η = ; (3) QUANTITATIVE INTERPRETATION OF PRECIPITATION RADAR 7R.3 MEASUREMENTS AT VHF BAND Edwin F. Campos 1*, Frédéric Fabry 1, and Wayne Hocking 2

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1 7R.3 QUANTITATIVE INTERPRETATION OF PRECIPITATION RADAR MEASUREMENTS AT VHF BAND Edwin F. Campos 1*, Frédéric Fabry 1, and Wayn Hocking 1 Dpartmnt of Atmosphric and Ocanic Scincs, McGill Univrsity, Montral, Qubc, Canada Dpartmnt of Physics, Univrsity of Wstrn Ontario, London, Ontario, Canada 1. INTRODUCTION Quantitativ intrprtation of prcipitation masurmnts in th VHF band implis th xtraction of th prcipitation signal out of th full powr spctra, as wll as th propr convrsion of this signal into rflctivity factors. In addition, a propr rlationship btwn radar rcivd powrs and scattrrs crosssctions (i.., a radar quation) is rquird. W will xplor this aspcts hr, utilizing simultanous obsrvations of co-locatd vrtically pointing radars, oprating in th VHF and X bands, as wll as drop-sizdistribution masurmnts at ground.. METHODS.1. Th radar quation Standard forms for this quation assum an avrag scattrr cross-sction pr unit volum (or radar rflctivity) that is constant within th sampling volum of a givn rang gat. Undr ths conditions, th radar quation valid for vrtically pointing radars is givn by th following xprssion: A PTx a D Pr 16π max η π [ F( θ, φ) ] H + L π r H φ 0 θ 0 r sinθ dθ dφ dr ;(1) whr P r is th rcivd powr (in Watts), A is th radar antnna ffctiv ara (in m ), P Tx is th transmittr powr (in Watts), a is th antnna fficincy, F is th antnna pattrn (or polar diagram), D max is th antnnapattrn maximum dirctivity, η is th radar rflctivity (in m -1 ), r is th rang (in mtrs), L is th transmittd puls lngth (in mtrs), H is th lowr hight within th rang gat (in mtrs), φ is th azimuth angl, and? is th znith angl. Equation (1) is appropriat whn daling with radars that hav a narrow transmittd bam and high rang rsolution. Howvr, this rlation may not b valid for radars with antnna pattrn having significant sid lobs * Corrsponding author addrss: Edwin F. Campos, McGill Univrsity, Dpt. of Atmosphric & Ocanic Scincs, 805 Shrbrook Strt Wst, Montral, Qubc, Canada. H3A-K6; -mail: campos@zphyr.mto.mcgill.ca. (.g., th McGill VHF radar). Th rason is that th radar will rciv additional powr from scattrrs locatd at th sam distanc but in a diffrnt dirction than th main-bam rang-gat. Thrfor, it is th following radar quation th on to b solvd: A PTx a D Pr 16π H + L π max π r H φ 0 θ 0 η ( r, θ, φ) [ F( θ, φ) ] r sinθ dθ dφ dr.() This projct will thn xplor th ffct of using quation (), instad of (1), whn computing profils of quivalnt rflctivity factor. Th nxt sction will dscrib th mthod usd to accomplish this objctiv... VHF rain signal from X-band powr spctra Rcall that th radar rflctivity, for Rayligh scattrrs, can b xprssd as (.g., Rinhart, 1997): 5 π Κ Z η ; (3) 4 18 λ 10 whr K is th dilctric factor, and? is th wavlngth of th radar transmittd puls (in mtrs). Z is th rflctivity factor (xprssd in mm 6 m -3 ). By convntion (.g., Smith 1984), K is takn qual to 0.93 (th valu corrsponding to liquid watr at nar 0ºC, and wavlngths in th S band). Thrfor, Z Z, th quivalnt radar rflctivity factor, is usd. This convntion is adoptd bcaus whn radar masurmnts ar mad, on is oftn not crtain of th hydromtor phas or composition. Thrfor, from an original fild of Z w can driv a fild of? [using (3)], and thn apply ithr (1) or () in ordr to obtain th radar rcivd powr. Oprationally, on th othr hand, th radar will masur rcivd powrs, and ths will hav to b convrtd into quivalnt rflctivity factors. This can b don asily from (1), which is an approximation, but it cannot b don dirctly if th mor ralistic quation () is usd (notic that hr? is within th intgral). Howvr, w wr abl to obtain rflctivity factor valus through quation () by using th following mthod. First, a fild of Z is input in quation (3) in ordr to obtain? ( K 0.93 is usd hr). Thn, w input this rflctivity in () to obtain a ralistic radar rcivd powr (for a givn radar rang gat). Nxt, w input th rcivd powr from () into (1) in ordr to obtain an

2 avrag (within th radar gat) rflctivity, η. An avrag quivalnt rflctivity factor (for th radar gat) is thn obtaind by rarranging (3) as follows: λ Z η. (4) 5 π Κ Th procdur is rpatd for all th radar rang gats. At th nd, w wr abl to stimat th ffct of quation () by comparing th output Z and th original Z filds. For simplicity, w usd Z filds that wr variabl only with hight (i.., on-dimnsional filds). Thrfor η η ( higth ) η( r cosθ ). (5) As wll, w usd for F th valus plottd in Figur 1, which com from a simulation of th McGill VHF antnna polar diagram (providd by Mardoc Inc., th buildr of this radar). Th valus in Tabl 1 wr also usd, and th antnna ffctiv ara was computd from (.g., Skolnik, 1990) D λ A max. (6) 4π Notic that th intgrals in quations (1) and () wr computd numrically. Thrfor, th accuracy of th programs usd for intgration rquird som prvious tsting. (Th rason is that computations wr vry snsitiv to th antnna pattrn rsolution.) Consquntly, w first gnratd a synthtic Z profil that was constant in hight, put this profil in quation (3) to obtain?, input this rflctivity into (), and solvd this quation numrically. In principl, th rflctivitis rsulting aftr numrical intgration ( Z ) hav to b th sam than th input (Z), but it is not so if a too coars rsolution is usd for th numrical intgration. Thrfor, w gradually incrasd th intgration rsolution until th output rflctivity qualld th input rflctivity (s rd continuous and dashd lins in Figur 4). Aftr validating th intgration programs, w gnratd two diffrnt profils of rflctivity, and thn considrd thir rang variation whn computing th Z valus through quation (), i.., using th abov dscribd mthod. Th first profil was a synthtic rflctivi ty linarly dcrasing with hight. Th scond profil corrspondd to a tn-minuts smoothd rflctivity-factor masurd by a high-rsolution vrtically pointing radar (i.., th McGill VPR, dscribd by Zawadzki, Fabry, and Szyrmr, 001). Th rsults ar prsntd in Sction 3. Tabl 1. McGill VHF Radar paramtrs Paramtr Valu Tx wavlngth (?) 5.77 m Pak Tx powr (P Tx) 40 kw Antnna fficincy ( a) 0.54 Maximum Dirctivity (D max) Transmittd puls lngth (L) 1 km.3. VHF rain signal from VHF powr spctra Figur 1. Antnna pattrn of th McGill VHF radar. Th automatic sparation of th rain signal from th total VHF rcivd powr rprsnts an intrsting challng in trms of radar signal procssing. On on sid, VHF radar Dopplr spctra masurd during rain vnts prsnt clarly sparatd mods corrsponding to th clar air (th slowst) and rain (th fastst) signals. On spctrum xampl is prsntd in Figur, which corrsponds to obsrvations by th McGill VHF radar at a rang gat btwn.5 and 3.0 km hight (agl, abov th ground lvl). This spctrum has a population of scattrrs paking at 3.5 Hz (i.., a Dopplr vlocity of about 10 m/s, typical magnitud for raindrop fall vlocitis), and a slowst population paking at 0.5 Hz (i.., a Dopplr vlocity of 1.4 m/s, a typical downdraft for Montral). In fact, it is not rar to obsrv rain spctral paks bing as strong (or vn highr) than th clar air pak. On th othr hand, part of th clar air signal is oftn ovrlappd into th rain spctral rang.

3 Thrfor, th powr-dnsitis (S) calibration quation for th i-th spctral bin is givn by 1 A S cal i + n f sky ( f ) B S ( f ). sky out i (8) Th scond stp consists in finding th clar-air spctral pak. To do this, w sarch for th four largst powr dnsity valus locatd into th spctral rang btwn -0.8 and 0.8 Hz (w hav noticd that th clar-air pak is usually locatd hr). If ths 4 points do not li within lss than 0.81 Hz of ach othr, thn w stop th procdur and conclud that no clar-air signal can b rtrivd. Othrwis (if th four points li within ) w comput th avrag frquncy for ths points, and th frquncy bin for th clar-air pak, f j, will b th on closr to this avrag frquncy. Th rd lin in Fig. indicats th clar-air pak obtaind for this particular cas. Figur. Dopplr powr spctrum obsrvd in rain by th McGill VHF radar. For this xampl, on Sptmbr 9, 004, at 15:9:51 UTC, th bam is pointing vrtically and th rang gat is within.5 and 3.0 km. Th rd vrtical lin corrsponds to th clar-air spctral pak. To dal with this challng, w dvlopd a mthod for xtracting th rain signal out of th total Dopplr powr spctra, which is valid for any vrtically-pointing VHF radars. This mthod has bn dvlopd from an mpirical basis (i.., try and rror), and it is dscribd as follows. W start with th raw spctra (non-calibratd, xprssd in arbitrary units, au, pr spctral bin) masurd by th McGill VHF radar. For a givn rang gat, a spctrum is obtaind vry 35 sconds, for a spctral rang is within and 10.0 Hz, and a spctral bin rsolution of Hz. Also notic that th ground cluttr signal has alrady bn rmovd by a notch filtr at nar 0 Hz (s Hocking 1997 for dtails on th Dopplr powr spctra drivation). W calibrat ths spctra by using a rlationship of th form P B A P out sky + ; (7) sky cal whr th subscript out corrspond to th radar raw output, th subscript cal corrsponds to th calibratd powr, and th subscript sky corrsponds to th valus drivd from a sky-nois calibration (s Campos t al. 005, in this issu). Figur 3. Prcipitation spctrum xtractd from th Dopplr powr spctrum in Fig and qn. (9). During th third stp, w subtract th clar air signal to th rcordd Dopplr powr spctrum, and th rmaining spctrum will thn b th on corrsponding to prcipitation. W us th fact that th clar-air spctrum is normaly distributd; thrfor, th clar-air signal at n spctral bins to th right of th clar-air pak should b th sam (on avrag) than at n spctral bins to th lft of th clar-air pak. W will not xpct to

4 hav prcipitation signal to th right of th clar-air pak, but only to th lft of this pak (sinc prcipitation Dopplr vlocitis in th vrtical ar always smallr than ddis Dopplr vlocitis). Thrfor, it is saf to assum that th rain powr dnsity is givn by ( f ) S( f ) S ( f ) S prcip j i j i j + i ; (9) whr S prcip (f n ) is th prcipitation Dopplr powr dnsity at th n-th spctral bin (in Watts pr Hz), S(f n ) is th raw Dopplr powr dnsity at th n-th spctral bin (in Watts pr Hz), j is th spctral bin corrsponding to th clar-air pak, and i is any givn spctral bin. Figur 3 prsnts th rsult of applying qn. (9) to th Dopplr powr spctrum in Fig.. From multipl obsrvations of th prformanc of this mthod with ral data, w hav stimatd that th largst Dopplr frquncy valid for th rain spctrum is locatd at 1.0 Hz to th lft of th clar-air pak, i.., f ( f 1.0) Hz prcip [ 10.0, j ]. (10).4. Datast rquirmnts In ordr to apply th mthods dscribd abov, VHF radar masurmnts nd to b takn during an vnt of wid sprad prcipitation, having a mlting lvl much highr than th lowr radar rang gat (so that w can guarant rain masurmnts at last at th vry first rang gats). For th McGill VHF radar, th lowst rang gat is btwn.5 and 3.0 km hight, and it is rar that wid sprad prcipitation ovr Montral prsnts bright bands abov this hight. Howvr, w managd to collct a vry xcllnt prcipitation vnt that fulfills ths rquirmnts. It corrsponds to th passag of th rmnants of hurrican Francs ovr th radar sit, on Sptmbr 9, RESULTS Th ffct of a variabl rflctivity fild in th radar quation was studid by gnrating a synthtic profil of rflctivity, whr Z dcrass logarithmically linar with hight (10 dbz pr km, th typical valu w can obsrv in snow ovr Montral). This Z profil was usd as input for obtaining Z through th mthod dscribd in th prvious sction. Figur 4 prsnts th input Z profil in black dashd lin, and th output Z profil is also plottd in a black continuous lin. Th rsults indicat a dpning in th Z slop with hight. A small undr stimation (of Z with rspct to th input Z) is obsrvd blow th 4km lvl, and th diffrnc btwn input and output bn gratr at highr rangs (raching 9 dbz at 9 km). Figur 4. Exampls using synthtic data as input. Rd lins corrspond to a validation tst with input of constant Z. Black lins corrsponds to a linarly dcrasing profil of Z. Dashd lins ar input profils (Z) and continuous ar output profils ( Z ). A scond xprimnt aimd to prsnt mor ralistic conditions by using a ral profil of rflctivity. For this, hight profils of th quivalnt radar rflctivity factor, masurd by th McGill VPR, ar usd. Th VPR was co-locatd with th McGill VHF radar, and its datast hav a tim rsolution of about 30 sconds and a rang rsolution of about 75 mtrs. W smoothd ths VPR masurmnts by taking, for ach particular rang gat, th 10-minuts mdian valu. Sinc th VPR oprats at X band, prcipitation attnuation has to b considrd. W thn had to rcalibrat th VPR masurmnts, and this was don by comparing th quivalnt rflctivity factors drivd from Drop siz distributions (DSDs). Th DSD masurmnts whr takn at ground by a Prcipitation Occurrnc Snsor Systm (POSS, dscribd by Shppard, 1990), collocatd with th VPR and th VHF radars. For th vnt on Sptmbr 9, 004, a tim lag of -1.5 minuts was addd to POSS obsrvations. This lag was dtrmind from th lowr panl in Figur 5, whr th maximum of th cross-corrlation function btwn POSS drivd and VPR masurd quivalnt rflctivity factors is locatd at zro lag whn -1.5 minuts ar addd to POSS tims. Th uppr panl in Figur 5 prsnts th tim sris (aftr th POSS tim lag corrction) of simultanous masurmnts by POSS (black continuous lin) and VPR (rd dashd lin). Th POSS valus (Z POSS) corrspond to th 10-minuts mdian at a hight of about mtrs (abov ground lvl, agl), whil th VPR valus (Z VPR) corrspond to th 10-minuts mdian at 450 mtrs agl (th lowst rang gat). Th undrstimation (du to attnuation)

5 by th VPR is clar. Thrfor, a VPR calibration factor, which compnsats for th rain attnuation at X band, is obtaind from ( t i ) ( t ) Z POSS Cal. Factor Mdian 6.70 Z VPR i.(11) Figur 6. VPR calibration from POSS DSDs drivd Z. For comparison, th rain signal masurd by th VHF radar is also plottd, as blu lins, in Figur 7. This rain signal was obtaind from th mthod dscribd in sction.3. For th cofficints in quation (7), w usd quation (11.c) by Campos t al. (005), i.., P cal ( 1.667x10 ± ) [ W ] 0 P ( 1.695x10 ± )[ W / au] + out (1) 4. DISCUSSION Figur 5. Lowr: Cross-corrlation function btwn POSS and VPR obsrvations. Uppr: Radar rflctivity factors simultanously masurd by th McGill VPR radar and th POSS drop siz distributions. In both panls, 1.5 minuts hav bn subtractd to POSS tim sris. Figur 6 th data from th uppr panl in Fig.5 as a scattr plot (139 pairs in total). Th black lin corrsponds to th hypothtical cas whn Z POSS and Z VPR ar qual, and th rd lin corrsponds to th cas whn th calibration factor in (11) is multiplid to Z VPR. Notic how this rd lin is locatd in th obsrvations cloud, which validats th us of (11) as attnuation corrctor. Th black dashd lin in Figur 7 prsnts th profil of quivalnt rflctivity factor masurd by th VPR, aftr all corrctions from th prvious two paragraphs hav bn applid. W usd this VPR profil as th input Z, for th algorithm dscribd in sction., in ordr to obtain Z (plottd as th rd histogram -lik lin). Notic that Z is in fact a simulation of th VHF Z from X band Z obsrvations. From this study, it is found that th spac-variabl rflctivity has a rlvant ffct on th radar quation only abov th mlting lvl. Abov ths hights, th sid lobs of th antnna polar diagram ar collcting nhancd powr from scattrrs locatd in th brightband (i.., rangs in th sid-lob dirction corrsponding to bright-band hight). From ths rsults, w also xpct that rain-only quivalnt rflctivity factors will b about th sam at X band than at VHF band (whn X band mas urmnts ar corrctd for attnuation). Thrfor, it is valid to us quation (1) for quantitativ masurmnts of rain by VHF radars. Howvr, th xprssion () has to b considrd whn daling with snow quantitativ masurmnts at VHF band. Diffrncs in Z from X and VHF band obsrvations can b du mainly to (a) incorrct radar absolut calibration, (b) K? 0.93, (c) ffct of th spacvariabl rflctivity and antnna sidlobs, (d) inaccurat simulation of th antnna polar diagram F, () non-uniformity of th raindrop fild obsrvd by th VHF radar (s Fabry, 1996, for implications). As futur work, all ths sourcs of rror should b xplord individually mor in dtail. For th singl rflctivity profil prsntd in Fig. 7, th comparison btwn th simulatd and masurd VHF rain signals (i.., rd and blu lins in Fig. 7) prsnts a bias in th ordr of 14 dbz. W associat this bias to th valus usd in quation (1), i.., th radar calibration. Thr is, howvr, good agrmnt in trms of th shap of VHF rflctivity profils blow th mlting lvl (i.., blow 4.5 km hight in Fig. 7). Basd on this agrmnt, this comparison can b usd as a calibration mthod for VHF ST radars, if th VHF radar

6 obsrvations ar xprssd in arbitrary units (of th analog-to-digital-convrtr in th rcivr). This nw calibration mthod rquirs th analysis of mor rflctivity profils, which will b undrtakn as futur work. Shppard, B., 1990: Masurmnt of Raindrop Siz Distributions Using a Small Dopplr Radar. Journal of Atmosphric and Ocanic Tchnology, 7, Skolnik, M.I., ditor, 1990: Radar Handbook, Scond Edition. McGraw-Hill. Availabl on th Intrnt at 1. Smith, P.L., 1984: Equivalnt Radar Rflctivity Factors for Snow and Ic Particls. Journal of Climat and Applid Mtorology, 3, Zawadzki, I., Fabry, F., Szyrmr, W., 001: Obsrvations of suprcoold watr and scondary ic gnration by a vrtically pointing X-band Dopplr radar. Atmosphric Rsarch, 59-60, Figur 7. Simulatd VHF Z profil (continuous rd lin) from obsrvd X band Z profil (black dashd lin). Th corrsponding VHF obsrvations ar plottd as blu lins. Co-locatd and simultanous radar obsrvations, at X and VHF bands, wr takn on Sptmbr 9, 004, at 15:30 UTC. Rfrncs Campos, E.F., Hocking, W., and Fabry, F., 1997: Powr Calibration of VHF Stratosphric-Troposphric radars. Prprints of th 3 nd Confrnc on Radar Mtorology, Albuqurqu, NM, USA, P1R7 in this issu, Amrican Mtorological Socity, Boston, MA, USA. Fabry, F., 1996: On th dtrmination of scal rangs for prcipitation filds. Journal of Gophysical Rsarch, 101, D8, 1,819-1,86. Hocking, W. K., 1997: Systm dsign, signal -procssing procdurs, and prliminary rsults fro th Canadian (London, Ontario) VHF atmosphric radar, Radio Scinc, 3(), Rinhart, R.E., 1997: Radar for Mtorologist, Third Edition. Rinhart Publications, Grand Forks, ND, USA, 48 pp.