Doppler Weather Radar

1522 PROCEEDINGS OF THE IEEE, VOL. 67, NO. 11, NOVEMBER 1979 Doppler Weather Radar

RICHARD J. DOVIAK, SENIOR MEMBER, IEEE, DUSAN S. ZRNIC, SENIOR MEMBER, IEEE, AND DALE S. SIRMANS

Abstmct-The Doppler weather radar and its signals are examined wavenumber = 2r/A from elementary considerations to show the origin and development of a parameter proportional to raindrop's refractive usefulweather echo propertiessuch as signal-to-noise ratio (SNR), range conelation, signal statistics, etc. We present a form ofthe weather index radar equation which explicitiy shows the echo power loss due to finite one-way propagation loss due to scatterand receiver bandwidth and how it is related to the range weighting func- absorption tion. Echoes at adjacent range samples have a correlation that depends echo power loss due to finite bandwidth receiver on receiver bandwidth-transmitter pulsewidth productas well as sample spacing. Stochastic Bragg scatter from clouds is examined, but experi- number of echo samples along sample-time axis; mental work is required to determine if this echo power is larger than mean molecular weight incoherently scattered power. Section III presents theration between integer; also refractive index Doppler powerspectrum and the distribution of reflectivity and velocity thermal noise power within a resolution volume.A new formula that relates spectrum width number of scatterers to the shear of radial velocities as well as turbulence, signal decorrela- tion from antenna rotation, and signal p- binses is presented. echo power of resolution volume centered at? The estimation of power spectral moments IS renewed and properties power delivered to the antennasystem of the most commonly used algorithms are discussed. Section V high- pulse repetition time lights some of the considerations that need to be made for Doppler point target echo power at the antenna port radar observation of severe thunderstorms. Echo coherency is shown instantaneous weather echo power (W) to limit the pulsed Doppler radar's unambiguous range and velocity measurements. Side anddual Doppler-radar teChniques for wind mean weather echo power at sample time-range measurements are reviewed. Observations of thunderstorms show tor- delay rs nado cyclones, and clear air measurements in the boundary layer reveal output of radar receiver turbulence and waves. range from radar 1,2 to grid point range from source to target or resolution volume NOMENCLATURE location prefilter ampli'tude unambiguous range cTJ2 filter output amplitudeof ith scatterer 6-dB range width of resolution volume scatterer's weight per unit volume spatial covariance of A (r*) receiver-filter bandwidth, 6-dB width in Hz autocovariance at lag TI propagation speed, 3 X lo* m * s-l correlation of samples spaced along range time refractive index structure constant power spectrum in frequencydomain diameter of the antenna system expected echo sample power separation of radars for dual radar system power spectrum in velocity domain for resolu- structure functionof refractive index tion volume center atT normalized range. weighting function normalized power spectrum normalized one-way power gain orradiation signal-to-noise ratio pattern air temperature (K) maximum measured antenna gain; gravitational time lag constant pulse repetitiontime (PRT) or sample time weighting function of resolution volume interval prefilter echo amplitude,inphase and quadrature dwell time to resolve target location in FM-CW components radar inphase andquadrature phase signal at filter mathematical symbolrepresenting a pulse: U = 1 output when 0 < t < r;otherwise it is zero attenuation rate due to droplets (m-'); also an radial velocity field at a point integer Nyquist velocity A/4Ts gaseous attenuation rate mean Doppler velocities corrected fortarget fall- shear along 8, 4J, and r directions speed at data points for radars 1,2 pulse pair estimate of Doppler velocity Manuscript received November 28,1978;revised July 30,1979. This mean Doppler velocity at a grid point work was partially supported by the FAA under Contract DOTIFA76 meanDoppler target velocities measured by WAI-622 (RC360205), the NWS under Contract 8AA80901. the NRC under Contract RC370503, and the ERDA under Contract RD840520. radars 1, 2 The submission of this paper was encouraged after review of an advance mean terminal velocity of drops in resolution proposal. The authors are with the National Severe Storms Laboratory, NOM, volume Norman, OK 73069. horizontal wind speed

0018-9219/79/1100-1522$00.75 O 1979 IEEE DOVIAK et al.: DOPPLER WEATHER RADAR 1523

radial component of velocity (Doppler velocity) [ 51. A morerecent article appeared in the PROCEEDINGS prefilter receiver output voltage [113]. echo signal voltage Because the angular resolution A6 in degrees (”) at wavelength weather echo voltage sample at 7 = 7, X is well approximated by A9 70 AID where D is the diam- resolution volume eter of theantenna system [ 161, it is evident thatremote echo samples along sample-time axis radio sensing, even at microwave frequencies, is characterized cylindrical wind components by pool spatial resolution compared to opticalstandards. One vertical wind speed essential distinguishing feature favoring microwaves is its vertical velocity of tracers property to see inside rain showers and thunderstorms, day or ith scattererrange weight due to receiver filter night. Rain andcloud doattenuate microwave signals, but reflectivity factor slightly (for X > 0.05 m) compared to the almost complete angular coordinate; antenna rotation rate; rate extinction of optical signals. Scattered signal strength can be of frequency change in an FM- CW radar related to rainintensity, and time rate of change of phase air density; phase (Doppler shift)is a measure of raindrop radial speed. wind direction Development of high power and high gain klystron amplifiers range-time sample spacing in the 1950’s made practical the generation of microwsves that range over which samples are averaged are phase coherent pulse to pulse,a requirement for pulsed two-way half-power beamwidth Dopplerradars if velocities of other than first time around target reflectivity cross section per unit volume (first trip) echoes are to be measured [ 891. Radar signals are (m-l) phase coherent from pulse to pulse if the distance (or time) angle between incident and scatterdirection between wave crests of successive transmitted pulses is fixed beamwidthbetween half-power points of one orknown. Magnetron oscillators, phase incoherent pulse to way antenna pattern pulse, can only be used for Doppler measurements of targets radar beam elevation and azimuth angles in hori- beyond the first trip if provision is made to store phase for zon coordinates (4) = 0 at true north);also angu- time durations longer than thepulse repetition time (PRT). lar position of scatterer relative to beam axis The first reported use of a Doppler radar to observed weather radar wavelength (m) was made byBrantley and Barczys in 1957 [ 191. A rapid structure wavelength development of Doppler techniques followed. Boyenval [ 171 wavelength of wind fluctuations deduced the drop size distribution of Rayleigh scatterers from perpendiculardistance from axis of cylindrical the Doppler spectrum while Probert-Jones and Harper[961 coordinate system used vertically pointed antenna and storm motion to produce backscatter cross section a vertical cross section [ 101. Zenith-pointing Doppler radars ud,ur,us, ut spectrum width dueto drop fallspeed differ- can be used to estimate vertical air velocities as a function of ences, antenna rotation, shear, and turbulence height and time, can yield data from which one can sometimes total spectrum width of Doppler spectrum infer the nature of the hydrometeors (snow, rain, or hail), and U2 mean square value of I or Q in some instances, yield data for calculating hydrometeor size use,us@, u,, spectrum widths contributed by shear along 6, distributions [ 1 1 . ] 9, and r, respectively These earliest observations of radial velocities used analog 0; ,u$ second moment of the two-way antennapattern spectrum analyzers or filter banks that have economical utility 0: second moment of the range weighting function for, at most, observations in a few resolution volumes. Atlas 7 pulsewidth [4] recognized the utility of scanning storms horizontally to 7s time delay betweentransmitted pulse and the map radial velocities on a plan-position indicator (PPI) type echo sample. display and Lhermitte [ 811 accurately assessed requirements for the development of a viable pulsed Doppler radar. These I. INTRODUCTION early investigators foresaw real-time severe storm and tornado ADARS were developed to detectand determine the warnings from pulsed Doppler observations of storm circula- range of aircraftby radio techniques, but as they be- tions and their predictions were to be verified a few years later Iw came more powerful, their beamsmore directive, re- by several investigating teams [24],[25], [42], [45]. The ceivers more sensitive, and transmitterscoherent, they also first remote measurement of tornadic wind speed was accom- found highly successful applicationsin mapping the earth’s plished in 1958by Smith and Holmes [ 1121 using a3-cm surface andatmosphere, and their signals have reached out continuous wave (CW)Doppler radar. into space to explore surface features on our planetary neigh- Real-time reflectivities displayed on PPI have been available bors. Recently pulsed Doppler radartechniques have been to radar meteorologists since the mid-1940’s. The PPI shows applied to map severe storm reflectivity and velocity structure reflectivity distributionson conicalsurfaces as theantenna with some astounding success, particularlyshowing, inreal beam sweeps in azimuth at constant elevation angle. But real- time,the development of incipient tornado cyclones [24], time Dopplervelocity mapping was a goal that eluded re- [42],[45]. Theradar beam penetratesthunderstorms and searchers until the late1960’s. clouds to reveal the dynamical structure inside of an otherwise Contrary to reflectivity estimation which only requires echo unobservableevent. This inside look will helpresearchers sample averaging to reduce statistical fluctuations, mean veloc- understand the lifecycle and dynamics of storms. The first ity estimation requires sophisticated data processing. Probably detection of storms by microwave radar was made in England the long development and cost of Doppler processors (to esti- in early 1941. An excellent historical review of the early de- mate velocities simultaneously at all resolution volumes along velopments in radar meteorology can be found in Atlas’ work the beam)lay principally in preoccupation withpursuit of 1524 PROCEEDINGS OF THE IEEE, VOL. 67, NO. 11, NOVEMBER 1979 spectrum measurements, from which the mostinteresting arepresently engaged in a joint experiment, the purpose of moments (meanvelocity andspectrum width) need to be which is to demonstrate the utility of the Doppler radar for extracted. severe storm warnings and establish guidelines for the design One of the first -Doppler spectrum analyzers that could in- of next generationweather radars [ 251. We anticipate that deed generate velocity spectra in real time for each contiguous the new radars with Doppler capability will go in production resolution volume is describedby Chimera [33],and this in the 1980's and believe that this paper will acquaint the elec- machine, called a velocity indicating coherent integrator, pro- trical engineering community with some specifics of Doppler cessed with a single electronic circuit the echo signals to gen- weather radar, weather echo data processing, and meteorologi- eratespectrum estimatessimultaneously at all resolution cal interpretation. volumes. Another machine, called the coherent memory filter (CMF), employing the same principles was developed [ 621 for 11. THEDOPPLER WEATHER RADAR AND ITS SIGNALS weather radar observations and used by researchers at the Air Fig. 1shows in a simplified block diagram the principal Force Cambridge Research Laboratories(AFCRL).' This components of a pulsed Doppler radar. The klystronamplifier, machine produced the first real-time maps of velocity fields on turned on and off by the pulse modulator, transmitsa train of a PPI [3]. high "peak" power microwave pulses having duration T of In the early seventies Sirmans and Doviak [ 1081 described a about 1 ps with spacing at the PRT designated as T,, the sam- device that generates digital estimates of mean Doppler velocity pling time interval. The antenna reflector, usually a parabola of weather targets. This device, a phase change estimator, cir- of revolution, has a tapered illumination in order to reduce cumvented spectral calculations and digitally processes echoes sidelobe levels. Weather radarsmeasure a wide range of in contiguous resolutioncells at the radar data rate. volumetric target cross sections; the weakest (about lo-'' m2/ The need to obtain the principal moments economically and m3) associates with scatter from the aerosol-free troposphere, with minimum variance, and have these in digital format (to the strongest with cross sections (3 X lo-' m-') of heavy rain. facilitate processing and analysis withelectronic computers) Needless to say, antenna sidelobes place limitationson the has ledresearchers to use covariance estimatetechniques weather radar's dynamic range and can lead to misinterpreta- popularly known as pulse pair processing described in Section tion of thunderstorm heights [41] and radial velocity measure- V. Hyde and Perry reported an early version of this method ments [ 1221. [ 721, but it was first used by ionosphere investigators at Jica- The backscatter cross section ob of a water drop witha diam- marca [ 1231.Independently andat aboutthe same time eter Di small compared to A (Rayleigh approximation, i.e., Rummler [ 1021, [ 1031 introduced it to the engineering com- Di < x/16) is munity. Soon the advantages of pulse pair (PP) processing be- came evident, and scientists at several universities and govern- mentlaboratories began implementing this signal processing technique on the Dopplerweather radar [ 831, [ 881, [91], [llOI. where lK12 is a parameter, related to the refractive index of A single Doppler radar maps a field of radial velocities. Two the water, that varies between 0.91 and 0.93 for wavelengths such radars spaced apart to view the winds nearly orthogonally between 0.01 and0.10 m and is practically independent of can be utilized to reconstruct the two-dimensional wind field temperature [ 1 1, p. 381. Icespheres have (KI2 values of in the planes containing the radials [2], [ 821. With help of about 0.18 (for a density 0.91 7 g/cm3 ) which is independent the air mass continuityequation the third wind component of temperature as well as wavelength in the microwave region. can be estimated and thus the total three-dimensional wind There is an abundance of experimental and theoretical work field within the storm may be reconstructed. This is most sig- that relates particle cross section to its shape, size relative to nificant as it will enable one to follow the kinematics during wavelength when Di 2 A/16, temperature, and mixture of birth, growth, and dissipation of severe storms and thus per- phases (e.g., water-coated ice spheres). These works are well haps understand storm initiation andevolution. It may even reviewed by Battan [ 11 ] and Atlas [ 51. provide the answer as to why some storms reach great severity Were it not for electromagnetic energy absorption by water while others undersimilar conditions do not. or ice drops, radars with shorter wavelength radiation would Doppler radars are not limited to the study of precipitation be much more in use because of the superior spatial resolution. laden air. The kinematic structure of the planetary boundary Short wavelength (e.g., A = 3 cm) radars suffer echo power loss layer (PBL) hasbeen mapped even when particulate matter that can be 100 times larger than radars operated withA 2 10 does not offer significant reflectivity [47] . Coherent process- cm [ 121. Weather radar meteorologists are not only interested ing can often improve the detection of weather echoes [67]. in the detection of weather but also need to make quantitative Measurement at VHF [ 601 and UHF [ 9 I, [ 361 suggests height measurement of target cross section in order to estimate rain- continuous clear air returns to over 20 km, and experiments fall rate. Thus it is important to consider losses that aregreater with a moderately powerful radar at S band consistently show than a few tenths of a decibel. reflectivity in the first kilometer or two [30]. Besides attenuation due to rain and cloud droplets, there is Although the Doppler radar became a valuable tool in meteo- attenuation due to energy absorbed by the atmosphere's mo- rological research, it has not yet been transferred to routine lecular constituents, mainlywater vapor and oxygen.This operational applications. As a matter of fact, several govern- gaseous attenuation rate kg is not negligible if accurate cross ment organizations (The National Weather Service, Air Weather section measurements are to be made even at x = 10 cm when Service, Air Force Geophysical Laboratory, Federal Aviation storms are far away (r 2 60 km) and beam elevation is low Administration and the National Severe StormsLaboratory) (eOcf) [IS]. The above considerations lead to the radar equation for a Presently the Air Force Geophysics Laboratory. single hydrometeor having backscatter cross section ob, and DOVIAK et al.: DOPPLER WEATHER RADAR 1525

+GET

Fig. 1. Simplified Doppler radar block diagram. located at angles (e, 4) from the antennaaxis mixer outputs are

where P, is the power delivered to the antenna port, g the maximum measured gain, f2(8,4) the normalized radiation function, and I = exp - [J(kg + k) dr] the one-way loss factor due to gaseous kg and droplet k (both cloud and precipitation) the inphase Io(t) andquadrature Qo(t) components of the attenuation.The measured gain, a ratio of far-field power modulating signal. For convenience we ignored losses (i.e., set density S(e, 4 ) to the density if power was radiated isotropi- b = 1) and used a fi factor in (2.5) so that the sum of aver- cally, accounts for losses associated with the antenna system age power in Z and Q channels equals input power A2/2aver- (e.g., radome, waveguide, etc.). aged over a cycle of the microwave signal. If r increases with time, the phase y = -4nr/i + $ decreases A. The Doppler-Echo Waveform (Inphase and Quadrature and the time rateof phase change Components) When there is a single discrete target, the echo signal voltage V(t) replicating the transmitted electric field waveform E is proportional to it; is the Doppler shfit. It is relatively easy to see from (2.6) that, for usual radarconditions (i.e., T s and weathertarget V(t,r) =Aexp [j2nf(t- 2r/c) +j$] U(t- 2r/c) (2.3) velocities of the order of tens of m * s-l, the change in signal where 2r is the total path traversed by the incident and scat- phase is extremely small during the modulating envelope tered waves, A the prefilter echo amplitude, c velocity of light, U(t - 2r/c). Thus we measuretarget phaseshift over a time, and U = 1 when its argument is between 0 and r, otherwise it T, sz lo-' s fromecho to echorather than duringa pulse is zero.After detection and filtering (to remove the carrier period. Because of this the pulse Doppler radar behaves as a frequency f and harmonics generated in the detectionprocess), phase sampling device; samples are at t = T, + (n - 1)T, where we obtain a signal r, is the time delay between the nth transmitted pulse and its echo, andis denoted as range time because it is proportional to range (i.e., T, = 2r/c). It is convenient to introduce another time scale, designated sample time, wherein time is incremented in discrete T, steps rnodulat2g signal diffegnce after t = 7,. Echo phase andamplitude changes are usually frequency signal examined in sample-timespace at the discrete instants (n - 1)T, for a target at range time 7,. However, there have been (2.4) efforts to measure phase change within a time r in order to if receiver bandwidth is sufficiently large. Thus heterodyning eliminate aliasing problems that plague observations of storm and detection serve only to shift the carrier frequency without systems 1501. affecting the modulation envelope (for simplicity Fig. 1 shows The receiver's filter (Fig. 1) response is usually a monotoni- homodyning wherein STALO frequency is the same asthe cally decreasing function of frequency and its width B6 is best transmitted frequency, i.e., fa E 0). A Doppler radar usually specified as the frequencies within which the response is larger has two mixersin order to resolve the sign of the Doppler than one-fourth of its highest level-its 6-dB width [ 1 191. shift; in one the STALO signal is phase shifted by 90" prior to The larger is B6, the better is the fidelity of the echo pulse mixing so that the detected andfiltered output of this mixer is shape, but noise power increases in proportion to B6. Band- equal to (2.4) butphase shifted by n/2. The actual signal from widths can beadjusted to optimize signal detection perfor- the mixer is the real part of (2.4) and for the homodyne re- mance [ 1341 , but optimization causes receiver bandwidth loss ceiver (or after a 2nd mixing step to bringfA to zero) the two that should be part of the weather radar equation [go]. Fur- 1526 PROCEEDINGS OF THE IEEE, VOL. 67, NO. 11, NOVEMBER 1979 thermore, filterresponse is associated witha spatial weight average larger than the first! We estimate it in the following along the range-time axis whose form is just as important to way. the radar meteorologist as the antenna pattern weight is along Consider adensity function A (7)that describes the scat- angulardirections. We considerthese inthe weather radar terer’s weight per unit volume. Then analogous to (2.7) equation (Section 11-C).

B. Weather Echo Samples Weather echoes are composites of signals from a dense array of hydrometeors, each of-which can be considered a point which is theFourier composition of A(?) along the radial target. In this section we show the origin of the statistical direction (for bistatic radars the compositionis analyzed along properties of the echo samples V(T,) and instantaneous power the mirror direction). Hence the intensity of the scattered sig- P(T,) and discuss incoherent, coherent, and Bragg scatter. The nal depends directly upon the Fourier contentof the scatterer’s echo voltage sample V(7,) is a composite cross section per unit volume [ 1 18I. V, is the volume over which A(T‘) contributes significantly to V(T,)-e.g., the resolu- exp (j477rilA) (2.7) ~(7,)=EA,. tion volume: see Section 11-Cl. i When A (7)is an random functionand can only be described of discrete echoes from all the scatterers, each of which has a in a statistical way, it can be shown that the sample-time aver- weight Ai determined by the scatterer’s cross section ob, the age of P(T,) is radiation pattern f4(8, $), and receiver bandwidth-transmitted pulsewidth product B6r. These latter weighting functions 1- P(7,) = :J exp (-j4np/A)R(p’)dV, + A(?) determine a resolution volume in space wherein targets signifi- -12 I cantly contribute to the echo sample at 7,. The echo sample I power averaged over an r - f cycle is proportional to - exp (j4nr/~)d V,~Z (2.10a) where

R(p’)E [A(;;) - 2(7)1 [A*(?’) - 2*(7‘)](2.10b) is the spatial covariance of A (7)for a statistically homoge- neous medium, p’ E7 - T‘, and the overbar is a sample-time average. To arrive at (2.10) we assumed thatthe resolution volume size is large compared to the scales over which R (3) has significant value. The first term in (2.10a) is due to fluctu- The above is the instantaneous echo power P(7,) for one trans- ations in the density of scatterers while the second term is mission and N, is the number of scatterers. If scatterers within steady and comes from the mean structure of the density (i.e., a resolution volume move randomly a significant fraction of a specularscattering). If particle positionsare uncorrelated, wavelength (e.g., A/4) between successive transmissions, each R (p’) is a Dirac delta function havinga weight so that the first successive echosample V(7,) (spaced Ts)will be essentially term in ma)is equal to incoherentscatter [ 1061. If in uncorrelated. In order to make measurements of the scatterer’s addition A (7)is constant and the radial extent of resolution mean radialspeed, thetime between successive samples T, volume large compared to wavelength, steady backscatter from must be small so that contiguous echoes, at fixed delay are T,, the mean structure is negligible. correlated. Scatter in anydirection comes fromFourier components The first sum in (2.8) is a constant independent of scatterer’s having a structure wavelength A, related to radio wavelength position and is portional to the zeroth momentof the Doppler spectrum, whereas the second represents the fluctuating por- A, = A/2 sin (8,/2) (Bragg’s Law) (2.1 1) tion of theinstantaneous power andcontains the Doppler velocity information. Although the second sum can be signifi- where 8, is the angle between the incident and scattered-wave cantly larger than the first (it has N,(N, - 1)) contributions directions (e, = 71 for monostatic radar). While it is customary compared to N, for the first term) for some echo samples, its to define Bragg scatter as being from a periodic structure in average over many successive samples (i.e., sample-time average) the meandensity profile, one can definea stochastic Bragg approaches zero for spatially uniform distributionsbecause the scatter if it arises from the shape of the correlation function. sample-time average of theexponential term tends to zero. Thus the first term in (2.10a) contains the incoherent scatter The first sum is then the mean power P(7,). An accurate esti- and stochastic Bragg scatter,. Chernikov [31] has determined mate of this term is important because it relates to themeteo- the relative strengths of Bragg andincoherent scatter and rological estimates of liquid water in the resolution volume. related it to the spatial covariance of cloud liquid water con- 1) Incoherent and Bragg Scatter: Because radar meteorolo- tent. He shows conditions of side scatter where stochastic gists relate echo power to the first term in (2.8), it is impor- Bragg scatter is much larger than incoherent scatter and hence tant to be aware of the conditions under which the sample- might be important for electromagnetic interference from rain time average of the 2nd term is negligible. The first term showers. represents incoherent scatter because its power is proportional Bragg scatter is commonly ignored in studies of precipitation to the number of scatterers; the time average of the second backscatter, but it would be significant if the liquid water’s term represents coherent scatter [ 1061. If scatterers are not covariance function indicated scale sizes less than a few meters. independent, but have their positions correlated, we then have Although stochastic Bragg scatter may be negligible for precip- astatistically varying nonuniformscatterer density. Inthis itation backscatter, it is not for echoes from clear air. Indeed case it is possible to have the 2nd term of (2.8) give a time clear air radar echoes are a result of Brag scatter because A (r) DOVIAK et al.: DOPPLER WEATHER RADAR 1527 has fluctuations at scales equal to X/2 although A (F) might be -tr independent (i.e., nosteady echoes). Incoherentscatter from air molecules at microwave frequencies is usually many orders of magnitude smaller than stochastic Bragg scatter. 2) Signal Statistics: The Z and Q components of the echo's sample are random variables if scatterers' positions change in an unpredictable way. Because Z, Q are comprised of a large number of contributions, each of which is not a significant portion of the whole, we can invoke the central limit theorem [ 94, p. 2661 to deduce thatZ and Q amplitudes have a Gaussian probabilitydistribution with zeromean. Thus I and Q are jointly normal [94, p. 1821 random variables and Davenport I 1 and Root [ 38, p. 1601 have shown that Z and Q from a narrow- 0 1.0 2.0 SO band Gaussian process have zero correlations. Thus the prob- BANDWIDTH WCSEWIOTH PRODUCT B,T ability distribution of Z and Q is Fig. 2. Receiver signal power loss L, (dB) and normalized 6-dBrange width, 2r,/cz, of the resolution volume versus receiver bandwidth- pulsewidthproduct. Receiver frequency transfer is Gaussianand 1 echo pulse rectangular. Prob (I,Q) = -exp (-I2/2u2 - Q2 /2u2) (2.1 2) 2TU' where u' is the mean-square value of I (equal for Q). Because shown [95] that P(rs) = (I' + Q'), we see from (2.12) that instantaneouspower is exponentially distributed and its mean value is F = 2a'. .e: f4@, q5) sin 6 de d# = - (2.17) 8ln2 C. The Weather Radar Equation We can now relate the sample-time averaging of echo power where 8 is the 3-dB width(in radians) of the one-way pattern. P(rs) to the radar parameters and target cross section. The The range weighting function W(r) can be expressed as a prod- contribution to average echo power at the filter output from uct of a receiver loss factor 10 and a weighting function fw(r) each scatterer is whose peak is normalized to unity in order to have a form 1 analogous to theproduct of gain squared gz and pattern pi = -A; (2.13) function f4(e,#). That is 2

which can be directly expressed in terms of radar parameters and target cross section through use of (2.2). Thus the sample- time average power at range-time delay rs is where 1, is the echo power loss due to finite receiver band- width. When the receiver has a Gaussian frequency transfer and the transmitted pulse is rectangular, fw(rs, r) contains error func- where W(ri) is a range weight determined by the filter band- tions [49]. The numerical integration of (2.1 8) withthese width and transmitted pulsewidtt. functions, definedby Abramowitz and Stegun [ 1 1, gives a We nowconsider anelemental volume AV thatcontains L, = - 10 log I, dependence on the B6r product shown in Fig. many scatterers. The summation of ubi over this volume nor- 2. Nathanson and Smith [ 901 examined the exactly matched malized to AV defines the target reflectivity q filter receiver (i.e., a sin x/x filter frequencytransfer for a (2.15) rectangular pulse) and deduced I, to be 1.8 dB. We have as- sumeda Gaussian receiver frequency transfercharacteristic and rectangular pulse shape. These two chosen characteristics Replacing the sum by an integration because particles are as- often approximate those met in practice. For condition B6r = sumed closely spaced compared to the scale of the weighting 1corresponding to apractical matched filter, the numerical functions we have the following form for the weather radar integration of the function Wz(r),shows L, = 2.3 dB, or about equation 0.5 dB more than that obtained with a filter transfer perfectly matched to the rectangular pulse. Thus a practical form of the weather radar equation is

(2.19) (2.16)

In the above it is assumed f4 (e, d)W2(7) has a scale (resolution where henceforth r is to be used in place of ro. This extended volume dimensions) such that the reflectivity and attenuation form of the weather radar equation shows not only thedepen- can be considered constant over the region which contributes dency of echo Power upon commonly used radar Parameters, most to F(rs). Range ro is the distance at which W'(r) is but also its relation to receiver bandwidth. Furthermore, in maximum and is assumed much larger than the extent over the limit of B6r>> 1, I, + 1 (see Fig. 2) so (2.19) is in agree- which W'(r) has significant weight. When antennapatterns ment with the Robert-Jones radar equation [95] used widely arecircularly symmetric andwith Gaussian shape, it canbe by radar meteorologists. 1528 PROCEEDINGS OF THE IEEE, VOL. 67, NO. 11, NOVEMBER 1979

1) The ResolutionVolume: The 6-dB width of f4(e,4) (the two-way antenna pattern function) is taken to represent the angular width of the resolution volume V,. In an analo- gous manner we define the 6-dB width of fk to represent the range resolution r6 of V,. The rangewidth r6 versus B6r is also shown in Fig. 2. 2) ReflectivityFactor: Radarmeteorologists need to relate a reflectivity Q, general radar terminology forthe scatter cross I? O.*- sectiondensity, to factorsthat have meteorological signifi- cance. If scattering particles are know to be spherical and have diameters small compared to wavelength (i.e,, the Rayleigh ECHO SAMPLE SPACING 8Ts/T approximation), then we can substitute (2.1) into (2.1 5) to Fig. 3. Normalizedcorrelation of echosamples spaced along the range- obtain the relation time axis.

(2.20) chosen small (i.e., not “matched” to r) in order to observe meteorologicalevents in a larger rangeinterval with fewer where samples along the range-time axis. This approach becomes more advantageous when real-time data processing equipment (2.21) limits simultaneous observations to a few range-time samples (as is sometimes thecase for real-time Doppler spectral proces- sors) and pulsewidth cannot be increased. If B6 is “matched” is the “reflectivity factor.” Either q or Z can be used in the to r or, as in many meteorological radars, large compared to radarequation, but radarmeteorologists have optedfor 2. r-l, then dimensionally small meteorologicalevents such as When backscattering cross section can not be simply related to tornadic vortices can be missed by samples spaced further than size, e.g, scatterers having large (compared to wavelength) di- the range extent of the resolutionvolume or else smaller inter- ameters, or liquid-ice mixes, equation (2.20) is used to define vals will be observed causing longer time to interrogate the the equivalent reflectivity factor 2,. entire storm. When the transmitted pulse is rectangular and The use of reflectivity factor alone really doesnot relate receiver response Gaussian, the correlation of samples along radar echo power to meteorologically significant variables such range time is shown in Fig. 3 [ 1491. The figure shows that as rainfall rate or liquid water content because there is one when B6 is more than twice r-l, the echo sample correlation more essential ingredient that needs to be known in addition is principally controlled by pulsewidth, whereas when B6 is less to phase (i.e., IKI’). This is the dropsize distribution. When than 0.5 r-l, it is controlled by the receiver-filter 6-dB band- drop size distribution is specified by two moments, Ulbrich width. and Atlas [ 1201 have shown how rainfall rate and liquid water aloft are related to the remotely measurableparameters: re- 111. DOPPLER SPECTRAOF WEATHER ECHOES flectivity and attenuation. The Doppler spectrum is a power weighted distribution of D. Signal-to-Noise Ratio (SNR)for Weather Targets the radial velocities of the scatterers thatmostly lie within the Consistent with the previous assumptions, it can be shown resolution volume.Not only does the power weight depend that the SNR for weather targets depends uponB6r as [49] on the reflectivity of scatterers, but it also depends on the weighting given to scatterers by the antenna pattern, the trans- co r21, mitted pulsewidth, and thereceiver filter. A derivation leading SNR=- - (2.22) Y2 to a relationship between the velocity and reflectivity fields, 867 theresolution volume weighting function, and the Doppler where C, contains constants pertaining to the radar. We im- spectrum was first put forward by Synchra [ 1171. Our deriva- mediately note that if B6r is a constant, then SNR is propor- tion takes a somewhat different route but neverthelss leads to tional to the square of the transmitted pulsewidth. It can be identical final results. shown that maximum SNR is obtained as 867 + 0. Therefore, in contrast to point target measurements, we do not obtain a A. Power Weighted Dism’bution of Velocities maximum of weather SNR at B6r z 1. However, even though To begin with, we consider scatterers that produce a radial SNR increases monotonically decreases, resolution 16’ as B~ velocity field u(Fl) and areflectivity field 61). Let the worsens. One can define an optimum 867 as that value which resolution volume center be at a location 7 (Fig. 4) with the maximizes SNR for a given resolution. A more formal deriva- corresponding weighting function I@, T1): tion leads to the following conclusionconcerning weather radar design: the optimum system consists of a matched Gaus- I(?, 7;)= c1f 4(e, 4) w 2(r1)/r: (3.1) sian filter and pulse which together yield the desired resolution [134]. where C1 is a constant that can be obtained from (2.16); also q now hasscales small compared to V, dimensions. E. Correlation of Echo Samples Along Range Time We locate a surface of constant velocity U@~)and seek the Sample spacing along the range-time axis is usually chosen so total power contribution from scatterers in the velocity range thatthere are independent estimates of reflectivity and/or u to u + du. This contribution will obviously be a summation radial velocity along the beam. Both r and B6 determine the of powers from the volume between the two surfaces u and correlation of these estimates and sometimesBb is deliberately u + du. It is convenient to choose for the elemental volume DOVIAK et al.: DOPPLER WEATHER RADAR 1529

stationary weather phenomena and averaged the respective spectra, he would obtain, on the average, the result given in (3.4). Beware, thisdoes not imply that aDoppler spectrum uniquelyspecifies the velocity and reflectivityprofile in a resolution volume. On the contrary, a variety of reflectivity- velocity combinations may yield identical Doppler spectra. For calculation of the mean velocity andthe spectrum width, the normalized yn@, u) version of (3.4) is used.

Note that the integral in the denominator is the total power and can be obtained from the volume integral of (3.3). The sum of (3.3) throughout the volume is proportional to the weighted average reflectivity 5jv) Fig. 4. Parameters andthe geometr that contribute towards the weather signal power specpn; q(4) and u(3) are the reflectivity F@)=ij@)Jj%,Fl)dV (3.6) and radial velocity fields, r 's the center of resolution volume. The weighting functions in azimuth and range are indicated by dashes. where the product dsl ds2 dl where dsl and ds2 are two orthogo- Jj(il)IF, Ill dV nal arc lengths, at a point il, tangent to u(il)= const (Fig. 4). The third coordinatedl is perpendicular to the surface of u ij(i)=yxF- (3.7) d2 = lgrad u(il)1-' du. (3.2) The integral value in thedenominator of (3.7) is obtained The elemental volume contributes an increment of power in from (2.19) for weather radar parameters usually met in the velocity interval u, u + du proportional to practice. Now the mean Doppler velocity is defined as &(u) = q(il)I(i,il)lgrad u(i1)I-l dsl dsz du. (3.3) - Finally, the integral over the. surface A of constant u gives u(i)= usnF,u) du (3.8) thetotal powerin the velocity range u, u +du and is, by 1.. definition,the product of power spectrum density and du. which is a combination of reflectivity (power) and illumina- That is, tion function weighted velocity and could be quite different F(u) = s(?,u) du fromthe I(i,il) weighted velocity. Likewise the velocity spectrum width a,(r) is obtained from

=[fv(il)I(?,il)krad uF1)l-' dsl ds2 du. (3.4) ui(i)= [v - ~(?)]2$n(f:u) du. (3.9) The overbar in (3.4) denotes the mean (unnormalized)I power I: spectrum. The relationshipbetween thepoint velocities u(il) andthe This last equation is fundamentaland is worthy of more power weighted moment C(7) is obtained by substituting discussion. Firstthe area A consists of all isodop surfaces both (3.4) and (3.5) into (3.8) (surfaces of constant Doppler velocity)on which the radial velocity is a constant u, Le., it is a union of such surfaces. At each pointon the surface the reflectivity is multiplied with the corresponding weighting function. The gradient term adjusts theisodops contribution according to theirdensity (i.e., the closer the isodop surfaces are spaced the smaller is the weight applied to the spectral components in the velocity Unlike the pulsevolume averaged reflectivity,this is the interval between two isodops). average of point velocities weighted by both reflectivity and the illumination function. Similarly reduces to So far we have considered a deterministic velocity field and (3.9) a natural question to pose is how does g(i, u) relate to the {~~U~~I)~G'I)I~,~~)dV velocities of real scatterers whose relative positions change from pulse topuke. Aheuristic argumentabout this is as - -2u (I)-t (3.11) follows. Bothpoint reflectivity 7)(Tl) andpoint velocity a,2 = J~TRF~)IF,71) d V ~(71)constantly change in time due to relative random mo- tions of air (small scale turbulence).Equation (3.4) which and corresponds to a weighted deviation of velocities from the was calculated for deterministic flow is valid on the average averaged velocity. forturbulent flow as well. Thissimply means that if one The meanvelocity (3.8) depends onthe distribution of observed under identical macroscopic conditions a statistically scatterers cross section within the resolution volume and its 1530 PROCEEDINGS OF THE IEEE, VOL. 67, NO. 11, NOVEMBER 1979 weighting functions. Thus iic) cannot in general be equated relates the Doppler spectrum S(u) to thepower spectrum S(f) to aspatial mean velocity. However, if winddistribution is via the equation symmetricalabout the resolution volumecenter and reflec- tivity is uniform,then E@) can be considereda very good SCU) = 2 stn. (3.15) approximation of the true radial component of wind at the h volume center. But hydrometeor fall velocity must either be insignificant or known (see Section VI-Al). When thespectrum or the autocovariances are known, all Asimplification of (3.4) occurs when the radial velocity pertinent signal parameters can be readily obtained. field and the reflectivity are height invariant. Then the power Neither s(u) nor R(T1) are available; they must be estimated spectrum reducesto [ 132 ] from the ensemble of samples V(rs)spaced T, apart. Because radars observe any resolution volume for a finite time, one is faced with estimating at a given range m,/2 the spectrum and its parameters from a finite numberM of time samples V(nT,). We shall henceforth delete the range-timeargument r,, and V(nT,) will designatean ensemble sample atthe implicit (3.12) delay rs and TIincrements in stepsof T,. Samples V(nT,), oftenmultiplied by a weighting factor Theinner integral sums thecontributions along the line W(nT,), are Fourier transformed and the magnitudz squared s =s(xl,yl) on which u(xl,yl)is constant.Because ds = of this transformation is a power spectrum estimate S(k) com- [dx: + dy: ] ‘I2 theintegration is asurface integral with monly known as the periodogram elementarea ds dzl. Both Mxl,yl) and lgradu(xl,yl)l areindependent of z butthe illumination function may 1 M-1 depend on it. At each point XI, y1 along a strip of constant g(k) = W(nT,)V(nT,) exp (-j2nkn/M) (3.16) u, the reflectivity is multiplied with the corresponding weight- n=o l2 ingfunction. To account for contributions of other infini- tesimalstrips within theresolution volume,integration is where k, n are integers. performed along the third (z-axis) dimension. Equation (3.12) Thefinite number of time samples from which the peri- was used to compute spectra of model tornadoes and meso- odogram is computed limits velocity resolution and creates an cyclones.These compared well withactual measurements undesirable “window effect.” Namely, one may imagine that 11321, 11331 (see also Section VI-B3). thetime series extends to infinitybut is observed through It can be shown with the help of (3.1 2) that when wind a finite length window. The magnitude squared of the data shear and t) are constant across V,, the power spectrum follows window transform is referred to as the spectral window and the weighting function shape. Because Gaussian shape approxi- is significant because its convolution with the true spectrum mates well the range and angular weighting patterns, we may equals the measured spectrum. infer, when weather spectra are Gaussian, that reflectivity and An illustration of a weather signal weighted with a uniform radial velocity shear are somewhat uniform within 5. window and one with a von Hann (raised cosine) window(Fig. 5) shows considerable difference in the spectral domain espe-

B. Estimating- Doppler.- Power Spectra cially in spectral skirts. Since the vonHann window has a In order to measure the power weighted distribution of gradual transition between no data and data points, its spectral velocities, frequency analysis of V(r,) is needed and can be windowhas a less concentrated mainlobe and significantly accomplished by estimating its power spectrum. It is impor- lower sidelobes. The resulting spectrum retains these proper- tant to bear in mind that the frequency analysis is performed tiesand enables us to observeweak signals to over 40 dB along the sample-time axis for samples V(rs)at fixed rs. Thus below the mainpeak. This is verysignificant when one is we have &Crete samples V(nT,), spaced Ts apart,of a con- trying to estimate the peak winds Of tornadoes Or Other tinuow random process. Next we shall make general severeweather [ 1281within the resolutionvolume; power in statements concerning spectral analysis of continuous random spectral “skirts” dueto highvelocities is rather weakand signals. would be masked by the strong spectral peaks seen through The power spectrum is the Fourier transform of the signal’s the sidelobesunless a suitable window is applied.The ap- autocovariance function. parent lack of randomness of coefficients in the spectralskirts for the rectangularly weighted data is due to thelarger correla- tion between coefficients. This correlation is attributed to the strong spectral powers seen through the nearly constant level window sidelobes [ 1281. The example on Fig. 5 is from a tornadic circulation with where TI is a time lag. translation. In this case the broad spectrum results from high Theautocovariance function of astationary (statistics do speedcirculatory motion within the resolution volume. The not changeduring thetime of observation) signal is found envelope shape lsin x/x12 is readily apparent for the rectangu- from the timeaverage lar window (at negative velocities), and the dynamic range for spectrumcoefficients is about 30 dB. This is in contrast to , rTl2 over 45 dBof dynamic range with the von Hann window which also better defines the true spectrum and the maximum velocity (60 m * s-‘). For visualclarity an estimate of the mean power from a 5-point running averageis drawn on Fig. 5. Because V(t) has zero mean, autocorrelation and autocovari- Besides the window effect which is intimately tied to signal anceare identical [94]. Notethat conservation of power processing, there are a number of spectral artifacts due to the DOVIAK et al.: DOPPLER WEATHER RADAR 1531

DATE 052077 TIME 185350 RZIRUTH 6.1 ELEVATION 3.1 RLTITUDE 1.952 RANGE 3Ll.758 KM GATE 03 S/N 31 dB (b) Fig. 5. Power spectra of weather echoes showing statistical fluctuations in spectral estimates denoted by x’s. (a) RET signifies spectra of echo samples unweighted whereas (b) HANN signifies samples weighted by a von Hann window. Solid curves are five point running averages of spectral powers. Thisspectrum is froma small tornadothat touched down in DelCity, OK, at 35 km from the Norman radar. radar hardware.These are discussed in several references in the radar’s spherical coordinate system. Thecomponents [126], [129], [1311. 0: and 02 are related to the radar and meteorological param- eters [891 as C. Velocity Spectrum Width, Shear, and Turbulence U; = (ud0 sin e)z (3.18) The velocity spectrum width(i.e,, thesquare root of the second spectral moment about the mean velocity) is a func- tionboth of radar systemparameters such as beamwidth, (3.19) bandwidth, pulsewidth, etc., andthe meteorological param- etersthat describe thedistribution of hydrometeor density where is the two-wayhalf-power beamwidth inradians for and velocity within the resolution volume [ 5 1. An excellent an assumed circularly symmetricantenna having a Gaussian explanation andassessment of each can be found inWaldteufel’s distribution of power. The width ad, is caused by the spread work [ 1221. Relative radial motion of targets broadens the in terminal velocity of various size drops falling relative to the spectrum. For example, turbulence produces random relative air contained in V,. Lhermitte [80] has shown that for rain, radial motion of drops. Wind shearcan cause relativeradial udo is about 1.O (m - s-l) and is nearly independent of drop target motions as will differences in fall speeds of various size size distributionand rainfall rate.The elevation angle 6 is drops. There is also a contribution to spectrum width caused measured to beam center, and CY is the angular velocity of the by the beam sweeping through space (i.e., the radar does not antenna in radians per second. In terms of the usually spec- receive echoes from identical targets on successive samples). ified one-way half-power beamwidth dl This change in resolutionvolume V, locationfrom pulse to = (3.20) pulse results in a decorrelation of echo samples and conse- e2 fie,. quent increasein spectrum width 0,. The echo samples will The wind shear width term u, is composed of three contribu- be uncorrelated more quickly (independent of particle motion tions, i.e., inside V,) thefaster the antenna is rotated.Thus spectrum width increases in proportion to the antennaangular velocity. u,’ = 0,’o + u& + u& (3.21) If each of the above spectral broadening mechanisms are where each term is due toradial velocity shear along the eleva- independent of oneanother, the total velocity spectrum tion, azimuth and radial directions, respectively. Assumptions width U, can be considered as a sum of u2 contributed by behind (3.21) are that shear is constant within the resolution each [ 701. That is, volume and that the weighting function is product separable a: = (7,’ + 0: + u; + 0: (3.17) along 0, $, and r directions. Let &e, &+ be shears in the 0, Q directions anduse (3.9) to obtain where u,’ is due to shear, 0: to antenna rotation, U; to dif- ferent drop size fall speeds, and a: to turbulence. The signifi- .,‘e + u&, = (roe ke )’ + (ru+k+)’ (3.22) cance of the total width U, for weather radar design is dis- where ui and u$ are defined as second moments of the two- cussed in Sections IV and V. way antenna pattern in the indicated directions. A circularly It should be noted that (3.17) does not show a beam broad- symmetric Gaussian pattern has ening term defined by Nathanson [89] because we have elected to define shear in terms of measured radial velocities 0; = 0; =e:/16 ~n2. (3.23) 1532 PROCEEDINGS OF THE IEEE, VOL. 67, NO. 11, NOVEMBER 1979

wasless than4 m s-’ , the aircraftexperienced only light PROBABILITY THAT WIDTHEXCEEDS ORDINATEVALUE turbulencein over 50 thunderstormpenetration flights. ANT. BEAM WIDTH e, =.e* - 15- - Accurate estimates of turbulence and shear, as well as rain and .) flights safer allow shouldhail hazards, throughpro- showers duced by thunderstorms.E by duced For most situations, the combination of radar characteristics 6‘ and meteorological parameters result in negligible contribution =- 10- - c from all mechanismsexcept ke and ke shear of the radial 0 3 velocity,and turbulence [ 1071. Because thetwo angular s shearscan be determined directly from the angular depen- K denceof the mean radial velocity,the component due to c B 5- - turbulence canbe extractedfrom spectral width. A value of 0 In ke shearequal to 1 X s-’ is suggestedby Nathanson R 135krn [ 891, yet NSSL‘s Doppler velocity fields show ke shears of about1 to 2 X s-’ which typifymesocyclone regions of 0 tornadic storms. 0.1 0.5 I 2 5 IO 20 40 60 80 90 95 98 99 99.8 CUMULATIVE PROBABILITY X IV. DOPPLERMOMENT ESTIMATION Fig. 6. Cumulative probability of unbiased spectrum widths for echoes from three tornadic storms.Spectrum widths are derived from The pulsedDoppler radar should supply (for each radar

. ,- spectrum (this is an indicator of water content in the resolu- tion volume), 2) the mean Doppler velocity or the first mo- Following arguments that led to (3.22), one can show that ment of the spectrum normalized to the zeroth moment, and constant radial gradientof shear kr contributes 3) spectrum width u,, the square root of the second moment c$,= u: k: (3.24) aboutthe first of thenormalized spectrum, a measure of to the width, where 0: is the second central moment of the velocity dispersion within the resolution volume. intensity weighting function in range. For a rectangular trans- TheDoppler spectrum’s zero and second moment can be mitted pulse,Gaussian inputfilter and under “matched” estimated also withincoherent radarsemploying envelope conditions (i.e., B6r = 1) the last equation reducesto detectors [ 1041. By far the mostAused spectp moment is the zeroth or echo power estimate P(rS). The P(7J values of U$ = (0.34 k, c~/2)~. (3.25) meteorologicalinterest may easily spana range of lo9 and The width ut due to turbulence is somewhat more difficult often the choice of receiver hinges upon the cost to meet this to model. When turbulence is homogeneousand isotropic large dynamic rangerequirement. Logarithmic receivers are within the resolutionvolume, widths can betheoretically quite effective in accommodating such a large dynamic range, related to eddy dissipation rates [ 521. thus the Doppler radar may sometimes have a separate loga- Doviak et 41. [48] have made measurements of total spec- rithmicchannel for reflectivity estimation, whereas a linear trumwidths u, in severe tornadicstorms and Fig. 6data channel is well suitedfor velocity measurements. Moment show a median width value of about 4 m * s-’ and about 20 estimates utilize samples of a randomly varying signal and the percent of widths larger than 6 m * s-’ . They have deduced confidence or accuracy with which these estimates represent that these large widths are most likely due to turbulence that the true moments directly depends on the SNR, on the distri- is not homogeneous and isotropic suggesting the presence of bution of velocities withinthe resolution volume, on the energetic eddies of scale size small compared to their radar’s receiver transfer characteristics, and on the numberof samples resolution volume. For these experiments dl = 0.8’ and r = 1 processed M. Inthe caseof weather echoes, single sample ps; so weatherradars, not having betterresolution, should estimates have too large a statistical uncertainty to give mean- obtain similar width distributions in severe storms. ingfuldata interpretation. Thus a large number of echo Strauchand Frisch [ 1161 have measured widths up to 5 samples must be processed to provide the required accuracy. m s-’ in aconvective store (3cm wavelengthradar, beam- To obtain a quantitative estimate of F(r9),samples must be width 0.9’, range up to 55 km).It is significant that those averaged over a period long compared to the echo decorrela- maximumswere in the transitionregion between up and tiontime which is thereciprocal of spectrum width. The downdrafts and closeto thereflectivity core. probability density and moments of the averaged output and It is extremely important to relate widths to severe turbu- of the input power estimate can be derived from the known lence so that radarscan give reliable measureof turbulence weatherecho statistics and the receivertransfer function hazardous to aircraft.Analysis by J. T.Lee at the National [ 851. The output signal Q of radar receivers can have one of SevereStorms Laboratory (NSSL) [78] suggestsa strong many functional dependencies upon the signal applied to the connectionbetween spectral width and aircraft penetration receiver’s input. The problem is to estimate RrS)from sample measurements of turbulence. His data show that when aircraft averages of Q. The estimation is complicated because Q is not derived gust velocities exceeded 6 m - s-’, corresponding to linearly related to P(rs)(except f?r square law receiver). That moderateor severe turbulence,the spectral width exceeded is, when mean output estimates Q are used with the receiver 5 m - s-l in every case for aircraft within 1 km of the radar iransferfunction (i.e., Q versus P(T~))to obtainestimates resolutionvolume. Not all storm regions of largespectral firs), we generate biasesand have larger uncertainty in the widthproduce aircraft turbulence. Furthermore, when U, estimates P(7J than ifwe averaged P(rJ directly [ 1071, DOVIAK et al.: DOPPLER WEATHER RADAR 1533

[ 1271. Because considerable correlation mayexist from 2 sample to sample, the variance reduction achieved by averaging SNR -0dB is less than it would be for independent samples [ 801. The 2,o .6 f FFTINS degree of correlation between samples is a function of radar parameters (i.e., wavelength, PRT, beamwidth,pulsewidth, etc.) and the meteorological status (e.g., degree of turbulence, shear, etc.) in the resolutionvolume. Estimate variance can be reduced, at the expense of resolu- tion, by averaging along range time as well as sample time. Because the resolution volume’s range extent is usually small compared to its angular width, averaging over a range interval .02 .04 .06 .OB .IO .I2 .I4 Ar usually results in a more symmetrical sample volume with NORMALIZED SPECTRUM WIDTH little degradation of the spatial resolution. Range-time averag- Tv/Vo ing the output of a linear or logarithmic receiver introduces a Fig. 7. Standarddeviation of FFT and covariance PP meanvelocity systematic bias of the estimatecaused by reflectivity gradients, estimatorsat two SNR’s. Normalization parameter theNyquist velocity va and square root of number of samples Me NS signifies which will limit the maximum Ar useful for averaging [ 1001. estimate after noise power subtraction; 15 dB is after application of a Nevertheless, we have a reasonable latitude available in choos- 15-dB threshold below spectralpeak. Gaussian signal spectrum and ing Ar. white noise are assumed. Only Doppler radar can provide first spectral moment esti- mates, but at the expense of considerable signal processing. An ML unbiased estimator’ of R(T,) [ 871 The algorithmic structure of amaximum likelihood (ML) h 1M mean frequency estimator is in general unknown, but impor- R(T,) = - V[(n + l)T,I V*[nT,I (4.1) tant special cases have been documented in the literature. For M n=l instance, when a pure sinusoid is immersed in white noise, the formsthe basis of the algorithm foran estimate of mean ML algorithm calls for a bank of narrow-bandfilters; the velocity given by center frequency of a filter with maximum output is then the ii desired estimate [ 661. Discrete Fourier transform processing generates,conveniently, a bank of parallel filters but is not used in the ML sense to extract the mean frequency because weather signals have considerable bandwidth. Rather, a straight- where 2va = h/2T, is the unambiguous velocity span (Nyquist forward power weighted mean frequencyprovides the estimate. interval). The covariance argument is an unbiased estimate of Miller and Rochwarger [ 871 and Hofstetter [ 71 I have estab- the first moment for symmetrical spectra [ 141, a condition lished the autocovarianceargument as a ML estimatorfor usually satisfied by meteorological signals. certain conditions.This estimator is popularlyknown as General statistics of covariance estimates for statistically the PP algorithm. Itis ML when pulse pairs are indepen- independent sample pairs with a Gaussian signal covariance dent, i.e., when the covariance matrix of time samples is function and white noise are given by Miller and Rochwarger tridiagonal withthe same off diagonal elements. Also, as [871. Equallyspaced samples, formingcontiguous pairs in shown by Brovko [ 201, the optimality of PP extends to a first- which each sample is common to two pairs, of a time signal order Markov sequence in case the white noise is negligible. having a Gaussian spectral density and white noise are treated Second moment estimators are of necessity more complex by Berger and Groginsky [141and both correlatedand un- and, therefore, their optimum properties are more difficult to correlated pairs are treated by Zmik [ 1301. Statistical proper- establish.Estimates based onFourier methods and PP pro- ties of the covariance argument estimator are also shownin cessing have proven to beuseful, andit is knownthat for Fig. 7. Satisfactory estimation of mean velocity can be made independent PP’s, the PP width estimator is ML [71], [ 871. with input spectrum widths up to about 0.4 of the Nyquist These twomethods of spectral momentestimation are dis- velocity. However, uncertainty of theestimate increases cussed in detail in the remainder of this section. rapidly for larger widths, requiring long dwell times for quanti- tative estimates.This can be seen fromthe exponential A. Mean Velocity Estimation-Doppler First Moment growth of variance at large widths [ 1301. 1) Fast Fourier Transform: The FFT algorithm is used to evaluate the discrete Fouriertransfrom (3.16) (341. Mean velocitycalculation bythe spectral density first moment usually involves some method of noise and ground clutter removal. More commonmethods are tkreskoldingby power { (;)’ + 2 (1 - exp [-8(2na,TS/X)’ I) + orfrequency [ 1091 or noisesuppression bysubtraction of 4n3’fP.Ts1- expected noise power fromthe spectral density coefficient (4.3) [ 141. Performance of two FFT meanvelocity estimators is shown in Fig. 7 for Gaussian signal spectra and white noise. In addition to performing well with populations having wide 2) Covariance or Pulse-Pair Estimator: The complex covari- widths, the PP estimator is superior (interms of estimate ance and the spectral density constitute a Fourier transform standard deviation) to the FFT at low SNR (Fig. 7). One of pair and thus by the moment theorem, the moments of the spectral density correspond to the derivatives of the complex ’This is strictly true if successive pairs give independent estimates of R(Ts). It has notbeen shown that similar propertiesensue for cor- covariance evaluated at zero lag. related sample pairs. 1534 PROCEEDINGS OF THE IEEE, VOL. 67, NO. 11, NOVEMBER 1979

.3 r

0 .z3' . NORMALIZED TRE SPECTRUM WIDTH, U& 0.2 Q4 0.6 0.8 Fig. 8. Expectation of spectrumwidth by FFT-TH (normalized to Nyquistvelocity) versus truewidth for simulated Gaussian spectra WORYILIZED SPECTRUM WIDTH, Q,/V, and white noise; a sliding threshold equal to SNR was applied below Fig. 10. Standard deviation of width estimate by PP from contiguous the spectral peak. PP's. At large and narrow widths the SD increases rapidly. Range of normalizedinput widths where estimates are precise is fromabout 0,06 to 0.6. Because this is a perturbation solution, it takes a large number of pairs M forthe results towards the origin to bevalid. acually at zero width the standard deviation is proportional to M-d rather than M-s [ 1301.

A S is the signal powerestimate obtained from the complex video signal after subtraction of the known noise power N. Statistics of this estimator for independent PP's was examined by Rummler [ 1031. A related estimator x .03- .m .w .06 .I .2 NORMAL SPECTRUM WIDTH O;/h Fig. 9. Standarddeviation of widthestimate by FFT-TH versus true spectrum width. does not depend on spectrum shapeprovided that the width is the major advantages of this estimator, though, is the ability sufficiently smaller than the Nyquist interval. Velocity spectra to operateon pairs ofsamples as opposed tothe equally associated with weather echoes have a wide range of spectrum spaced pulse train required for straightforward FFT analysis. widths, but since it has been our experience that shapes are It is worthmentioning that the PP algorithm is aspecial mostlyGaussian, version (4.4) is recommendedbecause it case ofBurg's maximumentropy method; i.e., when the eliminates theasymptotic (i.e., M-tm) biascaused bythe weather signal is modeled as a fiit-order autoregressive pro- fiite difference approximation for the derivatives. Perturba- cess, theequation that locates the spectrum peakfrom the tion analysis on Gaussian spectra shows both (4.4) and (4.5) forward and backward prediction coefficients is the same as have identical variances and very similar numberof sample (M) (4.21, [231. dependent bias. Forthe most part, the bias is not serious because it is proportional to , as is the variance. Fig. 10 B. Spectrum Width Estimation-Second Moment About M-' illustrates the standard deviation of width (4.4) when the auto- the Mean covariance is calculatedfrom contiguous pairs. Noteagain 1) FFT Width: A spectrumwidth estimate is thesquare that coherency limits the usefulrange of input spectrum width root of thesecond moment about the meanvelocity. In to about 0.6 of the Nyquistvelocity, while noise increases practice this computation usually involves some type of noise errors at narrow widths [ 1301. .-Very similar results are ob- suppression.Thresholding thespectrum by powertends to tained if noncontiguous or independent pairs are considered. bias systematically the width estimate low since part of the signal spectrum as well as noise is removed [ 11 01. The ex- C. EROTSin Estimated Moments pected width estimate by FFT is given in Fig. 8 as a function of truewidth withGaussian spectra and a threshold at the Besides inherent uncertainties due to thestochastic character SNR below the spectrum mode. The standard deviationof the of weather signal, the moments are subject to biases generated FFT width estimate is shown in Fig.9. at various stages of processing and errors due to extraneous 2) Covariance Techniques for EstimatingWidth: Like the spurious signals. Jitterin the oscillator chainbroadens the mean frequency, the second spectral moment can be estimated spectrum and so does the clipping prior to orin the analog-to- directly without recourse to Fourier transform. For Gaussian digital converters. This andother nonlinearities generate spectraand when PP's in autocovariancecalculations are harmonics. DC offsetsand line frequency pickup bias the independent,one canshow using the results ofMiller and meanand width but canusually be controlled byproper Rochwarger thatthe following spectrumwidth estimate is design andground clutter filters [ 1101. Imbalances inthe ML [ 871 phase and amplitudes of the video signals create undesirable imagespectra. If theamplitudes arebalanced to within10 percentand the phase to betterthan 5 percent,the image peak is more than 25 dB below the signal [ 1291. DOVIAK et al.: DOPPLER WEATHER RADAR 1535

-rT T 1 L COMPLEX MULTIPLICATION I PRODUCTS TRUNCATED INDIVIDUALLYTO IO BITS

SUMMATION OF REAL AND IMAGINARY COMPONENTSIMAGINARY I I- 7 SUMSSCALED TO 9 BITS SUCH THAT LARGER, T IXLI, IS 128 dX~d255,OR 8LSB'S OF SUM I CONVERSIONBINARY TO SIGNED /T DIVISION OF SMALLERBY LARGER - 7 BITQUOTIENT

VECTORLOCATED TO WITHIN ONE OCTANT; 3 BIT VECTORLOCATED IN OCTANT ; 7 BIT ADDRESS, 8 BIT OUTPUT

VECTORLOCATED INCOMPLEX PLANE Fig. 11. A flow diagram for hardware implementation of the covari- ance mean frequency estimator.

D. Hardware Implementation of Covariance Mean Estimators Oneaspect commonto any hardwired implementation is the digital quantization. Reduction of the length of input or output words orany intermediate numericalresult has the advantage of decreasing theamount of hardwareand thus both cost and complexity. One method of implementing thecovariance mean estimator in hardware is just straightforward expansion of the algorithm (4.1) into a series of real operations (Fig. 11). To select digital processing parameters it is convenient to describe the quantiza- tion effects statistically frcm which bias and variance due to quantization can be easily modeled. The bias, for fixed point arithmetic with a 6 bit word and the usual assumption of uni- form density across the digital class, is zero for round off and digital class width2-b, for truncation. The quantization NORMALIZED TRUE VELOCITY , ?no, variance is 2-zb/l 2. Word length is established on the basis of Fig. 12. Bias of meanvelocity estimate with truncation of complex quantization variance relative to the inherent signal variance productterms to inputword length. Results apply to both full at each point in the computation.See the following example. precision calculations after the multiplier and hardwired calculator. Consider the flow diagram of Fig. 11.Two of the digital parameters which should be selected primarily from required performance (meteorological)criteria are input word length, SIGNAL RMS '6 onthe basis of expected signal dynamic range, andoutput ,03125 word lengthsuch that this quantizationstandard deviation (SD) is small compared to the requiredSD of the Doppler estimate.The output word specifies the arc tangenttable outputincrements and thus the table input increments. Internal n word lengths are then specified by these two vari- ance boundaries. For the scheme and wordlengths on Fig. 11, thetrunca- tion that dominates quantization variance occurs in the digital multiplier. The product is truncated to make the quantization SD compatible with estimate SD. Care must be taken in select- Fig. 13. Ratio of standard error of mean velocity estimate with trun- ing this product length to preserve the dynamic range of the cated product terms UN to standard deviation of estimate with pro- duct of 16 bits and full precision, ulsp. NH is the difference (num- input because a significant estimate bias (comparable to out- ber of bits) between input work length and product terms in hard- put estimate SD) at lowlevel signals appears when the product ware scheme. Np is the difference with full precision. is truncated to the input word length. An example for the Fig. 11 scheme is on Fig. 12. This bias increases sharply as the sure of the variance introduced by producttruncation is product length is truncated to the input word length. A mea- reflected in the ratio of the SD of mean frequency estimate 1536 PROCEEDINGS OF THE IEEE, VOL. 67, NO. 11, NOVEMBER 1979

... ii'i ! 70 i: !i:nI 856 321 1 "5 " "=n1 n no 0 no no n on 0 no nn 0 no n no 0 no n no 0 nn no n no n 00 n 00 n no n no no 0 nn n no n nn nn 0 no 0 no 0 nn n on nn 0 no n nn nn n on n on n no 0 no no n nn 0 no no 0 no n 00 0 no no n no 0 on no 0 no 0 no n 00 n no 0 on no 0 on no ...... n I ?I ... I- -va -0 vPP

Fig. 14. Scattergram of hardware pulse pair velocity and power spectrum derived velocity for the same resolution volumes in a storm SIN power ratios are between zero and 15 dB. withtruncated hardwire logic to the estimate SD usingfull A. Ambiguities computer precision (Fig. 13). The sharp increase in estimate To illustrate the point, Fig. 15 is a conglomerate of thunder- SD for product truncation from 10 bits to 8 bits coupled with storm celk as seen by a WSR-57 radar. One of the cells pro- the bias increase at this point indicates that a product word duced a tornado that was tracked with a nearby Doppler radar length of input wordplus twobits is agood compromise (see Section VLB2b). The Doppler radar's unambiguous range between data load and quantization variance for this scheme. extends only to 11 5 km (Fig, 16) as opposed to the 900 km An example of the performance of NSSL's hardwired PP on the WSR-57. Although the Doppler radar range is ambig- processor is shown in Fig. 14. This is a scattergram of mean uous beyond 1 15 km, the tornado producingcell at 150 km in velocities in thunderstorms estimated by the hardwired proces- this particular case is not completely overlaid (obscured) with sor versus the mean velocity at the same radar pulse volumesas echoesfrom other trips (cT,/2 intervals). Furthermore, be- derived by FFT analysis. The SNR for this data set is between cause the radar is fully coherent from pulse to pulse, velocity 0 dB and 15 dB, and the computer labels along the axes are measurement is possiblewithin unobscured regions of this the offset binary numbersused in the numerical data handling. second-tripstorm. Target range becomes ambiguous when Velocity is given on the auxiliary axis where 34 m * s-l is the true range exceeds r, = cT,/2, but these ambiguitiescan be Nyquist velocity of NSSL's Doppler radar. resolved if different trip echoes are not overlaid so as to be scrambled. V. CONSIDERATIONS FOR OBSERVATIONSOF Targetvelocities are ambiguous because we cannot distin- SEVERETHUNDERSTORMS guish between real Doppler shifts and those aliases spaced in This section examines limitations, due to range-velocity am- frequency by the pulse repetition frequency. Thus the range biguities, in pulsed Doppler radar observations of severe storms velocity product and presents techniques to mitigatethese restrictions. Radar waveformdesigns [39],formulated to removeambiguities r,u, = cX/8 (5.1) when targets are discrete and finite in number or when cross sections do not span a large dynamic range, do not work well typifies the ambiguity resolution capabilities of conventional with weather targets which are distributed quasi-continuously (i.e., uniformpulse spacing) Doppler radars. The equation over large spatial regions (tens to hundreds of kilometers), and shows the advantage of longer wavelengths, but other factors whose echo strengths can span an80dB power range. may control this choice. DOVIAK et 01.: DOPPLER WEATHER RADAR 1537

Fig. 15. WSR-57 PPI display of thunderstorm cells on a tornadic day (181704 CST April 19,1976). Gray shadings (dim, bright, black, I dim, etc.) represent dB2 levels differing by about 10 dB2 starting at dsb -A -A 40 b Ib 2L 3b 4b ;o ' Bb 17 dBZ. Range marks are 100 km apart, elevation angle = 0.0'. Un- MEAN DOFRER VELOCITT ,3 m s-' ambiguousrange is 910 km. Theboxed area outlines a tornadic storm cell whose mesocyclone signature was detected in real time by Fig. 17.Relative frequency of occurrences(percentage) of mean NSSL's Norman (NOR) Doppler radar which is nearly colocated with Dopplervelocity for threetornadic storms. Data samples are uni- the WSR-57. formlyspaced through most of the convective cell. Note the large spread of radial velocities which needsto be measured unambiguously.

tion of the exponential term because sampling was ignored) was derived by Denenberg 1401 for variance in mean velocity estimates computed from Doppler spectra. This leads one to consider the inequality h - 2 211f.7, (5.3a) 2 TS or, in termsof the unambiguous range Ch - > 2110, (5.3b) 4ra as a condition to maintain signal samplecorrelation. Width estimate variance also has an exponential increase when cor- Fig. 16. Same storm system (181635 CST) seen with the Doppler radar rectionsare made to accountfor bias [ 1301. Requirement having 115-km unambiguous range. 10-log Z brightnesscategories (5.3) means that u, limits the largest unambiguous range for (dim, bright. . .) start at 10 dBZ and increment at about IO-dBZ steps.The 10-log Z scale applies only to fmt-trip echoes. Some a given wavelength, whereas (5.1) restricts ra only if ambigui- range-overlaidechoes can berecognized by their radially elongated ties due to velocity aliases need to be suppressed by choosing shape. The box outlines the same area as in Fig. 15. Range marks are a large vu. As shown below, there are methods to resolve 20 km apart.Part of the tornadic storm is obscured by anearby (30-60-km range) storm. velocity aliases (provided (I,is sufficiently small (see Section V-D4)). Thus (5.3), a necessary condition to maintain signal sample correlation or echo coherency, must dictate r, for a B. Echo Coherency Considerations chosen h or vice versa, not (5.1). Spectrum width U, is dic- In principle one can choose Tslarge enough so no second or tated by the weather. Unless spectrum widths are less than a higher ordertrip echoes would ever be received, but this fewmeters per second, it is unlikely that 10 cm or smaller choice is limited in that signal samples spaced Ts apart must wavelength radars will eliminate (i.e., by having an ra 2 500 be correlated for precise Doppler shift measurement. Correla- km) overlaid storm echoes. tion exists when [ 51 x C. Dism'bution of Velocity in Severe Storms - >> u, Histograms of mean velocity shown in Fig. (17) illustrate 2Ts what is to be expected from severe storms when viewed by a where u" is the velocity spectrum width of echoes at range r. narrow beam (0.8') antenna [481. Some 20 000 sample Condition (5.2) merely states that Doppler width should be points from resolution volumes near ground to about 10 km much smaller thanthe Nyquistinterval X/2T,. Correlation are used, and the estimates are from PP's. The center of the decreases appreciably when 2u,Ts/X 2 (2n)-' and the variance velocity distributions (Fig. 17)are displaced relative to one in mean Doppler velocity estimates u^ increases exponentially anotherand to zero due,in part, tostorm motion. More as can be seen from (4.3). A similar formula (with the excep- importantthan the mean motion,or peakradial speeds, is 1538 PROCEEDINGS OF THE IEEE, VOL. 67, NO. 11, NOVEMBER 1979

Fig. 19. A possible spacing oxsa!ppling gates for a staggered PRF sys- tem to obtain covariance R, ,% estimates at two different lags.

power can be decreased, possibly by as much as 20 dB [441. This may make insignificant the occurrence of pesky overlaid echoes. Fig. 18. Signal diagram for orthogonally polarized (V,H) samples of a A disadvantage of spaced pairs is that a longer time will be pulse pair; Pt is transmitted power and Pr, received meanpower. required to collect sufficient number M of sample pairs so that Weather-type targets are assumed to produce the echo pattern of Pr velocity estimate variance is reduced to acceptablelimits. However, one does not need the same M as in the case of uni- the larger than 50 m s-’ spread in velocities for any of these formly spaced pairs becausesample pairs are less correlated storms. ConventionalDoppler radars may require an un- [ 1301.Furthermore, by changing the carrier frequency be- ambiguous velocity span of at least +35 m - s-l in order to tween pairs of pulses, we can space pairs closer, thus signifi- limit velocity aliasing to a few percent orless. cantly increasing the numberof sample pairs without worsening obscuration.Another disadvantage of thistechnique is that D. Extension of Maximum Unambiguous Range and Velocity ground cluttercanceler design is more comp^licated. In the absence of practical methods to simultaneously elimi- 4) Staggered PR T: A velocity estFate u is obtained crom nate range-velocity ambiguities, several schemes have been PP estimates of complexzov&riance R1 at lag T,, while u2 is devised to alleviate the aliasing problem. A list of those more derived from covariance R2 for pairs sp^aces %2. To decrease promising follows. estimate variance, covariance estimate R1(or R2)is an average I) Random Signal Radar: This radar uses random modula- of M sample pair covariance estimates (R (or R i))(Fig. 19). tion of a CW signal [29] and requires continuous calculations Because Gl, 62 are associated withdifferent Nyquist co- of the autocorrelation for each range of interest. This exten- intervals,velocity aliases may give significantly different sive signal processing, the lack of any “clear area” [39], and estimates-differences that can be attributed to the different the need for two antennas has limited the use of this radar in aliases [48] . Mean velocity $asez can be resolved so long as meteorological research. theexpected difference E[u2 - u1 ]remains unambiguous. 2) Phase Diversity: If an r,, ua combination can be found However, errors inAres%lving aliases may occur because one which will provide the desired range coverage with acceptable cannotestimateE[v2 - ul] withzerovariance [lll]. velocity aliasing, the phase of multitrip signals withequally To illustrate what can‘ be gained with staggered PRT, con- spaced transmitted pulses can be decorrelated by retention in sider the two unambiguous Nyquist velocities ual = 16 m * s-’ the localoscillator, transmitter phase informationfor only and urn = 24m - s-l. Themaximum unambiguousNyquist one system period while transmitter phase is randomly shifted (composite) velocity u, becomes f48 m * s-l. Now a single from pulse to pulse. Alternately the transmitter phase history PRF radar witha Nyquist velocity urn = +35 m - s-l has an un- may be shifted by one or two system periods in another re- ambiguous range of 100 km if wavelength is 10 cm and 50 km ceiver and measurements thus made in thesecond or third-trip if it is 5 cm. Onthe other hand, the staggered PRT radar regions. Echoes outside the selected trip are then incoherent would have the unambiguous range increased by a factor of and when overlaid one over the other only produce an effec- um/urn 1.5 or an increase in area by 2. This improvement is tive SNR decrease that does not bias mean velocity estimates. obtained at the expense of dwell time. This technique is especially suited for good magnetron trans- In summary, a staggered PRT technique can significantly in- mitters and has been used on several systems [ 83 ],[ 271. crease the unambiguous range, and the maximum unambigu- 3) Spaced Pairs: The use of spaced PP’s of orthogonally ous velocity urn can easily be made large so that velocity alias- polarized samples (Fig. 18)for weatherradars can reduce ing is of little importance. occurrence of overlaid echoes [44]. This technique has been As of now, there is no conclusive study to determine whether successfully adapted to radars measuring ionospheric motions spaced pairs (withor without polarization diversity)would [ 1231. Also, the Wave Propagation Laboratory hasimple- give Doppler radarssufficient immunity from velocity-range mented it (without orthogonal polarization diversity) in their ambiguities in severe storms and whetherthis approach is more 3-cm Doppler radar (271. (or less) advantageous than staggered PRT, phase diversity, or The first advantage of the methodis that, withT2 suffidiently random signal radar. Because the intrapair spacing Tl in the large, overlay is limited to targets in contiguousrange invervals space pair technique is forced to be small to achieve the same (cTl /2) or trips. That is, first- and second-trip targets can have unambiguous velocity as obtainedwith staggered PRT, we overlaid echoes as well as second- and third-trip targets, etc., undoubtedly expect a larger occurrence of scrambled echoes but there is no echo overlay for targets in Tit- and third-trip for thespaced pair method. intervals or second and fourth, etc. Second, all overlaid echoes 5) Dual-Sampling Technique: Even thoughobscuration of are incoherent if T2 is sufficiently large (i.e., T2 >>h/4~,). Doppler signatures by overlaid multiple trip echoes might not Thus they do not bias velocityestimates and only serve to be eliminated, thereis reason to present the observer a velocity decrease the effective SNR. When the pulses of apair are data field wherein range to datumis unambiguous and velocity orthogonally polarized, (polarization diversity), overlaid echo values are credible (i.e,, are notin error due to scrambled DO PPLER WEATHERDOVIAK et d.: DOPPLER RADAR 1539

trip echoes is small if ru > 130 km, butthere could be a significant improvement in decreasing obscuration of second- trip storms. In any case, an advantage of incorporating a stag- gered PRT is the automaticde-aliasing of velocity estimates.

VI. OFSERVATION OF WEATHER The great utility of storm observation with centimeter wave- WdL SAWLIffi %HEM AAA length pulsed Doppler radar derives from its capacity to map Fig. 20. Dual sampling technique, whereR, ,rZ, ,R,, . . . ,are covariance reflectivity (q) and mean radial velocity inside the storm’s measurements (at equal lags) whose average is used to derive mean V Dopplervelocity estimates. We depict only fust and second-trip shield of clouds, A three-dimensional picture of a single storm echoes and assume T2 = 2T1. Clearing period T2 removes multiple- takes about 2 to 5 min of data collection time not only be- trip echoes from reflectivity estimation in a contiguous T2 interval. cause of antenna rotation limitations, but also because a large number of echoes from each resolution volume needs to be echoes). This can be accomplished by taking echo reflectivity processed in order to reduce the statistical uncertainty in the samples during an interval T2 sufficiently long to remove, for q and iT estimates.Although stormstructure can change practical purposes, all overlaid echoes and have this sampling significantly during this’ periodwith distortion of the radar period interlaced with another whose PRT is short enough to image of thetrue reflectivity and velocityfields, highly allow coherent measurements for velocity estimates (Fig. 20). significant achievements have been madein depicting the By interlacing the velocity estimation period MTl with one for structure andevolution of the thunderstorm. But the meterologically interesting variables are not nor reflectivity (T2),we can have nearly colocated resolution - 8 volumes for velocity and reflectivitymeasurements. The u, but parameters such as rainfall rate (on the ground) and figure shows one block of samples that contain M = 3 covari- wind. Pulsed Dopplerradar mostoften measures the radial ance estimates and one reflectivity estimate(for eachrange speed of hydrometeors, not air, and in certain situations such bin). To reduce velocity and reflectivity estimate variance, we as vertically directed beams, thesespeeds can differ signifi- need to average covariance and reflectivityestimates for K cantly from the radial component of wind. Likewise, surface blocks. In order to have all n-tripechoes samples (n = 2 in rainfall rate estimates are not easily related to q, andoften Fig. 20) in one Tl period,sampling should start in the nTl radar reflectivitymeasurements are supplemented by surface intervalbecause thenth multiple-trip echo wiU notappear rain gages [181, [371. until then. Dual samplingprovides echo reflectivitymeasure Incoherent radars map q but, if the radar’s resolution volume without range ambiguities so we can determine, through com- is sufficiently small andreflectivity estimates are accurate, parison of echo powers, those velocity data that are signifi- these radars can track prominent reflectivity structures to map cantlycontaminated by scrambledmultiple-trip echoes and winds that steer cells [ 351. However, Doppler radar measures, eliminate them from observation. Furthermore, we can assign practically instantaneously, velocities in each resolution correct ranges tothe surviving valid velocity data. Sucha volume and hence can provide better resolution of the velocity dual-sampling system is in operation at NSSL (N= 7, K = 8, field. ?‘I = 768 ps, T2= 4T1)and velocityfields displayed in real time are not rangeambiguous. The mostpersistent obscura- A. Dual Doppler Radar tion to plague the dual sampling radar is caused byground A single Doppler radar mapsa field of velocities that are clutter echoes overlaid ontothe second trip (i.e., ground directed toward (or away from) the radar. A second Doppler clutter seen just beyond ru) as well as ground clutter within radar, spaced far from the first, produces a field of different the first trip: Ground clutter obscurationcan be lessened with radial velocities because the true velocities are projected on cancelers and by displacing (through changes in Tl) the second- different radials. Thetwo radialvelocity fields can be vec- trip ground clutterring away from the stormof interest; other- tonally synthesized to retrieve the two-dimensional velocities wise, we have a permanent ring of about 10-1 5 km in range in the plane containing the radials [ 21. It is customary to ac- wherein the Doppler radar is blinded. complish the synthesis on common grid points to which radar A closely related method of increasing the range to which data are interpolated. Radial velocities in each of the radar’s reflectivity can be resolved unambiguously is to transmit co- resolution volumes surrounding a grid point are not measured herent signals at two different frequencies 02,ol each at dif- simultaneously but are separated in time up to the few min- ferent PRT’s so that simultaneous reception is possible. The utes required for each radar to scan the common volume. The long PRT yields large unambiguous range for reflectivity esti- respective resolution volumes are also usually quite different mates while the high PRT is used for velocity estimation. This in size and orientation. technique and its signal processing are analogous to the dual- Nevertheless, useful estimates of wind can be made on scales sampling technique described above: its advantage is a reduc- of air motion large compared to the biggest resolution volume tion of the acquisition time and possibility for better canceler dimensions if the velocity field is nearly preserved over the design. periodrequired fordata collection.Targets such as water The dual-sampling mode can accommodate a staggered PRF drops having small mass quickly respond to horizontal wind during the velocity estimation period to allow increase both forcesand faithfully tracethe wind.Stackpole [ 1141has in ru and u, at the expense of increased data acquisition time shown that, for energy spectrum of wind scales following a (i.e., less scans-about 30 percent less-per observation period) -5/3 law to at least 500 m, more than 90 percent of the rms for given velocity estimate accuracy.These might be impor- wind fluctuations are acquired by the drops if their diameters tant gains for a dual-sampling radar, but an issue is whether are less than 3 mm. When radar beams are at low elevation obscuration is significantly decreased. Studies with simplified angles, target terminal velocities (i.e., the steady-state vertical storm models suggest that the decrease in obscuration of fmt- velocity relative to the air) give negligible error in the radial 1540 PROCEEDINGS OF THE IEEE, VOL. 67, NO. 11, NOVEMBER 1979

4 or a Fig. 21. Cylindrical coordinate system used fordpl ah2data analysis. The radars are located at points 1 and 2, and a,,, as, aa are the unit normals defmingdirection of thethree orthogonal velocity com- ponents.

OKINGFISHER KLGUTHME

a MINCO 0 TUTTLE

0-m 0- lp , 2? , .prmj Fig. 22. Map of central Oklahoma with reflectivity field (shaded areas) and winds relative to the ground at a height of 1 km. Velocity scale vector is shownalong eastern comerof box. This velocity is the maximumat this height.Reflectivity contours (dashed lines) are labeled as log Z (from Ray et ul. [ 971). wind component.At high elevation angles these velocities whereare themean Doppler target ve1ocities.measured by need to be considered. radars 1, 2 at data points. To estimate Ut for each resolution 1) Reconstruction of Wind Fields: The dual Doppler radar volume, one could use the empirical expression [ 71 technique to derive quasi-horizontal winds was first illustrated by Lhermitte [ 821 and later extended by Frisch et al. [ 531 to Ut= 2.65Z0."4 (F).s-l][m (6.2) display all threeCartesian wind components. Wind field de- termination is greatly simplified if the synthesis is performed where theparenthetical term is a correction suggested by in cylindrical coordinates with an axis being the line connect- Foote and duToit [ 5 11 to account for height dependent air ingthe two radars. That is, radial velocities atdata points density y, and Z is the reflectivity factor. This relation well (centers of resolutionvolumes) are interpolated to nearby representsexperimental data over a large range of Z (Le., grid pointson planes having a common axis (the COPLAN 1 < Z < lo5 nun6 m-3) for regions of liquid water, but large technique) [82]. Cartesian wind components can be derived errors, up to several meters per second, in Ut estimates can be from these synthesized cylindrical components. Although one caused byerroneously relatingregions of hail with a Ut, Z could solve directly for Cartesian wind components, this relationappropriate for liquidwater. Usually there is little necessitatesa solution of an inhomogeneous,hyperbolic orno information to identify these regions uniquely,and partial differential equation to derive vertical wind [ 21. errorsin vertical wind wz can result. However, ithas been The cylindrical coordinate system is illustratedin Fig. 21. shown for typical arrangements of storms relative to the two The mean Dopplervelocity needs to be correctedfor the radar placement, that the error in wz is significantly smaller scatterers terminalvelocity relative to the air in which they than errors inUt [46]. are located. The corrected radial velocity estimate is The estimated radial velocities u~,~of the air can be inter- polated to uniformly spaced grid points in planes at angle 01 u~,~= + Ut sin e (6.1) to thehorizontal surface containingthe baseline. Interpola- DO PPLER WEATHERDOVIAK et al.: DOPPLER RADAR 1541 tion fiters the data andreduces variance [46]. The cylindrical factor Z within a resolution volume, 3) use of incorrect ut, wind components in the p,s plane are related to Fl, Fz, as Z relationship, 4) inaccuracies in resolution volume location, 5) increase of vertical velocity variance with height owing to (s + d)rlFl + (s - d)r2Fz wp = (6.3) error in derivative estimatesin thecontinuity equation, 6) 2 dp nonstationarity of the storm duringa data collection scan, and rlFl - rlFl 7) echoes received through sidelobes that contaminate signals ws = (6.4) associatedwith the resolutionvolume. How these errors 2d affect the estimates of horizontal and vertical wind is dis- where El ,2 are the interpolated Doppler velocities of air. cussed in [ 461. The wind component wa, normal to the plane, is obtained by solving the continuity equation incylindrical coordinates: B. Observations with a Single Doppler Radar Although the Doppler radar measures only the radial wind component,its spatial distribution can signify important meteorologicalevents such as tornado cyclones. Moreover, with an appropriateboundary condition [ 971. The mass high straight winds, if not across the beam, can be measured density y is given by as well as turbulent regions. Thus a single Dopplerradar offers good promise for severe weather warning and in our Y = YO exp[-gMp sin a/(RT)I(6.6) view will mostlikely become theoperational tool ofthe National Weather Service in the near future. and g is the gravitational constant (9.8 m . s-~),M the mean molecular weight of air (29 gmol-'), T the absolute tempera- I) Linear Wind Measurements- Velocity Azimuth Display: When the antenna beam is scanned in azimuth @ while eleva- ture (K),and R the universal gas constant (8.314 J * mol-' * K-'). Appropriate values of yo, T can be obtained from sur- tion angle 8 is fixed, theradial velocity has the q3 dependence face site and upper air soundings. The Cartesian components ur = sin e + Uh cos e cos ($ - 6 - n) (6.7) of wind can then easily be obtained from wp, ws, wa. 2) Observation of a Tornadic Storm: The first dual Doppler where 6 is the wind direction, w the vertical velocity, and uh radar observations of a tornadic storm were made on April 20, the horizontal speed of tracers in the resolution volume. The 1974, with NSSL's IO-cm radars. These radars provide a large w and uh velocities are readily computed from data in a veloc- unambiguous range andvelocity capability well suited for ity azimuth display (VAD) [84] under the assumption that air observation of the large and severe thunderstormsthat fre- is in pure translation. (The VAD is a display of radial velocity quent the high plains of the United States. at a single range location versus azimuth.) Then (6.7) hasa Fig. 22 locates thetwo radars,one at Cimarron airfield sinusoidal dependence on @;thus amplitude and phase of the (CIM), OklahomaCity and the second (NRO)at NSSL's sine curve are measures of uh and 6 at the height r sin 8 of headquarters,Norman, OK. It also shows thehorizontal the sampling circle. Vertical motion produces a dc offset of wind field synthesized using a slight modification of the the sine wave. However,when wind is nothorizontally above outlined scheme [97].3 Streamlines have been drawn homogeneous, equation (6.7) is no longer purely sinusoidal. in addition to velocity vectors, whose length is proportional Caton[281 showed how divergence can be determined to wind speed. The curvature inthe streamlinesshows ap- from VAD data. Browning and Wexler [22] carried the preciablelocal vorticity in the region nearthe grid point analyses even further by assuming the wind field waswell (30.0, 24.0). representedby a linear velocity field [68, p.1981 over the To view the storm's kinematic structure at several altitudes circle of measurement. Under thisassumption there are the mean wind at eachheight is subtractedfrom the wind four basic fields of motion that convey air: pure translation, vectorat each grid point. This perturbation velocity field vortical,divergent, and deformative. Fourier analysis of is displayed at two heights in Fig. 23(a). Cyclonic circulation (6.7) for linear wind reveals that of these four motions only is apparentat the grid point (30, 24), wherevorticity was the vortical onecannot be measuredby the VAD method. noticed in Fig. 22. Inflow into the tornadic cyclone is shown The average component of (6.7)is proportional to mean at an altitude of 3 km, and divergence and outflow are ap- horizontal divergence DIV uh plus mean E, that is parent at 7-km height. High reflectivity factor (60-dBZ) re- gions are located on the downwind side. These velocity r Ur d@= - cos 8 (DIV uh ) + sin 8 (6.8) fields arein general agreement withpresent storm models, 2 W particularly in the weak echo region where both imply a strong updraftnortheast of the circulation(Fig. 23(b), X=33, where t3 is the average of verticalvelocity on the sampling Y = 27.0 km). A downdraft (Fig. 23(b), X = 33, Y = 22.5 km) circle of radius r. The first harmonic component gives uh and is found to thesouth.west of the circulation. 6, andthe second measures deformation. By inserting the 3) Errors in Synthesized Wind Fields: The wind fields de- mass continuityequation into (6.8), we canthen solve for rived from dual Doppler radar measurement have errors that vertical wind if we have an estimate of target terminal veloc- arise from several sources. Some of these are: 1) variance in ity averaged over the circle. If the target is refractive index the mean Doppler velocity and Z estimates due to the statisti- fluctuations (see Section VI-C), then clearly terminal velocity cal nature of the weather echo, 2) nonuniform reflectivity is zero. Thus a single Doppler radar can measure thethree components of wind averaged over a sampling circle of radius r and produce a vertical profiie of wind. 'Terminal velocitycorrections were obtained from interpolated re- flectivityfactor values and a slightlydifferent or, Z relationship was 2) Severe Storm CycloneObservations and Presentations: used. Because the radar maps the distribution of Doppler velocity 1542 PROCEEDINGS OF THE IEEE, VOL. 67, NO. 11, NOVEMBER 1979

3 KM - 20.0 n/s

0 Ilrllrr.lllllllrlrlrllllll'll1 0 3 6 3 12 15 1E 21 Fl 27 30 33 36 33 92

*X

(b) Fig. 23. (a) Perturbation wind and reflectivity factor (log 2) at 2 dif- ferent altitudes. Mean wind at each altitude is shown in lower right comer.Velocity scale is in upper right comer. Distances are in km from an origin s = -1 0, p = 15 km (s = 0 is the midpoint of the base line) and abscissa is parallel to 8. (b) reflectivity and velocity fields in the x, z plane for two planes. The mean horizontal velocity in each vertical plane was removed and is indicated in the upper left of each plane. Reflectivitycontours are labeled as log Z (from Ray e? aL [971). insidethe storm, significantmeteorological events (unseen tions connected with straight lines (Fig. 24). This pattern has from outside) such as tornado cyclones should produce tell- been observed many times, and an example is shown in Fig. talesignatures. Donaldson [42] stipulatedcriteria whereby 25(a)for a tornadic storm that did considerable damage to avortex can beidentified from single radarobservations. Stillwater, OK, in 1975 [21], [24], [ 1331. Fig. 25(b) shows Briefly, there must be a localized region of persistently high contours of Doppler u, for the same storm, and we immedi- 25X lom3s-l azimuthalshear (i.e., thevelocity gradient ately see the striking correlation of large u, withsignificant along an arc at constant range) that has a vertical extent equal radialvelocity shear. Regions of large u, may also indicate to orlarger than its diameter. the presence of strong turbulence. It can be shown that nontranslating cyclones have isodops a) PPI weather display: Reflectivityfactor is nowrou- forminga symmetric couplet of closed contourswith equal tinelydisplayed by the National Weather Service radarson number of isodopsencircling positive and negative velocity the PPI scope and by some television stations on a color PPI maxima(Fig. 24). If theinner portion of thevortex is a display. While reflectivity cannot be reliably used for tornado solidly rotating core, its tangential velocity linearly increases detection, it has proved valuable for hydrological studies and with radius to a maximum. Outside this maximum, the veloc- severe weather warnings. Those warnings are primarily based ity decreases (roughly) inversely with the radius. The isodop on reflectivity values, stormtop heights, and sometimes on contours of such a combined Rankine vortex are circular sec- circulatory features or hook echoes. DOVIAK et al.: DOPPLER WEATHER RADAR 1543

\ 0-0.2 CIRCLE OF MAXIMUM TANGENTIAL WIND 70.4

ANTENNA PATTERN w. el * 2rt TO RADAR Fig. 24. Plan view of idealized isodop pattern for a stationary modified Rankine vortex located at range large compared to vortex diameter. 11 is Doppler velocity normalized to peak tangential wind. Radar is locatedtowards the bottom of the figure. Resolution volume, antennaand range weighting functions are depicted. The angular tilt (I determines radial inflow (u < 0) or outflow (a > 0).

Because PPI scopes are commonly used with radars, it is no surprise that weather researchers have begun displaying veloc- ity fields onthem. One of the earlydisplays was obtained when a mean velocity processor of the National Severe Storms Laboratory was mated to the PPI [ 1081. In this case bright- ness level signifies aparticular radial velocity category (e.g., n Fig. 26(b)). If the wind and reflectivity were uniform over a full360" of the scannedspace, the display would show in- tervals of radial velocity as angular sectors of different bright- ness for each velocity category. Strobed brightness as a func- tion of azimuth is used to differentiate positive from negative velocities (unstrobed brightness). NSSL's real-time Doppler velocity processor was operational during the Spring of 1973, generating the first velocitycon- tour maps of mesocyclonesignatures. The Marlow tornadic storm on June 4 produced aparticularly large mesocyclone, and its reflectivity and isodop signatures ate clearly shown in Fig. 26. This storm's reflectivity structureexhibits a hook echofeature suggesting mesocyclonic circulation, severe weather, and tornadoes. Although the hook echois identified usually as an appendage to the storm, it may, as in this case, be found within the storm. Average storm height during data collection was 16 km; motions of signature and storm were equal (280°j13 m . s-'), with both considerably to the right @) of meanenvironmental wind (250°/12.5 m * s-'). East of Fig. 25. (a) Doppler velocity field for the Stillwater storm at 1.5 km the highreflectivity core, alow reflectivity notchextends height. Grid spacing is 4fO m. Velocities are in m-s-' and contours well intothe storm. Storm motion relative to theradar is (iiodops)are in 5 m.s- steps.The mesocyclone is centeredat slight, and therefore, isodop patterns indicate radial velocity Y=97 km, X=36 km. The other shear region from 94 km north, 40 km east to thebottom of thefield was identifiedfrom dual relative to the storm. The nearly symmetric isodop signature Doppler data to be the low-level boundary (gust front) between storm indicates circularly symmetric cyclonic rotationconsistent inflow to the east of the shear line and outflowto the west. (b) Con- toursof constant spectrum width at 1.5 km aboveground. Values with that inferred from the reflectivity pattern. equal to or larger than 6 m.s-' in steps of 2 m.s-l are displayed for Asignature patternfor circularly symmetric convergence visual clarity.Large widths farther north are where the tornado is similar to thevortex pattern but rotated clockwiseby mesocyclone formed; the other region of large width is embedded in the gust front. Interpolation grids are spaced at 400 m. 90' [451.The reflectivityspiral suggests convergence,a view supportedby the clockwise angular displacement of isodop maxima about the vortex center (190°, 82 km). Color tion volume, and large second moment magnitudes have been displays of reflectivity, velocity and spectrum width allows, judgedpotentially important as tornado signatures.Suffi- in real time,an easier quantitative evaluation andbetter ciently intense azimuthal shear is the feature used to locate resolution of cyclones. the tornado vortex signature (TVS); this has been correlated b) The multimoment Doppler display: Large changes in with many tornadoes [ 2 1 I. Large spectrum width is another the first Doppler moment from resolution volume to resolu- distinguishing feature of a tornado predicted by Atlas [4] and 1544 PROCEEDINGS OF THE IEEE, VOL. 67, NO. 11, NOVEMBER 1979

(a) (b) Fig. 26. Storm reflectivity (a) and isodops(b) displayed on PPI at 21 15 CST. The elevation angle is 1.9', range marks correspond to 60, 80, 100 km. Reflectivityfactor categories are: dim (<21 dBZ),bright (21-31), black (31-44), dim (44-57), and bright (>57 dBZ).Velo- citycategories are dim <13 ms-'), bright (13-21), andbri&test (>21).' Positive radial velocities are angularly strobed in brightness. Mesocyclone signature b between 194'-203' and 73-90 km.

Fig. 27. Themultimoment Doppler display of a mesocyclone. Each arrowcontains information of the 3 principalDoppler spectrum moments for a resolution volume. For interpretation of arrows see insert in upper right comer (arrow length is proportional to received power,arrow direction to velocity and arrowhead size to Doppler spectrumwidth). Abscissa is azimuthand ordinate scale denotes range (km) from radar. Housekeeping information is at top of screen.

Lhermitte [ 8 1] (see Fig. 5). The various signatures of meso- direction and its zero position. A horizontal arrow pointing cyclones and tornadoes maybe revealed at once by using three leftcorresponds to the Nyquist velocity (k34 m * s-'). As separate displays or, as described below, with a single display the velocity increases beyond k34 m . s-' , the arrow rotates called a multimoment Doppler display (Fig. 27). smoothly through the Nyquist limits and appears as a lower To presentsimultaneously thethree principalDoppler velocity of opposite sign (e.g., 38 m * s-' appears as -30 moments for each resolution volume, a field of arrows is dis- m * s-I). In these displays, the radar is always towardthe playedwhere arrow length is proportional to the logarithm bottom of the figure so that arrows in the upper half of the of echo power,arrow direction to velocity, and arrowhead circle denote flowaway fromthe observer, whereas arrows size to spectrum width (see insert on Fig. 27 and [26]). Zero in the lower half denote flow towardsthe radar. Thusthe velocity is a horizontal arrow pointing right andnonzero field of arrows in Fig. 27 illustrates quite nicely the signature velocities are proportional tothe angle betweenthe arrow of circulation(centered at 187'-Az 70-km range) or con- DOVIAK et al.: DOPPLER1545 WEATHER RADAR

cording to their reflectivity, isodopdensity, and antenna pattern illumination (see Fig. 24). Only those targets whose spectral power is above the receiver noise level may be posi- tively identified. However, it has been our experience, that tornadoes in aresolution volume offer enough reflectivity, due to debris andhydrometeors, that a large velocity span can be observed. Presently it is not known whether tracers moving at thepeak tornado wind speed can be resolved. When centered on the’beam axis, the Rankine vortex model predicts a bimodal spectrum which was verified experimentally several times 11321, [ 1331. Shownon Fig. 29 are Rankine model vortex spectra matched to data by a least squares fit. Theexamples of spectra are fromthe Stillwater maxitor- nado. Deduced from the fitting are maximum velocity of 92 m * s-l and tornado diametex of 300 m. The deduced maxi- mum velocity is larger that the radar’s unambiguous velocity (k34.4 m s-’); therefore, aliasing was introducedin the .-17:53 m f model spectrum and the estimatesare indirect. Two spectra closest to the tornado were simultaneously least

CYCUNE SIGNATURE squares fitted. Simulated model vortex spectra andreal spectra MDIUUOE.TORNADO PAW show very good agreement not only for two where the fit was *-lolvn- I-Srni.+ made but also for adjacent gate locations (Fig. 29). Resolution volumes corresponding to any of these simulated spectra are Fig. 28. Mesocyclonesignature track for the Falconhead tornado showing damage path location relative to the cyclone signature center assumed to have uniform reflectivity within the volume. Dif- (April 19, 1976). The reflectivity field of the storms for that day is ferences in echo power from each resolution volume are ac- shown on Figs. 15 and 16, where the boxed areas contain the storm counted for by forcing each simulated spectrum to have power. that produced the only tornado on this day. equal to its matching data spectrum. Asymmetry of spectral peaks (Az 21.1’; range 104.136 km) about zero velocity sug- gests that targets were centrifuged outward with a velocity of vergence (188’ Az and 75-km range). The r$al component 13m.s-’. of stormmotion 10 m s-l from 225 ) has been sub- (SM: In view of the variety of displays and signatures associated tracted from all velocities. with tornadoes, it is natural to ask which technique is most A tremendous advantage is obtainedwith Doppler radar promising fortheir detection. A project was established in- because it can sortout among many stormsthe ones that volving theEnvironmental Research Laboratories,National have intensecirculation andhence potential for tornado Weather Service, Weather Service, and Air Force Geophys- development. Fig. 15 showsa large stom systemcomposed Air ical Laboratory to conduct experiments that should provide of many individual convective The multimoment display cells. some answers. Operations were conducted during the Spring depicts the principal moments in a sector of space that can of 77 and 78. It became apparent that mesocyclone circula- be placed over any storm so that the principal moments can tions showed very nicely on the color display of radial veloc- be simultaneously examined for evidence of significant mete- ities. Multimoment display inconjunction with the color orological phenomena. Each storm can be systematically in- display proved most suitable for TVS recognition. Using terrogated and forthe example shown in Fig. 15only the criteria discussed in Brown et al. [ 211, the project scientists storm outlined by the box produced a tornado. This tornado’s were able to detect a number of tornadoes [ 25 1. As a matter mesocyclonesignature was trackedfor almost anhour, and of fact, all tornadoes that occurred within a range less than Fig. 28 shows theposition of the signature relative to the 115 km (the radar’s unambiguous range) were detected. The damage path. Even thoughthe storm was in the Doppler average lead time was about 20 min. It was established that radar’s secondtrip, the beamwidth was sufficientlysmall circulation starts at mid levels (6-8 km) and works its way (0.8’) for tracking at ranges to 170 km when the signature towardthe ground. At further ranges signatures of small was fiit noticed. tornadoes are lost due to poor resolution; however, large de- Although the Doppler radar had an unambiguous range of structive tornadoes were detectedup to 240km using the 115 km on this day, storm distribution was such that none beam of NSSL’s Dopplerradar. Spectrum widthand were range-overlaid onto mesocyclone, thus allowing an 0.8O this shearalone have not been reliable indicators of tornadoes unobscured measurement of its velocity signature. The over- since turbulent areas in storms exhibit large widths and could laying ofstorms due to small unambiguous range (see Sec- be easily mistaken for tornadoes. Based on those experiences, tion V) associated with Doppler radars can result in obscura- tion of signatures. it is believed that an operationaltornado detection system will involve interaction between humanoperators and an 3) DopplerSpectra of Tornadoes: In1961 Smith and Holmes reported a tornadospectrum that was obtained automated scheme whereby the velocity pattern of a tornado witha CW Doppler radar [ 1121. It was twelve years later cyclone is recognized. that a tornado was first observed by a pulsed Doppler radar [132]. C. Pulsed Doppler Observation of Clear Air Wind Fields Radar views principally thatportion of circulation which Whenever turbulence mixes air in which there aregradients lies within the resolution volume so that tracers moving with of potential temperature and water vapor density, the turbu- the same velocities contribute to spectrum components ac- lence causes spatial fluctuations in the refractive index n. The 1546 PROCEEDINGS OF THE IEEE, VOL. 67, NO. 11, NOVEMBER 1979

lpm n.1 M 9. lpvumld M 1.

M 9.

6 *I

Fig. 29. Spectra from three consecutive azimuthal and range locations; Stillwater tornado. Dots show spectrum estimates from recorded time series data weighted with a von Hann window. Solid lines are threapoint running averages. Dashed lines are simulated spectra. The mean square difference between data and simulated spectra is simultaneously minimized for two spectra (Az-21.1' and range of 103.5 and 104.1 km). Resolution volume depth is 150 m, range gate spacing 600 m, and antenna beamwidth 0.8'. The tornado is located between the two upper middle gates. Tornado parameters obtained from these two fitted spectra are used to compute the remaining 7 simulated spectra. Height above ground for these spectra is 640 m. fluctuations are small (e.g., one part in a million). Neverthe- lated toreflectivity 17 as [ 631 less,sensitive microwave radars detect the very faint echoes returned from these irregularitiesin what otherwise (without turbulence) would be a smoothly changing n with negligible backscatter. Thus if we know the structure constantof refractivity, we can, Fluctuations in temperature,humidity, and pressure need through use of (6.11) and (2.19), determine the echo power to be described in a statistical manner. Thus correlation and scattered by refractive index irregularities. its Fourier transform, the power spectrum, are used to charac- Echoes from clear air have been seen almost from the incep- terize the spatial variability of Tatarski [ 1181 related the n. tion of radarobservations. These angel echoes were first velocity spectrum of turbulence scales to the correlation and mystifying, but often were actually associated with birds and spectrum of refractivity scales. Furthermore, Tatarski demon- insects. Clear air echoes, not related to any visible object in strated that, although there is a hierarchy of scales that pre- the atmosphere, were conclusively proven to emanate from re- vail in turbulentflow, those scales equal to A/2 contribute fractive indexfluctuations through useof multiwavelength most to backscatter. radars at Wallops Island [ 631. Simultaneous measurements of The parameter needed to obtain the backscatter cross sec- refractive index fluctuations and reflectivity corroborated this tion from statistically inhomogeneous media is the structure finding [ 1051. Of course, radar studies were preceded by an function DA,,defmed as enormous amount of measurement using troposphericscatter communication links which often depend upon clear air re- DA,, ([n(r + Ar) - n(r)12)(6.9) fractive index fluctuations to providereliable wide-band cir- where ( ) denotesensemble average. Whenever DA,, is as- cuitsbetween distance points [981 (see also aspecial PRO- sumed independent of r (i.e., the statistical properties of An CEEDINGS OF THE IRE, ScatterPropagation Issue, Oct. are spatially uniform) we refer to the refractivity fluctuations 1955). as being ZOCQZZY homogeneous. Tatarski [ 1181 has shown for In the 1960's ultrasensitive incoherent radars were used to scales within the inertial subrange of atmospheric turbulence remotelydetect and resolve clear air atmosphericstructure (i.e., scales from a fewmillimeters to tensor sometimes a and these studies are well reviewed by Hardy and Katz [64]. few hundred meters) that These radars 'showed meteorological phenomena such as con- vective thermals [73],[75], sea andland breeze [861, and DA,,= C:(Ar)'I3 (6.10) Kelvin-Helmholtz waves [ 691. Doppler processing of coherentradar echoes can improve where Ci is the refractive index structure constant and is re- target detection by at least an order of magnitude [671 and DOVIAK et al.: DOPPLER WEATHER RADAR 1547 hence medium resolution weather radars could have a detec- tioncapability matching that often associated with large aperture(18-27-m diameter) antennas used withincoherent radars. Furthermore,a coherent radar provides a way in whichground clutter can be distinguishedfrom moving at- mospheric targets and allows data acquisition at closer ranges, therebytaking advantage of the r-2 dependencein echo power. Chadwick et a2. [30] have monitored Ci values inthe 015 w planetary boundary layer and have concluded after one year I of observation that winds can always be measured to several hundredmeters height with moderately sensitiveradars. 0 Clear air wind measurement has practical significance because K)4S 10-17 lo-lS lo-18 10-1' pulsed Dopplerradars under development by the FAA also c,' ( m-'") could measure wind shear hazards near airports forall weather Fig. 30. Rofde of average structure constant cn2 due to temperature conditions [ 791. fluctuations Terms height h. Smooth lies are minimum radar detectable c, at range R using NSSL's Doppler system as configured Harrold and Browning [65] found that radar can delineate on 4/19/76 (dashed line) and for system if reconfigured (solid line)to the upper limit of convection, prior to precipitation, showing provide lower systemnoise temperature and higher transmitted that it is deeper in some regions than others. Some of these power. (From Ochs and Lawrence [ 931 .) areas of deepconvection persist for several hours and, if showersdevelop, they occur within, andonly within such regions.Important economic advantage can be achieved if fractive indexstructure constant was consistendy above radars can serve the dual purpose of locating weather hazards lo-' m-2/3 over land and ocean within the measured height to aircraft and predicting the locationof incipient showers. interval. A sample profie of averaged Ci due to temperature Mappingboundary-layer wind over large areashas double fluctuations over land in November 1971 is shown in Fig. 30. significancefor severe studies because 1) it allows early ob A bistatic radar was used to resolve weak scatter in the upper servation of thunderstorm development-hence, storm genesis trdposphere from strong scatter in the lower 1-2 km of the can be followed from the very beginning of cumulus develop troposphere 1431. These measurements inferred a continuum ment;2) the capability of Dopplerweather radars to map of scatterers up 7 km having a Ci value that could be as large clear air windmakes possible monitoring thunderstorm out- as m-2/3which compares with those values kmat3 flow and inflow whichis usually precipitation free. reported byOchs and Lawrence. Crane's [361monostatic Strongestfluctuations in refractive indexoccur where radar analysis for several da s of observations show that Ci turbulence mixes large gradients of mean potential tempera- is largerthan 1 X mm27 for heightsup to 15 km inthe ture and specific humidity [ 1251. Gage et a2. [54] measured clear air. Theseresults suggest Dopplerweather radars with theheight distribution of forwardscattered signal strength minimum detectable structure constant of lo-" m-'/' might to show good correlation between it and gradients of poten- beable tomonitor continuously the wind throughoutthe tial temperatureat high altitudeswhere water vapor con- troposphere. tributions can be ignored. The tropopause is a region where 2) Frequency Modulated Continuous Wave Radar: The FM potentialtemperature always increaseswith height, and it but CW radar transmits a continuous sinusoidal signal whose was first detected by radarin 1966 [61. More recently Van frequency f = at is linearly swept with a period Ts. A target Zant et a2. [ 1211, using backscatter results from a vertically at range r returnsa signal which is mixedwith that being pointed VHFradar, obtained consistent agreement between transmitted, to producea difference frequency fa propor- rawinsonde inferred Ci and radar measured Ci for the clear tional to (2ra/c)(see Section IEA). Thus fa is a measure of air above the moist boundary layer. targetrange, and ranging accuracy is determined by the For manyyears ionospheric scientists have beenemploy- dwelltime Td available to measure fa. Fornear targets Td ing high powerVHF and UHF radars to observe theiono- is nearly equal to the time T, required to change the micro- sphere.Recently these radars at Jicamarca, Peru [ 1241; wave frequencyby A f = aT,. The decisive advantage of Arecibo, Puerto Rico; Chatanika, AK [8] ; and Lindau, Ger- FM-CW radar is that range resolution can be increased simply many [ 101I have lowered their sights to examine echoes in by spanning a larger frequency A f without decreasing average thenonionized stratosphere and troposphere. A 6-m VHF transmitted power. pulsedDoppler radar specifically designed fortropospheric Spatialresolution of the order of 1m canbe achieved studies was recently assembled at Sunset, CO, and is making easily withan FM-CW radar to showdetailed structure of continuousobservations of winds inthe troposphere [60]. wave phenomenain the clear air planetaryboundary layer Some advantages of the VHF radar are that it can sometimes (PBL). High resolution PBL probing with FM-CW radar was differentiatebetween scatter from hydrometeors and re- first demonstrated by Richter[991 and clear air echoes fractive indexfluctuations [61], and it mightbe able to usually arefound in layers, oftenwith wave-like structure detect coherent scatter (second term of (2.10a)) from mean 1571. refractive index gradients at inversion layers[ 551. The FM-CW radar is sufficiently coherent for nearby targets. I) Height DLtnbution of Refkactive Index Structure Con- For the first time, Strauch et al. [ 1151 exploited this property stant: In1971 Ochs and Lawrence [93] made temperature to measureDoppler shift of FM-CW radarechoes from the fluctuation measurements using a sensor mounted on an &- opticallyclear boundary layer. Linear sawtooth modulation craft and sounded the atmosphere in the lower 3 km. Their of the microwave frequency gives the FM-CW radarcharac- work demonstrated that the temperature contribution to re- teristics that have analogy to the pulsed Dopplerradar: the 1548 PROCEEDINGS OF THE IEEE, VOL. 67, NO. 11, NOVEMBER 1979

\ '.-.-/ Fig. 31. PPI sector scan of echo power from clear air seen by the NRO Fig. 32. The dual Doppler-radar area (dashed lines) withinwhich the Doppler radar.Bright area of high power is aligned roughly parallel angle subtended by the radials from the Cimarron (CIM) and Norman to mean wind and bands are spaced about 4 km apart. Range marks (NRO) radars liesbetween 30' and 150'. The outlinedbox is the are 20 km apart. EL = 1.6' (April 27,1977,1447 CST). region wherein Doppler velocities were synthesized for detailed wind analyses. The wind speed and directionis a mean over 1.24-km depth of the PBL. reciprocal of A f can be considered analogous to an effective pulse length for the pulse radar, while Ts is equivalent to the PRT. Reflectivity is estimatedfrom digitally averaged log-video Because the FM-CW's peak-to-average power ratio is unity, samples at 762 contiguous resolution volumes (spaced 150 m) we immediately infer that resolution can be increased without along the beam. Doppler velocities are measured for each of changingthe effective per-pulse transmittedenergy; not so these volumes using the PP autocorrelation algorithm. Radial withpulsed radar where decrease in T reducestransmitted velocityestimates were obtainedfrom 255 contiguous PP pulse energy.Therefore, per-pulse SNR is decreasedin pro- samples.Data are acquired at rate of about 5 radial@, portion to the square of pulsewidth, whereas the equivalent andwith the rotation rate of 2 Is, thereare about 2 inde- SNR of the FM-CW radardecreases linearly with effective pendent data points/beamwidth(0.8'). pulse width (i.e., Af-'). This unquestionableadvantage of Sectorsstepped in elevation and scanned in azimuth by the FM-CW vanishes when considering pulsed Doppler radar eachradar encompass the primary synthesis region. Each andcoherent echoes. As a matter of fact,the pulsed radar beam scanned the same sector six times, startin! at 0.5' ele- may prove more advantageous [67]. vation,and tiltedup in 0.5' incrementsto 3.0 . With each 3) Observations of Turbulence and Roll Vortices: Clear air radar, the tilt sequence of sector scans was begun at the same winds in the planetary boundary layer (PBL, i.e., surface to time. The synthesized wind field and its perturbations from heights of about 1.5 km have beensynthesized from dual the mean are shown inFig. 33. Dopplerradar measurements [59], [761, whereinchaff was Berger and Doviak [ 131 have examined the spatial spectra dispensed over large areas to provide suitableecho levels S(K), where K is the wavenumber of the synthesized winds detectionand processing). Doviakand Jobson[471 showed on this convectively dry day, and have compared results with first results of two Doppler radar synthesized wind fields in those spectra obtained from anemometers located on the tall the PBL clear air where only the diffuse and intrinsic scatterers tower. The spectra follow a 513 power law in the wavelength of the medium wereused as targets.They observed mean A range of 1 to 8 km inagreement with spectra of tower wind fields at low height to have qualitative agreement with winds. Spectral analyses of clear air longitudinal wind fluctua- meanwind measured (withconventional anemometers) near tions using chaff anda single Dopplerradar have been re- the surface. ported by Chernikov et al. [32]. Their spectra, extending to On April 27, 1977, a day marked by strong nondirectional 3-km wavelengths, also show a 513 power law dependence in shear and curvature in the wind profile, NSSL's Doppler radar agreement with results on this day. However, O'Bannon [92] echo power measurements showed evidence of clear air con- shows spectra on another day when such a power dependence vective streets (Fig. 31),an observation that should signify is not evident. thepresence of rollvortices. First radar detection of clear A sample set of spectra, multiplied by K and plotted on a air thermal streets was reported by Konrad [ 741. semilogarithmic paper to show turbulence intensity per wave- Fig. 32 locates the radarsat CIM and NRO as well as a444-m band, is displayedin Fig. 34.The u componentspectra in meteorologicallyinstrumented tower. The 25 km X 25 km the y direction have a peak at 4-km wavelength which per- region(solid lined area) is wheresynthesized dual Doppler sisted over the one hourof data collection. radar winds were analyzed in detail and is referred to as the Various theories (e.g., [ 581, [77]) suggest that roll vortices primary synthesis region. However, reasonably accurate winds tend to form parallel to the mean wind in a strongly heated can be synthesized from Doppler data in the entire area en- PBL having a large unidirectionalshear. When vertical pro- closed bythe dashed lines. The winds were fairly uniform files of horizontal velocity have curvature, Keuttner [77] pre- from the southwest on this day, but there were small perturba- dicts rolls to have ahorizontal spacing of2.8 timestheir tions from the mean wind having a magnitude of about one depth. The strong 4-km wave in the y direction of the u com- order less than the mean wind itself. As is evident in Fig. 32, ponent might be the convectiverolls predicted by theory. the x direction and u component of wind are along the mean Fig. 35(a)depicts perturbation winds at one of six levels windand the y directionand u componentnormal to the synthesizedfrom a tilt sequence of eachDoppler radar. A mean wind. bandpassfiiter was appliedin the y directionto emphasize DOVIAK et al.: DOPPLER WEATHER RADAR 1549

27 APR 1377 DUAL DOPPLER MINDS 1'4'46'45-1'4'4335CST HEIGHT 1 .O Kn lOflPS - 144645 - 144935 CST ..-- 1.25 km AGL 27 APRIL 1977 -- 1.0

'1 2 '4 6 10 2 9 6 10'

3w (a) 16.0K/2r7 I I/Kfll 27 CST RPR 1377 DURL DOPPLER WINDS 1'4'46'45-1'49935 Fig. 34. A plot versuslog whichshows distribution of HEIGHT 1 .O nn of KS(K) K power (1446:45-1449: 36 CST). Note large power in velocity fluc- tuations atwavelength A = 4 km.

OURL DOPPLER WINDS 27 FlPR 1377 IY4645-177935 CST HEIGHT .75 KN !OMPS '1,

(b) Fig. 33. (a) Dual radar synthesized wind field at 1.0 km above ground. (b) Wind with its mean removed. Synthesized wind fields were low- pass fiitered once in X and Y direction with a 3-point Shuman fdter 1131. the 4-kmwave featurefor visual display. A low-pass fiter was applied inthe x direction alongwhich nodominant wavelength was noted. Fig. 35(b) is a vertical cross section at X = 25.5 km perpen- dicular to the mean wind. Vertical velocities were derived by integrating the mass continuityequation using wind fields fromthe six horizontal surfaces. Vertical grid spacing is 250 m. Readily apparent are counter-rotating vortices (roll Fig. 35. Horizontal (a) and vertical (b) cross sections of bandpass fd- vortices) having approximately 4-km wavelength whose maxi- tered wind data that highlights the clear air roll structure seen in the spectra displays of unfdtered data. mum vertical velocities are of the order of 1 m * s-l. Further- more, the ratio of roll spacing to height is 2.6 in good agree- ment with that predicted by theory [ 771. Gilmer et al. [56] 1550 PROCEEDINGS OF THE IEEE, VOL. 67, NO. 11, NOVEMBER 1979

144810 - 145140 CST 27 APRIL 1977 HEIGHT = I.Okm

// n\\\

-e X-LAG (KVI X-LAG IKMI (a) (b) FTg. 36. Cross correlation coefficients c (asa function of horizontal spatial lag) of (a) the u component (in the direction of mean wind) for two windfields, at 1.0 km AGL,synthesized 3% min apart. The mean wind speed is 14.9 m-s-' and the medium times for the two fields are 1448:lO CST and 1451:40.CST (b) the Y component (transverse to the mean wind). A spatial lag corresponding to mean windadvectioh during the 3% min timeinterval maximizes both c(u) and c(v). analyzed aircraft gust probe data collected on this day during mentationthat show the full extent of the~qrablem nor is the time of the radar observation. They also detect a promi- thereany foreseeable solution. Thus it appearsthat storm nent peak in power density at a wavelength of about 4 km in observers will have to accept some limitations in Doppler radar the y direction. weather measurements. 4) Time Correlation of Clear Air Wind Perturbations: The Dual Doppler-radar observations of the kinematic structure cross correlated (in space and time) wind fields (synthesized of severe storms and the planetary boundary layer (PBL)agree 3-1/2 min apart) at 1.0 km above ground level (AGL) show with theoretical models but much investigationis still required. that correlation is maximized for a translation equal to mean The Doppler weather radar shows promiseof greatly increasing wind advection c fig.^ 36) proving that these convective kine- our knowledge of thunderstorms and the planetary boundary maticfeatures advect with the mean wind. O'Bannon [92] layer on scales notbefore possible. Furthermore, we can has traced the time evolution of clear air eddies using wind monitor significant mesoscale phenomenawhich are of im- synthesizedfrom asequence of dualDoppler-radar data portance to air traffic safety, air pollution control, and (per- acquired 30 s apart. His results demonstratethat clear air hapsmost important) we may be able to see the triggering eddy fields of kilometer scales have a time scale of at least impulses of severe storms that each year cause such destruc- 10 min and, as with results on April 27, 1977, correlation is tion. Increased power and sensitivity of weather radars may maximizedwhen eddy winds are displaced to accountfor soon result inthe meteorologist being able to observe the advection by the mean wind. wind structure and its evolution throughout the troposphere. Theimportant advances in meteorological observations VII. CONCLUSIONS broughtforth by theapplication of Dopplertechniques to The introduction of Doppler frequency shift measurement weather radars will continute in the future. However, there capability into weather radars has opened new horizons for ex-is room for further improvement in the radar systemto reduce ploration by atmosphericscientists. The astounding success thedeleterious effect of ambiguities while lessening data achieved with these radars in detection of thunderstorm cy- acquisition time for observation of severe storm convection, clones well in advanceof tornado formation should cause incor-shear, and turbulencein clear or precipitation laden air. poration of coherent systems in new radars used for operation by the national services. Advances in digital signal processing ACKNOWLEDGMENT and display techniques have allowed economical development The authors appreciate the support they have received from of real time presentation of the three principal Doppler spectraltheir colleagues in theEnvironmental Research Laboratories moments. The techniques are here and constantly improving, (ERL), Air Force Geophysics Laboratory, National Weather and we believe that important new scientific disclosures and Service,Energy Research and Development Administration, newoperational applications in the areas of windmeasure- National Research Council, and Federal Aviation Administra- ment and tornado detection are forthcoming. tion, not only in the preparation of this paper, but also for Dopplerradars at centimeter wavelengths do not have a the continuing development of Doppler weather radar tech- sufficiently large velocity-range ambiguityproduct lava to nology. We are particularly indebted to Dr. R. Strauch of the match that required to observe, withoutobscuration, severe Wave PropagationLaboratory (ERL) and a reviewer whose convective storms. There is no comprehensive data and docu- carefulstudy of thismanuscript has significantly improved DOVIAK et al.: DOPPLER WEATHER RADAR 1551 the text. We are indebted to Joy Walton for her efficient [27] W. C. Campbell and R. C. Strauch,“Meteorological Doppler Ms. radar with double pulse transmission,” in Preprints 17& Conf. and accurate typing andcareful editing of the manuscript. Radar Meteor., Amer. Meteorol. Soc., Boston, MA 02108, pp. 42-44, Oct. 1976. REFERENCES [ 281 P. A.F. Caton, “Wind measurement by Doppler radar,” Mete- orol. Mag.,vol. 92, pp. 213-222, 1963. [ 11 M. Abramowitz and I.A. Stegun, Handbook of Mathematical [ 291 R.B. Chadwick and G. R. Cooper, “Measurement of distributed Functions, Nat. Bur. Standards,Appl. Math. Ser. 55, 2nd targets with the random signal radar,” IEEE Tmm. on Aerosp. printing,Supp. Doc.,U.S. Gov. 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