DTIC ADA464013: The Evaluation of Uncertainties in [UTC-UTC(k)]

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INSTITUTEOFPHYSICSPUBLISHING METROLOGIA
Metrologia43(2006)278–286 doi:10.1088/0026-1394/43/3/010
The evaluation of uncertainties in
−
[UTC UTC(k)]
WLewandowski1,DMatsakis2,GPanfilo3,4 andPTavella4
1BureauInternationaldesPoidsetMesures,Se`vres,France
2UnitedStatesNavalObservatory,USA
3PolitecnicodiTorino,Italy
4IstitutoElettrotecnicoNazionaleGalileoFerraris,Italy
E-mail: [email protected],[email protected],panfi[email protected]
[email protected]
Received24November2005
Published19May2006
Onlineatstacks.iop.org/Met/43/278
Abstract
Thisworkpresentsastudyofthedeterminationofuncertaintiesin
[UTC−UTC(k)]neededforpublicationintheBureauInernationaldes
PoidsetMesures’s(BIPM’s)CircularTandtheKeyComparisonDatabase,
asrequiredbytheMutualRecognitionArrangement. Inthefirstpartofthe
paper,ananalyticalsolutionbasedonthelawofthepropagationof
uncertaintyisderived. Inthesecondpart,thesolutionisverifiednumerically
usingthesoftwareusedbytheBIPMforthegenerationofUTC.
(Somefiguresinthisarticleareincolouronlyintheelectronicversion)
1. Introduction In this work, a study of the determination of the
uncertainties of [UTC −UTC(k)] is presented. In the first
Coordinated Universal Time (UTC), the worldwide time part of the paper, an analytical solution based on the law of
standard, is disseminated monthly through publication of the propagation of uncertainty is derived. The second part
[UTC −UTC(k)] in Circular T by the Bureau Inernational
presentsanumericalverificationoftheanalyticalresultsusing
desPoidsetMesures(BIPM).UTC(k)isarealizationofUTC
thesoftwareusedforthegenerationofUTC.
bylaboratoryk. Thesevaluesbefore2005havebeenpublished
withouttheiruncertainties,buttherulesoftheCIPMMutual
RecognitionArrangement(CIPMMRA)requiretheevaluation
2. ThealgorithmsofEAL,TAIandUTCandthe
ofthisuncertainty. Thispaperreportsthefirststepstowards
workinghypothesis
computingtheseuncertainties.
UTCisderivedfromInternationalAtomicTime(TAI)by
Theuncertaintyof[UTC−UTC(k)]canbederivedusingthe
the addition of leap seconds; while TAI is derived from the
Free Atomic Timescale (EAL) through the addition of pre- generalequationoftheFreeAtomicTimescale(EAL),which
announced frequency steers determined by comparison with isdefinedusingtheALGOSalgorithm[1–3]as
a weighted set of primary frequency standards. EAL is a
worldwideweightedaverageofalargenumberoffree-running, (cid:1)N
effectively uncalibrated, frequency standards [1–3]. The EAL(t)= wi[hi(t)+h(cid:1)i(t)], (1)
uncertaintyinthedeterminationofEAL,TAIandUTC,assteps i=1
intherealizationofTerrestrialTime(TT),isaffectedbythree
where N is the number of the atomic clocks, w the relative
major elements: clock variations, the means of comparisons i
weight of the clock H, h(t) is the reading of clock H at
ofremoteclocks(timetransfer)andthetime-scalealgorithm. i i i
The uncertainties of time transfer are particularly significant time t and h(cid:1)i(t) is the prediction of the reading of clock Hi
over averaging times of up to a few tens of days and also to guarantee the continuity of the time scale. The weight
influence the uncertainty in the access of the participating attributedtoagivenclockreflectsitslong-termstability,since
laboratories k to UTC (in other words, the uncertainty of theobjectiveistoobtainaweightedaveragethatismorestable
[UTC−UTC(k)]). inthelongtermthananyofthecontributingelements[4,5].
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The Evaluation of Uncertainties in [UTC-UTC(k)]
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Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
Theevaluationofuncertaintiesin[UTC−UTC(k)]
Theweightsoftheclocksobeytherelation Because[UTC−EAL]dependsonlyonpre-determined
leapsecondsandfrequencysteersthatdonotadduncertainty,
(cid:1)N
and because UTC(k) is deterministically derived from the
wi =1. (2) laboratory clock readings, the uncertainties of [UTC −
i=1 UTC(k)]areidenticaltotheuncertaintiesof[TAI−UTC(k)],
Subtractingthesamequantityfrombothsidesofequation(1), [EAL−UTC(k)] and [EAL−Hk]. Thus, the uncertainties
of the links between laboratories are the only source of the
(cid:1)N (cid:1)N (cid:1)N uncertaintyof[UTC−UTC(k)]. Inthefollowingsectionsits
EAL(t)− wh(t)= w[h(t)+h(cid:1)(t)]− wh(t).
i i i i i i i propagationwillbestudied.
i=1 i=1 i=1
Usingequation(2)andrearranging,
2.1. Thelawofpropagationofuncertainty
(cid:1)N (cid:1)N According to [6], the uncertainty in the x (t) can be found
w(EAL(t)−h(t))= wh(cid:1)(t). (3) j
i i i i using the law of the propagation of uncertainty. Defining y
i=1 i=1 asagenericquantityindirectlymeasuredbymeansofdirect
Setting measurementsoftheinputquantityxi,
x(t)=EAL(t)−h(t), (4)
i i y =f(x ,x ,....,x ).
1 2 M
itisclearthatequation(3)isoftheform
Theexpressionofthelawofthepropagationofuncertainty
(cid:1)N (cid:1)N
wx(t)= wh(cid:1)(t). (5) [6]isgivenby
i i i i
i=1 i=1 (cid:1)M (cid:4)∂f (cid:5)2 M(cid:1)−1 (cid:1)M ∂f ∂f
u2 = u2 +2 u , (9)
ThesoftwarepackagetermedALGOSisusedintheTime y ∂x xi ∂x ∂x (xi,xk)
SectionoftheBIPMtogenerateUTC.Weightsaredetermined i=1 i i=1 k=i+1 i k
fromthevarianceofmonthlyaveragefrequencies,subjecttoa
where the first term corresponds to the effect of the
maximumvalue[5]. ThedatausedbyALGOStaketheform
uncertainties on the input quantities x and the second term
i
ofthetimedifferencesbetweenreadingsofclocks,writtenas
accountsforthecorrelationbetweenthem.
x (t)=h (t)−h(t). (6) The uncertainty of the quantity xk = [EAL −UTC(k)]
i,j j i
defined in equation (8) plays here the role of the indirect
Equation(5)inconjunctionwiththeN−1equations(6)results quantityyandtheuncertaintycontributionsareonlyduetothe
inasystemwithN equationsandN unknowns: measurementnoiseofthelinksxi,j(t). Applyingequation(9)
(cid:2)(cid:3) (cid:3) tothemodel(8)yields
xi(Nit=)1−wxixji((tt))==xi,jNi(=t1)wih(cid:1)i(t). (7) u2xj =(cid:1)N (cid:4)∂∂xxj (cid:5)2u2xi,j +2N(cid:1)−1 (cid:1)N ∂∂xxj ∂∂xxj u(xi,j,xk,j)
i=1 i,j i=1 k=i+1 i,j k,j
Thesolutionis
(cid:1)N N(cid:1)−1 (cid:1)N
xj(t)=EAL(t)−hj(t)=(cid:1)N wi[h(cid:1)i(t)−xi,j(t)]. (8) = i=1wi2u2xi,j +2 i=1 k=i+1wiwku(xi,j,xk,j), (10)
i=1 whereu2 =u2 .
xj EAL−hj
ThedifferencebetweenanyclockH andEAL(8)depends The weights of the clocks are available from the BIPM
j
onweights,clockpredictionandmeasuredclockdifferences. website,andtheuncertaintiesoflinksbetweentheclocks[7]
The clock H may also represent a UTC(j) time scale; arepublishedinCircularT (seetheBIPMTimeSection’sFTP
j
therefore,x (t)mayalsobeinterpretedas serverathttp://www.bipm.org/jsp/en/TimeFtp.jsp).
j
The propagation of uncertainty (10) could as well be
xj =EAL−UTC(j) expressed by a matrix formulation using a multivariate
weighted approach [8,9]; the scalar approach presented in
havingdroppedthetimeinstanttinthenotationforsimplicity. thispaperallowsonetoclearlyunderstandhowthedifferent
Thepredictionsandtheweightsarefixedbyappropriate componentscontributetothecombineduncertainty.
algorithmsbasedonpastclockbehaviour,and,inequation(8), ItcanbedemonstratedthattheALGOSalgorithmwould
they can be considered as time-varying deterministic generate the same results if each laboratory’s clocks were
parameters. Suboptimalestimationoftheseparameterswould replacedbyasingle‘equivalent’clockwhosereadingwasthe
affecttheuncertaintyofTAIasarealizationoftheTT,butthey weightedaverageoftheindividualclockreadingsandwhose
donotaffecttheknowledgeofthedifferencebetweenEALand weightinEALwasthesumoftheindividualclockweights.
clockHj;themeasuresxi,jarethustheonlycontributorstothe Therefore,thecomputationscanbesimplifiedbysummingthe
uncertaintiesinxj. Thecontributionoftheuncertaintygiven weightsoftheclocksateachlaboratoryasfollows:
by measures of clocks located inside the same laboratory is
considerednegligible,andintermsoftheALGOSalgorithm N(cid:1)Labk
itcanbeignoredbecauseitisinseparablefromthenoiseofthe WLabk = wi, (11)
clock. i=1
Metrologia,43(2006)278–286 279
WLewandowskietal
where N is the number of clocks of the considered studyisinprogressandapreliminaryevaluationindicatesthat
Labk
laboratory k and all the clocks at a laboratory are treated as the final uncertainty would change to a small extent. Some
one equivalent clock whose reading is the weighted average detailsarereportedinthelastsectiondevotedtofuturework.
ofindividualclockreadings. Theformalismofequation(10) Inthiswork,thenoiseaffectingdifferentnonoverlapping
couldstillbeappliedwithoutthesesimplifyingassumptions,as linksisassumedtobeuncorrelated. Correlationscanandwill
thedoublesummationwouldaccountforthe100%correlation appear when the same intermediate link appears in multiple
ofthetimetransfernoisebetweenclocksinthesamelab. links,aswillbeshownintheexamplebelow.
2.2. Extensionofthecomputation 2.4. Examplesofapplicationofthemethod
Itispossiblebutnotnecessarytoapplyequation(10)toevery Someexamplesareprovidedinordertoillustratetheapplied
laboratory. Iftheuncertaintyforanyoneclockisknown,for procedure. Ineachexample,theuncertaintyofeachclockH
k
example, x = EAL −h , the evaluation of uncertainty on with respect to UTC increases as the noise of the laboratory
j j
x =EAL−h maybeobtainedbyusingthesecondequation klinkincreases,butalsodecreasesastheweightoftheclock
i i
in(7)andapplyingthepropertyofthevariancetoobtain increases.
u2 =u2 +u2 +2u . (12) Case A1. Let us consider the simple case of only two
xi xj xi,j xj,xi,j clocks(labelled1and2)formingUTCandmaintainedinthe
The last term is the covariance of the measures x and samelaboratoryk. Externallinksarenotneeded. Thecaseis
i,j
thequantityx =EAL−h . Sincealltheclocksareincluded depictedinthesketchbelow.
j j
in EAL, the measure x will produce a nonzero covariance
i,j
termbycouplingtothesamemeasuresx ,whichenterinto
i,j
theEALdefinitionasmanytimesasthereareclocksinsideor
behindthelaboratoryi,obtaining
u =u(cid:3) =−u(cid:3)
xj,xi,j ( N(cid:2)=1w(cid:2)(h(cid:1)(cid:2)−x(cid:2),j),xi,j) ( N(cid:2)=1w(cid:2)x(cid:2),j,xi,j)
N(cid:1)eqlabi
=− w u2 =−W u2 (13) TheensembletimeUTCwouldbecomputedwiththese
(cid:2) xi,j eqi xi,j
(cid:2)=1 twoclocksinsidethelaboratoryk. SupposeclockH1realizes
awlshoerienNcleuqdlainbigisththeecelqouckiveaxletnertnnaulmtobethraotflcalboockrastionryla,bbourtawtohryicih, oUfTeCq(uka)t.ioTnh(e1u0n),cwerotauilndtybeofxk =[UTC−UTC(k)],bymeans
are connected to UTC through laboratory i. The equivalent
u2 =u2 =w2u2 . (15)
weight Weqi can be defined as the sum of the Neqlabi clock [UTC−UTC(k)] [UTC−h1] 2 x1,2
weights. This will be explained further in the next section,
Theuncertaintyofthemeasurex isnegligiblebecause
whereexamplesaregiven. Inthiscase,equation(12)becomes 1,2
theclocksarewithinthesamelaboratory. Thus
u2 =u2 +u2 −2u2 W . (14)
xi xj xi,j xi,j eqi u2[UTC−UTC(k)] ≈0. (16)
Equation(14)givestheuncertaintyofeachclockH,given
i Case A2. Consider two clocks (labelled 1 and 2), but
the uncertainty of any clock H and the uncertainties of the
j maintainedintwodifferentlaboratories(kandl,respectively)
chainsofmeasureslinkingclockH toclockH .
i j connectedbyonemeasurementlinkx .
k,l
2.3. Correlationsandanticorrelations
The BIPM provides the link uncertainties in Circular T
(section 6), but every time a link x appears in a multiple
i,j
linksuchasx = x +x itisnecessarytoestimatethe
i,k i,j j,k,
correlationswithrespecttox ,andthisisthemeaningofthe
i,j
lastcovarianceterminequation(10).
A more refined evaluation should take into account the Inthiscase,theuncertaintyofthemeasurementlinkwould
differentcorrelationpropertiesoftheseverallinks,including benon-negligibleandthefinaluncertaintyforUTC(k),realized
TWSTFT,GPS/Glonasscommonview(CV)andMeltingPot inlaboratorykwithclockH ,wouldbe
1
(also termed All-in-View) [10,11]. For example, in some
casestheuncertaintyisdominatedbycalibrationuncertainties u2 =w2u2 =(1−w )2u2 . (17)
betweenlaboratoryiandlaboratoryj,whileinothercasesthe [UTC−UTC(k)] 2 xl,k 1 xl,k
linkuncertaintyreportedinBIPMCircularTismostlydueto Equation(17)showsthatincreasingtheweightofclock
thenoiseofthereceiversatbothsites. H from laboratory k decreases the uncertainty of [UTC −
1
Correlations will always occur in situations where the UTC(k)].
samereceiverorsystemisusedtolinktwodifferentexternal Case B1. Consider three clocks (labelled 1, 2 and 3),
laboratories. Theanalysisoftheseeffectsrequiresmoredetails maintained in two different laboratories (k and l) connected
thanarereadilyavailableforcorrelationofthelinks. Further byameasurementlinkx accordingtothesketchbelow.
k,l
280 Metrologia,43(2006)278–286
Theevaluationofuncertaintiesin[UTC−UTC(k)]
Assumingthetwolinksx andx areuncorrelated,one
k,l l,m
obtainsfromequation(10)
u2 =w2u2 +w2[u2 +u2 ]+2w w u
[UTC−UTC(k)] 2 xl,k 3 xl,k xm,l 2 3 (xl,k,xm,k)
=(w +w )2u2 +w2u2 , (21)
2 3 xl,k 3 xm,l
where the correlation between x and x is due to the
k,l k,m
commonpath(l,k):
For site k, with UTC(k) realized by clock H , the
1
uncertaintyon[UTC−UTC(k)]wouldbe u(xl,k,xm,k) =u2xl,k. (22)
u2 =w2u2 +w2u2 +2w w u This example shows that, for uncorrelated links, the
[UTC−UTC(k)] 2 x2,k 3 x3,k 2 3 (x2,k,x3,k) weightoflaboratoriesbehindapivotlaboratoryisaddedtothe
=(w +w )2u2 =(1−w )2u2 , (18)
2 3 xl,k 1 xl,k weightofthepivotlaboratoryitselfandthisisthemotivation
sinceu22,k = u23,k = u(x2,k,x3,k) = u2l,k,becauseclocksH2 and boefhthinedptihveotdleafibnoirtaiotonroyfi‘peqluusivtahleenwtewigehigtshot’fWtheeqicaloscthkesbweehiignhdt
H areinthesamelaboratory.
3
thatpivotlaboratory,whichwasintroducedinequation(13).
CaseB2. WecomputetheuncertaintyforUTC(l)realized
Thelastterminequation(21)takesintoaccountthenoisefrom
withclockH insidelaboratorylforasystemofthreeclocks
2 thepivot(laboratoryl)totheremotelaboratorym,weighted
(labelled1,2and3)inthreedifferentlaboratories(k,landm)
bytheweightofclocksinlaboratorym.
connectedaccordingtothesketchbelow.
In the case of links correlated through site-based noise,
with the same assumptions leading to equation (20) and
assuming that the noise of the receiver in laboratory l is
cancelledwhileconnectinglaboratorymtolaboratorykwith
contemporaneous measurements, the same equations would
apply,and
u2 =(w2+w w +w2)u2 . (23)
[UTC−UTC(k)] 2 2 3 3 xl,k
Thiswouldbeexpectedforlinksdominatedbysite-based
noise, because the site-based noise of each site combines in
such a way that all possible combinations of links yield the
If the links between laboratories (l,k) and laboratories sameresults.
(l,m) are uncorrelated, then the double summation in
equation(10)iszeroand 2.5. FormulaefortheUTCnetwork
u2 =w2u2 +w2u2 . (19) Startingfromtheexamplesabove,theuncertaintycorrespond-
[UTC−UTC(l)] 1 xk,l 3 xm,l ingtothenetworkoflinkscurrentlyusedforthecomputation
ofUTCasreportedinfigure1canbeevaluated.
If the links between laboratories (l,k) and laboratories
Figure1showsthatUSNO,NISTandNICT1,andNTSC2
(l,m) are correlated through site-based noise in the receiver
actas‘intermediate’pivotsandthecentralroleofthePTB.
in laboratory l, but the total noise in both links is the same
Since the PTB plays a central role, equation (10) can
and equally contributed by the two laboratories: u =
(xl,k,xl,m) be used to obtain the uncertainty for [UTC − UTC(PTB)]
0.5u2 =0.5u2 and
l,k l,m considering the equivalent weights W for the intermediate
eqi
u2 =w2u2 +w2u2 +2w w u pivot laboratories USNO, NIST, NICT and NTSC. The final
[UTC−UTC(l)] 1 xl,k 3 xl,m 1 3 (xl,k,xl,m) expressionturnsouttobe
=(w2+w w +w2)u2 (20)
1 1 3 3 xl,k N(cid:1)1+1 (N1(cid:1)+N2)+1
Comparing equation (19) with equation (20), the effect u2[UTC−UTC(PTB)] = Wi2u2xi,UTC(PTB) + Wk2u2xk,UTC(USNO)
of considering partial correlations between the link noises i=2 k=N1+2
becomesevident. (N1+N(cid:1)2+N3)+1 (N1+N2(cid:1)+N3+N4)+1
CaseB3. ThesameconfigurationasCaseB2,exceptthat + Wk2u2xk,UTC(NIST) + Wk2u2xk,UTC(NICT)
theuncertaintyforlaboratorykistobeevaluated. Theclocks k=(N1+N2)+2 k=(N1+N2+N3)+2
inlaboratorymareconnectedtolabkthroughthepivotlabl +W2 u2 +W2 u2
withx =x +x : eqUSNO xUTC(PTB),UTC(USNO) eqNIST xUTC(PTB),UTC(NIST)
k,m k,l l,m +W2 u2 +W2 u2
eqNICT xUTC(PTB),UTC(NICT) eqNTSC xUTC(PTB),UTC(NTSC)
+W2 u2 (24)
JACT xUTC(NTSC),UTC(JACT)
HereN =25isthenumberoflaboratoriesdirectlylinked
1
toPTBwithonesinglelink(IEN,AOS,SP,LT,etc)excluding
1 NICTisthenewnameforCLR.
2 All the acronyms appearing in this paper are in agreement with the
list published by the BIPM Time Section at http://www.bipm.org/jsp/en/
TimeFtp.jsp?TypePub=publication.
Metrologia,43(2006)278–286 281
WLewandowskietal
North America Europe Asia
ORB DLR IFAG CH LDS JV SP
NIMT NMLS BIRM
NRC
SMU KRIS
OP
NMC
APL NIM
SG
IEN UME
USNO NAO
CRL
PTB NPLI
TP INPL
NMIJ
CNMP CNM
NIST NPL OMH
VSL CAO NTSC TL
JATC
ROA DTAG BEV AOS PL LT SU NIMB JATC SCL HKO
NIS
ONBA TCC IGMA ONRJ CSIR AUS MSL
South America Africa Oceania
ORGANIZATION OF THE INTERNATIONAL TIME LINKS MJaurlyc2h0 024004
Laboratory equipped with TWSTFT GPS CV single-channel link
TWSTFT GPS CV single-channel back-up link
TWSTFT by Ku band with X band back-up GPS CV multi-channel link
Laboratory equipped with dual frequency reception GPS CV multi-channel back-up link
GPS CV dual frequency link
GPS CV dual frequency back-up link
Figure1. InternationaltimelinksusedforcomputationofUTC(July2004).
the four laboratories acting as intermediate pivots; N = 4 mean value equal to the obtained measurement value a and
2
isthenumberoflaboratorieslinkedtoUSNO(namelyNRC, standarddeviationequaltou .
xi,j
CNMP,APL,ONBA);N3 = 4isthenumberoflaboratories The Monte Carlo simulation consists of picking up
linkedtoNIST(namelyTCC,ONRJ,CNM,IGMA);N4 =13 different values for the measure xi,j from its statistical
isthenumberoflaboratorieslinkedtoNICT(SG,TL,NAO, distributionandcomputing[UTC−UTC(k)]withthedifferent
MSL,etc)andWeqi areequivalentweightsoflaboratoriesas simulated measure values. This gives an indication of the
introducedinequation(13). variabilityoftheresults[UTC−UTC(k)]duetothevariability
TheequivalentweightoftheUSNOis,forexample, of the measures x ,which have been shown to be the only
i,j
sourceofvariabilityand,hence,ofuncertainty.
WeqUSNO =WNRC+WCNMP+WAPL+WONBA+WUSNO. Thesimulationproceededbyfirstevaluatingtheeffecton
[UTC−UTC(k)]ofonlyonenoisylink(alltheotherlinksare
Similarexpressionsholdfortheotherintermediatepivots, consideredwithnegligiblenoise);thenthenoisewasinserted
NIST,NICTandNTSCandWLabwasdefinedinequation(11). link by link. This ‘experimental’ variability was compared
Oncetheuncertaintyof[UTC−UTC(PTB)]isevaluated, withtheexpectedtheoreticalresultscomingfromtheanalytical
theexpression(14)canbeusedtocomputetheuncertaintyof estimation(10).
[UTC−UTC(k)]foreveryotherlaboratoryk.
3.1. Exampleofasinglelink
3. AMonteCarlosimulation
The effect of the uncertainty of the link between the PTB
Theanalyticalapproachpresentedintheprevioussectionwas and the LT laboratories using the data of July 2004 was
testedbyaMonteCarlosimulationusingtheBIPM’ssoftware evaluated. LaboratoryLThasoneclockofpercentageweight
ALGOS,onthefullsetofclockandtimetransferdatausedto wLT =0.119;thesquareduncertaintyofthelink[UTC(LT)−
generateCircularTpublishedinJuly2004. Forthatmonth,the UTC(PTB)] is u2[UTC(LT)−UTC(PTB)] = 27.25 ns2 and on MJD
pivotlaboratorieswerePTB,USNO,NIST,NICTandNTSC. 53144[UTC(LT)−UTC(PTB)]wasfoundtobe−242.8ns.
Since the source of uncertainty in [UTC −UTC(k)] is A large number, N, of values of [UTC(LT) − UTC(PTB)]
the links whose uncertainty u is listed in Circular T, a was generated normally distributed around the central value
simulationwasperformedassumxii,jngthateverylinkmeasureis [UTC(LT)−UTC(PTB)]=−242.8nsandwithavarianceof
describedbyarandomvariablewithaGaussiandistribution, u2[UTC(LT)−UTC(PTB)] =27.25ns2.
282 Metrologia,43(2006)278–286
Theevaluationofuncertaintiesin[UTC−UTC(k)]
70 0.08
nit 60 unit 0.07
distribution/ arbitrary u 345000 distribution/ arbitrary 0000....00003456
e e
ativ 20 ativ 0.02
Rel Rel
10 0.01
0 0
-16.75 -16.74 -16.73 -16.72 -16.71 -16.7 -16.69 -16.68 -16.67 205 210 215 220 225 230 235 240 245 250
[UTC – UTC (PTB)]/ns [UTC – UTC (LT)]/ns
Figure2. Theoreticalandexperimentaldistributionofthevalues Figure3. Theoreticalandexperimentaldistributionofthevalues
[UTC−UTC(PTB)]consideringonlyonelinkwithnoise. [UTC−UTC(LT)]consideringonlyonelinkwithnoise.
Usingequation(10),withonlythelinkPTB/LTcorrupted 0.09
withnoise
0.08
u2[UTC=−U(cid:4)TC01(.P10T10B9)](cid:5)=2×wL2(T2u72[.U2T5Cn(LsT2))−=UTC3(.P8T5B)×] 10−5ns2. bitrary unit 00..0067
ar
n/ 0.05
o
Thus uti
u[UTC−UTC(PTB)] =0.0062ns, distrib 0.04
e 0.03
andfromequation(14) v
ati
u[UTC−UTC(LT)] =5.21ns. Rel 0.02
0.01
Alternatively, using equation (10) directly referenced to
0
laboratory LT, which corresponds to having the total clock -30 -25 -20 -15 -10 -5 0 5 10 15
weight in PTB with the exception of the LT clock, the same [UTC – UTC(NIST)]/ns
theoreticalresultwasobtained:
(cid:4) (cid:5) √ Figure4. Theoreticalandexperimentaldistributionofthevaluesof
u[UTC−UTC(LT)] = 1− 0.119 × 27.25ns2 =5.21ns. [UTC−UTC(NIST)].
100
Using N = 10000 simulated measurement values, a 3.2. Completesystem
normaldistributionofvalues[UTC−UTC(PTB)]wasobtained
withameanvalueequalto−16.7087nsandstandarddeviation The Monte Carlo results are reported here for the complete
equal to 0.0063ns. The value [UTC −UTC(PTB)] for that system of links, using N = 20000 simulated data for any
date published in Circular T was equal to −16.7ns and link. Figure 4 shows the close agreement, for example,
the standard uncertainty from the above computation was betweenthetheoreticalandexperimentalresultsforthevalues
u[UTC−UTC(PTB)] = 0.0062ns. Therefore, the mean value of[UTC−UTC(NIST)].Intable1theanalyticalresultsare
and the standard deviation of [UTC −UTC(PTB)] obtained comparedtothesimulationsassumingthateverylinkhadthe
by simulating the noisy link PTB/LT corresponded to the noisecorrespondingtotheuncertaintylistedinCircularT.The
publishedvalueandtotheexpectedtheoreticaluncertainty. agreementbetweenthetwoestimationmethodsisevidentand
ForthelaboratoryLTanormaldistributionofmeanvalue givesconfidencetotheanalysis.
equal to 226.161ns was obtained, with a standard deviation
of 5.2ns. The published value in Circular T was [UTC −
UTC(LT)] = 226.1nsandthestandarduncertaintyfromthe 4. ApplicationtothecomputationofCircularT
abovetheoreticalcomputationwasu[UTC−UTC(LT)] =5.21ns.
The results are depicted in figures 2–4, where the The formulae developed in section 2.5 for the evaluation of
histogram reports the results obtained from the Monte Carlo [UTC−UTC(k)] uncertainties have been applied using the
simulation,whilethesolidline(inred)representstheexpected link uncertainties as listed in section 6 of Circular T. They
statisticaldistributionofthevaluesaccordingtothetheoretical have been used operationally since January 2005, reporting
analysis. theuncertaintiesinCircularT.ResultsforCircularTno205
Ascanbeseen,theanalyticalandsimulationresultsagree. ofJanuary2005arereportedinfigure5.
Metrologia,43(2006)278–286 283
WLewandowskietal
Table1. AnalyticalandnumericallyestimateduncertaintiesofalltheUTCparticipatinglaboratoriesconsideringeverylinkwithnoise.
u[UTC−UTC(k)]/ns u[UTC−UTC(k)]/ns
k Analyticalmethod Numericalmethod k Analyticalmethod Numericalmethod
AOS 5.6 5.6 NIS 20.3 20.6
APL 5.7 5.7 NIST 5.0 5.0
AUS 7.3 7.3 NMC 20.7 20.8
BEV 5.5 5.5 NMIJ 7.0 7.0
BIRM 20.6 20.6 NPL 5.3 5.2
CAO 21.2 21.0 NPLI 20.3 20.4
CH 5.4 5.3 NRC 15.2 15.1
CNM 21.0 21.1 NTSC 7.1 7.0
CNMP 8.4 8.4 OMH 20.2 20.5
CSIR 20.3 20.6 ONBA 8.9 8.9
DLR 5.4 5.4 ONRJ 21.2 21.3
DTAG 10.6 10.5 OP 5.4 5.4
HKO 7.3 7.3 ORB 5.3 5.4
IEN 2.4 2.4 PL 5.4 5.3
IFAG 5.3 5.3 PTB 1.9 1.9
IGMA 20.8 20.9 ROA 5.4 5.4
INPL 10.9 11.0 SCL 11.6 11.6
JATC 21.2 21.3 SG 20.5 20.6
JV 20.7 20.6 SMU 20.7 20.7
KRIS 7.1 7.2 SP 10.4 10.5
LDS 20.3 20.2 SU 6.1 6.1
LT 5.6 5.6 TCC 21.2 20.9
MSL 20.7 20.8 TL 6.7 6.7
NAO 20.6 20.7 TP 5.8 5.8
NICT 4.4 4.4 UME 25.1 25.1
NIM 20.4 20.3 USNO 2.3 2.3
NIMT 20.7 20.9 VSL 5.3 5.3
CIRCULAR T 205 ISSN 1143-1393
2005 FEBRUARY 15, 16h UTC
BUREAU INTERNATIONAL DES POIDS ET MESURES
ORGANISATION INTERGOUVERNEMENTALE DE LA CONVENTION DU METRE
PAVILLON DE BRETEUIL F-92312 SEVRES CEDEX TEL. [phone] FAX. [phone] [email protected]
1 - Coordinated Universal Time UTC and its local realizations UTC(k). Computed values of [UTC-UTC(k)].
From 1999 January 1, 0h UTC, TAI-UTC = 32 s.
Date 2004/05 0h UTC DEC 30 JAN 4 JAN 9 JAN 14 JAN 19 JAN 24 JAN 29 Uncertainty/ns
Notes
MJD 53369 53374 53379 53384 53389 53394 53399
Laboratory k [UTC-UTC(k)]/ns u u u
A B
AOS (Borowiec) -21.2 -14.8 -18.7 -13.6 -11.2 -12.1 -17.9 1.6 5.4 5.6
BEV (Wien) 84.9 87.5 90.2 90.2 94.8 96.9 99.9 1.5 5.3 5.5
CH (Bern) -27.4 -25.2 -27.4 -31.1 -28.6 -23.8 -21.1 0.8 5.3 5.4
DLR (Oberpfaffenhofen) 1.4 -3.1 -7.8 -14.1 -19.1 -22.3 -31.3 0.8 5.4 5.5
IEN (Torino) -58.2 -71.8 -77.9 -79.8 -78.5 -84.6 -87.5 0.7 2.2 2.3
LT (Vilnius) 486.0 519.1 547.4 549.6 528.4 533.6 542.7 1.6 5.3 5.5
NICT (Tokyo) 4.5 -0.7 6.8 6.1 -1.7 1.1 3.9 1.2 4.3 4.5
NIST (Boulder) 3.5 2.6 1.3 1.5 0.8 -0.6 -1.7 0.6 5.1 5.1
NPL (Teddington) 8.0 11.2 14.6 18.5 22.7 25.7 33.7 0.7 2.2 2.3
NTSC (Lintong) 17.7 20.4 26.7 31.1 27.4 19.6 19.1 2.7 6.5 7.0
ONBA (Buenos Aires) -923.0 -967.0 -1200.8 -1407.0 -1666.2 -3418.7 -3392.1 5.0 7.3 8.8
ONRJ (Rio de Janeiro) 155.1 153.1 162.1 159.6 161.7 158.4 176.0 5.0 20.6 21.2
OP (Paris) 41.7 45.8 49.4 43.6 40.3 39.8 36.4 0.6 2.1 2.2
ORB (Bruxelles) -42.9 -41.2 -39.3 -37.5 -35.2 -34.5 -32.3 0.8 5.3 5.4
PL (Warszawa) -3.9 -1.8 -5.7 -6.5 -4.5 -4.1 -7.8 1.5 5.2 5.4
PTB (Braunschweig) -0.6 1.2 2.5 3.0 5.5 5.9 4.7 0.4 1.9 1.9
ROA (San Fernando) -63.1 -62.0 -68.3 -67.9 -67.1 -65.8 -74.1 0.8 5.3 5.4
UME (Gebze-Kocaeli) 599.8 595.1 593.2 603.0 608.3 611.2 612.5 15.0 20.1 25.1
USNO (Washington DC) -4.9 -4.7 -5.1 -4.6 -3.5 -3.5 -2.5 0.5 2.2 2.3
VSL (Delft) -76.9 -77.0 -77.9 -77.4 -56.4 -41.8 -21.2 0.7 2.2 2.3
Figure5. AsampleofthefirstsectionofCircularTno. 205showinguncertaintiesin[UTC−UTC(k)].
284 Metrologia,43(2006)278–286
Theevaluationofuncertaintiesin[UTC−UTC(k)]
In Circular T, the last column (see figure 5), denoted andalltheexternalclockswouldbeseenthroughthatdominant
by u, corresponds to the combined standard uncertainty of noise independently of their location. If different satellite
[UTC −UTC(k)] for any laboratory k. The values u and schedulesareusedintherelevantlinks,uncertaintiesintime
A
u arethecomponentsofthecombinedstandarduncertainty transfer using GPS CV are largely, though not entirely, site-
B
u originating from type A and type B evaluations listed in based. EvenforCVobservationsmadeusingeveryavailable
section6,respectively. Thefollowingrelationshipholds: satellite, closure violations will only arise if simultaneous
(cid:6) satellite observations are recorded at only two of the three
u= u2 +u2. sites. In such cases, orbit misestimation and receiver noise
A B
will contribute uncertainties, and any azimuth or elevation-
Itisnotedthattheapplicationofequations(24)and(14) dependent asymmetries in the multipath environment would
separately to the type A and type B evaluations is permitted cause both uncertainties and biases. Since calibration is
only when all the other quantities involved are constant, in achievedbyanall-skysamplingthatissystematicallydifferent
particularthenumberofclocksandtheirweightsareconstant; from the sky-sampling of CV, a different multipath will
thisiscurrentlytrueonlywithineachseparatemonthofUTC also lead to uncertainties. Despite these noise sources, the
computation. closurerelationlargelyholdsforCV,andthelargestsourceof
Theapplicationoftheabovetheorytotheactualcaseof uncertaintyistypicallyduetovariationsofthereceiversystem
[UTC−UTC(k)]suggestsanadditionalexperimentaltestto thatarecommontoalldata.
checkthevalidityoftheobtainedestimates. Infact,underthe For Two-Way Satellite Time and Frequency Transfer
assumptions that the values of [UTC−UTC(k)] are almost (TWSTFT), the noise is again largely site-dependent. Some
constantorveryslowlyvaryingintime,whichisareasonable closureviolationscanoccurbecausetheobservationsbetween
assumption in case of the UTC(k) time scale realized with pairs of sites are typically made with different codes and
hydrogenmasers, onecanascribetheobservedinstabilityof slightly different frequencies. The largest source of closure
thevalues[UTC−UTC(k)],atafive-dayobservationinterval, errors is probably due to the fact that the received signals
mostlytothecombinedeffectoftimetransfernoiseratherthan are shaped by the product of the transmitting and receiving
toclocknoise. ByestimatingtheAllandeviationortheTime bandpasses, while the delay and certain noise components
Deviation(TDEV)ofthequantity[UTC−UTC(k)]onehas suchasthecable-dependentmultipathcansystematicallyvary
thusaroughideaofthecombinedtimetransfernoiseofallthe over the bandpass [12]. While TWSTFT closure violations
linksinTAIasobservedfromlaboratorykandthereforegetsan are seen at the 1ns level in the data sent to the BIPM, they
independentestimateoftheu uncertaintyforthatlaboratory. couldbereducedthroughbaseline-dependentcalibrations. In
A
AsimpletestevaluatingtheTDEVforafewtimelaboratories the cases where a TWSTFT system is calibrated with GPS,
gavereasonablygoodagreementwiththecalculatedvaluesof the uncertainties in the calibration are determined by the
u reportedhere. uncertaintiesintheGPScalibration.
A
Special situations arise when one site is a pivot site,
connected to some sites by one technique and other sites by
5. Futureextension
a different technique. By means of an illustration, we can
consider a very simplified situation in which every site is
Torefinetheevaluationofuncertainties,itisnecessarytohave
directly linked to one central pivot site, either by AV GPS
moredetailsthanarereadilyavailableaboutthecorrelationof
orTWSTFT;thiscaneasilybegeneralizedtodescribemore
the links. This is largely because, as noted in the examples,
complextopologies. Wewillassumethatvariationsbetween
the correlated noise between two sites should include the
contribution from site-based uncertainties at sites along the any two links are completely uncorrelated with respect to
path of links and because the correlation between two links variationsatthelinkextremities,butthatthelinksare100%
with a common site that uses the same equipment for both correlated with respect to variations of the equipment at the
linksshouldincludethesite-basednoisecontributionfromthat centralpivotsite,providedbothlinksarebyeitherTWSTFTor
site. Thisisdifficulttoestimatewhencalibrationsaredoneby GPS.LetusalsoassumeabiasBexistsintheGPSequipment
linksratherthanbysites. Incomputationsassumingsite-based atthepivotsite. Inthiscase, itiseasytoshowthatthebias
noise, which supplemented the link uncertainties reported in wouldaffectthoselaboratoriesklinkedbyGPStothepivotas
CircularT,itwasfoundthatonthewholetheeffectsofthese follows:
simplificationsaresmall. (cid:3)[UTC−UTC(k)] =(1−WG)B
Furtherextensionofthisevaluationwouldbebasedonthe
following consideration. If all time transfers were achieved where WG is the sum of all the weights of the laboratories
usingasinglesystempersiteandifallsourcesofnoisewere linkedtothepivotsitebyGPS.
site-based,suchasmostlyhappensforMeltingPotGPS,also Thebiaswouldaffectthepivotlaboratoryandthoselinked
termedAll-in-View(AV)[10,11],thenallpossiblelinkswould toitbyTWSTFTasfollows:
obeythefollowingclosurerelation:
(cid:3)[UTC−UTC(k)] =−WGB.
x (t)+x (t)+x (t)=0. (25)
i,j j,k k,i
Underthenormalcircumstancesdescribedinthispaper,
In this situation, the noise of each site’s time transfer the existence of any biases would not carry any significant
systemwouldbeindistinguishablefromthenoiseofitsclocks statistical implications, as they would be directly related to
andthedominantuncertaintywouldbegivenbythesitenoise, thetabulateduncertaintiesinthelinksthemselves. However,
Metrologia,43(2006)278–286 285
WLewandowskietal
theaboveequationsillustratethedependenceofTAIandUTC [2] GuinotBandThomasC1988EstablishmentofInternational
upontheequipmentatanycentralpivot,whichmaynotalways AtomicTimeAnnualReportoftheBIPMTimeSection1
[3] TavellaPandThomasC1991Comparativestudyoftimescale
follow the Gaussian behaviour assumed here, particularly in
algorithmsMetrologia2857–63
thecaseofequipmentfailure.
[4] ThomasCandAzoubibJ1996TAIcomputation: studyofan
alternativechoiceforimplementinganupperlimitofclock
6. Conclusions weightsMetrologia33227–40
[5] AzoubibJ2001Arevisedwayoffixinganupperlimittoclock
weightsinTAIcomputationWorkingDocument
This paper presents a study of the determination of the
CCTF/01-14ofthe15thMeetingoftheCCTF,Se`vres(see
uncertainties in [UTC − UTC(k)]. An analytical solution http://www.bipm.org/cc/AllowedDocuments.jsp?cc=CCTF)
is derived from the law of the propagation of uncertainty, [6] ISO1993GuidetotheExpressionofUncertaintyin
taking into account that leap seconds and deterministic Measurement(Geneva:InternationalOrganizationfor
Standardization)
frequency steering of EAL do not affect these uncertainties.
[7] AzoubibJandLewandowskiW2002Uncertaintyoftime
The analytical results were verified through Monte Carlo
linksusedforTAI2002Proc.34thAnnualPreciseTime
simulations using the software used to generate UTC, and andTimeInterval(PTTI)Meeting(USNavalObservatory,
agreement was found, giving confidence in the analytical WashingtonDC)pp413–24
estimation. [8] JointCommitteeforGuidesinMetrologyGuidetothe
ExpressionofUncertaintyinMeasurement(GUM)
A more detailed analysis is in progress, including full
Supplement2: ModelswithanyNumberofOutput
inclusion of site-based noise contributions, all available
Quantities(tobepublished,seehttp://www.bipm.org/
calibration information, more details for the correlation en/committees/jc/jcgm)
betweenthelinks,methodsforoptimizingthelinkstructure, [9] HahnJandTavellaP2000Atimescaleforsatellite
given uncertainty information, non-Gaussian behaviour and navigationsystems: whyandhow? Int.J.Satell.Commun.
18305–24
different correlation properties of uncertainties due to
[10] MiranianMandKlepczynskiW1991TimeTransfervia
calibrationorduetorandomnoise.
GPSatUSNOProc.4thInt.TechnicalMeetingof
ION-GPS-91(InstituteofNavigation,Alexandria,VA)
References pp215–22
[11] JiangZandPetitG2004TimetransferwithGPSsatellites
All-in-viewProc.ATF2004(Beijing)pp236–43
[1] ThomasC,WolfPandTavellaP1994TimeScalesBIPM
[12] DavisDanddeJongG1992privatecommunication
Monographie94/1
286 Metrologia,43(2006)278–286