APRIL1990 BRIAN E. SHEPPARD 255Measurement of Raindrop Size Distributions Using a Small Doppler Radar BRIAN E. SHEPPARDAtmospheric Environment Service, Downsview, Ontario, Canada(Manuscript received 2 May 1988, in final form t5 September 1989)ABSTRACT A small X-band bistatic Doppler radar originally developed for use in automated weather stations as a Precipitation Occurrence Sensor System (POSS) can also measure real time raindrop size distributions. In contrastto large-scale pulsed Doppler radars this system is continuous wave and the measurement volume is relativelysmall. The drop size distribution is retrieved from the Doppler power spectrum by using an iterative inversionmethod. The sampling requirements for a representative average power spectrum are estimated. The effect ofwinds on the measurement is discussed. The POSS measures rates that are consistent with conventional gaugesassuming a 6 dB transmission loss due to water on the radomes. Average drop size distributions measured instratiform rain are typically negative exponential for diameters greater than 0.7 mm. One-minute averages inrain showers show the multimodai character of the distribution.1. Introduction Knowledge of raindrop size distributions is necessaryto model the microphysics of precipitation, to establishempirical relationships between radar measurementsand rainfall parameters and to determine atmosphericattenuation of electromagnetic radiation. The oldesttechniques for counting raindrops using dyed filter paper or photographic images are labor intensive. Morerecent devices such as electromechanical counters (Jossand Waldvogel 1967) and optical imaging devices(Knollenberg 1970; Illingworth and Stevens 1987) haveautomated this process. Doppler radars have been usedto measure drop size distributions (DSDs) in large volumes of air (Rogers 1967 ). The Atmospheric Environment Service (AES) ofCanada has developed a small Doppler radar systemfor the observation of precipitation occurrence, type,and intensity in an automated observing network(Sheppard and Wu 1985). This system, shown in Fig.1, is known as the Precipitation Occurrence SensorSystem (POSS). It measures the Doppler velocityspectrum of scatterers in a small volume of air immediately above the sensor. The spectral mode is usedto identify the type of precipitation, e.g., snow, rain orhail. The spectral power is used to calculate the intensityof the precipitation. This paper describes the application of the POSS to the measurement of DSDs. Theprocessing of the Doppler signal differs significantlyfrom that of large-scale pulsed radars. Corresponding author address.' Brian Sheppard, Data AcquisitionSystems Branch, Atmospheric Environment Service/EnvironmentCanada, 4905 Dufferin Street, Downsview, Ontario, M3H 5T4, Canada.2. Instrumentation The POSS radar is a low power, continuous wave,X-band, bistatic system. The transmitter and receiverare housed separately and mounted on a frame 45 cmapart (Fig. 2). Their antennas are identical smoothwalled rectangular pyramidal horns protected from theenvironment by flat radomes made of the transparentmaterial LEXANr. The horn aperture dimensions are7.6 and 9.4 cm. The semiflare angle of both planes is26-. The electric field in the aperture is horizontallypolarized and distributed as the TEl0 mode of a rectangular waveguide for both the transmitting and receiving antenna. The beam axes are oriented 20- fromthe vertical so that they intersect midway between thetwo horns, 31 cm above the horizontal plane throughthe center points of the radomes. The size of the samplevolume is a function of raindrop diameter. The hornseparation and orientation minimizes the effects of impact on, and runoff from the radomes while maintaining sensitivity to small drop sizes. Two aluminum platesmounted vertically between the transmitter and receiver also increase the isolation between the radomes. The transmitting Gunn diode continuously radiatesat 10.525 GHz (?~ = 2.85 cm). The nominal outputpower is 100 mW. The scattered signal from the atmosphere is detected by the receiver and homodynedwith the transmitted signal via semirigid coaxial cable.The resultant Doppler voltage is amplified by a 84 dBgain amplifier with a pass band frequency range from35 Hz to 1 KHz.3. Doppler signal characteristics An approximate expression for the average powerreceived by a bistatic radar from a single scatterer inc 1990 American Meteorological Society256 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOI.UME7FiG. 1. Photograph of the POSS Doppler radar.the antenna far field, assuming no atmospheric transmission losses, is adapted from Skolnik ( 1980): Pr = PttwGt(gl)Gr(R2)3'2a 64~r31Rl 121 R212 (!)where/~, and fit are the average received and transmitted powers respectively, Lw is the transmission lossdue to water on the radomes, RI and R2 are the po:sitionvectors of the scatterer relative to the transmitter andreceiver respectively (Fig. 3), Gt(Ri ) and Gr(g:!) arethe far field antenna gains in the direction of the scatterer for the transmitter and receiver respectively, ~ isthe transmitted wavelength, and ~ is the scatteringcross section. For a spherical particle, ~ is a fu~ctionof its diameter and complex refractive index, )~, andthe scattering angle (Mie 1908). The scattering angleis defined as the angle between the position vectors R~and -R2. The Doppler signal phase ~b relative to the cartierphase depends on the pathlength traveled by the: scattered radiation from transmitter to receiver: ---2,~(Igxl + IR21)/X. (2) In the far field, the surfaces of constant pha~;e areprolate spheroids with focal points at the phase centersof the transmitter and receiver as shown in Fig. 2;. Thefrequency fo of the Doppler signal equals the rate atwhich the scatterer.intersects the spheroidal sudhces: ft~ = Vq,. V/2~r (3)where Vq, is the gradient of the phase angle, "." indicates the scalar product, and V is the velocity vector. [~~ D OPPLER VELOCITY SPECTRUM 'TSBC80/24 MICROCOMPUTERFIG. 2. Schematic of the POSS Doppler radar.APRIL 1990 BRIAN E. SHEPPARD 257l~o. 3. Vector diagram of Doppler frequency generation in the POSS measurement volume.The gradient is a function of the location in the measurement volume. During the measurement samplingwindow, a single drop moving at a constant velocityproduces a frequency that depends on its position andvaries with time. For example, the frequency producedby a drop falling vertically from the top of the volumeto the top of the shield plates decreases by 30%.4. Sensor calibration The combined antenna gain Gt Gr in ( 1 ) is measuredin the near field. A Cartesian coordinate system is defined with origin midway between the transmitter andreceiver in a horizontal plane containing the centersof the antenna radomcs (Fig. 3 ). The x-axis is directedfrom receiver to transmitter and the y-axis is perpendicular to this, both in the horizontal plane. The zdirection is zenith pointing. Grid points are locatedwith 5 cm spacing in the volume defined by I xl < 20cm, I y l < 20 cm, and z < 115 cm. A drop is released from the z = 120 cm plane ateach grid point with [ x] ~< 20 cm and 0 ~< y ~< 20 cm.Symmetry about the x-axis is assumed. Figure 4 is anexample of the Doppler signal produced by a 2.6 mmdiameter drop falling from rest along the central verticalaxis of the volume (x = 0 cm, y = 0 cm). The frequencyand amplitude increase as the drop accelerates towardsthe radar. The amplitude is directly proportional to thescattered electric field. The time scale is converted to52-5 0 0.2 0.4. 0.6Time (.econde)FIG. 4. Doppler signal from a 2.6 mm diameter drop accelerating, due to gravity,along the central axis of the measurement volume from a point 1.2 m above the POSS.258JOURNAL OF ATMOSPHERIC distance using the integral equations for an accelerating drop given by Wall ( 1975 ). The measurement volume is represented by elemental cubes centered on each 5 cm grid point. For the purpose of determining the av erage combined antenna gain in the cube centered at (x, y, z), the amplitude in the range from (x, y, z + 2.5 ) cm to (x, y, z -.2.5 ) cm is averaged. At each grid point, a is calculated from Mie theory using the software of Wiscombe (1979). The test drops are. assumed spherical. The complex refractive index is computed from the formula of Ray (1972) at a tem perature of 20-C. The real and imaginary components are 7.96 and -2.13, respectively. The drop diameter is determined by a weight measurement. A fixed scatter ing angle of 150- is assumed for all grid points after verification that this approximation resulted in a max imum system calibration error of 3% for all diameters. Using ( 1 ), the combined antenna gain is calculatedat each grid point. Figure 5 gives the gains in dB relativeto the maximum for both the x-z and y-z planes. Out'side the volume I xl ~< 20 cm, l yl ~< 20 cm, and z~< 115 cm the theoretical far-field gain is calculatedusing computer programs for smooth-walled pyramidalhorn analysis (Sletten 1988 ). The antenna phase centers are also calculated using these programs. A range of transmission losses, Lw in ( 1 ), was mea sured in laboratory tests by depositing water drops on the radomes in a variety of manners. A fine mist of very small drops had little effect on the measured power. When drops were splashed on the radomes, cre.ating puddles of water similar to those observedAND ,OCEANIC TECHNOLOGY VOLUlViE7during rainfall, the resultant power loss was 6 dB. Whenthis value was used in (1) the integrated rainfallamounts calculated from POSS measurements a~eedwith those measured by conventional gauges (see section 6). From the above measurements, the average Dopplerspectrum generated by drops of any size, traversing thevolume with any velocity vector, can be simulated.. Thesignal frequency is given, by (3) and the amplitude from( 1 ). The initial simulation assumed a drop falling atterminal velocity with no winds. The terminal velocitydrop size relationship is from an empirical fit by Atlaset al. (1973) to data from Gunn and Kinzer ( 1949)':u(D) = 9.65 - lO.30e--'6-(4)where u(D) is the terminal velocity in m s-~ and D isthe diameter in mm. The Doppler voltage generated by a drop falli[ng atterminal velocity during the 64 ms sampling windowis Fourier transformed to a power spectrum as described in section 5a. It is assumed that there is anequal probability of finding a drop at any grid positionat the start of the sampling window. The spectra associated with each of these positions are averaged todetermine the simulated volume-averaged power spectrum for a given drop diameter. The outer boundaryof the measurement volume is defined by the surfaceof initial grid points for which the rms Doppler voltage,averaged over the 64 ms window, equals the resolutionof the analogue to digital conversion described in secCombined antenna gain (riB) Combined antenna gain (dB) -20 --10 0 10 20 --20 -10 0 10 20 80 O 80 80 80 50 N 50 50 N x (cm) Y FIG. 5. Combined antenna gain (dB) measured in the x-z and y-z planes.A?mL 1990 BRIAN E. SHEPPARD 259tion 5a. Figure 6 gives the volume as a function of dropdiameter. The symbols are plotted at the drop diameterscorresponding to a terminal velocity resolution of 0.23m s-l. The simulation is performed for 37 drop diameters from 0.35 mm to 6.0 mm. If a sufficient number of drops of each size is sampled(section 5c), then the measured Doppler power densityspectrum is fOma, N(D)V(D);~(f, D)dD (5) S(f) = aOmlnwhere N(D) is the number concentration per unit volume per unit size interval of drop diameter D (mm-~m-3), V(D) is the measurement volume (m3), ~q(f,D) is the volume-averaged Doppler power density atfrequency f from a single drop, and Dmin = 0.35 mmand Dmax = 6.0 mm are the minimum and maximumdetectable diameters in the size distribution.5. Signal processinga. Doppler power spectrum The maximum Doppler frequency produced by rainin the absence of wind is less than 700 Hz. To avoidaliasing, the analogue Doppler voltage is digitized at2.048 KHz. The resolution of the analogue to digitalconversion is 12 bits. A time series v(t) of 128 samplesis measured in 1/16 second. A Hanning window H(t) isapplied to this data to reduce spectral leakage in thecomputation of the discrete Fourier transform. Themagnitude of the kth Doppler frequency componentS( k fo) isS(kfo) = I1/M ~ H(mTs)v(mTs)e-~2~m~'/'ul2 m=0 (6)where M is the number of samples (128), Ts is thesampling interval (0.5 ms),fo is the spectral resolution 24 22 2Oi FIG. 6. The POSS measurement volume per second. The symbolsare plotted at diameters corresponding to increments in the terminalvelocity of 0.23 m s-~.(16.0 Hz) and j = V--L--~. The spectral resolution corresponds to a resolution in the Doppler velocity component of 0.23 m s-1. The Fast Fourier Transform is computed in aboutone second. The number of spectra required to obtaina representative average is discussed in section 5c. Ana priori average noise spectrum is measured when thereis no precipitation and is subtracted from the averagespectrum.b. Retrieval of the drop size distribution Equation (5) is an inversion problem where a measurement of S(f) is used to estimate N(D). The procedure used to solve this system of equations is an iterative method of Chahine (1972). Equation (5) isapproximated by 42 S~= ~Ak~N~ k=0to63 (7) j=6where & is the measured power in the frequency range,J~ -fo/2 to J~ +f0/2 andfo is the spectral resolution( 16 Hz), Ak~ = ~q~(J~, D~)V(D~) is the weighting function determined from the simulation, D~ is the diametercorresponding to the terminal velocity u(D~) = 0.23jm s-l in (4), and ~ = N(D~)zXDj represents the numberof drops per unit volume in the diameter interval corresponding to an interval of 0.23 m s-~ centeredon u(Dj). The initial estimate of N~ makes no assumptionabout its distribution. It is determined by simplifying(7). Each column of A~ is reduced to a single nonzero'value by assuming that the total power scattered fromdrop size Dj is found at the mode of the power spectrum~qs(J~, D~). Equation (7) can then be solved directly: 63 ~o = Sin/Z ,~k~ (8) k=0where N~- is the initial estimate of ~ and $m is thepower measured at the mode frequency of the simulated volume-averaged power spectrum ~qs(J[, D~). The iteration of (7) is terminated when the rainfallrate changes from the previous iteration by less than0.1%. This usually occurs in less than 10 iterations andis relatively insensitive to Nil. The weighting functions are determined from thecalibration procedure described in section 4. Some examples, normalized by D6, are given in Fig. 7 for different frequencies. The weighting function widthsbroaden as the frequency increases. The larger diameterdrops generate a broader range of Doppler frequenciesthan the smaller sizes because their larger scatteringcross sections allow detection in regions where thereare larger variations in V~. -. The effect on theweighting functions of adding a horizontal wind component was calculated by the same procedure. Figure7 gives examples of the weighting functions for a 5 ms-l wind in directions parallel and transverse to the260 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME? 1.8 ,x (0,0) m s-1 576 Hz 1.7 X (-5,0) m s-1 1.6 V (0,-5) m s-1 ' 288 Hz 0.9 144 Hx 0.7 0.5 0.4 0.3 0.2 0 4 6Diamete~ (mm) Do. 7. Examples of notarized wei~ting functions at four s~ctral frequencies in theab~nce of wind "(0, 0) m s-" and for a 5 m s-~ wind speed in directions along "(-5, 0) ms-" and across "(0, -5) m s-" the ~nsor's axis.sensor's axis. The ','wind weighting functions" (WWF)are broader than the "no wind weighting functions"(NWWF) because V~. - increases on the windwardside of the measurement ,volume and decreases on theleeward side. The effect of the wind on the retrieval ofthe DSD is discussed in section 7. The inversion algorithm was tested for conditionsof no wind using the Marshall-Palmer (M-P) model(Marshall and Palmer 1948) for stratiform rain givenbyN(D) = Noe-'~ (9)where N(D) is the number concentration of drop diameter D, per.unit volume, per unit size interval(mm-~ m-3), No = 8000 'mm-~ m-3, A = 4.1,xMpD-0'21mm-~, and RMe is referred to here as the "M-P rate"in mm h-~. The "mass flux rate" Rfis calculated using10000 1000 100 10 1 0.1 ' -- M-P 4.0 mrn h-1 A Initiol estimoteI i i I i i i 0 1 2 3 4 Diorneter (mm) FIG. 8. The DSDs retrieved by inversion of a simulated Doppler spectrum 'generated usingthe NWWF and the M-P DSD for a M-P rainfall rate of 4 mm h-~. The initial estimate DSDis calculated using (8). The iterative solution of (7) uses the NWWF.APRIL 1990 BRIAN E. SHEPPARD 261 ;rj~ Rf=- Z N~Dj3uj (10) 6 Jmi,where Jmin and jmax are the indices corresponding tothe minimum and maximum detectable diameters inthe DSD, N~ is the number of drops per unit volumein the jth diameter interval, and uj is the terminal velocity. Note that if(4) is used for uj, the mass flux rateis greater than the M-P rate. The volume-averaged power spectrum generated bya M-P distribution for a M-P rate of 4 mm h-1 wassimulated using the NWWF in (7). This spectrum wasthen inverted using the same weighting functions toretrieve a DSD. Figure 8' shows good agreement between the M-P and retrieved DSDs except for a smalloverestimation at the smallest diameter sizes. The initial estimate distribution determined from (8) is alsogiven. The broadening of the retrieved DSD caused by theinversion algorithm is shown in Fig. 9. The volumeaveraged power spectra generated by four single dropsof different diameters were simulated using the NWWFin (7). The four spectra were averaged. The use of theiterative inversion to solve (7) significantly reduces thebroadening in the DSD compared to the initial estimateusing (8).c. Sampling requirements The sampling requirements for this system are different from other methods such as filter paper and impact counters which are independent of the drop location in the measurement volume. These are analyzedby Joss and Waldvogel (1969). For POSS there is alarge variability in the received power from differentlocations due to variations in the combined antennagains and the R-2 factors in (1). The sampling requirements are determined by the number of dropsnecessary for the measured power spectrum to represent the simulation average for the volume. For a first-order estimate it is assumed, as in thederivation of (8), that each diameter produces powerat only one frequency. The effect of this assumption isseen in Fig. 8 by comparing the initial estimate of theDSD and the iterative solution of(7). The total power(zero moment) of the Doppler spectrum is 63 Sj= Z Ss(J~, Dj) (11) k=0where &(J~, D~) is the simulated Doppler spectrum fora single drop of diameter D~. The conditional probability density function (PDF) for a single dropp(Sjl 1 )is estimated from a histogram of S~ at each grid locationin the simulation volume. The corresponding cumulative probability distribution is given in Fig. 10 forfour drop diameters. For example, for a 1 mm diameterdrop, there is a 87% probability that the power is lessthan the mean. The conditional PDF of the averagepower from n~ drops, p(S~l nj), is given by the (nj - 1 )fold convolution ofp(Sj[ 1 ). Assuming that the probability p(n~) of finding njdrops in the measurement volume is a Poisson distribution with mean and variance t~ (Joss and Waldvogel1969), then the joint PDF of Sj and nj, p(Sj, nj) = p(nj)p(S~lnj) (12)100001000 100 10 1 0.1 0.01 0.0010.00010.5 mmt 1.0 mm ~ Itemt ve ]n-or~iorl ~1.7 mm//~ initial.... i i 0 1 2 3 Diameter (ram) FIG. 9. The broadening of the DSD caused by the inversion algorithm. The average Dopplerspectrum from four single drops of different sizes is generated and inverted using the NWWF.The "initial estimate" DSD is calculated from (8).262 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME7100 700 0.76 mm $0 50 o.~ 0:3 'I ~ lO Fraction of mean power FIG. 10. Cumulative probability distribution of power measured during the sampling window(64 ms) for a randomly selected location in the measurement volume. The units on thehorizontal axis are fractions of the mean Power averaged over the measurement volume.and the marginal PDF,p(Sy): ~ p(S~, ny) (13)n?0can be determined. The p(S~) is positively skewed sothat its mode, i.e., the most probable average powerfrom ~ drops, is less than its mean power 4- As ~increases, the mode approaches the mean. Figure 11gives the ratio of the mode to mean as a function of ~for four drop diameters. We define the sample size asadequate if this ratio exceeds 95%. For a 1 mm diameterdrop, this requires about 360 drops. The minimumnumber concentrations (mm-~ m-3) required to meetthis criterion are determined for a specified diameterand averaging period. In Fig. 12, the curves labeled"MSR" (minimum sampling requirement) give', thesevalues for ,three averaging times of 10, 50 and 500 stconds. For these times, t~- is calculated assuming aM-P DSD (RMe = 4 mm h-i). The mode powers ofp(S~) (Fig. 11 ) are used to generate a simulated Dopplerspectrum. Figure 12 gives the DSD retrieved usirtg (8).The underestimation increases with increasing diameter and decreasing averaging times because in both1 O0go80 70 60 50 40 30 2010 0o 1.7 ,,,~ /. / / /A 31m ~ 1~0 ~ 140 ~ ,10~00 ~ 10000Mean n~mber of drops in measurementFIG. 11. The ratio of mode to mean power for the PDF given in (13) of the average power measured from fi~ drops in the measurement volume.APRIL1990 BRIAN E. SHEPPARD 263E7EouZ10000 1 ooO, oo '~ M~.~ ~: N ~ ~ MSE 50 s ~ ,o ,o s 500 0.1 i I I I II I 1 2 3 4 Diometer (ram) F~G. 12. The effem of unde~mpling on the invemion ofa M-P DSD (R~ = 4 mm h-~)for three different avem~ng perils. The cu~es la~l~ "MSR' (minimum ~mpling requirement) ~ve ~e minimum number con~ntration of drops required to redu~ the bi~ toless than 5% for the~ avem~ng perils.cases t~ is decreasing. The fractional bias in the massflux rate due to undersampling is calculated from theDSDs in Fig. 12 and is given in Fig. 13. The variance in the retrieved DSD (a~v~)2 is equalto the variance of p (S~-), assuming (8). From (10) theresultant variance in the mass flux rate isO.R2 = g ~ (o'~'Dj3z~j)2.(14)The fractional standard deviation in rainfall rate isgiven in Fig. 13. For an averaging time of 50 secondsand for rates greater than I mm h-~, the standard deviation is less than 20%.6. Raindrop size distribution measurements The POSS, Joss-Waldvogel Disdrometer (Joss andWaldvogel 1967), and conventional rain gauges were 0.9 a: Bias 10 s ~ b: Bias 50 s 0.8 ~ c: Bias 500 s ~ ~ d: S.D. 10 s 0.7 ~ ~ e: S.D. 50 s 0.6 ~ '"--~~. 500 s 0.,5 ~ 0.4 0.3 0.2 ~ 0.1 f 0.1 ~ 1 100Rote (ram h-l)FIG. 13. The fractional bias due to unde~mpling, and the fractional s~nd~d deviationin rainfall rote, as a function of the rate and avera~ng ~fiod, assuming a M-P DSD.264 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME7compared during the summer of 1987 at the AES RadarFacility at King City, Ontario. The sensors were separated by about 5 m. The Joss-Waldvogel Disdrometer (JWD) consistsof a sensor head and signal processing electronics, bothmanufactured by Distromet, Basel, Switzerland. Thetransducer in the sensor head is a thin aluminum surface covering a styrofoam body. When raindrops impact this 50 cm2 surface, the downward displacementof the body induces a voltage in a sensing coil. Fordrops falling at terminal velocity, the output voltage(U) is proportional to a value that lies between thepeak force and the total momentum transferred to thesurface (Joss and Waldvogel 1977):U=kDn (15)where k is the proportionality constant and D is thedrop diameter (mm) and 4.3 > n > 3.1. A pulse height analyzer (model AD-69) sorts thepeak amplitudes into 20 channels of drop diameterintervals in mm: 0.3-0.4, 0.4-0.5, 0.5-0.6, 0.6-0.7,0.7-0.8, 0.8-1.0, 1.0-1.2, 1.2-1.4, 1.4-1.6, 1.6-1.8,1.8-2.1, 2.1-2.4, 2.4-2.7, 2.7-3.0, .3.0-3.3, 3.3-3.7,3.7-4.1, 4.1-4.5, 4.575.0, 5.0-max. The number ofimpacts in each channel is output digitally every30 seconds. The DSD in mm-I m-3 is calculatedfrom (4). The conventional gauges used in the comparisonwere a tipping bucket and a Universal Belfort WeighingRain Gauge with measurement resolutions of 0.25 and0.1 mm, respectively. Their outputs were recorded every 10 minutes and converted to rates in mm h-~. Forcomparison purposes the POSS and the JWD rates wereaveraged for the previous 10 minute period. A rainevent (14 July 1987) with two periods of "heavy"showers is given in Fig. 14. The average wind speed,measured at 10 m, was 5 m s-l across the axis of thePOSS. At the POSS measurement height of 2.5 m, thespeed is estimated as 3 m s-~ (Sevruk 1988). TheseWWF are used for the POSS spectral inversions. ThePOSS and JWD rates are plotted as continuous traceswhile the conventional gauges are plotted at 10 rainuteintervals. In general the POSS and the JWD are withinthe range of the conventional gauges. Compared to theJWD, the POSS overestimates rates less than 6 mmh-~ and underestimates higher rates. The accumulatedamounts for the 7 hour period were 22.2 mrct, 22.3ram, 22.3 ram, 20.1 mm for the POSS, the JWD, thetipping bucket and the Belfort, respectively. If the: POSSNWWF are used, the accumulated amount increasesto 24.8 mm. Figure 15 shows DSDs inverted from POSS spectra,averaged for about one minute, for the period from0800 to 0900 UTC given in Fig. 14. The number concentrations of drop diameters greater than 3 mm increase as the rate increases after 0830 UTC. Modes inthe DSDs are evident. Figure 16 compares the DSDs measured by the, POSSand the JWD averaged over a one-hour period. Thetipping bucket and Belfort gauges measured rates of1.3 and 1.5 mm h-~, respectively. The POSS and theJWD measured 1.2 and 1.9 mm h-~, respectively. Theaverage wind speed was 2 m s-~. The M-P DSD= 1.2 mm h-~ and Rf = 1.4 mm h-~) is given forreference. These average DSDs are typical of thosemeasured during the experiment in light rainfall ratesand in light winds. The POSS distribution is negative17161514.15121110 9 8 7 6 5 4 3 2 1 0-- JWD TB BELFORT7 FIG. 14. Comparison of 10-minute average rainfall rotes measured by the POSS, the JWD,the tipping bucket (TB) and the Belfort Weighing Rain Gauge for the period from 0700 to1400 UTC 14 July 1987.APRIL 1990 BRIAN E. SHEPPARD 265 E - 09:00 E e~ ~ 08:~0 o - 08:40 L-~ (J~ ~- ~ k. ~ - 08:20 ~ J- ~? ~8:~0 ~~o Co~ 08:00 C ~2 ~G. 15. DSDs inveRed from Doppler s~ctra averaged for a~ut on~ miRutc for th~ pe~ from 0800 to 0900 UTC 14 July 1987.exponential while the JWD shows characteristic modesat 0.65, 1.1 and 1.95 mm. The modes detected by thePOSS in short time averages are usually smoothed bylonger averaging. These observations led to an investigation of the calibration of the JWD's signal processing electronics (model RD-69). It was concludedthat modes observed at these diameters were sensorartifact introduced by the compression amplifier in theRD-69 circuitry (Sheppard 1990). While the JWD gavegood estimates of rainfall rates, this artifact confusedthe comparison of the multimodal characteristics ofDSDs.7. The effects of wind In general, the wind causes a redistribution of thepower in the Doppler frequency spectrum. For analysis,the horizontal wind vector is resolved into componentsV~, along the direction of the transmitter-receiver axis,and Vy across the axis. The Doppler frequencies forthe x and y components, given by (3), are V qxVx/2;rand V,I, yVy/2;r, respectively. The surfaces of constantphase are ellipses in the x-z plane (Fig. 3 ), and circlesin the y-z plane. The effect of the Vx component whenaveraged over the measurement volume is a small decrease in the mode frequency in the Doppler spectrumfrom the value in the absence of wind. This results ina slight increase in the mode diameter of the normalizedweighting functions (Fig. 7). The Vy component haslittle effect on the mode. The resultant "wind weightingfunctions" (WWF) are broader than the corresponding"no wind weighting functions" (NWWF). The effect of this broadening on the inversion ofsimulated Doppler spectra was tested following the approach of section 5b. The Doppler spectrum is generated for a M-P DSD (Rr = 4.7 mm h-~) using theWWF for a 5 m s-I wind direction along the sensor'saxis. The spectrum is inverted using the same WWFand the NWWF. Figure 17 compares the M-P and10000 ~x~ O POSS~1000-E g~ +JWD h 1i 100 1 0.1 ~ 1 2 3 Diameter (mm) FIG. 16. Comparison of DS~ me~ured by the POSS and ~e JWD averag~ for one hour.The M-P DSD (R~ = 1.2 mm h-~, R~ = 1.4 mm h-~) is ~ven for reference. The me~uredrates were: tipping bucket = 1.3 mm h-~; Belfo~ = 1.5 mm h-~; POSS = 1.2 mm h-~; JWD= 1.9 mm h-L The wind s~ (measured at 10 m) was 2 m s-~.266JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY10000 -- M-P 4.0 mm h-1~l:: 1000' ~ O NWWFT~ 100~ lOou_~z 1 0.1 ~ ~ ~ ~ ~~ 0 1 2 3 4 Diameter (mm) I~G. 17. Comparison of DSDs retrieved by inversion of a simulated Doppler spectrumusing the WWF corresponding to winds of 5 m s-~ along the sensor's axis and the NWWF.The spectrum is generated using the WWF and the M-P DSD (RM~ = 4 mm h-l).VOLUME 7retrieved DSDs. The NWWF increasingly overestimatethe number concentrations as the drop- diameter decreases from 1.5 mm. The WWF reduce the range ofoverestimation to diameters less than 0.5 mm. Theflux rates calculated using the relationship in (4).are8.8 and 4.9 mm h-I for the NWWF and WWF DSDs,respectively. Figure 18 gives the same comparison fora 5 m s-' wind across the sensor's axis. Here theNWWF and the WWF overestimate the number concentrations for drop diameters less than 1.0 and 0.5mm, respectively. The rates are 6.9 and 4.9 mm h-l,respectively. Large average vertical wind speeds are uncommonnear the earth's surface. Winds towards the surface (+)shift power to higher frequencies and winds away from,the surface (-) shift power to lower frequencies. Figure19 shows the effect of vertical winds on the inversionof Doppler spectra generated by a M-P DSD (Rf = 4.7mm h-I ). In both cases the retrievals use the NWWFsince it is unlikely that the vertical wind speed will be 10000~ 10007- ~ 100 ~o0.1~ -- M-P 4.0 mm h-1 D NWWF I 2 3 Diameter (ram)FIG. 18. AS in Fig. 17 but for wind direction across the sensor's axis.APRIL1990 BRIAN E. SHEPPARD 267 100007~ 100 ~o 0 1 2 ,3 Diameter (ram) FrO. 19. Comparison of DSDs retrieved by inversion of simulated Doppler spectra usingthe WWF for vertical winds of +0.5 m s-~ and -0.5 m s-~. The spectra are generated usingthese WWF and the M-P DSD (RM~, = 4 mmknown in practice. The retrieved rates are 3.2 and 7.2mm h-~ for +0.5 and -0.5 m s-~ vertical winds, respectively. Turbulence will broaden the Doppler spectrum and reduce the resolution in the retrieved DSD.8. Conclusions and comments The POSS measures real-time raindrop size distributions. The rainfall rates calculated from these distributions are consistent with conventional gauges ifthe transmission loss due to water on the radomes is6 dB. Additional development is necessary to keep theradomes dry, or to assure that the wetting process results in a constant loss. Although no assumptions aremade about the DSD model for the inversion algorithm, average distributions measured in stratiform rainare typically negative exponential for diameters greaterthan 0.7 mm. The number concentrations at small diameters frequently exceed the M-P model for themeasured rainfall rate. This is partially caused by thewind and by low frequency signals produced whendrops impact the radomes. Measurements in this rangewill be compared to a Particle Measurement System2DG laser imaging probe during the Experiment onRapidly Intensifying Cyclones over the Atlantic (ERICA) at Canadian Forces Base Shearwater, Nova Scotia(December 1988 to February 1989). Horizontal winds cause significant overestimationof the number concentrations of drops less than 1 mmin diameter if the Doppler spectra are inverted usingweighting functions assuming no wind. POSS estimatesof rainfall rates are in better agreement with conventional gauges if weighting functions specific to the windconditions are used. A redesign of the antenna system may improve thesystem performance. A reduction of the beam widthscould limit the measurement volume to a region wherephase surfaces are more nearly horizontal. Low frequency power produced by horizontal winds would bereduced. In addition, narrow antenna beams wouldalso improve isolation between the transmitter and receiver and reduce the effect of impacts on the radomes.Although a smaller measurement volume would decrease the number of drops sampled per second, itwould reduce the variance in the scattered power fromdifferent locations in the volume. At present, a 10-minute averaging period is requiredto reduce the underestimation of rate, due to sampling,to less than 10% in a light rainfall rate of 1.0 mm h-~.For higher rates the required averaging time is less.The signal processing electronics have recently beenre-designed to increase the processing speed, reducingthe averaging time requirements by a factor of five. Acknowledgments. Ken Wu of the Data AcquisitionSystems Branch of the AES designed and developedthe bistatic POSS system hardware and his efforts madethis work possible. 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