<![CDATA[Algorithms for International Atomic Time]]>

0885–3010/$25.00 © 2010 IEEE

140 IEEE TransacTIons on UlTrasonIcs, FErroElEcTrIcs, and FrEqUEncy conTrol, vol. 57, no. 1, JanUary 2010

Abstract—This article reviews the creation and technical evolution of atomic time scales. In particular, we focus our attention on the method of calculation and the characteristics of International Atomic Time (TAI), and show how it is dis-seminated at the ultimate level of precision.

I. Introduction

The algorithm for the calculation of International atomic Time (TaI) has been designed to guarantee

the reliability, long-term frequency stability, frequency ac-curacy, and accessibility of the scale. It relies critically on the methods of clock comparison, which can act to the detriment of a highly precise time scale. The International Bureau for Weights and Measures (BIPM), with the sup-port of the timing community, devotes much effort to de-veloping and improving these methods. In this paper we present various issues related to time scales: we start in section II with the historical evolution of the time scales; in section III we present the different methods used to de-termine time links used in the calculation of TaI; section IV introduces and briefly explains TaI algorithms; and the final section deals with the dissemination of interna-tional time.

II. Evolution of Time scales and Their associated Interval Units

The construction of the first cesium frequency standard at the national Physical laboratory (Teddington, UK) in 1955 marked the beginning of the atomic era in frequency referencing and timekeeping. It was quickly recognized that the cesium transition could serve as a reference for frequencies. The adoption of this transition for the defini-tion of the second was more difficult, but did not raise fun-damental objections. However, the acceptance of a time scale built by accumulating atomic seconds was not easily accepted; one counter-argument was that atomic time dif-fered from dynamic time (which is, of course, true), but there was also the ancestral feeling that time is defined by the motion of celestial bodies. a fundamental difference between dynamic time scales, based on planetary motions, and atomic time scales, is that atomic time results from integration over frequency and that uncertainties are also integrated, leading to an unlimited departure from an ide-

ally integrated time. In contrast, astronomical times are based on observations of positions of celestial bodies, with limited uncertainties, decreasing as a consequence of obser-vational progress. at some time, the error in atomic time should exceed that in the reading of astronomical time. In spite of the opposition, TaI was formally adopted in 1971, and a compromise between this continuous time scale and the apparent rotation of the celestial bodies was accepted with the definition of coordinated Universal Time (UTc). The progress of atomic time standards narrows the limits of what can be considered a local phenomenon or a local measurement, leading to a situation where the structure of an atomic clock cannot be seen as local. Today the model-ing of a frequency standard itself must be considered in the framework of general relativity.

In the 1950s, when astronomical time scales were the standard for timekeeping, the precision of measurements did not require the application of relativistic theories. The goal of astronomers was to produce a time scale for world-wide use that was the best possible representation of ab-solute newtonian time.

A. Universal Time (UT)

World-wide unification of time started in 1884, with the adoption of the Greenwich meridian as the origin of terrestrial longitudes, and an associated universal time [1]. The name “universal time” (UT) indicates any time scale based on the rotation of the Earth and referred to the Prime Meridian. The use of universal time was recom-mended by the International astronomical Union (IaU) in 1948, although it had been in practical use since the end of the 19th century. To deal with the irregularities of UT, 3 kinds of “universal times” have been defined: UT0 is the time derived from astronomical observations, affected both by the motion of the rotation axis on the Earth relative to the crust (known as polar motion), and by the irregular rate of the rotation of the Earth. These 2 effects are independent, and can be clearly separated and studied. By eliminating the effects of the polar mo-tion from UT0, a form of universal time denoted UT1 is obtained; this is defined such that it is proportional to the rotation angle of the Earth in inertial space [2]. UT1 suf-fers from the irregularities of the rotation of the Earth, in-cluding secular deceleration and decade fluctuations. UT2 is derived from UT1 by eliminating the seasonal effects in the rotation of the Earth. It is a smoothed version of UT1, and rarely used.

since its creation in 1988, the International Earth ro-tation and reference systems service (IErs) has been charged with monitoring the irregularities of the Earth’s

Algorithms for International Atomic TimeGianna Panfilo and E. Felicitas arias

(Invited Paper)

Manuscript received June 22, 2009; accepted august 30, 2009. The authors are with the International Bureau for Weights and Mea-

sures (BIPM), sèvres, France (e-mail: [email protected]).digital object Identifier 10.1109/TUFFc.2010.1390

rotation, as was done in the past by the Bureau Interna-tional de l’Heure (BIH).

The variation in the length of the day, meaning the difference in duration of the rotational day from 86 400 “standard” seconds, is of the order of milliseconds. after the arrival of atomic clocks, these differences could be determined with precision.

The unit of time (the second), was then defined as the duration of 1/86 400 of a mean solar day. Time metrol-ogy was developed in astronomical observatories and a time determination consisted of observing star transits to obtain the data necessary to calculate corrections to the clocks that had a rate different from the sidereal rotation. These corrections were applied to the clock readings to obtain local time, and after adding the longitude, univer-sal time. By extrapolation, the clock provided universal time in real time. clearly, local realizations of universal time were different, and the need for an organization to centralize these activities was evident. after the end of World War I, the BIH was established at the Paris obser-vatory. one of the responsibilities of the BIH was, until 1988, to unify the time. Based on the contributions of the observatories, the BIH calculated the readings of an aver-age clock, and referred the time signals emissions from the observatories to it.

B. Ephemeris Time (ET)

confirmation of the irregularities of the Earth’s rota-tion came from the study of the motion of planets and of the Moon. discrepancies were detected when comparing an ephemeris calculated using the uniform time of the newtonian theory with the observed positions dated with a universal time scale, and it was concluded that these discrepancies arose from the non-uniformity of the Earth’s rotation. In the early 1930s it seemed clear that the time embedded in the dynamical equations of the solar system was a representation of uniform time, and, after some 50 years of difficult research, Ephemeris Time (ET) was de-fined in 1950 [3], [4] on the basis of the orbital motion of the Earth. The poor precision in positioning the sun with respect to the stars made it impossible to exploit the good uniformity of ET in acceptable delays and better precision of reading was obtained by defining a secondary ephemeris time by the motion of the Moon. However this motion, perturbed by ocean tides and geophysical phenomena, re-quires an empirical calibration against the fundamental ET which, although it extended over centuries, strongly limited the uniformity.

a new definition of the second was consequently associ-ated with the orbital motion of the Earth, and it became a fraction of a particular tropical year; the tropical year is the duration between 2 transits of the sun through the vernal equinox. Its duration was chosen such that it cor-responded to the average duration of the second of mean solar time over the century. When this definition was ad-opted, in 1960 [5], the second of ET was shorter than the second of mean solar time by 1.4 parts in 108.

In practice the determination of ET was rather com-plex. ET was determined in post-real time, in the form of a correction to be applied to universal time, based on the principle of the uniformity of the time argument in newtonian theory. The difference between a calculated ephemeris and the corresponding observation at the date measured in universal time, served to represent the correc-tion that should be applied to UT to derive ET. The use of ET remained limited to astronomical dynamics, and it was never used as a time scale world-wide.

during the short life-time of ET there was an inconsis-tency between the time scale used in practice and the unit of time. although the unit was associated to dynamical time, the practical time scale remained UT. This situation persisted until the adoption of atomic time.

C. Adoption of TAI

The General conference on Weights and Measures (cGPM) decided in 1968 [6] that “the second is the dura-tion of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.” This definition was completed in 1997 by declaring that it refers to a cesium atom at rest at a thermodynamic temperature of 0 K.

The unification of time on the basis of the atomic time scale of the BIH was recommended by the Internation-al astronomical Union (IaU) in 1967, the International Union of radio sciences (UrsI) in 1969, and the Interna-tional radio consultative committee of the International Telecommunication Union (ccIr) in 1970. Finally, in 1971, the 14th meeting for the cGPM [7] introduced the designation International atomic Time and the universal acronym TaI (as had been used to designate BIH atomic time since 1955).

a clear problem at that time for the universal accep-tance of TaI was to provide time to those users that re-quired astronomical time, including for sea navigation and other domestic applications. There was still a need to maintain 2 different time scales, and to avoid the enor-mous confusion that this might create, Universal coor-dinated Time (UTc) was defined by the International radio consultative committee of the International Tele-communication Union (ccIr, presently the International Telecommunication Union, radiocommunication sector, ITU-r), and its use was endorsed by the 15th meeting of the cGPM in 1975 [8].

In spite of objections, atomic time was increasingly used. nevertheless, TaI was never disseminated direct-ly and UTc, approximating UT1, continues to rule the world because it is needed in real time for some specific applications, including astronomical navigation, geodesy, telescope settings, space navigation, and satellite tracking. The definition of UTc evolved with an increasing toler-ance for the time offset [UT1 − UTc]. since 1972, UTc differs from TaI by an integer number of seconds, changed when necessary by insertion of a leap second to maintain |UT1 − UTc| < 0.9 s. although this system works well,

141panfilo and arias: algorithms for international atomic time

leap seconds are increasingly cumbersome and introduce an ambiguity in dating events when they occur. This leads to the fact that continuous time scales parallel to TaI, but with time offsets of an integer number of seconds, have been created, which puts at risk the unification of time. With the progress made in communications, other means of providing UT1 in real time can be conceived and the future of UTc is being discussed [9], [10]. TaI and UTc have numerous applications in time synchronization at all levels of precision, from the minute needed by the general public, to the nanoseconds required in the most demanding applications. The case of the Global Position-ing system (GPs) is typical [11]. The time scale of the system could be totally independent from TaI; however, the GPs authorities found it convenient to closely syn-chronize GPs Time with UTc(Usno) modulo one second and hence with TaI modulo one second. since 1998, the time difference between the 2 has mostly been between –30 and +20 ns with excursions up to +40 ns. In addition, GPs disseminates a good approximation to UTc, easily available at all levels of precision from the second to a few nanoseconds. similar features will be adopted for Galileo, the future European satellite positioning system.

TaI is the basis of realization of time scales used in dynamics, for modeling the motions of artificial and natu-ral celestial bodies, with applications in the exploration of the solar system, tests of theories, geodesy, geophysics, and studies of the environment. In all these applications, relativistic effects are important.

D. The Requested Properties of TAI

The phenomenon giving origin to a time scale should be reproducible with a frequency that is, ideally, constant. But this is never exactly the case, so we must be able to identify the causes of its variation, and to eliminate or at least minimize them. The realizations of the second of the International system of Units (sI) [12] differ from the ideal duration specified in its definition; in the pro-cess of constructing a time scale we should be capable of reducing these differences. one solution is to average; we can average in time, but taking into account that because the measurements are not simultaneous, a well-adapted process is essential. alternatively, we can average over the realizations of the second in different laboratories; in this case there are 2 sources of uncertainty: the individual experiments that converge to a realization, and the algo-rithm used to fabricate the scale.

TaI is an integrated time scale built by accumulation of seconds, which means that uncertainties are also ac-cumulated. This was one of the criticisms at the time of discussing the adoption of atomic time in replacement of ephemeris time. The main concern was the increasing discrepancy in the long-term, which must be taken into account in some applications. There was a doubt about the higher accuracy of TaI compared with ET.

The following elements are necessary for the construc-tion of an integrated time scale, such as TaI:

1) a periodic phenomenon; 2) a definition of the unit, derived from the frequency

of the phenomenon; 3) realizations of the unit, that is frequency standards,

called clocks if they operate continuously, and; 4) an algorithm of calculation adapted to the required

characteristics of the scale.

a time scale is characterized by its reliability, frequency stability and accuracy, and accessibility.

The reliability of a time scale is closely linked with the reliability of the clocks whose measurements are used for its construction; at the same time, redundancy is also requested. In the case of the international reference time scale, a large number of clocks are required; this number is today about 350, most of which are high-performance commercial cesium atomic standards and active auto-tuned hydrogen masers.

The frequency stability of a time scale represents its capacity to maintain a fixed ratio between its unitary scale interval and its theoretical counterpart. one means of estimating the frequency stability of a time scale is to calculate the allan variance [13], which is the 2-sample variance designed for the statistical analysis of time series, and depends on the sampling interval.

The frequency accuracy of a time scale represents the ability of its unitary scale interval to reproduce its theo-retical counterpart. after the calculation of a time scale on the basis of an algorithm conferring the required fre-quency stability, the frequency accuracy can be improved by comparing the frequency (rate) of the time scale with that of primary frequency standards, and by applying, if necessary, frequency (rate) corrections.

The accessibility to a worldwide time scale represents its ability to provide a way of dating events for everyone. This depends on the precision that is required. We con-sider here only the ultimate precision, which requires a de-lay of a few tens of days to reach the long-term frequency stability required for a reference time scale. Besides, the process needs to be designed in such a way that the mea-surement noise is eliminated or at least minimized; this requires a minimum number of data-sampling intervals.

The instability of TaI, estimated today as 0.4 × 10−15 for averaging times of 20 d to 40 d, is obtained by process-ing clock and clock comparison data at 5-d intervals over a monthly analysis, with a delay to publication of about 15 d after the last date of data reported in the official document called BIPM Circular T [14] (see Fig. 1). In the very long-term, over a decade, the stability is maintained by primary frequency standards and is limited by the ac-curacy at the level of parts in 1015 assuming that the pres-ent performances are constant.

III. an Essential Tool: clock comparisons

The calculation of a time scale on the basis of the read-ings of clocks located in different laboratories requires the


use of methods of comparison of distant clocks. a prime requisite is that the methods of time transfer do not con-taminate the frequency stability of the clocks; in fact in the past they were often a major limitation in the con-struction of a time scale.

The uncertainty of clock comparisons today is between a few tens of nanoseconds and a nanosecond for the best links, a priori sufficient to allow a comparison of the best atomic standards over integration times of a few days. This assertion is strictly valid for frequency comparisons, where only the type a (statistical) uncertainty affects the process. In the case of time comparisons, the systematic uncertainty (type B) coming from the calibration should also be con-sidered. In the present situation, calibrations contribute an uncertainty that surpasses the statistical component, and which can reach 20 ns for uncalibrated equipment (see Ta-ble I). It can be inferred that repeated equipment calibra-tions are indispensable for clock comparison.

a network of international time links has been estab-lished by the BIPM to organize these comparisons. It is a star-like scheme with links from laboratories to a pivot laboratory (currently the Physikalisch-Technischen Bunde-sanstalt, PTB, in Braunschweig, Germany).

It should be noted that participating laboratories pro-vide time transfer data in the form of a comparison of their UTc(k) with respect to another time scale or to another local realization of UTc.

A. Use of GPS for Time Transfer

The use of GPs satellites in time comparisons intro-duced a major improvement in the construction and dis-semination of time scales. It consists of using the signal broadcast by GPs satellites, which contains timing and positioning information. It is a one-way method, the sig-nal being emitted by a satellite and received by specific


Fig. 1. an excerpt of section 1 of BIPM Circular T.

equipment installed in a laboratory. For this purpose, GPs receivers have been developed and commercialized to be used specifically for time transfer. The common-view method proposed in the 1980s by allan and Weiss [15] re-lies on the reception by several receivers of the same emit-ted signal. It is still in use for clock comparisons because it eliminates the instability of the satellite clocks. The russian satellite system of global navigation, Glonass, is not yet used on a routine basis for time comparison in TaI, but it is being introduced.

GPs transfer data are provided in the form of the dif-ference between a clock in a laboratory [generally the one realizing UTc(k)] and the GPs time. The GPs receivers developed for time transfer include a time interval counter allowing them to provide these differences.

Thanks to new hardware and to improvements in data treatment and modeling, the uncertainty of clock synchro-nization via GPs has been reduced from a few hundreds of nanoseconds at the beginning of the 1980s to 1 ns today. an important issue with respect to receiver’s hardware is the capability of locking the internal reference to an exter-nal clock from the laboratory.

old single-channel, single-frequency c/a code receiv-ers are being replaced in time laboratories by multi-channel receivers, which allow the simultaneous obser-vation of all satellites over the horizon. The effects of ionospheric delay introduce one of the most significant errors in GPs time comparisons, in particular in the case of clocks compared over long baselines. dual-frequency receivers installed in some of the participating laborato-ries permit the removal of the delay introduced by the ionosphere, thus increasing the accuracy of time transfer. GPs observations with single-frequency receivers used in regular TaI calculations are corrected for ionospheric delays by making use of ionospheric maps produced by the International GPs service (IGs). all GPs links are

corrected for satellite positions using IGs post-processed precise satellite ephemerides.

The GPs links obtained using dual-frequency receiv-ers, denoted GPs P3 [16], [17], provide ionosphere-free data and allow clock comparisons with nanosecond sta-tistical uncertainty or better. This kind of link uses GPs code measurements only. Better results are expected with the use of the carrier phase combined with the code; this method is in an experimental phase currently at the BIPM (GPs PPP [18]).

B. Two–Way Satellite Time and Frequency Transfer

after about 25 years of experimentation, the method of 2-way satellite time and frequency transfer (TWsTFT) began to be used extensively in TaI at the beginning of the 21st century [19], [20]. The TWsTFT technique utilizes a telecommunications geostationary satellite to compare clocks located in 2 receiving-emitting stations. Two-way observations are scheduled between pairs of laboratories so that their clocks are simultaneously compared at both ends of the baseline. The clocks are directly compared, us-ing the transponder of the satellite. This two-way method has the advantage over the one-way method of eliminat-ing or reducing some sources of systematic error, such as ionospheric and tropospheric delays, and the uncertainty in the positions of the satellite and the ground stations. The differences between 2 clocks placed in the 2 stations are directly computed. The first TWsTFT link was intro-duced into TaI in 1999. since then, the number of labora-tories operating 2-way equipment has increased, allowing links within and between north america, Europe, and the asia-Pacific region. Until mid-2004, intervals of 5-min measurements were made on 3 days per week, impeding the technique from reaching its highest potential perfor-mance, which is sub-nanosecond uncertainty. With the


TaBlE I. characteristics of some of the Time links in TaI.

link1 lab 1/lab 2 Technique2 ua,3 ns uB,4 ns

calibration type5 lab 1/lab 2

distance, km (approx.)

nIsT/PTB GPs Mc 1.5 5.0 GPs Ec/GPs Ec 10 000nIsT/PTB TWsTFT 0.5 5.0 Bc(GPs Ec) 10 000oP/PTB GPs sc 2.5 5.0 GPs Ec/GPs Ec 600oP/PTB GPs P3 0.7 5.0 GPs Ec/GPs Ec 600oP/PTB TWsTFT 0.5 1.0 lc(TWsTFT) 600Usno/PTB GPs P3 0.7 5.0 GPs Ec/GPs Ec 7000Usno/PTB TWsTFT 0.5 1.0 lc(TWsTFT) 7000onrJ/PTB GPs sc 4.0 20.0 na/GPs Ec 10 0001conventional acronyms [nIsT (national Institute of standards and Technology, Boulder, co), PTB (Physikalisch-Technischen Bundesanstalt, Braunschweig, Germany), oP (observatoire de Paris, Paris, France), Usno (United states naval observatory, Washington, dc), onrJ (national observatory in rio de Janeiro, rio de Janeiro, Brazil)] for the laboratories as used in the BIPM reports.2GPs Mc = GPs multi-channel c/a data; GPs sc = GPs single-channel c/a data; GPs P3 = GPs multi-channel dual-frequency P code data; TWsTFT = two-way satellite time and frequency transfer data.3ua is the statistical uncertainty evaluated by taking into account the level of phase noise in the raw data.4uB is the uncertainty of the calibration. 5Ec indicates equipment calibration, lc(technique) indicates a link calibrated using the mentioned technique, Bc(technique) indicates a link calibrated using the mentioned technique to transfer a past equipment calibration through a discontinuity in the operation of the link. na = not available.

installation of automated stations in most laboratories, some of the TWsTFT link observations in TaI are now made at 2-h intervals, with the consequence of achieving an uncertainty below 1 ns.

C. Characterization of the Relative Delay of Time–Transfer Equipment and Evaluation of Link Uncertainties

Measuring the delays of the laboratory’s equipment for time transfer is fundamental for the stability of TaI and for its dissemination. campaigns for determining differential delays of GPs time equipment are organized by the BIPM to compensate for internal delays in lab-oratories by comparing their equipment with traveling BIPM GPs equipment. successive campaigns with BIPM traveling receivers have been conducted since 2001 with the result that about 65% of the GPs equipment used in TaI has been calibrated (see, for example, [21]–[23]). The situation for the TWsTFT links is rather different; the laboratories organize calibrations of the TWsTFT equipment with the support of the BIPM [24]–[27]; in this case the lowest type B uncertainty is obtained for a link calibrated by the TWsTFT technique [lc(TWsTFT) in Table I]. For those TWsTFT links where this calibra-tion is not available that of the corresponding GPs link is considered, thus increasing the type B uncertainty [in Table I, Bc(GPs Ec)].

The BIPM estimates type a (ua) and type B (uB) stan-dard uncertainties of all time links in TaI [28]. In Table I, some links have been selected to show examples of their values. The statistical uncertainty ua is evaluated by tak-ing into account the level of phase noise in the raw data; uB is the uncertainty of the calibration.

For 2 decades, GPs c/a-code observations have pro-vided a unique tool for clock comparisons in TaI, render-ing impossible any test of its performance with respect to other methods. The present situation is quite different; the introduction of the TWsTFT technique has allowed the opportunity of comparing the results of clock comparisons traditionally obtained with the GPs technique to those coming from an independent technique, and made the sys-tem more reliable. For the links where the 2 techniques are available, both GPs and TWsTFT links are computed; the best being used in the calculation of TaI, and the other kept as a backup. at present, 85 % of the links in TaI are obtained by using GPs equipment and about 15 % of the links are provided by TWsTFT observations. There still remain a small number of laboratories (6 % only) equipped with old-type receivers.

IV. The calculation of UTc and TaI

The time laboratories realize a stable local time scale using individual atomic clocks or a clock ensemble. clock readings are then combined at the BIPM through an al-gorithm designed to optimize the frequency stability and

accuracy, and increase the reliability of the time scale above the level of performance that can be realized by any individual clock in the ensemble. an efficient algorithm is necessary for the statistical generation of a time scale. alGos is the algorithm in use in the Time, Frequency, and Gravimetry section of the BIPM, which produces, monthly, the international reference UTc (coordinated Universal Time). The calculation of UTc using alGos is carried out in 3 successive steps:

The free atomic time scale Eal (Echelle atomique •libre) is computed as a weighted average of about 350 free-running atomic clocks spread world-wide. a clock weighting procedure has been designed to optimize the long-term frequency stability of the scale.The frequency of Eal is steered to maintain agree-•ment with the definition of the sI second, and the re-sulting time scale is International atomic Time TaI. The steering correction is determined by comparing the Eal frequency with that of primary frequency standards.leap seconds are inserted to maintain agreement with •the time derived from the rotation of the Earth. The resulting time scale is UTc.

different algorithms can be considered depending on the requirements on the scale; for an international refer-ence such as UTc, the requirement is extreme reliabil-ity and long-term frequency stability. UTc therefore relies on the largest possible number of atomic clocks of different types, located in different parts of the world and connected in a network that allows precise time comparisons between remote sites. Each month, the dif-ferences between the international time scale UTc and the local approximations UTc(k) in contributing time laboratories are reported, at 5-d intervals, in differed time in the official document called BIPM Circular T. several examples of [UTc − UTc(k)] are reported in Fig. 2.

The original algorithm alGos for defining Eal was developed in the 1970s [29]–[31] from the equation

EAL t w h t h ti i ii

N

( ) = ( ) + ¢ ( )[ ]=å

1

, (1)

where N is the number of participating clocks during the interval of calculation (one month), wi the relative weight of clock Hi, hi(t) is the reading of clock Hi at time t, and hi’ (t) is the prediction of the reading of clock Hi that serves to guarantee the continuity of the time scale. The weight attributed to a clock reflects its long-term stability, because the objective is to obtain a weighted average that is more stable in the long-term than any of the contribut-ing elements [31], [32]. The weights of the clocks obey the relation


w ii

N

=å =

1

1. (2)

subtracting the same quantity from both sides of (1) we obtain

EAL

t w h t w h t h t

w h t

i ii

N

i i ii

N

i ii

N

( ) - ( ) = ( ) + ¢ ( )[ ]

- ( )

= =

=

å å

å

1 1

1

.

Using (2) and rearranging,

w t h t w h tii

N

i i ii

N

= =å å( )- ( )( ) = ¢ ( )

1 1

EAL . (3)

setting x t t h ti i( ) = ( ) - ( )EAL , (4)

Eq. (3) takes the form:

w x t w h tii

N

i i ii

N

= =å å( ) = ¢ ( )

1 1

. (5)

The data used by alGos take the form of time differ-ences between readings of clocks, written as:

x t h t h ti j j i, .( ) = ( ) - ( ) (6)

Eq. (5) in conjunction with the N − 1 equations de-fined by (6) results in a system of N equations and N unknowns

w x t w h t

x t x t x t

ii

N

i i ii

N

i j i j

= =å å( ) = ¢ ( )

( ) - ( ) = ( )

ì

í

ïïïïï

îïïïïï

1 1

,

(7)

whose solution is:

x t t h w h t x tj j i i i ji

N

( ) = ( ) - = ¢ ( ) - ( )[ ]=åEAL , .

1

(8)

The difference (8) between any clock Hj and Eal depends on the clock weights, the clock frequency prediction, and the measured clock differences. The clock Hj may also represent a UTc(j) time scale; therefore, xj(t) can also be interpreted as:

x jj = -EAL UTC( ), (9)

where for simplicity we have dropped the time instant t from the notation.

The clock frequency prediction and weights are fixed by appropriate algorithms based on the clock behavior in the past, and in eq. (8) they can be considered as time-varying deterministic parameters. From (9), and following the steps described in the preceding section, the differ-ences [TaI − UTc(j)] and finally [UTc − UTc(j)] are evaluated.

In the next sections, the most important algorithms used in alGos will be presented. We present 3 different algorithms:

The weighting algorithm optimized to guarantee the •long–term stability of the time scale.


Fig. 2. [UTc − UTc(lab)] for Usno (United states naval observatory, Washington, dc), PTB (Physikalisch-Technischen Bundesanstalt, Braun-schweig, Germany), IT (Istituto nazionale di ricerca Metrologica, Torino, Italy), oP (observatoire de Paris, Paris, France), nIsT (national Institute of standards and Technology, Boulder, co), nPl (national Physical laboratory, Teddington, UK), nIM (national Institue of Metrology, Beijing, china), and nIcT (national Institute of Information and communications Technology, Tokyo, Japan) (MJd means Modified Julian date).

The prediction algorithm used to avoid time and fre-•quency jumps due to different clock ensembles being used in consecutive calculation periods.The steering algorithm used to improve the time scale •accuracy.

A. Weighting Algorithm

In time-scale algorithms, clock weights are generally chosen as the reciprocals of a statistical quantity which characterizes their frequency stability, such as a frequency variance (classical variance, allan variance, etc.) [30]. If strictly applied, this gives a time scale which is more sta-ble than any contributing element. In the Eal computa-tion, the weight attributed to clock is the reciprocal of the individual classical variance computed from the frequen-cies of the clock, relative to Eal s i T2 12, ,( ) estimated over the current 30-d interval (T) and over the past 11 consecutive 30-d intervals (12 consecutive 30-d intervals are considered). The weight determination thus uses clock measurements covering a full year. This reduces the weight of both clocks that are highly sensitive to seasonal chang-es and hydrogen masers that show a large frequency drift. It thus helps to improve the long-term stability of Eal.

as the frequency over the current interval of computa-tion is not known for the direct computation of s i T2 12, ,( ) an iterative process, including 4 iterations is used in al-Gos. Each iteration runs as follows:

1) system (7) is solved for the dates of the interval [ti, ti+T], using a given set of relative weights ωi. For the first iteration, {ωi} are those obtained in the previous computation interval after normalization. For follow-ing iterations, the {ωi} are those obtained from the previous iteration.

2) Frequencies y(ti + T) are estimated for each clock using the {xi(t)} from system (7).

3) Individual variances s i T2 12,( ) are estimated. 4) The relative weight of clock Hi is computed theoreti-

cally using a temporary value given by

ws

s

ii

kk

N

T

T

TEMP/

/

=( )

( )=å

1 12

1 12

2

2

1

,

,

.

5) The new weight ωi of clock is equal to ωi,TEMP except in 2 cases:

a) clock Hi satisfies the requirement set for the limi-tation of weight so it cannot contribute according to its full stability. This first condition establishes the upper limit of weight statistically necessary to make the time scale rely on the best clocks and yet avoid giving a predominant role to any one of them. This is linked to the requirement of reli-ability.

b) clock Hi shows abnormal behavior during the in-terval of computation so it cannot contribute. This second condition protects the time scale against an abrupt change in frequency of one contributing clock, for which the predicted frequency would be very bad. This is linked to the requirement of stability.

a key feature of case a) is that the resulting time scale is not necessarily more stable than the best contributing clocks when an upper limit of weight is set, the full quality of stability of these individual elements not being taken totally into account. The choice of a method to implement an upper limit of weight, as well as its value, thus plays an important role in the stability of the resulting time scale. The choice can be tested by the efficiency with which the stability of the scale is improved. In practice, the appli-cation of cases a) and b) calls for the choice of objective rules for the determination of the upper limit of weight and of a criterion for the detection of abnormal behavior. To limit individual clock contributions, preventing domi-nation of the scale by a small number of very stable clocks, a value of ωMaX is chosen, expressed as a fraction between 0 and 1. In alGos ωMaX is chosen as a function of the total number of clocks N, such that

wMAX =AN

,

where a is an empirical constant equal to 2.5.

B. Prediction Algorithm

In this section we present the prediction algorithm used in alGos [29]–[33]. In the generation of a time scale, the prediction of the atomic clock behavior plays an important role; in fact the prediction is useful to avoid or minimize frequency jumps of the time scale when a clock is added or removed from the ensemble or when its weight changes. considering 2 successive intervals of TaI calculation Ik−1 (t k−1, tk) and Ik (tk, tk+1) we impose several constraints on the prediction term at time tk to avoid or minimize time and frequency jumps in the resulting time scale.

alGos operates in post-processing, treating as a whole measurements taken over a basic period of t = 30 d or 35 d. For each one-month period, the results are the quanti-ties xi (tk + nT/6) [or xi (tk + nT/7)] with n = 0, ..., 6 (or n = 0, ..., 7) for each clock Hi. The least squares slope of these quantities is referred to as the frequency B t Ti I kk, +( ) of the clock Hi, relative to Eal, for the one-month inter-val Ik = [tk, tk + T]. The correction term [7] for clock Hi is the sum of 2 terms:

h t a t B t t ti i I k ip I kk k'

, , ,( ) = ( ) + ( ) -( ) (10)

where ai(tk) is the time correction relative to Eal of clock Hi at date tk:


a t t h t x ti I k k i k i kk, ( ) .( ) = - ( ) = ( )EAL (11)

B tip I k, ( ) is the frequency of clock Hi, relative to Eal, predicted for the period Ik = [tk, t] and (t – tk) = nT/6 (or nT/7) with n = 0, ..., 6 (or n = 0, ..., 7). at present, the predicted frequency B tip I k, ( ) for t included in Ik = [tk, tk + T] is kept constant for the whole one-month interval Ik = [tk, tk + T]. It is chosen to be the frequency Bi(tk) com-puted for the previous one-month interval I k−1 = [tk − T, tk], that is to say, the least squares slope of {xi(tk − T + nT/6)} [or {xi(tk – T + nT/7)}] with n = 0, ..., 6 (or n = 0, ..., 7). This is a one-step linear prediction written as B t B tip I i kk, ( ) = ( ) for t = tk + nT/6 (or t = tk + nT/7), n = 0, ..., 6 (or n = 0, ..., 7). The assumption that clock Hi is most likely to behave over the coming one-month inter-val as it did on the previous one is linked to the choice of the sample duration T.

C. Steering Algorithm

International atomic Time TaI is a realization of Ter-restrial Time TT, a coordinate time of a geocentric refer-ence system. TaI gets its stability from some 350 atomic clocks kept in some 68 laboratories worldwide and its ac-curacy from a small number of primary frequency stan-dards (PFs) developed by a few metrology laboratories. The frequency of Eal is compared with that of the pri-mary frequency standards using all available data, and a frequency shift (frequency steering correction) is applied to Eal to ensure that the frequency of TaI conforms to its definition. changes to the steering correction are expected to ensure accuracy without degrading the long-term (several months) stability of TaI, and these changes are announced in advance in the BIPM Circular T. The accuracy of TaI therefore depends on PFs measurements, which are reported more or less regularly to the BIPM. data from several PFs are combined to estimate of the duration of the scale unit of TaI [33], [34].

D. Procedure to Estimate the Duration of the Scale Unit of TAI

Time laboratories maintaining PFs report to the BIPM measurements of the frequency of the primary standard with respect to that of a clock participating in TaI over an interval elapsed between 2 dates where the standard has been operated more or less continuously (typically between 10 and 30 d). a report of the evaluation of a pri-mary standard j over an interval Tji contains, at present, the following information:

the interval of comparison, • Tji ;the time average of the frequency difference between •the reference and the PFs during the interval Tji, indicated by Wji;•uBj, the combined standard uncertainty from system-atic effects;

•uaij, the combined standard uncertainty related to the instability of the PFs;•ulink/lab, the standard uncertainty of the link between the PFs and the clock in the laboratory participating in TaI used to reference the frequency of the primary standard (if applicable);•ulink/TaI, the uncertainty of the link of the laboratory to TaI, as estimated by the BIPM.

The algorithm used at the BIPM to estimate the dura-tion of the scale unit of TaI [34] combines the individual calibrations of PFs and calculates the frequency of the time scale during a given interval (usually the month of calculation of Circular T).

The calibrations are usually referred to a local inde-pendent time scale. When using them to improve the ac-curacy of TaI, we should account for the transfer result-ing from the local time scale to the reference time scale (in this case Eal), and for the transfer of the frequency measurements from the various calibration dates to the period of interest T.

a frequency standard j carries out nj calibrations. If N is the number of standards considered, the number of available calibrations will be n jj

N=å 1 . We calculate the

rate of Eal over an interval T as:

y a Wjii

njjij

N=

== åå 11,

where Wji is the rate difference between Eal and the PFs j for a given interval Tji, and aji are the filter coefficients. The filter coefficients aji, are normalized and depend on:

1. the uncertainty of the evaluation i of the standard j.

2. the time separation in days between Tji and T 3. the instability of Eal, which transfers the evalua-

tion from Tji to T.

The same algorithm used to evaluate the frequency of Eal is used in post–processing to calculate another time scale called TT(BIPM), also a realization of terrestrial time TT [35]–[37]. TT(BIPM) is a time scale optimized for frequency accuracy. It is evaluated annually by mak-ing use of all available PFs data reported to the BIPM by national laboratories.

V. dissemination of the Time scales

The time scales TaI and UTc are disseminated every month by Circular T. access to UTc is provided in the form of differences [UTc − UTc(k)], thus at the same time making the local approximations UTc(k) traceable to UTc. since January 2005, the uncertainties of the dif-ferences have also been published [38], [39]. The values of the frequency corrections to TaI and their intervals of validity are regularly reported. This information is needed


for the laboratories to steer the frequency of their UTc(k) to UTc. Circular T provides wide access to the best real-ization of the second through the estimation of the frac-tional deviation d of the scale interval of TaI with respect to its theoretical value based on the sI second, calculated as explained above. The values of d for the individual con-tributions of the PFs are also published, giving access to the second as realized by each of the primary standards. access to GPs Time with an uncertainty of a few nanosec-onds and to Glonass Time with an uncertainty of a few tens of nanoseconds is provided via their differences with respect to TaI and UTc. Each monthly issue of Circular T provides information on the time links used for that particular computation together with their respective type a and type B uncertainties, and the technique used in the characterization of the time transfer equipment or link. also the daily differences between [UTc − GPs Time] and [UTc − Glonass time] are reported in Circular T. The ftp server of the BIPM Time, Frequency, and Gra-vimetry section [40] gives access to clock data and time transfer files provided by the participating laboratories, as well as the rates and weights for clocks in TaI in each month of calculation. This information is particularly use-ful for laboratories in the study of their clocks behavior. results for a complete year are published in the BIPM Annual Report on Time Activities [41], together with infor-mation about the equipment in contributing laboratories, time signals and time dissemination services, as reported by the laboratories to the BIPM. data used for the calcu-lation of TaI, Circular T, some tables of the Annual Re-port, and other relevant results and information are avail-able on the ftp server of the BIPM Time, Frequency, and Gravimetry section [40].

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[40] http://www.bipm.org/jsp/en/TimeFtp.jsp[41] BIPM Annual Report on Time Activities, published annually by the

BIPM.

Gianna Panfilo received the master’s degree in mathematics from the University of rome “la sa-pienza,” rome, Italy, and the Ph.d. degree in me-trology from the Politecnico of Turin, Turin, Italy, in 2006. she is a physicist at the Bureau Interna-tional des Poids and Mesures (BIPM), sèvres, France. she studies the algorithm alGos, main-tained at the BIPM and used to generate the In-ternational atomic Time (TaI) and the coordi-nated Universal Time (UTc). In particular, she

analyses the role of the prediction term in the generation of a time scale. she collaborates with the Working Group on Primary Frequency stan-dards (PFs) and on the Mutual recognition arrangement (Mra).

E. Felicitas Arias received the master’s degree in astronomy from the University of la Plata, ar-gentina, and the Ph.d. degree in astrometry, ce-lestial mechanics and geodesy from Paris obser-vatory, France, in 1990. she has been director of the Buenos aires naval observatory and is pro-fessor of the University of la Plata. since 1999 she has been the head of the time, frequency, and gravimetry section at the International Bureau of Weights and Measures (BIPM), sèvres, France. Her fields of activity are the space and time refer-

ences. she is a member of scientific organizations and unions such as the International astronomical Union, the International association for Ge-odesy, and the International Earth rotation and reference systems ser-vice.


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