Table of contents

This special issue of Metrologia presents selected papers from the Fifth International Time Scale Algorithm Symposium (VITSAS), including some of the tutorials presented on the first day. The symposium was attended by 76 persons, from every continent except Antarctica, by students as well as senior scientists, and hosted by the Real Instituto y Observatorio de la Armada (ROA) in San Fernando, Spain, whose staff further enhanced their nation's high reputation for hospitality.

Although a timescale can be simply defined as a weighted average of clocks, whose purpose is to measure time better than any individual clock, timescale theory has long been and continues to be a vibrant field of research that has both followed and helped to create advances in the art of timekeeping. There is no perfect timescale algorithm, because every one embodies a compromise involving user needs. Some users wish to generate a constant frequency, perhaps not necessarily one that is well-defined with respect to the definition of a second. Other users might want a clock which is as close to UTC or a particular reference clock as possible, or perhaps wish to minimize the maximum variation from that standard. In contrast to the steered timescales that would be required by those users, other users may need free-running timescales, which are independent of external information. While no algorithm can meet all these needs, every algorithm can benefit from some form of tuning. The optimal tuning, and even the optimal algorithm, can depend on the noise characteristics of the frequency standards, or of their comparison systems, the most precise and accurate of which are currently Two Way Satellite Time and Frequency Transfer (TWSTFT) and GPS carrier phase time transfer.

The interest in time scale algorithms and its associated statistical methodology began around 40 years ago when the Allan variance appeared and when the metrological institutions started realizing ensemble atomic time using more than one single atomic clock. An international symposium dedicated to these topics was initiated in 1972 as the first International Symposium on Atomic Time Scale Algorithms and it was the beginning of a series:

• 1st Symposium: organized at the NIST (NBS at that epoch) in 1972,

• 2nd Symposium: again at the NIST in 1982,

• 3rd Symposium: in Italy at the INRIM (IEN at that epoch) in 1988,

• 4th Symposium: in Paris at the BIPM in 2002 (see Metrologia40 (3), 2003)

• 5th Symposium: in San Fernando, Spain at the ROA in 2008.

The early symposia were concerned with establishing the basics of how to estimate and characterize the behavior of an atomic frequency standard in an unambiguous and clearly identifiable way, and how to combine the reading of different clocks to form an optimal time scale within a laboratory. Later, as atomic frequency standards began to be used as components in larger systems, interest grew in understanding the impact of a clock in a more complex environment. For example, use of clocks in telecommunication networks in a Synchronous Digital Hierarchy created a need to measure the maximum time error spanned by a clock in a certain interval. Timekeeping metrologists became interested in estimating time deviations and time stability, so they had to find ways to convert their common frequency characteristics to time characteristics. Tests of fundamental physics provided a motivation for launching atomic frequency standards into space in long-lasting missions, whose high-precision measurements might be available for only a few hours a day, yielding a series of clock data with many gaps and outliers for which a suitable statistical analysis was necessary to extract as much information as possible from the data. In the 21st century, the field has been transformed by the advent of atomic-clock-based Global Navigation Satellite Systems (GNSS), the steady increase in precision brought about by rapidly improving clocks and measurement systems, and the growing number of relatively inexpensive small clock ensembles.

Although technological transformations have raised the intensity and changed the details of the debates, the VITSAS conference showed that even the issues raised by the early symposia are still current. This selection of papers encompasses the full breadth of the VITSAS, including tutorials, laboratory-specific innovations and practices, GNSS applications, UTC generation, TWSTFT applications, GPS applications, small-ensemble applications, robust algorithms, and statistical measures that are either robust themselves or which reflect nonstationarity and robustness characteristics of the clocks.

The Editors of this special issue of Metrologia would like to express their thanks to the referees of the papers published here for all their hard work, to Drs Juan Palacio and Javier Galindo and the people of the ROA, and to all the attendees for the excellent symposium they have created.

The grey system and grey forecast are being increasingly widely used. As we know, a system could be a white system of which all the information can be known, or a black system of which none of the information can be known, or a grey system of which not all the information can be known. An atomic clock is, in fact, a grey system. We can only know the historical information of it, whereas there is some uncertain information for the future. In order to predict the trend of an atomic clock, a linear model or a non-linear model is used frequently, but in this paper a new model named the dynamic grey-autoregressive model is studied. Compared with the older methods, the dominance of the dynamic grey-autoregressive model is discussed in this paper, and finally, the new method is used to forecast the velocity of caesium clocks and hydrogen masers.

TWSTFT (Two Way Satellite Time and Frequency Transfer, TW hereafter) is a major technique used in TAI (International Atomic Time) generation. More than two-thirds of TAI clocks and almost all the primary frequency standards are transferred using TW. Up to now, the only geometry in TAI time transfer is single-link. However, the TAI TW time transfer data are highly redundant. In general, for an N-point network, there are N(N − 1)/2 independently measured links. Among them, only N − 1 will be used. We then have (N² − 3N + 2)/2 redundant links. As a function of N, the redundant measurements increase quickly (cf figure 1 and table 1). At present, for the European–American network N = 13, but only 12 out of a total of 78 measured links are used in TAI. For the Asia–Pacific regions, N = 8. Full use of the high redundancy is an effective way to improve TAI without new cost.

The sum of three TW links that form a closed triangle is the triangle closure. Theoretically a closure is expected to be zero if there are no measurement errors, namely the triangle closure condition. A non-zero closure is a true error and an index of the time link quality. A redundant link sets a geometric constraint. There are (N² − 3N + 2)/2 independent conditions in a network. In 2006, Jiang and Petit (Proc. EFTF 2006 pp 468–75) proposed a mathematical model to adjust the closures to zero by global network processing. In consequence, time transfer between any two points through any link(s) in the network gives exactly the same result with the same uncertainty. This is the so-called network time transfer.

In this paper, the author introduces his recent works on completing the network model by adding the calibration, the uncertainty estimation and the quality assessment using GPS PPP (time transfer by precise point positioning (PPP hereafter)) (Kouba and Héroux 2001 GPS Solut.5 12–28, Ray and Senior 2005 Metrologia42 215–32, Orgiazzi et al 2005 Proc. IEEE FCS 2005 pp 337–45, Defraigne et al 2007 Proc. EFTF 2007 pp 909–13, Petit and Jiang 2008 Int. J. Navig. Obs.2008 1–8). As an independent technique with higher short-term stability, PPP is then a good reference to evaluate the improvement in the network time transfer. The gain is at least 30%. The new method also gives a solution for the high redundancy in the TAI international TW time transfer network. The TAI software Tsoft is operational to perform the network time transfer.

The National Institute of Standards and Technology (NIST) operates 22 network time servers at various locations. These servers respond to requests for time in a number of different formats and provide time stamps that are directly traceable to the NIST atomic clock ensemble in Boulder. The link between the servers at locations outside of the NIST Boulder Laboratories and the atomic clock ensemble is provided by the Automated Computer Time Service (ACTS) system, which has a direct connection to the clock ensemble and which transmits time information over dial-up telephone lines with a two-way protocol to measure the transmission delay. I will discuss improvements to the ACTS servers and to the time servers themselves. These improvements have resulted in an improvement of almost an order of magnitude in the performance of the system.

I will describe the backup time scale system that I have constructed at the site of the NIST radio stations near Fort Collins, Colorado, and I will compare its performance with the primary ensemble in Boulder. The Fort Collins system is designed to be a backup for the Boulder system and is intended to support all of the NIST time services should the primary scale become unavailable for any reason. The backup time has a number of unique problems and requirements, and I will discuss the design considerations that I used to address these issues. The backup scale tracks UTC(NIST) in frequency with an uncertainty (measured by the Allan deviation of the difference) of about 1 × 10⁻¹⁴ by use of administrative steering that is applied not more often than once per week. The corresponding time deviation is less than 1 ns for all averaging times less than 1 week, and the peak time difference between UTC(NIST) and its backup realization is less than ±25 ns and is generally much better than this value. This is much better than would be needed for supporting the radio stations, the digital time services (ACTS, a time service that provides time in a digital format using dial-up telephone lines, and the Internet services) and the Frequency Measurement Service. Its frequency stability and time accuracy would not be adequate for the most demanding users of the Global Time Service and for international time and frequency coordination. The primary limitations to the performance of the backup time scale are caused by environmental perturbations, especially temperature and supply voltage, and the existing hardware could probably support all of the NIST services if the environment were improved.

TA(NIM) is being updated now. The new TA(NIM) uses an active hydrogen maser as a reference clock, and the frequency of TA(NIM) is steered by an algorithm to keep it the same as the NIM4 fountain clock, which is NIM's primary frequency standard. This paper introduces the generation of new TA(NIM), including the equations to predict maser frequency and the steering algorithm. The time difference and the frequency difference between the new TA(NIM) and UTC are also reported.

The frequency characteristics of two hydrogen masers at the National Time Service Center are studied in this paper by comparing the Allan deviation with the Hadamard deviation. The results show that the hydrogen masers exhibit a linear frequency drift (Lfd) within a period of about a few tens of days. The main purpose here is to test whether such hydrogen masers could contribute to a time scale composed of a clock ensemble and how to deal with the algorithm. It proves feasible and effective to remove the Lfds from the phase data of the masers according to the frequency prediction before data are added to the time scale computation.

The French atomic time scale TA(F), which is computed with data from about 20 industrial caesium standards located in nine French institutions, aims to provide a stable national frequency reference to the contributing institutions. To improve its stability, it was decided a few years ago to steer the time scale, which up to that date was free running, on the LNE-SYRTE primary frequency standards (PFS). The frequency of TA(F) was first slowly corrected monthly by an arbitrary given frequency offset with respect to TAI, to compensate the drift without disturbing the 30 d relative frequency stability of the time scale. Once close enough to the SI second, the time scale was steered monthly to the frequency data issued from the LNE-SYRTE PFS, in that way providing a more stable reference. We describe the steering applied to TA(F) and show the results in terms of relative stability with respect to TAI, or by comparing TA(F) with the SI second on the geoid as published monthly by the BIPM in its Circular T. When applying this steering during recent years, the departure over 30 d intervals of TA(F) from the SI second on the geoid was maintained inside the limits ±3 × 10⁻¹⁵. Within these limits, the TA(F) scale unit interval is in agreement with the SI second, a result which was made possible thanks to the four PFS currently in operation in the LNE-SYRTE.

We report on the Compass global satellite navigation system and its time reference system. China has sent three satellites into geostationary orbit since 2000, and the Compass Navigation Test System has been established. Compass time reference, named as BDT, is based on atomic time; BDT is derived from the atomic clock ensemble in Compass ground control centre and can be traced to the international time, UTC.

This paper proposes methods for calibration of two network time services—NTP servers and time-stamp authorities. The calibration is described in conformity with metrological principles like other time distribution systems. The authors have built up the calibration sets and tested them in cooperation with five European time and frequency laboratories. The paper also presents and discusses experimental results collected by measurement of time servers between involved laboratories.

The Time and Frequency Division of the Centro Nacional de Metrología (CENAM) has developed and implemented a time scale algorithm in order to give better characteristics of stability, accuracy and robustness to the UTC(CNM). The UTC(CNM) has been generated since March 1996 with no interruptions and since MJD 53 000 the |UTC − UTC(CNM)| time differences were smaller than 50 ns during more than 90% of the time. When time differences were bigger than 50 ns they were smaller than 80 ns; time stability of the UTC(CNM) was 4.3 ns for 5 days and 30 ns for 1 year. With the new method to generate the UTC(CNM), time differences |UTC − UTC(CNM)| no bigger than 25 ns, a time stability of 2 ns for 5 days and 15 ns for 1 year are expected. In this paper we report on the progress made at the CENAM Time and Frequency Division to generate the UTC(CNM) in terms of a virtual clock. We present and discuss preliminary results when the time scale algorithm is implemented on four industrial Cs clocks (two of them high performance clocks) and one active hydrogen maser.

The National Physical Laboratory, India (NPLI), is responsible for maintenance and development of the Indian Standard Time (IST) and also contributes through the GPS network to the generation of UTC coordinated by BIPM. Recently NPLI has made some special efforts to improve the uncertainty of UTC(NPLI) with respect to UTC. One basic clock ensemble algorithm is implemented to make the time scale of UTC(NPLI) by combining the intercomparison data of five caesium clocks. Some initial tests have been carried out to check the validity of this algorithm. The recent UTC − UTC(NPLI) values (shown in Circular T) and the corresponding frequency offset with respect to UTC have exhibited significant improvements. This paper describes these studies and presents the analysis of these observations.

We present the theory and the development of an algorithm used for locking a voltage controlled oscillator (VCO) to a hydrogen maser clock (HM clock) and then to a caesium clock (Cs clock). This algorithm is the heart of a composite clock using these three oscillators.

We present a composite clock based on servoing a voltage controlled oscillator (VCO) by both a caesium beam clock and a hydrogen maser. We wish to obtain a physical realization of a new reference timescale with an output signal which combines the long term stability of the caesium clock, the middle term stability of the H maser and the short term stability of the VCO. The system contribution at 100 MHz to the middle term frequency instabilities is expected to be around 10⁻¹⁵ at 1 s. This paper describes the dual-mixer time-difference (DMTD) system used to obtain the information needed for the control of the VCO.

Within the GIOVE Mission (GIOVE-M), two experimental satellites called GIOVE-A and GIOVE-B have been launched by the European Space Agency. This paper analyses the different issues involved in GPS/GIOVE interoperability for positioning and timing, including GGTO (the GPS to Galileo time offset) and timing biases, and presents practical experience and results related to EGGTO, the GIOVE-M experimental version of GGTO broadcast within the GIOVE navigation messages.

The generalized likelihood ratio test (GLRT) is a statistical fault detection method used for revealing fault occurrences in electronic systems. In this paper, the GLRT technique is analysed and customized for a rubidium frequency standard in order to reveal mean or standard deviation changes in the clock frequency. Experimental results are presented that confirm the effectiveness of the technique also when it is applied to data acquired from a rubidium clock. Monte Carlo simulations are shown in order to characterize the proposed method and to give a simple interpretation of the obtained results. The effectiveness of the GLRT has been already compared with standard tools such as the Allan variance (Nunzi E et al 2007 IEEE Trans. Instrum. Meas.56 523–8). In particular, the sensitivity of the method with respect to the jump size has been analysed. In this paper, the fault detection technique is characterized with respect to its readiness in terms of the number of samples employed for obtaining a failure occurrence when applied to clock frequency. Results obtained are employed for giving practical indications on the design of this failure test.

In this paper we use a mathematical model based on stochastic differential equations to predict the behaviour of atomic clocks. We consider several different cases with deterministic and random signatures and we obtain the best clock prediction together with its uncertainty. The results given by our mathematical model are compared with the results of the traditional method of extrapolating clock behaviour from the past data.

Wyye test our methods using experimental data from caesium and hydrogen masers maintained by the Italian time laboratory (INRIM), showing the good performance of our model.

The Ornstein–Uhlenbeck process is presented with its main mathematical properties and with original results on the first crossing times in the case of two threshold barriers. The interpretation of filtered white noise, its stationary spectrum and Allan variance are also presented for ease of use in the time and frequency metrology field. An improved simulation scheme for the evaluation of first passage times between two barriers is also introduced.

When an anomaly occurs in an atomic clock, its stability and frequency spectrum change with time. The variation with time of the stability can be evaluated with the dynamic Allan variance. The variation with time of the frequency spectrum can be described with the spectrogram, a time–frequency distribution. We develop a method that uses the dynamic Allan variance and the spectrogram to detect and to identify the typical anomalies of an atomic clock. We apply the method to simulated data.

The ability of the Allan variance (AVAR) to identify and estimate the typical clock noise is widely accepted, and its use is recommended by international standards. Recently, a time-varying version called Dynamic Allan variance (DAVAR) was suggested and exploited.

Currently, the AVAR is commonly used in applications to space and satellite systems, in particular in monitoring the clocks of the Global Positioning System, and also in the framework of the European project Galileo. In these applications stability estimation, either AVAR or DAVAR (or other similar variances), presents some peculiar aspects which are not commonly encountered when the clock data are measured in a laboratory. In particular, data from space clocks may typically present outliers and missing values. Hence, special attention has to be paid when dealing with such experimental measurements.

In this work we propose an estimation algorithm and its implementation in a robust software code (in MATLAB® language) able to estimate the AVAR in the case of missing data, unequally spaced data, outliers, and with long periods of missing observation, so that the Allan variance estimates turn out unbiased and with the maximum use of all the available data.

The frequency spectrum of the atomic clock noise can change with time for several reasons. We develop a method to represent the time–frequency spectrum of clock noise. The method can detect and identify clock anomalies, since they generate non-stationarities in the time–frequency spectrum. We apply the method to numerical and experimental data.

The detection of jumps in a frequency record is a challenging problem by either visual or mathematical means. The former takes considerable experience and judgment, and is therefore quite subjective, but has the advantage of providing insight into device behaviour. The latter is more impartial and consistent, and can be automated. In combination, mathematical jump detection can be applied for automatic clock testing and monitoring. If a jump is detected, the record can then be inspected visually before deciding on the action required. This paper describes frequency jump detection algorithms that are included in a program for frequency stability analysis.

I will discuss the three general methods that are commonly used to transmit time and frequency information: one-way methods, which measure or model the path delay using ancillary data, two-way methods, which depend on the symmetry of the delays in opposite directions along the same path, and common view, in which several stations receive data from a common source over paths whose delays are approximately equal. I will describe the advantages and limitations of the different methods including uncertainty estimates for systems that are based on them.

The two-state model provides an effective description of the atomic clock noise. We present the construction of the two-state model, which is useful to understand the statistics of clock noise. Furthermore, we derive the discrete-time version of the model, which can be used for numerical simulations. We illustrate our results through numerical examples.

Statistical and mathematical tools useful for atomic clock applications are reviewed to offer to the reader an overview of past work along with work still in progress for time scale and atomic clock algorithms.

The PDF file provided contains web links to all articles in this volume.