Statistical dynamics of tropical wind in radiosonde data

13 Pages·2011·0.65 MB·English

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
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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
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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
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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
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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
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