Table of contents

Time-scale formation, along with atomic time/frequency standards and time comparison techniques, is one of the three basic ingredients of Time Metrology. Before summarizing this Symposium and the relevant outcomes, let me make a couple of very general remarks.

Clocks and comparison methods have today reached a very high level of accuracy: the nanosecond level. Some applications in the real word are now challenging the capacity of the National Metrological Laboratories. It is therefore essential that the algorithms dealing with clocks and comparison techniques should be such as to make the most of existing technologies. The comfortable margin of accuracy we were used to, between Laboratories and the Field, is gone forever.

While clock makers and time-comparison experts meet regularly (FCS, PTTI, EFTF, CPEM, URSI, UIT, etc), the somewhat secluded community of experts in time-scale formation lacks a similar point of contact, with the exception of the CCTF meeting. This venue must consequently be welcomed.

Let me recall some highlights from this Symposium: there were about 60 attendees from 15 nations, plus international institutions, such as the host BIPM, and a supranational one, ESA.

About 30 papers, prepared in some 20 laboratories, were received: among these papers, four tutorials were offered; descriptions of local time scales including the local algorithms were presented; four papers considered the algorithms applied to the results of time-comparison methods; and six papers covered the special requirements of some specialized time-scale 'users'.

The four basic ingredients of time-scale formation: models, noise, filtering and steering, received attention and were also discussed, not just during the sessions.

The most demanding applications for time scales now come from Global Navigation Satellite systems; in six papers the progress of some programmes was described and the present and future needs were presented and documented. The lively discussion on future navigation systems led to the following four points:

• an overall accuracy in timing of one nanosecond is a must;

• the combined 'clock and orbit' effects on the knowledge of satellite position should be less than one metre;

• a combined solution for positioning and timing should be pursued;

• a 'new' time window (2 h to 4 h) emerged, in which the accuracy and stability parameters of the clocks forming a time scale for space application are to be optimized. That interval is linked to some criteria and methods for on-board clock corrections.

A revival of interest in the time-proven Kalman filter was noted; in the course of a tutorial on past experience, a number of new approaches were discussed. Some further research is in order, but one should heed the comment: 'do not ask too much of a filter'. The Kalman approach is indeed powerful in combining sets of different data, provided that the possible problems of convergence are suitably addressed.

Attention was also focused on the possibility of becoming victims of ever-present 'hidden' correlations.

The TAI algorithm, ALGOS, is about 30 years old and the fundamental approach remains unchanged and unchallenged. A number of small refinements, all justified, were introduced in the 'constants' and parameters, but the general philosophy holds.

In so far as the BIPM Time Section and the CCTF Working Group on Algorithms are concerned, on the basis of the outcome of this Symposium it is clear that they should follow the evolution of TAI and suggest any appropriate action to the CCTF.

This Symposium, which gathered the world experts on T/F algorithms in Paris for two days, offered a wonderful opportunity for cross-fertilization between researchers operating in different and interdependent communities that are loosely connected.

Thanks are due to Felicitas Arias, Demetrios Matsakis and Patrizia Tavella and their host organizations for having provided the community with this learning experience.

One last comment: please do not wait another 14 years for the next Time Scale Algorithm Symposium.

In recent years, a certain correlation detected between clocks at time and frequency laboratories has been the main reason behind the development of a revised version of the classical `Three-Cornered Hat' method as a mathematical tool to estimate the frequency instability of a set of clocks. The method has been formulated keeping in mind the possibility of dependence among clocks and has been generalized to an ensemble of N clocks [1–5]. It basically consists of the minimization of an objective function, the quadratic sum of cross-correlation coefficients, with the constraint condition established by the positive definiteness of the (estimated) absolute covariance matrix related to the clocks.

Later on, a matrix formulation to calculate the optimal weights in the case of correlated clocks was presented in [6]. The impact of this weighting procedure on the instability of the ensemble time was evaluated, on the basis of simulated clocks, when correlation was appropriately taken into account and in the case when it was neglected. That paper established, as a main conclusion, the mathematical treatment to be used in the usual case of weak dependence.

In this paper, we present a summary of the theory involved, and then apply the procedures to optimal weighting of clocks using real data clocks at Real Observatorio de la Armada (San Fernando, Spain). Finally, we discuss the results obtained.

The Polish independent atomic timescale TA(PL) was officially started on 4 July 2001. It is currently based on the indications of nine clocks from several Polish laboratories and Lithuania. The clocks at the laboratories are compared using TTS-2 multi-channel GPS receivers developed in cooperation with the Bureau International des Poids et Mesures (BIPM). The participating institutions are linked to the Central Office of Measures (GUM) in Warsaw. TA(PL) is computed as a weighted average of the participating clocks. This paper presents the clock ensemble, the data processing outline, and some experimental results.

The National Time Service Center (NTSC), Chinese Academy of Sciences, is responsible for the maintenance and development of the Chinese national timescale, UTC(NTSC). The main responsibilities include maintenance of UTC(NTSC), UTC(JATC) and the BPL long-wave and the BPM short-wave time transmission station. In this paper, we will mainly introduce recent advances in time and frequency measurement at NTSC, including the establishment of a local timescale, international time comparison and various time dissemination services in operation.

International atomic time (TAI) is an ensemble timescale based on a weighted average of clocks. The basic properties of the algorithm have been fixed since its inception: that is, the weight of a clock is inversely proportional to a variance measuring the instability of the clock, and it cannot exceed a maximum value. On the other hand, the procedure used to set the maximum weight has been subject to several changes over the years. While the most stable timescale would ideally be computed without an upper limit of weights, such an upper limit is introduced to bring reliability. Up to now, however, there has been no adopted measure of the reliability. In this paper, a quantitative estimator of the reliability is proposed, which therefore helps in choosing a weighting scheme. Different procedures for setting the maximum weight are examined in light of the application of the reliability criterion defined here. Tests using simulated and real data are presented. A weighting scheme to be used for TAI computation is proposed in which the reliability estimator is optimized.

A mathematical model for the clock phase and frequency deviation based on the theory of stochastic differential equations (SDEs) is discussed. In particular, we consider a model that includes what are called the `white and random walk frequency noises' in time metrology, which give rise in a mathematical context to a Wiener and an integrated Wiener process on the clock phase. Due to the particular simple expression of the functions involved an exact solution exists, and we determine it by considering the model in a wider theoretical context, which is suitable for generalizations to more complex instances. Moreover, we determine an iterative form for the solution, useful for simulation and further processing of clock data, such as filtering and prediction. The Euler method, generally applied in literature to approximate the solution of SDEs, is examined and compared to the exact solution, and the magnitude of the approximation is evaluated. The possible extension of the model to other noise sources, such as the flicker and white phase noises, is also sketched.

For the Galileo system it is required that a space clock time prediction be performed, covering the time interval (T_p) between two uploads. The time prediction accuracy of the space clock is therefore an important issue. The predictability of the Space Passive Hydrogen Maser (S-PHM) time error is evaluated by the RMS of the predicted time errors at the prediction time T_p: ΔT_RMS(T_p). A linear prediction model is used, corresponding to the absence of a frequency drift. The results show that: (1) the RMS time error can be evaluated from a priori knowledge of the clock's Allan deviation; (2) conversely, it is possible to extract the Allan deviation from the measurement of ΔT_RMS versus T_p; (3) the modelling of ΔT_RMS(T_p) of S-PHM based on the white frequency and flicker frequency noises appears to be particularly accurate: the difference between the model fit and the measured prediction accuracy is ⩽10 ps RMS for T_m = 24 h; (4) for T_p = 4 h, the performance of the S-PHM is a factor of 4.5 better than the performance of the space rubidium frequency standard (S-RAFS). This has a dramatic effect on the probability of the Signal In Space Accuracy: assuming a Gaussian distribution the probability of a predicted time error ΔT⩽1.5 ns for T_p = 4 h is 89% for the S-RAFS, while the same time error constitutes an absolute upper bound for the Galileo S-PHM.

The development within the International GPS Service (IGS) of a suite of clock products, for both satellites and tracking receivers, offers some experiences which mirror the operations of the Bureau International des Poids et Mesures (BIPM) in its formation of TAI/UTC but some aspects differ markedly. The IGS relies exclusively on the carrier phase-based geodetic technique whereas BIPM time/frequency transfers use only common-view and two-way satellite (TWSTFT) methods. The carrier-phase approach has the potential of very high precision but suitable instrumental calibration procedures are only in the initial phases of deployment; the current BIPM techniques are more mature and widely used among timing labs, but are either less precise (common-view) or much more expensive (TWSTFT). In serving its geodetic users, the essential requirement for IGS clock products is that they be fully self-consistent in relative terms and also fully consistent with all other IGS products, especially the satellite orbits, in order to permit an isolated user to apply them with accuracy of a few centimetres. While there is no other strong requirement for the IGS timescale except to be reasonably close to broadcast GPS time, it is nonetheless very desirable for the IGS clock products to possess additional properties, such as being highly stable and being accurately relatable to UTC. These qualities enhance the value of IGS clock products for applications other than pure geodesy, especially for timing operations. The jointly sponsored `IGS/BIPM Pilot Project to Study Accurate Time and Frequency Comparisons using GPS Phase and Code Measurements' is developing operational strategies to exploit geodetic GPS methods for improved global time/frequency comparisons to the mutual benefit of both organizations. While helping the IGS to refine its clock products and link them to UTC, this collaboration will also provide new time transfer results for the BIPM that may eventually improve the formation of TAI and allow meaningful comparisons of new cold atom clocks. Thus far, geodetic receivers have been installed at many timing labs, a new internally realized IGS timescale has been produced using a weighted ensemble algorithm, and instrumental calibration procedures developed. Formulating a robust frequency ensemble from a globally distributed network of clocks presents unique challenges compared with intra-laboratory timescales. We have used these products to make a detailed study of the observed time transfer performance for about 30 IGS stations equipped with H-maser frequency standards. The results reveal a large dispersion in quality which can often be related to differences in local station factors. The main elements of the Project's original plan are now largely completed or in progress. In major ways, the experiences of this joint effort can serve as a useful model for future distributed timing systems, for example, Galileo and other GNSS operations.

In this primer we first give an overview of stochastic models that can be used to interpret clock noise. Because of their statistical tractability, we concentrate on fractionally differenced (FD) processes, which we relate to fractional Brownian motion, discrete fractional Brownian motion, fractional Gaussian noise and discrete pure power law processes. We discuss several useful extensions to FD processes, namely, composite FD processes, autoregressive fractionally integrated moving average processes and time-varying FD processes. We then consider the statistical analysis of clock noise in terms of how these models are manifested via the spectral density function (SDF) and the wavelet variance (WV), the latter being a generalization of the well-known Allan variance. Both the SDF and the WV decompose the process variance with respect to an independent variable (Fourier frequency for the SDF, and scale for the WV); similarly, judiciously chosen estimators of the SDF and WV decompose the sample variance of clock noise. We give an overview of the statistical properties of estimators of the SDF and the WV and show how these estimators can be used to deduce the parameters of stochastic models for clock noise.

The existence of power-law noise structures in the measurements of atomic clocks has been well documented in the literature. Each of these power-law noises exhibits a spectral density proportional to 1/f^α at low frequencies. There is, however, another class of noise, termed fractionally integrated or long-memory noise, which too possesses spectral densities of this form. These fractionally integrated noises are analysed and applied to atomic timescales in this research. An alternative atomic clock model is developed and validated via simulation and live data tests. Estimators of clock rate and drift which account for the long-memory noise structures are derived and shown to produce both superior estimates of rate and drift and superior reports of the variability of these estimates. Estimation strategies which account for the autocovariance structures characteristic of fractionally integrated noise are also found to yield more powerful tests of hypotheses than do the short-memory techniques historically employed.

A family of second-difference statistics has been designed for use on regular but unevenly spaced two-way satellite time and frequency transfer (TWSTFT) time series. The characteristics of these new statistics are compared with those of the Allan variance (AVAR) and modified Allan variance (MVAR). The new statistics are used to estimate values of AVAR, MVAR and TVAR. Direct comparisons are made with estimates of these quantities based on an interpolated time series. Confidence interval determination is discussed. The application of the new variances to TWSTFT and geodetic GPS time transfer data is presented.

We study the case of an atomic clock whose parameters are time dependent. This results in a time variation of the frequency spectrum of the clock error that we are able to track using time–frequency techniques. The clock is modelled by a standard stochastic differential equation, and several cases of time-varying coefficients are studied, including, amongst others, sudden and periodic variations. The time–frequency technique used to estimate the instantaneous power spectrum is the sliding Welch's estimator. We present numerical simulations that prove the effectiveness of the approach.

The Kalman filter is a very useful tool of estimation theory, successfully adopted in a wide variety of problems. As a recursive and optimal estimation technique, the Kalman filter seems to be the correct tool also for building precise timescales, and various attempts have been made in the past giving rise, for example, to the TA(NIST) timescale. Despite the promising expectations, a completely satisfactory implementation has never been found, due to the intrinsic non-observability of the clock time readings, which makes the clock estimation problem underdetermined. However, the case of the Kalman filter applied to the estimation of the difference between two clocks is different. In this case the problem is observable and the Kalman filter has proved to be a powerful tool.

A new proposal with interesting results, concerning the definition of an independent timescale, came with the GPS composite clock, which is based on the Kalman filter and has been in use since 1990 in the GPS system. In the composite clock the indefinite growth of the covariance matrix due to the non-observability is controlled by the so-called `transparent variations'—squeezing operations on the covariance matrix that do not interfere with the estimation algorithm. A useful quantity, the implicit ensemble mean, is defined and the `corrected clocks' (physical clocks minus their predicted bias) are shown to be observable with respect to this quantity. We have implemented the full composite clock and we discuss some of its advantages and criticalities.

More recently, the Kalman filter is generating new interest, and a few groups are proposing new implementations. This paper gives an overview of what has been done and of what is currently under investigation, pointing out the peculiar advantages and the open questions in the application of this attractive technique to the generation of a timescale.

This is a study of three timescales formed from a Kalman filter operating on a model of a clock ensemble. The raw Kalman scale is unstable at short averaging times. The Kalman-plus-weights and reduced Kalman scales are stable at all averaging times. An optimality property is proved for the reduced Kalman scale.

Two-way satellite time and frequency transfer (TWSTFT) has been used operationally by the international time and frequency community for several years. This paper describes analysis techniques being developed at the National Physical Laboratory for the processing of TWSTFT measurements.

The measurements are modelled in terms of phase and normalized frequency offsets together with random errors considered to be a linear combination of the well-known noise types of white phase modulation, white frequency modulation and random walk frequency modulation encountered in time and frequency measurements. A method is described for solving the model to provide estimates of the phase and normalized frequency offsets, together with their associated uncertainties, at any measurement epoch. The random errors associated with the measurements are first characterized through the use of second-difference statistics. An analysis of the second-difference statistics is undertaken to estimate a covariance matrix for the time series of measurements. Finally, with the covariance matrix so obtained, generalized least-squares regression is used to provide unbiased and efficient estimates of the required offset parameters.