Atmos. Chem. Phys.,11,4177–4189,2011 Atmospheric www.atmos-chem-phys.net/11/4177/2011/ Chemistry doi:10.5194/acp-11-4177-2011 ©Author(s)2011. CCAttribution3.0License. and Physics Statistical dynamics of tropical wind in radiosonde data T.-Y.Koh1,2,3,Y.S.Djamil1,andC.-K.Teo3 1SchoolofPhysicalandMathematicalSciences,NanyangTechnologicalUniversity,Singapore 2EarthObservatoryofSingapore,NanyangTechnologicalUniversity,Singapore 3TemasekLaboratories,NanyangTechnologicalUniversity,Singapore Received: 1May2010–PublishedinAtmos. Chem. Phys.Discuss.: 1July2010 Revised: 21March2011–Accepted: 28April2011–Published: 6May2011 Abstract. Weibull distributions were fitted to wind speed deviationtodetectoutliers)ratherthanondynamics(e.g.by data from radiosonde stations in the global tropics. A sta- examining properties emergent from statistical mechanics). tistical theory of independent wind contributions was pro- This may be because it is hard to generalize a single global posed to partially explain the shape parameter k obtained statisticaldynamicsthatisapplicabletowidelydifferentcli- over Malay Peninsula and the wider Equatorial Monsoon maticzones. Adoptingtheformer“statisticalmathematical” Zone.Thisstatisticaldynamicalunderpinningprovidessome approach results in smaller regions with denser station net- justification for using empirical Weibull fits to derive wind workexertinggreaterinfluenceintheformulationofQCcri- speed thresholds for monitoring data quality. The region- teriaandthresholdsthanlargerregionswithmoresparsenet- allyadaptedthresholdsretainmoreusefuldatathanconven- work. ThetropicallandmassesinSouthAmerica,Africaand tionalonesdefinedfromtakingtheregionalmeanplusthree Southeast Asia are good examples of the latter regions and standard deviations. The new approach is shown to elim- the quality of radiosonde data from these regions requires inate reports of atypically strong wind over Malay Penin- somescrutinyevenafterQC. sulawhichmayhaveescapeddetectioninqualitycontrolof In weather forecasting, modern data assimilation tech- global datasets as the latter has assumed a larger spread of niques incorporate additional QC based on the model first- wind speed. New scientific questions are raised in the pur- guessfieldsandin-builterrormetrics.Sodatavaluesthatare suitofstatisticaldynamicalunderstandingofmeteorological toodifferentfromfirstguessesmayberootedoutbeforeas- variablesinthetropics. similation. However,inthetropics,thequalityoffirst-guess fieldsmaysometimesbesuspectbecausemodelperformance is known to be poorer and less data is assimilated prior to making the first guess. Therefore, a QC methodology de- 1 Introduction pendentonlyonthecollecteddataitselfandunderpinnedby statisticaldynamicalunderstandingmaybeuseful,atleastas Radiosondeobservationsprovidearguablythemostreliable anindependentcheckofdataqualitybeforedataassimilation long-term meteorological data, especially before the advent andtheirassociatedQCchecks. ofsatellites. Theyareusedforroutineweatheranalysesand forecasts,aswellasvalidationofsatelliteretrievals(e.g.Di- In the recent decades, there has been emerging interest vakarlaetal.,2006;Stoffelenetal.,2005). Unprocessedra- in Southeast Asia by the international community studying diosonde data contain many types of error (Gandin, 1988) the global atmosphere. Neale and Slingo (2003) pointed andmustpassthroughqualitycontrol(QC)beforeuse. Be- out that the diurnal cycle in the maritime continent is not cause radiosonde data are collected all over the world un- well-captured by general circulation models (GCM) de- dertheauspicesofWorldMeteorologicalOrganization,QC spite its importance to global circulation (Ramage, 1968). methods are usually global in perspective and statistical in Zhu and Wang (1993) showed that strong interactions ex- nature (e.g. Durre et al., 2006). The statistical methods are ist over Southeast Asia between the Asian-Australian mon- usually based on mathematics (e.g. by the use of standard soon and the intra-seasonal oscillations spanning the global tropics (Madden and Julian, 1971, 1994). There has been moreresearchfocusedonthisregion’sclimateandweather, Correspondenceto: T.-Y.Koh e.g. on El Nin˜o impacts: Hendon (2003), Juneng and Tan- ([email protected]) gang (2005); on Southeast Asian monsoon: Lau and Yang PublishedbyCopernicusPublicationsonbehalfoftheEuropeanGeosciencesUnion. 4178 T.-Y.Kohetal.: Statisticaldynamicsoftropicalwindinradiosondedata (1997), Chang et al. (2005); on tropical cyclones: Chang 2 Dataoverview et al. (2003); on sea-breeze circulation: Hadi et al. (2002); Joseph et al. (2008). For the benefit of global and regional TwicedailyradiosondeobservationsweretakenfromtheDe- atmospheric research, besides gathering more data using partment of Atmospheric Science, University of Wyoming non-conventionalplatformsinSoutheastAsia(KohandTeo, (http://weather.uwyo.edu/upperair/sounding.html). Seven 2009), it is timely to re-examine the nature and quality of stations situated on the Malay Peninsula (MP) in Southeast conventional radiosonde data from this region. This paper Asia (Fig. 1) were used for the first part of this study. The reportsourinvestigationsintothestatisticaldynamicsofra- peninsula spans a region of about 1200km by 400km ori- diosondewinddatawhileon-goingworkontemperatureand entedintheNW-SEdirection. ItisroughlythesizeofGreat humiditydatawillbereportedinfuturepublications. BritainorCalifornia,USA.Itrepresentsaconvenientlysized One may reasonably pose a general question: could the region for which statistical homogeneity might be expected statisticsofasetofwinddatabeunderstoodfromunderlying tounderlietheprevailingmesoscaleconvectiveweather.The regionalatmosphericdynamicsandtherebyprovidingabasis sevenstationsspanthepeninsulauniformlyandtogetherpro- for better quality monitoring? Unlike the global problem, a vide35yearsofdatafrom1973to2007withsomegapsin- statisticaldynamicalapproachissoundinprincipleherebe- terspersedin-between. Another235stationsbetween25◦N cause the statistical properties of regional atmospheres are and25◦Swereusedtotesttheextensionofthefindingsfrom well determined by a few controlling factors from the re- MPtotheglobaltropics. gion’s climate (e.g. ambient stratification, humidity profile Wind speed at 00:00UTC and 12:00UTC on eleven and prevailing wind pattern) and for the planetary bound- mandatory pressure levels (1000, 925, 850, 700, 500, 400, arylayer(PBL),fromthesurfacecharacteristics(e.g.eleva- 300, 250, 200, 150 and 100mb) were used for all stations. tion,roughness,temperature,wetness). Butthereisacaveat: For MP (Table 1), data was available for less than half the theunderlyingstatisticaldynamicsmustberevealedthrough timeat12:00UTCforPhuketandSongkhla,whiledatafrom data before QC; otherwise, data that could possibly reflect SepangandPhuketcoverlessthan20years. Overall,theav- newphysicalunderstandingmayhavealreadybeencategor- eragenumberofstationsonthepeninsulareportingperday, ically rejected by existing QC methods based on statistical outofseventotal,liesintherangeof3.8to4.4at00:00UTC mathematics. and2.6to3.1at12:00UTCforallpressurelevelsexcluding Literatureonthestatisticalcharacterizationofwindspeed 925mb(whichhaslessdatabecauseitwasonlyadoptedas has mainly focused on the surface layer (e.g. Takle et al., amandatorylevelinthe1990s). 1978; Labraga, 1994; Lun, 2000)andtoalesserextent, the Wind data were given to the nearest 1knot and so where PBL(e.g.Franketal.,1997).Mostoftheliteratureemployed relevant,binsizeinstatisticalanalyseswasspecifiedinunits theWeibulldistribution(Wilks,1995)tomodelwindspeed. of knots to avoid artificial clustering of data if otherwise Justus et al. (1978) demonstrated that Weibull distribution specifiedinunitslikems−1. Abinsizeofδv=2knotswas fitssurfacewindbetterthanthesquare-root-normaldistribu- used for the frequency histograms (e.g. Fig. 2) throughout tion used by Widger (1977). The Rayleigh distribution is this work because this was the finest resolution practically anothercommonlyusedempiricalfitforsurfacewind(Man- achievable. A large difference in the data frequency was well et al., 2002) but this distribution is only a special case notedbetweenodd-andeven-valuedv inknots, whichmay oftheWeibulldistributionwithshapeparameterk=2. The indicatethatsomestationsactuallymeasureinintegralnum- authorsareunawareofanypublishedcharacterizationoftro- berofms−1butrecordinknotsafterapplyingtheconversion posphericwindusingWeibulldistribution,butfoundRoney 1ms−1≈1.944knots. (2007) who fitted the Weibull distribution to lower strato- spheric wind soundings. In all the reviewed literature, no 3 Methodology quantitative explanation was attempted for why the Weibull distributionisagoodfittothewinddata. Thefrequencyhistogramfornon-zerowindspeedat850mb Theobjectivesofthisworkaretwo-fold: (1)toelucidate from all 7 stations on MP without quality control is shown the statistical dynamics of tropical wind by analyzing long- inFig.2. Measurementsofzerowindspeedwereignoredas termrecordsofrawradiosondedatafromselectedstationsin theymayactuallydenotecalmconditionorlightwindspeed Southeast Asia and extending the results to the wider trop- whichradiosonderecordsdonotresolve.Equation(1)below ics; (2) to assess the feasibility of using that statistical dy- showstheprobabilitydensityfunction(PDF)oftheWeibull namicalunderstandingformonitoringthequalityofregional distributionthatwasempiricallyfittedtothewindspeeddata wind data. It is hoped that the results presented would mo- ateachpressurelevel: tivate similar statistical dynamical studies in other tropical k(cid:16)v(cid:17)k−1 (cid:20) (cid:16)v(cid:17)k(cid:21) regionswithsparsedatacoverage. P(v;k,c)dv= exp − dv (1) c c c where v is the wind speed, c is the scale parameter and k is the shape parameter. Maximum Likelihood Estimation Atmos. Chem. Phys.,11,4177–4189,2011 www.atmos-chem-phys.net/11/4177/2011/ T.-Y.Kohetal.: Statisticaldynamicsoftropicalwindinradiosondedata 4179 Table1. Tableshowstheperiodforwhichradiosondewinddatawereavailableat00:00UTCandat12:00UTCwiththeprecisestartand enddatesinthecolumnDateRange. (NotethatdatafromKuantancameintwoperiods.) Withineachperiod,theproportionofdaysfor whichdatawereactuallyavailablelieswithintheshownpercentagerangeforallpressurelevelsusedinthisstudy,except925mbwhichwas institutedasamandatorylevelonlyinthe1990sandsothelowerpercentageatthislevelisshowninparentheses. Thetotalperiodspanned andthenumberofstationreportsperlevelaregiveninthelastline. Dataperiodslessthantwentyyearsordataavailabilitylessthan50% (excluding925mb)arehighlightedinbolditalics. 00:00UTC 12:00UTC Station DateRange PercentageofData DateRange PercentageofData Available Available Phuket 20Sep1988–31Dec2007 51%–64%(43%) 31Jul1990–4Oct1994 24%–46%(4%) Penang 1Jan1973–31Dec2007 77%–83%(38%) 1Jan1973–31Dec2007 74%–80%(36%) Sepang 16Jul1999–31Dec2007 90%–93%(93%) 17Jul1999–31Dec2007 86%–92%(92%) Changi 24Aug1980–31Dec2007 84%–88%(54%) 19Jul1983–31Dec2007 51%–55%(28%) Kuantan 2Jan1973–9Jan2000 65%–77%(26%) 25Oct1973–30Nov2000 59%–71%(24%) 5Feb2005–30Dec2007 4Feb2005–30Dec2007 KotaBharu 1Jan1973–30Dec2007 60%–80%(36%) 1Aug1975–30Dec2007 52%–75%(36%) Songkhla 1Jan1973–31Dec2007 67%–80%(21%) 1Jan1973–24Jun1998 37%–49%(1%) DateRange Numberof DateRange Numberof StationReports StationReports Allstations 1Jan1973–31Dec2007 48520–55719(26369) 1Jan1973–31Dec2007 33153–39944(16960) 0.14 0.12 0.1 y c n e qu 0.08 e Fr e v 0.06 ati el R 0.04 0.02 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Wind Speed/c Fig.2.Frequencyhistogramofthescaledwindspeedat850mbfor all7stationsonMPfrom1973to2007usingabinsizeof2knots. The frequency is normalized over the total number of measure- ments.ThethickcurveistheempiricallyfittedWeibulldistribution withshapeparameterk=1.67andscaleparameterc=12.9knots (or6.62ms−1).Theboldverticallinedenotesthethresholdvmax= Fig. 1. Location of radiosonde stations on the Malay Peninsula 52.1knots,beyondwhichwindspeeddataisflaggedaserroneous. (MP)usedinthisstudy. Therewasaveryweakdependenceofkandcontherange (MLE) was used to determine k and c (Wilks, 1995). The ofrawdata[0,v ]towhichMLEwasappliedforsufficiently fit exponentk willbeshowntobeindicativeoftheunderlying large v . This dependence is inherently weak because log- fit statisticaldynamics. likelihoodforasetofn datapointsisdefinedas: tot www.atmos-chem-phys.net/11/4177/2011/ Atmos. Chem. Phys.,11,4177–4189,2011 4180 T.-Y.Kohetal.: Statisticaldynamicsoftropicalwindinradiosondedata ln‘d=efnPtotlnP(v ) 4 Empiricalresults i i=1 ⇒PPv0v<iv>iv<fivtfilntlPnP(v(iv)i)<n(nv0(v<i>viv<fivt)fit)(cid:28)1, Mramaxeitmerusma-sliwkeellilhaosodthreesstihmoladtessfoorfwthiendscsapleeedanadt esahcahpelepvae-l in MP are shown in Fig. 3. All Weibull fits are good ac- forsufficientlylargevfit cording to the χ2-test at 90% confidence level. Within the PBL (which for this paper is taken to be 850mb and where v is such that P(v) decreases monotonically for 0 below), c increases with height but above the PBL, it is v>v and n refers to the number of data points. The con- 0 nearlyinvariantupto500mbwithavaluearound13knots. tributionofextremevaluestothelog-likelihoodisevidently In the upper troposphere, c increases sharply with height negligibleandsokandcestimatedbyMLEarenotsensitive from 15.4±0.1knots at 400mb to reach a maximum of toerroneouslylargevaluesofthedata. Nonetheless,inorder 42.2±0.2knotsat150mb. to proceed in practice, the value of v used for the Weibull fit k has the smallest value of 1.54±0.01 at 1000mb. The fitwasselectedbyminimizingthemeanabsolutedifference value of kincreases upward from 1.67±0.01 at 925mb to between the fitted distribution and the frequency histogram values somewhat bigger than 2 in the upper troposphere overallavailabledata. (Othercriteriaforchoosingv were fit (300mbto150mb). Similarvaluesofaround5/3arenoted testedwithnosignificantdifferenceintheresults.) atthetropopauselevel(100mb)andinthePBL(925mband The goodness of fit of the Weibull distributions to the 850mb). histograms was tested with χ2-statistics at 90% confidence The scaled threshold v /c shows the opposite vertical level: max trend from k, which is expected from the threshold being χ2=ntotXnb (pobs,b−pfit,b)2 peggedtonfit=1: alargerk impliesastrongerdecayinthe τ b=1 pfit,b hPaDsFthaetllaarrggeesvt/cvaalnudehoefn4c.e6aastm10a0ll0ermsbcaalenddtihsraersohuonldd.4vminaxth/ce wherep andp aretheobservedandfittedprobability PBLandatthetropopause. Itdecreasesupwardfrom3.9at obs,b fit,b ofwindspeedlyinginbinb,n isthenumberofbinsinthe 700mbtovaluesaround3intheuppertroposphere. b histogram and n is the total number of data points. τ is tot ascalefactorthatcompensatesforthelackofindependence 5 Theoreticalbasis among nearby data points in time and is taken as the criti- calnumberofdaysbeyondwhichthelagauto-correlationof The literature mentioned in Sect. 1 rarely justified the use dailywindspeedisnotsignificantat90%confidencelevel. ofWeibulldistributionbeyondthefactthatitdoesyieldre- χ2 definedaboveisprobablyanupperboundonthetrueχ2 alistic fits to the observations. Moreover, most work dealt becauseeachstation’smeasurementisnotindependentofthe withsurfacewindforwhichtheunderlyingassumptionsmay others.Thus,thisχ2-testisratherstringent,butitsufficesfor notbeapplicabletothetroposphereoreventhePBL.There- ourpurpose. fore, the statistical dynamical underpinning of the Weibull Theroot-mean-square(rms)velocityσ isgivenby distribution for near-equatorial wind must be sought anew σ2d=efZ ∞v2P(v)dv=c2Z ∞t2/kexp(−t)dt (2) from our understanding of statistical dynamics. The ver- tical profile of c is largely dictated by the climatology of 0 0 planetary-scaleHadleyandWalkercirculationandtheAsian- =c20(2+1) k Australianmonsoon. Theverticalprofileofkistheobjectof where 0 is the gamma function (Arfken, 2000). This im- studyinthissection. plies that c is constrained by the climatological wind speed 5.1 ApproachtoGaussianstatistics measuredbyσ foragivenk. Inthiswork,itwasfoundthat k∈[1.3,2.6] which implies [0(2/k+1)]−1/2∈[0.86,1.04]. Suppose the horizontal wind vector v may be decomposed Thusinpractice,c≈σ andcisagoodindicationofthecli- into numerous contributions associated with different tropi- matologicalwindspeed. calmeteorologicalphenomena: FromthePDF,theexpectednumbern (v)ofwindspeed fit X reportsinthebin[v,v+δv]ofthefrequencyhistogramwas v= v (3) n computedforthesizeofthedataset. Thewindspeedthresh- n old v was defined such that n (v)≤1 for v≥v . By max fit max For instance, v can arise from diurnally excited gravity thefittedWeibulldistribution,windspeedrecordslargerthan 1 waves(Rotunno,1983),v fromequatorialwaves(Wheeler v areunlikelytobereliableforthegivendatasetsize. It 2 max andKiladis,1999),v fromintra-seasonaloscillations(Mad- wascheckedthatv wassmallerthanv atalllevels,con- 3 max fit denandJulian, 1971, 1994; Waliser, 2006), v fromAsian- firmingthevalidityoftheWeibullfitincludingvaluesaround 4 Australian monsoon (Wang, 2006), v from Walker circu- v . 5 max lation (Katz, 2002) under inter-annual variations etc. One Atmos. Chem. Phys.,11,4177–4189,2011 www.atmos-chem-phys.net/11/4177/2011/ T.-Y.Kohetal.: Statisticaldynamicsoftropicalwindinradiosondedata 4181 pendix A shows that the Central Limit Theorem is still ap- (a) 100 plicable even in the case where the set of vn has members withnon-zerocovariance.) mb) 400 Thus,inthelimitoflargeN,windvelocityv followsthe e ( ur Gaussiandistribution, Press 700 P(v=(cid:20)vx(cid:21))d2v= 1 exp"−(cid:18)vx−v¯x(cid:19)2# (4) 1000 vy πc2 c 0 5 10 15 20 25 30 35 40 45 c(knots) " (cid:18)v −v¯ (cid:19)2# (b) ·exp − y y dv dv x y 100 c mb) 400 wherec2 isthevarianceofv andistwicethevarianceofv e ( x essur 700 or vy because the wind velocity anomaly is assumed to be Pr isotropic. Theisotropicassumptionissupportedbythevir- tualabsence,oratbest,weaknessofanisotropyfromobser- 1000 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 vations(e.g.Mori,1986;Ibarra,1995;KohandNg,2009). k (c) Forzeromeanwind, integratingoveralldirectionsθ, the 100 PDFforwindspeedvistheRayleighdistribution: mb) 400 Z2π 2v (cid:20) (cid:16)v(cid:17)2(cid:21) essure ( 700 P(v)dv= P(v)vdvdθ≈ c2exp − c dv (5) Pr 0 whichisalsoWeibulldistributionwithshapeparameterk = 1000 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 2. ItmustbeemphasizedthatthederivationofEq.(5)does V /c max not depend on the statistical dynamics of each contribution v (orw )exceptfortheempiricallyjustifiableassumptions n n Fig. 3. Plots of empirically fitted attributes of the Weibull distri- of isotropy. The assumption of zero mean wind will be ex- butionforwindspeedatdifferentpressurelevelsinMP:(a)scale aminedlater.Thus,thestatisticaldynamicsofmanyindepen- parameterc;(b)shapeparameterk,(c)scaledthresholdvmax/cfor dent contributions from tropical meteorological phenomena windspeed. Errorbarsforcandk areestimatedbyMLEat95% confidencelevel. Verticaldashedlinescorrespondtok=5/3,2in mayexplainwhyk≈2isobservedgenerallyintheuppertro- (b)andvmax/c=3,4in(c).Crossesin(b)denotetheoreticallower posphereinFig.3b. Infact,Rayleighdistributioncannotbe boundfork forwindanomalymagnitude. Asterisksin(c)denote rejectedbytheχ2-testat90%confidencelevelforlevelsbe- the threshold vm3sd (mean plus three standard deviations) at each tween500mband150mbinclusive. (Notethatthisdoesnot pressurelevel. meanthatRayleighdistributionisthebest-fitdistribution.) 5.1.1 DeparturefromGaussianstatistics mightevensplitthecontributionsamongsub-categories,dis- For the levels below 500mb and at 100mb, the χ2-test re- tinguishing between: land-sea and mountain-valley diurnal jects Rayleigh distribution at 90% confidence level. More- circulations; Kelvin and Rossby waves of different equiva- over, MLE fits of Weibull distribution did show k<2 for lent depth; monsoon cold surges and westerly wind bursts; thoselevelsandsmallbutsignificantdeviationsfromk=2in El Nin˜o – Southern Oscillation (ENSO) and Indian Ocean theuppertroposphere. Furthertheoreticalunderstandingfor Dipole (IOD). But the detailed cause of each v is not im- n such departure from Gaussian behavior is sought below by: portanttothefollowingargumentaslongastherearemore (1)examinationtheeffectofnon-zeromeanwind;(2)intro- thanafewindependentv contributingtov.Overalongtime n ducing Shannon’s entropy as a measure of non-Gaussianity suchas35years,thesetofvaluesthateachv takesmaybe n and explaining its variation in the lower and upper tropo- reproducedbytherealizationsofarandomvariablewithits sphere. own characteristic probability distribution. Note that this is NOTsayingthateachcontributionactuallyvariesrandomly 5.1.2 Non-zeromeanwind intime. Assumingeachrandomvariablevn inEq.(3)isindepen- In the presence of non-zero mean wind, Appendix B shows dent of the others, the Central Limit Theorem implies that thatforisotropicGaussianwindanomaliesasinEq.(4),the P thePDFof vn approachesGaussiandistributionastheto- PDFforwindspeedvis NvtaaolrtnieaunmtchebateortfhoefPrmavneadnoismoftvhPaerisvaunbmliesso,thfNet,hsieunmcvraeoraifsaemnsceewaniothfvonvu,ta.bnod(uAnthpde-. P(v)dv=exp −vcm22!·I0(cid:18)2vcm2v(cid:19)·2cv2exp −vc22!dv (6) n n www.atmos-chem-phys.net/11/4177/2011/ Atmos. Chem. Phys.,11,4177–4189,2011 4182 T.-Y.Kohetal.: Statisticaldynamicsoftropicalwindinradiosondedata 100 5.1.3 Shannon’sentropy (a) Rayleigh (b) vm/σ=0.49 200 0.18 vvkmm=2//σσ.2==00..6894 340000 Svahrainabnloenv’sisendterfionpeydfaosr a random two-dimensional vector σP(v/)00..46 k=2.9 pressure (mb) 567000000 Ent[P(v)]d=ef−ZaZllv P(v)lnP(v)d2v (7) 800 Among all probability density functions P(v) of unit vari- 0.2 900 MEMPZ ance,Shannon’sentropyismaximalfortheGaussiandistri- 0 1000 bution only (Artstein et al., 2004). Thus, the Central Limit 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 1.2 v/σ mean wind (non−dimensionalized) Theorem may be understood as an approach towards max- imal Shannon’s entropy. Small Shannon’s entropy denotes Fig.4. (a)TheRayleighdistribution(bolddashed)iscomparedto strongdeparturefromGaussianityinthedistribution. thePDFofwindspeedwhenthemeanwindvmisnon-zero(solid) Assuming that isotropy prevails, Appendix C shows that assuming Gaussian wind anomalies. Examples of Weibull distri- Shannon’s entropy of Weibull distributions of unit variance butionswithk>2(dashed)arealsoshownforcomparison. Wind speed v is normalized by its rms value σ for each distribution to isrelatedtotheshapeparameterk, facilitatecomparison. (b)Normalizedmeanwindvm/σ fortheset Ent[P(v/σ)]=ln(cid:16)2π(cid:17)−lnh0(2+1)i+(1−2)γ+1 (8) of7stationsonMalayPeninsula(MP)andmeannormalizedwind k k k um forEquatorialMonsoonZone(EMZ),whereu=v/σ foreach where 0 is again the gamma function (Arfken, 2000) and station. γ isthe Euler-Mascheroni constant(Whittakerand Watson, 1996). The expression shows that Shannon’s entropy has maximum value of E =lnπ+1 at k=2 and decreases where v is the magnitude of the mean wind vector and I max m 0 monotonically for k larger or smaller than two (graph not is the modified Bessel function of the first kind. When v m shown). Thus, Shannon’s entropy corresponding to the k- is zero, the underlined factor in Eq. (6) is unity, recovering values at 500mb and below in Fig. 3b was computed and Rayleigh’sdistribution. Thisfactorcomprisestwoterms: as showninFig.5. Shannon’sentropyatupperlevelswasnot v increases, the exponential function decreases while the m computedbecausetheeffectofstrongmeanwindonP(v/σ) modifiedBesselfunctionincreases. Thetwotendenciestend implies that Eq. (8) is not applicable to wind speed but to to balance for small v , but the latter wins out for large v m m windanomalymagnitudeonly. andsothePDFdepartsincreasinglyfromRayleighdistribu- FromEq.(8),tounderstandthevariationofk istounder- tion. standthevariationofShannon’sentropy. InAppendixD,we Theeffectofnon-zeromeanwindv onthePDFinEq.(6) m showthatnotwithstandingtheCentralLimitTheorem,when was computed andis shown in Fig. 4a, wherethe PDFs are the variances are non-uniform among the velocity contribu- expressedintermsofwindspeednormalizedbyitsrmsvalue tions v in Eq. (3), Shannon’s entropy of P(v/σ) can de- σ tofacilitatecomparison. Forv /σ<0.5,theeffectcanbe n m creasebyanamountasmuchas1E asthenumberofinde- neglected as P(v/σ) approximates that for Rayleigh distri- pendentcontributionsN increases,i.e.theapproachtoGaus- bution. While for v /σ >0.5 the effect of non-zero mean m sianityisnotmonotonicingeneral. ForlargeN,thetheoret- windissignificant,thePDFcaninpracticebeapproximated icallowerboundforShannon’sentropy,(E −1E),could byWeibulldistributionswithk>2(cf.dashedandsolidlines max beestimatedroughly(seeAppendixDfordetails). inFig.4a). ThisexplainswhyWeibullfitsarestillgoodeven The vertical trend in (E −1E) in Fig. 5 shows that inthepresenceoflargemeanwind. max even for large N, wind anomalies can have the most depar- The magnitude of the mean wind v in MP is shown in m turefromGaussianitybetween400mband150mb,whichis Fig.4b,normalizedbythermswindspeedσ overtheregion. At500mbandbelow,v /σ≤0.49,whichimpliesthemean consistentwithourdeductionattheendofSect.5.1.2. Thus, m k6=2 in the upper troposphere may arise in part from the windhasnegligibleeffectonP(v)andisnotthemaincause ofk6=2atthoselevels. At400mbandabove,v /σ≥0.84. non-uniformvarianceamongthewindcontributionsvn. For m So,usingFig.4a,WeibullfitstoP(v)wouldresultink=2.9 illustration, rough estimates of the theoretical lower bound for k for wind anomaly magnitude were computed from orlargeratthoselevels,ifthewindanomalieswereGaussian. (E −1E) by inverting Eq. (8) (crosses in Fig. 3b). The Thus, strong mean wind can explain why k tends to some- max decreasing effect on k by non-uniform variance appears to timesovershoot2atupperlevels,butitisalsoclearthatthe competewiththeincreasingeffectonkbystrongmeanwind windanomaliesarenon-Gaussianbecausek isconsiderably lessthan2.9(Fig.3b).Theremustbeanothercausefork6=2 in the upper troposphere, resulting in k close to and some- timesovershooting2. intheuppertropospherethatreducesthevalueofk. From 925mb to 500mb in Fig. 5, 1E is negligible be- cause the variance is roughly uniform among wind contri- butions v . But Shannon’s entropy is much less than E . n max Atmos. Chem. Phys.,11,4177–4189,2011 www.atmos-chem-phys.net/11/4177/2011/ T.-Y.Kohetal.: Statisticaldynamicsoftropicalwindinradiosondedata 4183 0 55 50 100 45 s) 200 knot 40 ata ( 35 D 300 (left) BL value eed 30 (right) max value Sp 400 theoretical bound nd 25 mb) observed al Wi 20 essure ( 500 Region 15 pr 10 600 5 700 0 0 50 100 150 200 250 300 Global Wind Speed Data (knots) 800 Fig. 6. The 850mb-wind speed over MP from Wyoming data 900 archiveafterscreeningwiththethresholdvmax(52.1knots)plotted againstthewindspeedfromIGRAdatabaseforthesamestations 1000 andperiod.Eachorderedpairdenotedbyacrossreferstothesame 0.01%-quantileinbothdatasets.Thestraightlineshowswherethe 2.07 2.08 2.09 2.1 2.11 2.12 2.13 2.14 2.15 2.16 Shannon’s entropy Ent[P(v/σ)] crossesshouldlieifbothdatasetshadthesamedistribution. Fig.5.Shannon’sentropyfortheWeibull’sdistributionsfittedtothe radiosonde windspeed datafrom MPat 500mb and below(solid A common statistical threshold to reject outlying data is line). Thewindspeedwasfirstnormalizedbyitsrmsvalue. The the mean plus three standard deviations. For a variable v, maximalShannon’sentropyandthevalueassociatedwithk=5/3 this threshold is denoted as v . In Fig. 3c, v is com- m3sd max (dotted lines) are shown. The theoretical bound (dashed line) for paredwithv ,wherethemeanandstandarddeviationare reductionfrommaximalShannon’sentropyiscomputedforwind m3sd computedfromtheMPdataonly. Atallpressurelevels,the anomalyusingEq.(8). thresholdv islargerthantheregionalv ,implyingthat max m3sd moreusefulregionaldataisretainedinourstatisticaldynam- This means N is not large enough for the wind anomalies icalapproachratherthanthecommonstatisticalmathemati- to approach Gaussianity. Artstein et al. (2004) proved that calapproach. when the velocity contributions vn have uniform variance, TheMPdatabelowourwindspeedthresholdvmaxiscom- Shannon’sentropyincreasesmonotonicallyasthenumberof paredwithdatafromIntegratedGlobalRadiosondeArchive contributionsN increases(seeAppendixD).Thisisconsis- (IGRA) (Durre et al., 2006) in Fig. 6. The finding is that tentwiththetrendinShannon’sentropyinthelowertropo- a theoretically sound regional data-monitoring strategy can sphere,suggestinganincreaseinthenumberofindependent identifyerroneouslyhighwindspeedthatescapesdetection contributionsN withheight. intheQCofglobaldatasets. Thisispossiblybecauseglobal QC assumes a larger spread of wind values than is valid within a specific region like MP. Similar large erroneous 6 Applicationtomonitoringdataquality wind speed in Indonesia reported over the Global Telecom- munication System (GTS) was also noted by Okamoto et The preceding understanding for Weibull distribution of al.(2003). wind speed supports the view that beyond the empirical threshold of validity of the distribution, v , wind speed In modern data assimilation systems, such as used max by European Centre for Medium-Range Weather Forecast data are likely to be dominated by noise and hence are sus- (ECMWF),east-westandnorth-southwindcomponentsare pect. Itfollowsnaturallytoapplysuchthresholdstomonitor analyzed separately and often assumed to follow Gaussian thequalityoftheradiosondedatafromMP.Fordemonstra- distributions of equal variance. The non-Gaussianity iden- tionpurpose,dataatthreemandatorylevels,850mb,500mb tified in the last section, especially in the lower tropo- and 250mb, were selected. Results showed that about half sphere where the mean wind is weak may be cause for re- a percent of 278711 available wind speed records at these examinationoftheseassumptions. Moreover,ineliminating threelevelsaresuspect. unrealisticwindspeed(e.g.v>v =52.1knotsat850mb max inMP,seeFig.6),theproposedQCmethodwouldraisethe www.atmos-chem-phys.net/11/4177/2011/ Atmos. Chem. Phys.,11,4177–4189,2011 4184 T.-Y.Kohetal.: Statisticaldynamicsoftropicalwindinradiosondedata ° 25 N ° 10 N ° 10 S ° 25 S ° ° ° ° ° ° ° 180 W 120 W 60 W 0 60 E 120 E 180 E Fig.7.All242tropicalradiosondestationsusedinthelatterpartofthestudy(includingthesevenstationsinMP):“+”signsdenotestations intheupper-level(500mbto100mb)westerlyzone;circlesdenotestationsintheupper-levelmixedwindzone;allothersymbolsdenote stationsintheupper-leveleasterlyzone. Withintheeasterlyzone, stationsaredenotedbytheirgeographicalregions(numberofstations showninbrackets): “x”sign=Africa(6);asterisk=SouthAsia(4);dot=SoutheastAsia(31);triangle=IndianOcean(2);diamond=West Pacific(6). qualityofdataassimilatedandincrementallyimprovemodel (a) Westerly Zone analyses and re-analyses in the tropics. This would eventu- 100 ally contribute to the quality of model first-guess fields so mb) 400 thattheycouldbeusedmorereliablytochecktropicalobser- e( ur vationsatthetimeofassimilation. ess 700 Pr 1000 7 ExtensiontoEquatorialMonsoonZone 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 In this section, the preceding statistical dynamical theory (b) Mixed Wind Zone fortheWeibulldistributionofradiosondewindandthedata 100 monitoringstrategydevelopedfromitaretestedfortheirrel- evancetoothertropicalregions. mb) 400 e( 242stationsacrosstheglobaltropics(includingMP)were essur 700 firstdividedintothreeclimaticzonesaccordingtothetime- Pr averagedzonalwindintheuppertroposphere(i.e.mandatory 1000 levelsfrom500mbto100mbinclusive): (a)westerlyzone: 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 everylevelshowswesterlymeanwind;(b)mixedwindzone: both westerly and easterly mean wind are present; (c) east- (c) Easterly Zone erly zone: every level shows easterly mean wind (Fig. 7). 100 The existence of the mixed wind zone in the equatorial belt and its significance to cross-equatorial propagation of e(mb) 400 Rossbywaveshavebeennotedbefore(WebsterandHolton, essur 700 1982). Onapressurelevel, eachstationisapointmeasure- Pr mentandwouldunder-sampletheunderlyingdynamics.The 1000 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 k-valueateachstationwouldbehavelikearandomvariable k itselfwithaprobabilitydistribution. Comparisonofthever- Fig.8. Scatterplotshowingthevaluesofkacrossverticallevelsat ticalprofilesofmediank-valuesinFig.8withFig.3bshows tropicalstationsinthethreeupper-levelclimaticwindzones. For thatthestatisticaldynamicsinthewesterlyandmixedwind eachzone:medianvaluesateachlevelareconnectedtoshowaver- zonesareprobablydifferentfromthatoverMP,butthestatis- ticalprofile;thedelimitedhorizontalbarsdenotetheinter-quartile ticsintheeasterlyzonewarrantfurtherinvestigation. range at each level. For the easterly zone, k-values of stations in Thek-valuesinWestPacific(diamondsinFig.8c)arecon- Africa (“x” sign), South Asia (asterisk), Southeast Asia (dot), In- sistently larger than most other values in the easterly zone dianOcean(triangle)andWestPacific(diamond)aremarkedwith andbearcloserresemblancetothoseinthemixedwindzone. thesamesymbolsasinFig.7. TheWestPacificstationsarealsotheonlyonesintheeasterly windzonethatdonotliewithinthemonsoonregionaccord- UnlikeMP,EMZspanshalftheglobeacrossvaryingcli- ingtofigure1.2ofRamage(1971).Therefore,theEquatorial matology of wind speed (Fig. 9a). So at each pressure Monsoon Zone (EMZ) is defined to encompass the stations level, wind speed v from each station must be normalized intheeasterlywindzoneexcludingtheWestPacificstations. byitsrmsvalueσ estimatedinEq.(2)beforethecombined datasetofu=v/σ canconstituteastatisticallyhomogeneous Atmos. Chem. Phys.,11,4177–4189,2011 www.atmos-chem-phys.net/11/4177/2011/ T.-Y.Kohetal.: Statisticaldynamicsoftropicalwindinradiosondedata 4185 uctofthelocalσ andtheregionalthresholdu derivedes- 100 100 max (a) (b) sentiallyfromkEMZ: theexpectednumbernfit(u,δu)≤1for u≥u usingabinsizeδuof2knotsdividedbythemean max mb) 400 mb) 400 σ ateachlevel. Awayfromthesurface,vEMZ islargerthan sure ( sure ( thevmEM3sZdforabout90%ormoreofthestatmioaxns,wherevmEM3sZd Pres 700 Pres 700 is the statistical mathematical threshold computed from the unionsetofallvmeasurementsinEMZforeachlevel. Asa threshold to flag off suspicious outlying radiosonde reports 1000 1000 in EMZ, vEMZ preserves more useful data than vEMZ be- 0 20 40 0 0.5 1 1.5 max m3sd σ (knot) σ causevEMZ capturestheregion’sstatisticaldynamicsandis EMZ max adaptedtothelocalwindclimatology. At1000mb,vEMZ is m3sd notasuitablereferenceforcomparisonaslocalv should m3sd 100 100 beusedinstead. (c) (d) Tounderstandthevaluesofk ,thecorrespondingShan- EMZ mb) 400 mb) 400 median non’sentropyforunormalizedtormsvalueof1isshownin e ( e ( (left) 25%−tile Fig.10. Asbefore,Shannon’sentropywasnotcomputedfor ssur ssur ((rleigfth)t )1 07%5%−t−ilteile 400mbandabovebecauseoftheeffectofstrongmeanwind Pre 700 Pre 700 (vrmEigM3hsZdt) 90%−tile (Fuigm.>9c0a.6n9d)hoenncPe(fuo/rσSEhMaZn)no(Fni’gs.e4n)t.roTphyeinerFroigr.b1a0rswfoerrekeisn- timated by generating another two sets of best-fit k by 1000 1000 EMZ 1.6 1.8 2 0 100 200 separately removing the stations with the top or bottom 5 kEMZ VmEMaxZ (knot) percentileofk-values(i.e.toporbottom2stations)andcom- putingthestandarddeviationamongthethreesetsofbest-fit Fig.9.(a)Scatterplotofrmswindspeedσ computedusingEq.(2) k . The theoretical lower bound for Shannon’s entropy, EMZ atall43stationsintheEquatorialMonsoonZone(EMZ).Theverti- (E −1E),couldalsobeestimatedroughlyforlargeN as max calprofileconnectsthemedianvaluesanddelimitedhorizontalbars before(seeAppendixDfordetails). denotetheinter-quartileranges. (b)ThermsvalueσEMZ ofnon- dimensionalwindspeedu=v/σ computedfromthefittedWeibull ComparedtotheMPresultsinFig.5,theEMZresultsin distribution of u in EMZ using Eq. (2). (c) The shape parameter Fig.10showthatthelowertroposphere(1000mbto500mb) kEMZ describing the fitted Weibull distribution of u in EMZ with is nearer to attaining maximal entropy because there is a errorbarsshown. Crossesdenotetheoreticallowerboundforkfor largernumberofindependentvelocitycontributionsN aris- windanomalymagnitude. (d)Thespreadofthresholdwindspeed ingfromspatialde-correlationwithinEMZ.Below850mb, (symbols)amongthestationsinEMZcomparedto(line). it appears that the non-uniformity of variance among the contributions across EMZ may be keeping Shannon’s en- tropyawayfromthemaximalvalue(dashedlineinFig.10). populationthatdescribablebyaWeibulldistributionofdis- Above 500mb, competing effects from non-uniform vari- tinctshapeandscaleparameters(k ,c ). Becausethe EMZ EMZ anceamongvelocitycontributions(thatdecreasek)andlarge combined dataset is the union of normalized subsets of rms meanwind(thatincreasek)tendtobalanceleadingtok≈2, value1,σ computedfromk andc shouldbe1for EMZ EMZ EMZ although nearer the tropopause the former effect seems to alargedatasetifσ ateachstationcorrectlycapturesthecli- dominate(crossesinFig.9c). Notethatk≈2doesnotimply matologicalwindspeed.FromFig.9b,weseethatσ ≈1 EMZ Gaussianity(buttheconversewouldbetrue). for all levels indeed. The largest difference of σ from EMZ one is only −0.055 and occurs at 1000mb, possibly due to thecomplicatinginfluenceoflocalterrainandsurfacechar- acteristics. 8 Summaryanddiscussion Itwasnotpossibletocarryouttheχ2-testfortheWeibull fitforthewindspeedintheEMZbecausethespatialautocor- Empirical Weibull distributions of wind speed were derived relationishardtoestimatereliablyfromanirregularstation byMaximumLikelihoodEstimateforradiosondedataspan- distribution. However,thefactthatσ ≈1isindirectbut ningmorethan30yearsfrom7stationsintheMalayPenin- EMZ clear evidence that the Weibull distribution is a good fit be- sula (MP) and from 43 stations in the Equatorial Monsoon cause otherwise, Eq. (2) would have yielded wrong values Zone (EMZ). The Weibull distribution is governed by two not only for σ but also for σ at each station in the first parameters: the shape parameter k is the key quantity in- EMZ place. vestigatedinthispaper; thescaleparameterc isdetermined The characteristic profile of k and the associated by a given k and the rms wind speed σ which is the re- EMZ thresholdforwindspeedvEMZareshowninFig.9canddre- sultofplanetary-scaleclimatedynamics. WindinEMZwas max spectively. Ateachstation,vEMZ wasestimatedastheprod- non-dimensionalized by the local σ to remove the effect of max www.atmos-chem-phys.net/11/4177/2011/ Atmos. Chem. Phys.,11,4177–4189,2011 4186 T.-Y.Kohetal.: Statisticaldynamicsoftropicalwindinradiosondedata 0 climatology. SuchanimproveddatasetfromEMZwouldul- timatelybenefitresearchandforecast. 100 The existence of non-Gaussianity in the troposphere ap- pears to be a natural consequence of non-linear dynamical 200 models. Sardeshmukh and Sura (2009) showed in an adi- abatic GCM that skewness and excess kurtosis (which are 300 identicallyzeroforGaussiandistributionandhencerepresent non-Gaussian behaviour) are associated mainly with small- (left) BL value 400 (right) max value scale turbulent fluxes. Interestingly, they also showed that b) theoretical bound the statistical relation between skewness and excess kurto- m ure ( 500 observed siscanbereproducedinlinearstochasticmodelswhenaddi- ess tive(state-independent)andmultiplicative(state-dependent) pr Gaussianwhitenoisesarecorrelated.Thismayhintatfurther 600 investigation of the non-Gaussianity identified in this work withlinearequatorialwavemodels. 700 Thecurrentworkalsoraisesspecificquestions: (1)What are the physical causes of the dominant velocity contribu- 800 tions? (2) Why does the number of independent velocity contributionsN seemtoincreasewithheight? (3)Doesthe 900 seemingly common value of k=5/3 observed in the PBL (850 and 925mb) and at the tropopause (100mb) in MP 1000 reflect any statistical dynamics occurring at local scales at 2.07 2.08 2.09 2.1 2.11 2.12 2.13 2.14 2.15 2.16 thoselevelsorisitmerecoincidenceinthisdataset?(4)How Shannon’s entropy Ent[P(u/σ )] EMZ do we understand the profile of k outside of the EMZ? (5)Whatdistributionsdotropicaltemperatureandhumidity Fig.10. Figure10showsShannon’sentropyat500mbandbelow for the Weibull’s distributions fitted to the non-dimensional wind follow and what are their underlying statistical dynamics? speedu=v/σ fromEquatorialMonsoonZone(solidline). uwas Thesequestionsandothersleavemuchroomforexcitingre- firstnormalizedbyitsrmsvalueσEMZ asthelatterisclosetobut searchintothestatisticaldynamicsofregionalatmospheres. notexactly1. Themaximalentropyandthevalueassociatedwith k=5/3(dottedlines)areshown, aswellasthetheoreticalbound (dashedline)forreductionfrommaximalentropy. AppendixA Non-zerocovariancebetweenvelocitycontributions geographic variation of climatology before empirical fitting oftheWeibulldistribution. If a particular v has non-zero covariance with another v , n m A statistical theory of independent physical contributions the two velocity contributions would not be independent. totheobservedwindwasproposedtoexplaintheobservedk However,itiseasytodefinetwonewvariablesasfollows: asfollows. w d=efv −cov(v ,v )[var(v )]−1v 1. The increase in the number of such contributions N n n n m m m causes Shannon’s entropy to rise and the value of k to wmd=ef(cid:8)1+cov(vn,vm)[var(vm)]−1(cid:9)vm approach2fromthelowertomid-troposphere. where the variance and covariance for vectors are defined 2. Intheuppertroposphere,Nislikelytobelarge. Butthe in e.g. Feller (1968). The contributions to v could be re- non-uniformity of variance among the velocity contri- expressedas butions prevents Shannon’s entropy from attaining the v +v ≡w +w maximal value and tends to decrease k, while strong n m n m meanwindtendstoincreasek. Thus,khasvaluesclose It is readily verified that the new random variables, w and n to2(EMZ)orsometimesmayovershoot2(MP). w , have zero covariance. In this way, the Central Limit m Best-fit Weibull distribution can be used to derive confi- Theoremmaybeappliedasbefore. dence thresholds for monitoring radiosonde wind speeds. The thresholds are generally larger than those obtained by takingthemeanplusthreestandarddeviations. Moredatais retained and data quality is improved because these thresh- oldsarebasedonanunderstandingofthestatisticaldynam- icsofnear-equatorialwindandtheyareadaptedtothelocal Atmos. Chem. Phys.,11,4177–4189,2011 www.atmos-chem-phys.net/11/4177/2011/