zhang 2012

Time Scales and Time TransformationsAmong Satellite Navigation Systems

Pengfei Zhang, Chengdong Xu, Chunsheng Hu and Ye Chen

Abstract The definitions of time scales and the time transformations are veryimportant for Global Navigation Satellite Systems (GNSSs) research. DifferentGNSSs use different internal reference time systems which include GPS Time,GLONASS Time, GALILEO System Time and BeiDou System Time. With thedevelopment of GNSSs, it is significant to set up the relationships of time scalesamong different GNSSs to improve the compatibility and interoperability. Thispaper concludes the definitions of different time scales and describes the timesystems in different GNSSs firstly. Secondly, the relationships among differenttime scales are analyzed according to the data published by Bureau Internationaldes Poids et Mesures (BIPM) in recent years, and time transformations amongdifferent time systems are presented. Finally, these transformations are tested andvalidated, and the result shows that they can satisfy the accuracy requirements formost users.

1 Introduction

Navigation satellites move around the earth in high speed, for example, the speedof a GPS satellite is about 3.9 km/s. The observation time error should be less than2:6 ls when the position error of the satellite is suggested to be \1 cm at that

P. Zhang (&) C. Xu C. HuSchool of Aerospace Engineering, Beijing Institute of Technology, No.5 SouthZhongguancun Street, Haidian District, Beijing 100081, Chinae-mail: [email protected]

Y. ChenSchool of Information and Communication Engineering,North University of China, No.3 Xueyuan Road, Taiyuan, 030051 Shanxi, China

J. Sun et al. (eds.), China Satellite Navigation Conference (CSNC) 2012 Proceedings,Lecture Notes in Electrical Engineering 160, DOI: 10.1007/978-3-642-29175-3_45, Springer-Verlag Berlin Heidelberg 2012

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moment [1]. 1 ls propagation time error of the signal will cause 300 m mea-surement distance error. If the distance needs to be measured in meter levelprecision, the time should be measured in nanosecond level precision [2]. Thesatellite signal transmitting time takes satellite navigation system time as a ref-erence; however, the time of the receiver takes Coordinated Universal Time (UTC)as a reference. So the two time scales should be unified to meet high accuracymeasurement requirements of signal propagation time. In addition, in the samesatellite navigation system, it is also very important to achieve time synchroni-zation among all the satellites for precision positioning [3]. With the developmentof Global Navigation Satellite Systems (GNSSs), positioning using differentGNSSs has become an inevitable trend. Different GNSSs use their own internalreference time systems, so time transformations among different time systemsshould also be considered. This paper introduces the definitions of different timescales including astronomical time, atomic time and UTC, and it presents therelationships among them firstly. Secondly, it describes the time systems in dif-ferent GNSSs including GPS Time (GPST), GLONASS Time (GLST), GALILEOSystem Time (GST) and BeiDou System Time (BDT), and then it shows therelationships between each time system and UTC. Finally, it derives time trans-formations which include both the integral part and the fractional part of thedifference among different time systems.

2 Time Scales

The establishment of a time scale should take both a time starting point and a timespan for consideration. The time span can make use of a periodic motion phe-nomenon which is repeatable observational, continuous and stable as a reference,such as the pendulum swinging, the earth rotation and the crystal oscillation [4].

2.1 Astronomical Time

Astronomical time includes Sidereal Time (ST), Solar Time (SOT), UniversalTime (UT) and Ephemeris Time (ET).

2.1.1 Sidereal Time and Solar Time

Both ST and SOT are time scales based on the earth rotation. An apparent solarday is defined as the interval between two successive returns of the sun to the localmeridian. Since the orbit which the earth rotates around the sun is not a circle, andthe earths axis is not strictly vertical with the rotating orbit plane, mean SOT isintroduced. An apparent sidereal day is defined as the interval between two

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successive returns of a fixed star which is far from the earth to the local meridian.In fact, an apparent sidereal day is non-constant, so mean ST is also introduced.Mean ST and mean SOT has the following relationships [1]:

1 mean solar day 24 mean solar hours 86400 mean solar seconds 1:002737 mean sidereal days 1

1 mean sidereal day 0:997270 mean solar day 86164:095563 mean solar seconds 2

Both ST and SOT depend on the longitude of the observers. Mean SOT ofGreenwich meridian is shorted for GMT. The earth was divided into 12 standardtime zones. The time of each zone is equal to the mean SOT of the centralmeridian in this zone, the time difference between each zone and GMT can bemeasured by integral number of hours.

Julian Day (JD) is used to record mean solar day, which starts at 12 h January 1,4713 BC. In order to operate the data conveniently, Modified Julian Day (MJD)was applied, the starting point is 0 h November 17, 1858 GMT [4].

2.1.2 Universal Time

UT has the same scale as mean SOT. According to the conditions whether theearth polar motion is modified or the seasonal change of the earth rotation rate iscorrected, UT can be divided into UT0, UT1 and UT2 [3].

UT0: Mean SOT of Greenwich meridian which is achieved by severalobservatories.

UT1: UT0 adds the observation influence Dk (up to 0.06 s) which is caused bythe earth polar motion.

UT2: UT1 adds the seasonal correction of the earth rotation rate DTs.

2.1.3 Ephemeris Time

ET is a time scale based on the earth revolution around the sun, and it is notinfluenced by the earth polar motion and the seasonal change of the earth rotationrate which are unpredictable. An ephemeris second is defined as 1/31556925.9747of the tropical year at 1900 [3].

ET UT2 DT 3DT can be only decided by observation, and it can not be deduced by any

formula. Obviously, this definition of the time scale is fussy and difficult tooperate. In 1976, the International Astronomical Union decided to replace ET with

Time Scales and Time Transformations Among Satellite Navigation Systems 493

Terrestrial Dynamical Time (TDT) and Barycentric Dynamical Time (TDB). In1991, TDT was renamed as terrestrial time.

2.2 Atomic Time

Atomic time is a time scale base on the resonant frequency of Cesium atoms. In1967, an atomic second was defined as the duration of 9, 192, 631, 770 periods ofthe radiation corresponding to the transition between two hyperfine levels of theground state of the Cesium133 atom [2]. The time scale which derives from thedefinition is called International Atomic Time (TAI). It is more stable and easier tooperate than ET, so the ephemeris second is replaced by the atomic second as thebasic unit of time measurement. The starting point of the TAI which is establishedby the United States Naval Observatory (USNO) is 0 h January 1, 1958 UT2. Thestarting point of TAI which is established by Bureau International de lHeure(BIH) is 34 ms earlier than that of USNO [1].

The definition progress of one second is shown in Table 1.

2.3 Coordinated Universal Time

TAI has nothing to do with the earth rotation. However, it is necessary for satellitenavigation to link the time with the earth rotation. Therefore, a compromise timescaleUTC is proposed. UTC is a time scale based on atomic second, and it is asclose as possible to UT1. It has been adopted since 1972. When the time differencebetween UTC and UT1 exceeds 0.9 s, one leap second (positive or negative) willbe added or subtracted to UTC such that UTC is the closest to UT1. By now, theleap seconds which have been introduced are all positive.

jUT1 UTCj\0:9 s 4UTC TAI 1 s n 5

where n is an adjusted parameter which is equal to the total leap seconds. TheInternational earth rotation and reference system service (IERS) are responsible fordefinitively bulletining n and the difference between UT1 and UTC [5]. The dif-ference between UT1 and UTC from year 1991 to 2011 is shown in Fig. 1.

Table 1 The definitionprogress of one second

Year One second definition

Before 1960 1/86400 mean solar day19601967 One ephemeris secondSince 1967 One international atomic second

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It can be seen that the value of [UT1-UTC] is discontinuous, because a leapsecond was introduced in the date when a inflection point appears in the curve sothat the difference between UTC and UT1 is no more than 0.9 s.

The latest time when the leap second was loaded is 23 h 59 min 59 s December31, 2008 UTC. It is to say the next two seconds of UTC are ordered as follows: 23 h59 min 60 s December 31, 2008 UTC and 0 h 0 min 0 s January 1, 2009 UTC. So far,n has been equal to 34 [5]. Figure 2 shows the leap seconds of UTC since 1972.

UTC is a paper time scale, but it is approximated by local physical repre-sentations UTC (k) through about 250 cesium atomic clocks and hydrogen

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-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

date(UTC)

(UT1

-UTC

)(s)

Fig. 1 Curve: [UT1-UTC] from year 1991 to 2011

72-1-1 77-1-1 81-7-1 85-7-1 90-1-1 94-07-01 99-01-01 09-01-010

5

10

15

20

25

30

35

40

date(UTC)

(UTC

-TAI

)(s)

Fig. 2 Curve: leap seconds of UTC


microwave blaster devices in national metrology laboratories and observatoriesthat contribute to the formation of the international time scales at the Bureauinternational des poids et mesures (BIPM) [6]. The difference between UTC andUTC (k) is provided by the publication in the monthly BIPM Circular T.

Figure 3 shows some examples about the difference between UTC and UTC(k).Among these local physical representations UTC (k), UTC (USNO) is the most

closest to UTC. The difference between UTC (USNO) and UTC is kept within10 ns. GNSSs rely on UTC (k) which is provided by local metrology laboratoriesand observatories, so atomic clocks technology with high accuracy is researched tobe more closer to UTC by all the local metrology laboratories and observatories.

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0

100

200

Date(UTC)

ns

UTC-UTC(USNO)

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-100

0

100

200

Date(UTC)

ns

UTC-UTC(SU)

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-100

100

200

Date(UTC)

ns

UTC-UTC(NTSC)

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0

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200

Date(UTC)

ns

UTC-UTC(OP)

04-11-5 06-7-23 08-4-8 09-12-24 11-10-30-200

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0

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Date(UTC)

ns

UTC-UTC(AOS)

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0

100

200

Date(UTC)

ns

UTC-UTC(NIM)

USNO: U.S. Naval ObservatoryNTSC: National Time Service Center of ChinaAOS: Astrogeodynamical Observatory

SU: Institute of Metrology for Time and SpaceOP: Paris ObservatoryNIM: National Institute of Metrology

Fig. 3 Curve: [UTCUTC (k)] from year 2004 to 2011

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3 GNSS Time

In order to meet the needs of precision positioning and navigation, each GNSScreates its own internal reference time system. All these systems, which includeGPST, GLST, GST and BDT, should be ensured as close as possible to UTC, eventhe leap second of UTC.

3.1 GPS Time

The length of a second in GPST, which is measured by the GPS satellite atomicclocks and the ground station clocks, is consistent with that of TAI. GPST is acontinuous time system, so it does not have to be adjusted by leap second. Thestarting point of GPST is consistent with 0 h 0 min 0 s January 6, 1980 UTC(USNO). An epoch in GPST is distinguished by the number of seconds that haveelapsed since saturday/sunday midnight and the GPS week number. The GPSnumber will return to zero when the weeks number are up to 1024. The totalnumber of seconds in a week is 604,800 [1]. The conversion formula betweenGPST and UTC is as follows:

GPST UTCUSNO t1 t2 UTC t3 t1 t2 UTC t1 t2 t3 UTC t1 Dt

6

t1 (t1 1s n 19s) which is 15 by now is the integral part of difference inseconds between GPST and UTC (USNO), it is caused by the continuity of GPSTand the leap second of UTC. t2 is the fractional part of difference in secondsbetween GPST and UTC (USNO). t3 is the fractional part of difference in secondsbetween UTC and UTC (USNO). The change of t3 is described in Fig. 3.Dt(Dt t2 t3) which is the fractional part of difference in seconds between UTCand GPST is broadcasted by the GPS satellite navigation message and publishedafterwards by BIPM [7]. The change of Dt from 2009 to 2010 is described inFig. 4.

From the above curve, the Dt from 2009 to 2010 is controlled within 20 ns.

3.2 GLONASS Time

The length of a second in GLST, which is based on UTC (SU) and is 3 h earlierthan UTC (SU) because of the specific characteristic of GLONASS, is consistentwith that of TAI. The GLST is not a continuous time system, so it has to beadjusted by leap second. And the navigation capability of GLONASS will beaffected because of the discontinuous time caused by leap second. The starting


point of GLST is 0 h 0 min 0 s January 1, 1996 UTC (SU). An epoch in GLST,which starts from the last leap year, is described by accumulated days and thenumber of seconds in less than 1 day. The maximum number of accumulated daysin one cycle is 1461, and the number of the total seconds which is started from themidnight between the previous day and the very day is 86400. The conversionformula between GLST and UTC is as follows:

GLST UTCSU 3h t01 UTC t02 3h t

01

UTC 3h t01 t02 UTC 3h Dt0

7

t01 is the fractional part of difference in seconds between GLST and UTC (SU).

Because GLST and UTC (SU) use the same timing method including its leapseconds, so there is no integral part of difference in seconds between them. t

02 is the

fractional part of difference in seconds between UTC and UTC (SU). The changeof t

02 is described in Fig. 3 Dt

0(Dt0 t01 t02) which is the fractional part of dif-

ference in seconds between UTC and GLST is broadcasted by the GLONASSsatellite navigation message and published afterwards by BIPM [7]. The change ofDt0 from 2009 to 2010 is described in Fig. 5.

From the above curve, Dt0 from 2009 to 2010 reaches hundreds of ns.Compared with Fig. 4, the magnitude of Dt0 is significantly larger than themagnitude of Dt.

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-15

-10

-5

0

5

10

15

20

date(UTC)the

frac

tiona

l par

t of d

iffer

ence

(UTC

-GPS

T) in

seco

nds (

ns)

Fig. 4 Curve: the fractional part difference [UTC-GPST] in seconds from year 2009 to 2010

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3.3 GALILEO System Time

The length of a second in GST which is a continuous time system is consistentwith that of TAI. The deviation of GST and TAI are controlled within 50 ns in95% time in one year [8]. The differences between GST and TAI, GST and UTCwill be broadcasted by GALILEO satellite navigation message.

To improve compatibility and interoperability with the GPS, the starting pointof GST is set to 0 h 0 min 0 s January 6, 1980 UTC (USNO). An epoch in GST issimilar to that of GPST, but the number of accumulated weeks will return to zerowhen the weeks number up to 4,096. The difference between GST and UTCincludes integral part and fractional part, as shown in formula (8):

GST UTC t1 Dt00 8t1 which equals to the t1 in formula (6) is 15 s by now, and it is the integral part

of difference in seconds between GST and UTC. Dt00 which is the fractional part ofdifference in seconds between GST and UTC will be broadcasted by the GALI-LEO satellite navigation message. By now, there is no public institutions bulletinthese values.

3.4 BeiDou System Time

The length of a second in BDT which is a continuous time system is consistentwith that of TAI, and there is integral part of difference in seconds between BDTand UTC. The starting point of BDT is set to 0 h 0 min 0 s January 1, 2006 UTC

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date(UTC)the

fract

iona

l par

t of d

iffer

ence

(UTC

-GLS

T) in

seco

nds (

ns)

Fig. 5 Curve: the fractional part difference [UTC-GLST] in seconds from year 2009 to 2010


(NTSC) which maintained by NTSC. An epoch in BDT is the same as an epoch inGPST [9]. The conversion formula between BDT and UTC is as follows:

BDT UTCNTSC t0001 t0002 UTC t

0003 t

0001 t

0002

UTC t0001 t0002 t

0003 UTC t

0001 Dt000

9

t0001 (t

0001 1s n 33s) which is 1(n = 34) by now is the integral part of dif-

ference in seconds between BDT and UTC. Dt000, which is generated from the timeand frequency system in master station and be published by NTSC, is the frac-tional part of difference in seconds between BDT and UTC. Dt000 is controlledwithin 100 ns. The change of Dt000 is given from June 12, 2010 to August 26,2010 in the reference [9].

4 Time Transformations Among GNSSs

With the development of GNSSs, it is significant to set up the relationships of timescales among different GNSSs to improve the compatibility and interoperability[10]. According to formulas (6)(9) about the relationships between every GNSStime and UTC, the relationships among four GNSSs can be calculated as follows.

(1) The conversion formula between GPST and GLST:

GPST GLST t1 3h Dt Dt0: 10(2) The conversion formula between GPST and GST:

GPST GST Dt00 Dt: 11(3) The conversion formula between GPST and BDT:

GPST BDT t1 t0001 Dt Dt000: 12(4) The conversion formula between GLST and GST:

GLST GST 3h t1 Dt0 Dt00: 13(5) The conversion formula between GLST and BDT:

GLST BDT 3h t0001 Dt0 Dt000: 14(6) The conversion formula between GST and BDT:

GST BDT t1 t0001 Dt00 Dt000: 15

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The difference in seconds among these GNSS times consists of integral partdifference and fractional part difference. The integral part difference will changeonly while the leap second is adjusted. The fractional part difference is broadcastedby navigation messages of GNSSs or bulletined by related time service centers. Sofar, the integral part of the difference in seconds among four GNSS times is shownin Table 2.

According to the formulas which are derived above, it is convenient for timetransformations among the four GNSS times and GNSSs simulation.

5 Conclusion

This paper introduces the time scales which are commonly used in GNSSs and therelationships among them. In addition, this paper describes the internal referencetime systems of the four GNSSs and derives the relationships among them.Figure 6 shows the transformations among time scales and GNSS times.

Table 2 The integral part ofthe difference in secondsamong four GNSS times

GPST GLST GST BDT

GPST 0GLST 15 s3 h 0GST 0 3 h15 s 0BDT 14 s 3 h1 s 14 s 0

UT0

UT1

UT2

Ts

ET

TAI

UTC

BDT

GSTGPST

GLSTFormula(7) Formula(9)

Formula(10)

Formula(14)

Formula(15)

Formula(11)T+

+

+

Formula(6) Formula(8)

Formula(4)

Formula(5)

Fig. 6 The transformations among time scales and GNSS times


GNSS times can convert to one another accurately according to the fractionalpart difference of seconds which is broadcasted by navigation messages andthe formulas which are derived in this paper. Thus, it is convenient for processingthe data of GNSSs. These formulas have been applied in GNSSs simulation, andthe accuracy of them can meet the requirements.

Acknowledgments This work was supported by the National High-Tech. R&D Program, China(No.2011AA120505) and the National Natural Science Foundation, China (No.61173077).

References

1. Misra, P., & Enge, P. (2006). Global positioning system, signals, measurements, andperformance. (2nd edn, pp. 8189). London: Artech House Publisher.

2. Arias, E. F., Panfilo, G., & Petit, G. (2011). Timescales at the BIPM. Metrologia, 48,S145S153.

3. Lewandowski, W., & Arias, E. F. (2011). GNSS times and UTC. Metrologia, 48, S219S224.4. Wang., A. (2010) GNSS measurement data processing. (pp. 2330). Xuzhou: China

University of Mining and Technology Press(In Chinese).5. IERS Bulletin D [DB/OL]. Retrieved from http://www.iers.org/nn_10970/IERS/EN/

Publications/Bulletins/bulletins.html?__nnn=true.6. BIPM Circular T [DB/OL]. Retrieved from http://www.bipm.org/jsp/en/TimeFtp.jsp?

TypePub=publication.7. BIPM [TAIGPS time] and [UTCGPS time], [TAIGLONASS time] and [UTC

GLONASS time] [DB/OL]. Retrieved from http://www.bipm.org/jsp/en/TimeFtp.jsp?TypePub=scale.

8. Chen, X., Fang, Y., Yin, J., Zhang, H. (2005). Galileo satellite navigation system. (p. 45).Beijing: Peking University Press (In Chinese).

9. Han, C., Yang, Y., & Cai, Z. (2011). BeiDou navigation satellite system and its time scales.Metrologia, 48, S213S218.

10. Liu, Q., Bao, H., Wang, H., Wang, Q. (2008). Time transformation and coordinatestransformation among GPS, GLONASS and GALILEO. Journal of Science of Surveying andMapping, 1315 (In Chinese).

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45 Time Scales and Time Transformations Among Satellite Navigation SystemsAbstract1Introduction2Time Scales2.1 Astronomical Time2.1.1 Sidereal Time and Solar Time2.1.2 Universal Time2.1.3 Ephemeris Time

2.2 Atomic Time2.3 Coordinated Universal Time

3GNSS Time3.1 GPS Time3.2 GLONASS Time3.3 GALILEO System Time3.4 BeiDou System Time

4Time Transformations Among GNSSs5ConclusionAcknowledgmentsReferences

zhang 2012

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