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1968 GLENN E. STOUT AND EUGENE A. MUELLER zt65Survey of Relationships Between Rainfall Rate and Radar Reflectivity in the Measurement of Precipitation GLENN i. STOUT AND EUGENE A. MUELLERIllinois State Water Sirvey, Urbana(Manuscript received 22 December 1967, in revised form 4 March 1968) ABSTRACT Numerous investigations have been made in the last two decades from both direct measurement of theradar reflectivity and the rainfall amount, as well as indirect measurements of the raindrop size spectra.Calculations of the reflectivity factor and rainfall rate from these spectra can be made and the relationshipsdetermined. Both methods are discussed and a summary of the relationships pesented. These relationships show differences in excess of 500% in rainfall rate at the same reftectivity. Theselarge differences are primarily associated with differences in geographic locality. In addition, there aresmaller differences on the order of 150% that can be attributed to different types of rain or different synopticconditions. Some data are available which are indicative of the differences in the relationship on a given day, depending upon the location within the storm which is sampled. This is briefly described and in only one caseout of 18 is there a significant difference. Estimates of the effects of evaporation, accretion and coalescence on the relationship are made and showsome of the reasons for the differences in the relationships noted at different geographical locations. Theaccuracy of the relationships is investigated with attention directed to the evaluation of total storm amounts.It is shown, in general, that the relationships introduce less uncertainty than the uncertainty in obtaininga radar measurement of the reflectivity.1. Introduction Many radar users have found problems in makingquantitative measurements of rainfall. In Alaska, radaroperators report that the reflectivity is high for light tomoderate rainfalls. In hurricanes, the reflectivity is lowwhile the rain rate is high. In the early 1950's eventhough radar meteorologists were working with 3.2-cmradar equipment and concerned about the attenuationfactor, it was evident that the radar-rainfall relationships changed from day to day and season to season andthat these changes were much greater than the attenuation losses. Since that time a number of investigators havedetermined Z-R relationships. The authors with thesupport of the U. S. Army have conducted raindropspectra measurements at a number of locations. Theresults of these investigations are presented in thispaper. Two different approaches to determination of arelationship have been recognized. These are a directapproach which consists of simultaneously measuringthe rainfall rate by rain gages and the radar reflectivityby radar, and an indirect approach of measuring theraindrop spectra and calculating both parameters fromthe spectra. The advantages and disadvantages of thetwo methods are discussed in this paper. In addition, itis hoped that by presenting the relationships in onelocation, a nmre efficient utilization of the relationshipscan be made by the radar meteorologist. Basic to any measurement of precipitation by meansof a radar is some form of a relationship between radarparameters and rainfall rate. The rainfall rate and thereflectivity factor are both functions of the raindropsize distribution. In the United States the radar meteorologist has informally adopted a unit called reflectivityfactor and designated the symbol Z to represent thisquantity. Some investigators have referred to Z as thereflectivity. Some confusion has arisen as the radarengineer uses reflectivity to represent a slightly differentquantity. The backscattering cross section of an objectis defined as the area which intercepts an amount ofpower in the incident beam which, if radiated isotropically, would yield a reflected signal strength at thetransmitter of the same magnitude as the actual objectproduces. The radar engineer's definition of reflectivityis the average sum of the radar backscattering crosssection per unit volume of space. It can be noted thatthe dimensions of this quantity is per unit length. Theradar meteorologist frequently uses Rayleigh's scattering law and removes the constants of wavelength andrefractive index, leaving a term of diameter of thesphere to the sixth power. If the sum of the diametersto the sixth power of the raindrops per unit volume(i.e., the refiectivity factor) is multiplied by the constants of wavelength and refractive index, the normalrefiectivity used by radar engineers results. Commonusage as introduced into the literature by Marshall466 JOURNAL OF APPLIED METEOROLOGY VOLuMETet al. (1947) has been to call the value of ZD thereflectivity Z. Thus, to avoid this confusion we willspeak of Z as the reflectivity factor. The common unitsof Z are mm m-3. Most work has been directed towardthe relationship between Z and rainfall rate R. The Z-R relationships are generally reported in theformZ=ARb.Many investigators have noted a tendency for departures in the data from the familiar Marshall-PalmerZ-R relationship. Some confusion has arisen from thisrelationship because of uncertainty as to which variableis treated as independent in the original analysis. Forthe use of the radar meteorologis.t who wishes to predictthe rainfall rate from measurement of the radar refiectivity, the reflectivity should be treated as the independent variable. If the rainfall rate is considered theindependent variable, the exponent is smaller and thecoefficient larger for the same data.2. Direct measurement of the relationship between radar reflectivity and rainfall rate One method for obtaining a relationship between theradar backscattering cross section and the rainfall rateis to actually measure both simultaneously. Thisobvious method has been attempted by several groups(Austin, 1964; Marshall et al., 1947; Hudson et al., 1952)with varying degrees of success. There are a number ofdisadvantages of such a straightforward method. Thefact that the radar invariably samples rain aloft, and therain gage samples the rain at the surface is one difficultyin the procedure. Austin and Williams (1951) attemptedto minimize this error by directing the radar beamdirectly over a rain gage located on a high point ofground. The radar antenna was directed as low aspossible without any ground return showing at therange of the rain gage. Most investigators havetempted to compensate for the time of fall of the raindrops by applying a time lag to the radar observations. A second problem associated with the elevated radarsample is the horizontal drift of the raindrops duringtheir fall from the radar beam location to the ground.In order to reduce these errors a network of rain gageshas been utilized by some groups so that the drift andtime lags could be incorporated in the analysis. Thesemethods certainly tend to increase the confidence of theexperiment, but there remains considerable doubtwhether the errors due to time lag and drift can becompletely eliminated by these techniques. A further disadvantage is the immense dlscrepancybetween the sizes of the samples. Neglecting the -erticalextent of the radar beam (this amounts to a loss in thetime resolution) and assuming common radar parameters of 1- horizontal beamwidth and 1 usec pulsewidth, the radar at 10 km samples a volume over anarea of about 2.6.10 m. The rain gage samples an areaon the order of 7.10-= m=. As the range increases, theradar area is increased proportionately. To reduce this difficulty one may use more than onerain gage under a radar volume, as was done byDimaksian et al. (1962). They used three networks atranges of 12, 22 and 32 km with 5, 9 and 12 gages alllocated within their respective radar areas. This yieldeda gage density of one gage per 0.04, 0.045 and 0.05 km,respectively. Thus, the measurement of the radar Zcould be related to the average rainfall rate from theaverage of at least five gages. Unfortunately, the.calibration results of this work have beeen directedtoward the calibration of a particular radar in terms ofdeflection of an A scope. Without specific knowledge of'the receiver and detector characteristics, it is not.possible to use these results elsewhere. The authorswere surprisingly successful as shown in a later paper(Dimaksian and Zotimov, 1965), where they reportthat "when the radar installation is sufficiently sensitiveto rainfall intensity, the estimate of total precipitationin an area will be more accurate than could be obtainedfrom a rainfall measuring network of practically any'density." Doherty (1963) performed a unique direct measurement which permitted a high confidence in the measurement of the radar scattering. In this experiment thereceiver was separated from the by 860 m,.and measurements of the direct transmission betweenantennas made it possible to eliminate the need for'knowing precisely the transmitter power and gain ofthe receiving and transmitting antenna. His results.shown in Table 1 indicate a much lower coefficient thanthat ordinarily found. He found higher A's as the .rainfall rate increased. His Doppler frequency records.indicated downdrafts on a number of occasions beforethe onset of rain. This would account for the low A. Wilson (1963), using data from a ll00-mi rain gagenetwork, obtained Z-R relationships for a number ofthunderstorms in Oklahoma. His procedure consistedof obtaining the best relationship, using least squares.method, between network average amounts from theradar and network average amounts from rain gages.In 4 of the 6 storms analyzed, his Z-R relationships didnot depart significantly in terms of his measurementerror from the frequently quoted, Z= 200 R.. Caton (1964) used a Doppler radar in conjunctionwith a rain gage to deduce the drop size spectra. Therain gage provided an average water flux at the groundlevel and the radar provided a frequency power spectrum. The drop size spectra were deduced from thesetwo measurements, and the reflectivity factor andrainfall rates were calculated from this spectrum. Hefound little change of Z (i db) between the meltinglevel and a Z= 240 R.3 in rain near the cloud base. Other investigators (Berjuljev et al., 1966; Aoyagi,1964) in USSR and Japan have reported Z-R relationsfrom radar measurements. Results are within the rangealready shown. These differences, which may be dueJuNE 1968GLENN E. STOUT AND EUGENE A. MUELLERT^BLE 1. Radar refiectivity, rainfall rate relationships from direct measurement.467Investigator Accuracy estimateGeographical Range of Z = A Rb (standardlocation applicability* A b deviation, db)CommentsDoherty (1963) Ottaw% CanadaBerjuljew et al. (1966)Valday, USSRWilson (1963) Norman, Okla.Aoyagi (1964) TokyoTRW 70 1.42 2.5not TRW 38.4 1.63 1.7R<iO 18.6 2.37 1.6R<20 25.9 2.02 1.7R<40 33.9 1.79 1.9R<60 38.2 1.69 2.0 340 1.5TRW 45 1.43TRW 241 1.45TRW 183 1.18TRW 141 1.72 100 1.4The exponent is as sumed equal to 1.5 and the coefficient determined from 2 y of rainfall.Extreme low coefficientExtreme high coefficientExtreme low exponentExtreme high exponentFor diffuse radar echoes* TRW, thunderstorm; R (ram hr-).somewhat to technique or measurement error, are alsothought to be real. One cannot model a rainstorm for theentire world.3. The relationship between refiectivity factor and rainfall rate from measurements of drop size spectra Many problems associated with direct measurementof radar return and rain gage rainfall rates, and comparison of the two results, can be eliminated by directmeasurement of the drop size spectra. However, newproblems arise. The most serious difficulty with thistype of measurement is that the volume in space inwhich the drops can be sampled is limited to a fewcubic meters. The assumption must then be made thatthese few cubic meters are representative of the 10* or10- ma sampled by the radar. A study by Mueller and Sims (1966a,b) indicates thatfor a sample at 5 it above ground level, a sample of 44 mais required to estimate the rainfall rate to within 10%with 95% confidence. It is also demonstrated in thesame paper that a smaller volume is adequate todetermine the Z-R'relationship, if an adequate numberof samples is included in the analysis. Thus, in thisanalysis using 1-ma samples, less than 12% of thevariance of data points around the regression line couldbe attributed to the sample size. When raindrop size spectra for a volume in space areused to determine the rainfall rate, the velocity of theindividual raindrops must be known. Likewise, thecalculation of the reflectivity factor from spectra on ahorizontal surface requires the velocity of the drops.The nearly universal acceptance of the terminal velocity, reported by Gunn and Kinzer (1949), was usedfor the velocity of the raindrops. This assumption isprobably quite reasonable near 'the ground, as isevidenced by the generally good agreement between theaverage rainfall rates from drop size spectra and therates from a rain gage. However, at the heights sampledby the radar, it is equally certain that the raindrops arenot moving with terminal velocity with respect toground because of the existence of either updrafts ordowndrafts. Vertical pointing Doppler radar measurements have confirmed that the drops are moving atvelocities with respect to earth that are different thanthe stagnant air terminal velocities. Calculations of the radar scattering from the drop sizespectra, assuming spherical drops, can be made by eitherthe Rayleigh scattering assumption or from the morecomplete Mie scattering, depending primarily on thewavelength of the radar under consideration. Since themajority of work is at a wavelength of 3 cm or longer,the simpler Rayleigh scattering is usually assumedadequate. Rayleigh scattering for 3-cm radiation differsfrom the Mie scattering by less than 2 db at rainfallrates of 400 mm hr-, and the difference is much less atlower rainfall rates. Table 2 is a list of Z-R relationships as determinedfrom drop size spectra from several different investigators and for different types of rains. Diem's (1966)observations were taken at a number of locations andexhibit a low exponent. Most of the rain was under12 mm hr-. It is not known whether he has used R or Zas the independent variable. The low coefficient for the orographic rains in Hawaiias first reported by Blanchard (1953) and later byFujiwara (1967) appear to be in order. The drops in theHawaiian upslope rainfall are very numerous and quitesmall, and rainfall rates are low.468 JOURNAL OF APPLIED METEOROLOGYTAuL 2. Radar reflectivity, rainfall rate relationships from drop size spectra.VoLu4 7 Standard error Z A Rb of estimateInvestigator A b of log R CommentsMarshall et al. (1947) 220 1.6Blanchard (1953) 31 1.71 16.6 1.55Fujlwara (1967) 80 1.38Hardy (1962) 312 1.36Imai (1960) [Japan] 700 1.6 300 1.6 200 1.5 80 1,5Diem (1966) 184 1.28 278 1.30 240 1.30 176 1.18 151 1.36 179 1.25 227 1.31 178 1.25 150 1.23 137 1.36Foote (1966) 520 1.81Dumoulin and 730 1.55 Gogolombles (1966) 255 1.45 426 1.5Mueller and Sims (1966a, b)286 1.43 0.198221 1.32 0.170301 1.64 0.136311 1.44 0.147267 1.54 0.142230 1.40 0.171372 1.47 0.153593 1.61 0.175256 1.41 0.163Canada, widely accepted and usedOrographic Hawaiian rain at cloud baseOrographic Hawaiian rain within the cloudOrographic Hawaiian rainArizona and Michigan rain with rates greater than 5 mm hr-One day of probably warm rainOne day continuous rainAir mass showersPre-warm front rainOverall average of different locationsEntebbe, Uganda (tropical)Lwiro, Congo (tropical)PalmaBarza, ItalyKarlsruhe, Germany, springKarlsruhe, Germany, summerKarlsruhe, Germany, fallKarlsruhe, Germany, winterAxel Heiberg Land,Tucson, Ariz.France, highest coefficientLowest coefficientAverage of all observations, 0.95 correlation coefficientFloridaMarshall IslandsOregonIndonesiaAlaskaNorth CarolinaIllinoisArizonaNew Jersey Dumoulin and Gogolombles (1966) performed anexperiment similar to Austin's with a radar directed atlow elevation angles over a single rain gage. Additionally, they obtained a number of drop size spectra inthe vicinity of the rain gage for which Z-R relationshipsare reported in Table 2. These results from the rain gagereadings and the radar measurements for identicalobservation times were compared. In general, theyfound good agreement between the rainfall ratesobtained from drop size spectra and the rain gages.However, when the radar Z was converted to a rainfallrate by means of the spectra-established Z-R relationship, a factor of at least 2 in rainfall rate remains.bumoulin and Gogolombles also indicated that theZ-R relationship shows a large variation with timeduring a storm.4. Discussion of the relationships and their vari ability It is immediately apparent from examination ofTables 1 and 2 that the constants of the relationshipsare widely variable. At the extremes one might comparethe differences in the Z value at 1 mm hr- betweenDoherty in Table 1 and Dumoulin's relationships ofTable 2. A difference of a factor of I0 (10 db) exists. If'one assumes a measured value of Z= 10, the rainfallrates calculated from these two relationships would bedifferent by a factor of 5. Thus, differences of at least500% in heavier rainfall rate exist between theserelationships. If one assumes a lower value of Z of 10then it is only a factor of 3 or 300%. It is probable thatsome of the differences may be partially due to variations in methods of obtaining the relationships. However, considerable differences exist using the sametechnique as the result of topography, geographicalvariation, rain type, synoptic type, thermodynamicstructure of the atmosphere, evaporation and, to someextent, coalescence. a. Geographical differences Results of studies conducted by Diem and by thestaff of the Illinois State Water Survey over a period of ]ts:s 1968 GLENN IT. STOUT AND T^L 3. Mean rainfall rat , o,.. (mtivity Mar mm-5 Flo:-, _shall ..... ua /s/ands Ore lndo- North 1 I 10 . uregon nesia Aid ' Caro. , - 0.6 0.5 0.6 0.6 L1.10s 1.0 1.6 1.I 1.1 1.2 3.5.10a 2.5 3.7 2.3 2. 1.6 1.1 '10 6.3 8.7 5.4 2.8 3.5 3.5.10 14.5 21.6 9.5 6.0 &2 104 1.1 'lOa 34.8 48.4 18.7 8.8 i4.4 7.8 68.5 90.5 29.5 17.7 3.5-I0a 167.1 65.7 38.7 1.1 10 247.7 % 70.0 87. 123.8 & &EUGENE A. MUELLER tOLINOis aSeveral years sho- -. cude 6000 se,nte,d m Tables 2 -a?ous climati reen/and t, )ra rom a n .... am 3. Diem, t dyed lter mca. Diem ok,..T-e of location[a . per tech' brained hb rom . ne el , tuque. Spectra b a Size sno,. '* reationshi, ...... device*'y.a OOtained fro-_'ere deduced fr l_ma .."urOgraphed the- - raindron c_ '" atop .... mme in a la .... aorops .* mera. This . . obtained ann .,e Pctures, the were taken for ?rag a logarp. . te Z- re] Sze spctm ,,,ulc, least-s Uar atluushlps calc e gst(uments Were - q es fit'; ulatea Y lsland ,, , llis, Or - ' '; Ma;...- ' - Geograh: . , tmm/hr paign. 'm , , nmsa: ..... k ." =Ogor I Juf-o, e ca/variations takjn'?" aPprOximat.Yrq"hn, N. C .'nnesm; . -fZ'RrelatiOnshis .... [ each site - 1300 1,,;- "o Chain relati,_t. P opectra on 9, - - e total .... .,m Sanm/,.. - , "mps wh;k the r=;-, ,.- ays. A* a,, - omple Consis, Were uuected by ,k *u Were de* .... - lie data h'o, eeoed 25 mm[ 7r and Cham: rmStem data als =.2... ,m samples sample durin, 2 ,agstaff renrefr I0 of the *? atnshO for dir .mimt large d;_ Since ,,- ..lUly and Au=u, ent one a 2 m. 'fi ?ntebbe. U,orynt dimatic a,o"rences At,. cue Climat , - [' -mOnth ramtallio.. .: sua, at - a. etw . -a, the r--' -t Oreg .... . . _, ?Vqent. k- . zlO " rama medium ,., -u to be ne--, Y smilar. FI ._ t ot ,u be noted ,f ?.*u=er s worta.._ -er-fmagnb.. yah .... mes o[ tb .... ry the sam rma


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