 408.91PP 64 0.5 9.5021 0.9408 1.4805 23.420 0.4869 60.720 0.8135 5.5993PP 64 0.2 10.521 0.7146 1.9955 72,335 0.7083 7.7234 0.9862 1.2602PML 32 5.0 7.3217 1.4801 0.5264 0.9266 0. i272 710441 1.1624 . 1.4976PML 32 2.0 6.7973 1.1595 0.6540 0.9087 0.1881 11.116 1.1656 2.2642PML 32 1.0 11.784 0.7045 1.2595 0.9412 0.3000 20.034 1.1652 4.2724PML 32 0.5 15.847 0.9673 0.7254 0.5898 0.4894 9.8004 1.1939 1.9903pML 32 0.2 78.297 3.1347 0.1774 0.4105 0.8895 6.4253 1.1645 1.2274PML 64 5.0 6.5687 1.6033 0.4342 0.9392 0.1046 14,667 1.2299 2.3947PML 64 2.0 9.5131 0.9380 0.9414 0.9247 0.1947 9.0614 1.3789 1.5785PML 64 1.0 24.172 0.5855 1.8912 0.9772 0.3054 16.589 1.2930 3.1727PML 64 0.5 23.960 0.8495 0.8965 0.5554 0.4869 10.613 1.2883 1.8341PML 64 0.2 134.62 2.8399 0.2370 0.3568 0.8924 6.7535 1.2535 1.0988MVYW 32 5:0 8.8394 1.8919 0.5558 0.7427 0.2210 20.737 1.0770 3.5032MVYW 32 2.0 . 18.395 0.8558 1.5665 0.8191 0.2751 15.984 1.1590 3.245 lMYYW 32 1.0 16.033 0.6886 1.6551 0.8194 0.3661 23.797 1.1440 5.0656MVYW 32 0.5 13.826 0.5801 '1.5710 0.772i 0.4875 10.140 1.1818 2.1000MVYW 32 0.2 0.89414 6.3789 0.06950 0.8038 0.7014 7.5787 1.1381 1.4224MV~W 64 5.0 13.11 1.1831 0.9328 0.8460 0.1788 26.273 1.2172 4.2762MV~W 64 2.0 7.174 1.2515 0.7070 0.7653 0.2626 32.158 1.2506 5.9408MVYW 64 1.0 13.75 0.5791 1.7506 0.9245 0.3587 18.894 1.2725 3.6550MyYW 64 0.5 730090 0.2226 23.980 1.8727 0.4873 il.000 1.2645 1.9591MVYW 64 0.2 0.3008E-6 0.5033 0.8209 334.31 0.7021 6.9962 1.2552 1.1475c0rrelation time is of the order ~f I us and the intervalbetween pulies is larger than 10 ms. - The log-likeliho0d function LK for K statistically independent data vectors z~ and fixed range resolution(fixed MTs) is (Scharf 1991 )L~(f, SK, w) = -Kln[ll:l(f~ Sn, w)l] K - KMln(~r) - ~ Z*m*l-~(f, Sn, W)Zm, (41) rn=lwhere z~m denotes the transpose of the mth data w:ctorz,, and S~c = S/K is the average signal power for eachpulse. The CRB for estimation of mean frequency af~cusing K pulses is (Scharf 1991 ) a~: = a~( St:) (42) K 'where a](S~:) is the CRB for one pulse with averageOCTOBER 1994 FREHLICH AND YADLOWSKY 12271.21.00.80.60.40.2~ ......., f~ = 0.5 ~0q 10- 10~ 102 103 104 10s 106 FIG. 8. The standard deviation g of the "good" ML estimates formean frequency and the fraction b of "bad" estimates as a functionof - for ~ = 0.5. The results of the simulation are given by the bestfit empirical models [Eqs. (39) and (40)] for M = 32 (solid), 64(dotted), and 128 (dashed). 1.0 g2=20.8~ 0.60.40.2- ~x~_.[~..~. i ,imuI i lira,I i Sllllll~ i ilJllllI i ill,,I i iiiii1[o.8 h0.6 ~0.4 - ~ 0.2 - : '10-~ 100 10~ 102 103 10~ l0s 106 FIG. 9. The standard deviation g of the "good" ML estimates formean frequency and the fraction b of "bad" estimates as a functionof - for fl = 2.0. The results of the simulation are ~ven by the ~stfit empirical models [Eqs. (39) and (40)] for M = 32 (solid), 64(dotted), and 128 (dashed).signal power Sg. The CRB for the mean frequency asa function of the total number of pulses but with fixedtotal energy q? is shown in Fig. 10. Multiple pulses arebetter for - > 10 and smaller ft. The performance ofthe ML estimator [the value of fthat maximizes Eq.(41)] for different K as a function of - is shown inFig. 1 1 for fl = 0.5 and in Fig. 12 for ~ = 2. For small,I~, the best performance is obtained by using one pulse.For large ~, performance is improved by dividing thetotal laser energy among many pulses. The effects aremost pronounced for small ft. For small q,, concentrating all the energy into one pulse produces the optimal signal for identifying the good estimates. For large- , there are only good estimates and more pulses resultin a decrease in the mean-frequency error because eachpulse produces a statistically independent estimate andthe error scales as K-l/2af(Sic). Since af(SK) changesslowly compared to K-~/2, performance improves withmore pulses. The CRB (see Fig. 10) provides an effective prediction of the qualitative behavior of the standard deviation of the good estimates g for all ~.6. Summary For the signal model of Eqs. (2), (4), and (5), theperformance of mean-frequency estimators is conveniently presented in terms of two basic parameters:the ratio of the signal energy per estimate to the spectralnoise level; and g, which is proportional to the numberof independent samples per estimate. For Doppler lidar,- is the number of coherently detected photoelectronsper estimate. For fixed - and r, the CRB's for unbiasedestimates of the signal parameters are independent ofthe number of samples per estimate M (Fig. 2). The6.0 -- I I I IllllI I I I llis,rI )- i/l~i'/l[ _ ~o'! /-~//o,'- q) = 10 ,. / /."/...-'" ~.. 4.0 - .' ./ / ." ~' .~ ~ .,' ~*' ,~ ~/ ,-' ~ ~ 0.0 I I I I ;~J I I I ~ I~llJ I L~.I II1 I J I J I IIIJ J I I J I IllJ I I I J I, JJ -- /' ,/ ~ 0.8~ ~=~00 /.' // ,~ - --........ 0.6 ~~0.4 0.2 ='==~=';== ' - - - ' 00 I i i i iiiii i I I i Iiiii I i i i i I I - I I I I I IIII I I I I I iiiI I I I i i ii 0.3 ~ ~i-~.~: ~5~2~ ~~0.2 0.1 .... ~;5~9~--.~~a~~ . . 0.0 ~ ~ ~ ~l I I I I,IIlll I ~ ~ ~M~ 10 100 10o0 Number of Pulses K F;o. l 0. The CRB for esfimatio~ of mea~ frequency ~or multiplep~lscs with ~x~ total en-;~y (fixed ~) a~d Q = 0.2 (mlid), 0.5(dotted), 1.0 (dashed), 2.0 (dot-dash), and 5.0 (dash-double dot).1228 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 11 1,~ ! IIIIII1~ I IIIIIIII I IIIIIIII I IIIIIIII ~ IIIIIIII I IIIIIIII I IIIIIII - M=32 0.6 0.4 ~' ~.%,. ~ _ ~ 0.2 - ~<'~'~:...~ ~ ~H.d ~ ~Hmd ~ ~m.I ~ ~H.d ~ ~Hmd ~ ~,md ~G 0.0 ~e~ ....... ~ ........ ~ ....... ~ ........ ~ ........ ~ ........ %~ ~ 0.8 ~X5 O.8 ~~ ~1 0.4 ~ O.~ d ~ ~m.~ ~ ~Hmd ~ ~,md ~ ~mm 0'~ 10- 10~ 10~ 10a ~ 0~ ~ 0* 10~ FIG. 11. The standard deviation gofthe "good" ML estimates formean frequency for Doppler lidar and the fraction b of "bad" estimates as a function of - for different numbers of pulses per estimatewith the total signal energy per estimate fixed and fl = 0.5. Theresults of the simulation am ~ven by the best-fit empidcal models[Eqs. (39) and (40)] for K = 1 pulse (solid), 5 pulses (dotted), and10 pulses (dashed).the uniformly distributed bad estimates. For fixed. M,the behavior of the fraction of bad estimates b is almosta universal function of q' with a weak dependence on[2 (Figs. 6-9); that is, the better estimators (ML, PML,AR, MV) perform equally well in identifying the caseswhere the random signal is not buried in the randomnoise. For fixed (I, and [2, as M increases (Figs. 8... 9 ),the fraction of bad estimates increases because the random spectral feature of the signal must be identifiedamong more competing noise spikes. For q' > 100 andfixed ~2, the standard deviation of the good estimatesnormalized by w is essentially independent of M andfor fixed range resolution and fixed signal energy:, thestandard deviation of the mean-frequency estimatorsis independent of the maximum observable frequencyFN. For fixed range resolution, fixed spectral wJ.dth,fixed transmitted pulse energy, and a backscatter coefficient/~ oc X-j, the velocity error is proportional tothe operating wavelength )x. The spectral width o:f thesignal is constant with operating wavelength when thewind fluctuations are negligible over the sensing volumeof the pulse. This is true for Doppler lidar with shortpulse lengths and typical atmospheric conditions. Fortypical Doppler radar conditions, the spectral width ofthe signal is determined by the wind fluctuations overthe sensing volume of the pulse. Then, the velocityerror is approximately proportional to X ~/2 for tixedPDF of a mean-frequency estimator is the best description of performance. For the better estimators, the PDFcan be accurately modeled as a Gaussian function centered at the true mean frequency and a uniformly distributed component of bad estimates (Fig. 3). Thismodel requires two parameters: the standard deviationof the good estimates g and the fraction of the estimatesb that are bad. The empirical functions Eqs. (39) and(40) provide a good approximation for g and b as afunction of q~ for fixed [2 and M. There are no simulation results for very small - (when b ~ 0.9) and theempirical model is an extrapolation. The parametersof these empirical models are listed in Table I for somecommon mean-frequency estimators. The fraction ofbad estimates can be reduced by neglecting the estimates that have low signal power (Rye and Hardesty1993c). However, this also reduces the number of goodestimates. Adaptive mean-frequency estimators havealso been proposed to reduce the detrimental effects ofthe fades in signal power (Dabas et al. 1993). For large (I,, the estimates are unbiased, the standarddeviation of the spectral domain and covariance-basedestimates are bounded by the periodogram CRB (Figs:6, 7), and the standard deviation of the ML estimatorapproaches the exact CRB (Fig. 4). For smaller [2, theML estimator has more than a factor of 2 improvementin performance compared to previous estimators (Fig.7). For ee < 100, all the estimators are biased due to 1.2 i iiiiiii 1.0 ~ = 2 M=32 0.6 0.4 0.2 0.0 ~ ~,1~i~,~%1 I IIIIIII I I IIIIII1 I I IlllllI I I IIIIIII I I IlllllI I ,I III 0.8 - ,,,\ 0.6 - ".\ 0.4 0.2 - ,.,~ 0-~ 100 10~ 102 103 104 105 106 I~G. 12. The standard deviation g of the "good" ML estimates formean frequency for Doppler lidar and the fraction b of "bad" estimates as a function of q~ for different numbers of pulses per estimatewith the total signal energy per estimate fixed and Q = 2.0. Theresults of the simulation are given by the best-fit empirical models[Eqs. (39) and (40)] for K = I pulse (solid), 5 pulses (dotted), and10 pulses (dashed).OCTOBER 1994 FREHLICH AND YADLOWSKY 1229range resolution and fixed signal energy (Rye and Hardesty 1993a). For Doppler lidar and for large ~, it is better totransmit many low-energy pulses instead of one pulsewith the same total energy. For small ~I,, the converseis true (Figs. 11, 12). The behavior of the performance of mean-frequencyestimators as a function of the two basic parameters q,and ~ simplify Doppler lidar design. In many cases,the range resolution of the measurement is determinedby the application. Assuming one independent velocitymeasurement per observation time and small fluctuations of the wind field over the sensing volume of thepulse (w is determined by the transmitted pulse profileand $2 = 0.45) fixes the pulse length and observationtime per estimate MTs. The maximum search velocity/)max = )k/2Ts determines Ts and M. The fraction ofbad estimates b and the standard deviation of the goodestimates (av = Xg/2) is then a function of one parameter q,. The minimum required pulse energy for a specified backscatter coefficient is determined from the c~that is required to produce a minimum number of goodestimates. Acknowledgments. The authors acknowledge usefuldiscussions with M. J. Kavaya, D. S. Zrnifi, and B. J.Rye. This work was supported by the National ScienceFoundation and the National Aeronautics and SpaceAdministration, Marshall Space Flight Center underresearch Grant NAG8-253 (Michael J. Kavaya, Technical Officer).APPENDIXSimulation of Doppler Radar and Lidar Data Complex Doppler radar and lidar data with a specified covariance can be generated with a simple computer algorithm. The complex Gaussian data vector zis produced by Ms-~ [ 2*rimk \Zk=m=0Z ymexp[--~-~-s J, (A1)where Ym are zero-mean complex Gaussian randomvariables withand Ym-~YkY~) = Yrn~k-m (A2)1 M~-~ 2~rimkI ~ Ra exp , (A3)Ms k=0 \ ~ss Jwhere Rk is the desired covariance of the stationarydata and RMs-~ = R~ is required to produce real Ym.The random vector z then has the properties(zaz~) = R~-t = (zaz~s_t), (A4)and the first points of the simulated data are correlatedwith the end points. Therefore, a simulation of lengthMs = 2M is required to generate an independent realization of length M. The length Ms must be longenough to ensure that all Ym are positive [see Eq. (A3)].Zrnifi (1975) and Sirmans and Bumgarner (1975) usedthis algorithm with Ym chosen to produce exact sampling of the spectrum instead of an exact sampling ofthe covariance. For Doppler lidar, the covariance ofthe data is specified by the transmitted pulse and lidarsystem parameters (Frehlich 1994b).REFERENCESAnderson, J. R., 1991: High performance velocity estimators for co herent laser radars. Sixth Topical Meeting on Coherent Laser Radar: Technology and Applications, Snowmass-at-Aspen, CO, OSA, 216-218.Bilbro, J. W., C. DiMarzio, D. Fitzjarrald, S. Johnson, and W. Jones, 1986: Airborne Doppler lidar measurements. Appl. Opt., 25, 2952-2960.Capon, J., 1969: High resolution frequency-wavenumber spectrum analysis. Proc. IEEE, 57, 1408-1418.Dabas, A., P. Salamitou, D. Oh, M. Georges, J. L. Zarader, and P. H. Flamant, 1993: Lidar signal simulation and processing. Seventh Topical Meeting on Coherent Laser Radar Applications and Technology, Paris, France, CNES, CNRS, ESA, Ecole Poly technique, OSA, 221-228.Doviak, R. J., and D. S. Zrni~, 1993: Doppler Radar and Weather Observations. 2d ed. Academic Press, 562 pp.Frehlich, R. G., 1994a: Cramer-Rao bound for Gaussian random processes and applications to radar processing of atmospheric signals. IEEE Trans. Geosci. Remote Sens., 31, 1123-1131. ,1994b: Coherent Doppler lidar signal covariance including wind shear and wind turbulence. Appl. Opt., 33.--, and M. J. Kavaya, 1991: Coherent laser radar performance for general atmospheric refractive turbulence. Appl. Opt., 30, 5325-5352.Gal-chen, T., M. Xu, and W. L. Eberhard, 1992: Estimations of atmospheric boundary layer fluxes and other turbulence param eters from Doppler lidar data. J. Geophys. Res., 97, 409-418.Hardesty, R. M., 1986: Performance of a discrete spectral peak fre quency estimator for Doppler wind velocity measurements. IEEE Trans. Geosci. Remote Sens., 24, 777-783.Helstrom, C. W., 1968: Statistical Theory of Signal Detection. Per gamon Press, 470 pp.Henderson, S. W., C. P. Hale, J. R. Magee, M. J. Kavaya, and A. V. Huffaker, 1991: Eye-safe coherent laser radar system at 2.1 um using Tm,Ho:YAG lasers. Opt. Lett., 16, 773-775. , P. J. M. Suni, C. P. Hale, S. M. Hannon, J. R. Magee, D. L. Bruns, and E. H. Yuen, 1993: Coherent laser radar at 2 um using sofid-state lasers. IEEE Trans. Geosci. Remote Sens., 31, 4-15.Huffaker, M. R., T. R. Lawrence, M. J. Post, J. T. Priestley, F. F. Hall Jr., R. A. Richter, and R. J. Keeler, 1984: Feasibility studies for a global wind measuring satellite system (Windsat): Analysis of simulated performance. Appl. Opt., 23, 2523-2536.Kavaya, M. J., S. W. Henderson, J. R. Magee, C. P. Hale, and R. M. Huffaker, 1989: Remote wind profiling with a solid-state nd:yag coherent lidar system. Opt. Lett., 14, 776-778.Kay, S. M., and S. L. Marple Jr., 1981: Spectrum analysis--A modern perspective. Proc. IEEE, 69, 1380-1419.Levin, J. M., 1965: Power spectrum parameter estimation. IEEETrans. InJJ Theory, H, 100-107.Mahapatra, P. R., and D. S. Zrni6, 1983: Practical algorithms for mean velocity estimation in pulse Doppler weather radars using a small number of samples. IEEE Trans. Geosci. Electron., 21, 491-501.1230 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUI~,~E 11Marple, S. L., Jr., 1987: Digital Spectral Analysis with Applications.Prentice-Hall, Inc, 492 pp.May, P. T., and R. G. Strauch, 1989: An examination of wind profiler signal processing algorithms. J. Atmos. Oceanic Technol., 6, 731 735.--, T. Sato, M. Yamamoto, S. Kato, T. Tsuda, and S. Fukao,1989: Errors in the determination of wind speed by Dopplerradar. J. Atmos. Oceanic Technol., 6, 235-242.Menzies, R. T., 1986: Doppler lidar atmospheric wind sensors: A comparative performance evaluation for global measurement applications from earth orbit. Appl. Opt., 25, 2546-2553. , M. J. Kavaya, P. H. Flamant, and D. A. Haner, 1984: At mospheric aerosol backscatter measurements using a tunable coherent CO2 lidar. Appl. Opt., 23, 2510-2517.Miller, K. S., and M. M. Rochwarger, 1972: A covariance approach to spectral moment estimation. IEEE Trans. Inf Theory', 18, 588-596.Pearson, G. N., and B. J. Rye, 1992: Frequency fidelity of a compact CO2 Doppler lidar transmitter. Appl. Opt., 31, 6475-6484.Petheram, J. C., G. Frohbeiter, and A. Rosenberg, 1989: Carbon dioxide Doppler lidar wind sensor on a space station polar plat form. Appl. Opt., 28, 834-839.Post, M. J., and R. E. Cupp, 1990: Optimizing a pulsed Doppler lidar. Appl. Opt., 29, 4145-4158.Press, W. H., B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, 1986: Numerical Recipes. The Art of Scientific Computing. Cambridge University Press, 818 pp.Rye, B. J., and R. M. Hardesty, 1993a: Discrete spectral peak esti mation in incoherent backscatter heterodyne lidar. I. Spectral accumulation and the Cramer-Rao lower bound. IEEE Trans. Geosci. Remote Sens., 31, 16-27. , and --, 1993b: Discrete spectral peak estimation in inco herent backscatter heterodyne lidar. II. Correlogram accumu lation. IEEE Trans. Geosci. Remote Sens., 31, 28-35. , and ,1993c: Cramer-Rao lower bound-limited Doppler estimation using discrimination. Seventh Topical Meeting on Coherent Laser Radar Applications and Technology, 217-.220.Scharf, L. L., 1991: Statistical Signal Processing. Addison-Wesley, 524 pp.Shirakawa, M., and D. S. Zrni6, 1983: The probability density of a maximum likelihood mean frequency estimator. IEEE Trans. Acoust., Speech, Signal Processing, 31, 1197-1202.Sirmans, D., and B. Bumgarner, 1975: Numerical comparison of five mean frequency estimators. J. Appl. Meteor., 14, 991-1003.Srivastava, V., M. A. Jarzembski, and D. A. Bowdle, 1992: Com parison of calculated aerosol backscatter at 9.1- and 2.1-t~m wavelengths. Appl. Opt., 31, 1904-1906.Van Trees, H. L., 1968: Detection, Estimation, and Modulation The ory, Part I. John Wiley and Sons, Inc., 697 pp.Whittle, P., 1953: The analysis of multiple stationary time series. J.Roy. Statist. Soc., 15, 125-139.Zrni6, D. S., 1975: Simulation of weathedike Doppler spectra, and signals. J. Appl. Meteor., 14, 619-620.---, 1979: Estimation of spectral moments of weather echoes. 1EEE Trans. Geosci. Electron., 17, 113-128.