The Coordinated Universal Time (UTC)

In this section we present the clock frequency prediction algorithm currently used in the calculation of UTC. The algorithm for the atomic clock frequency prediction has been updated in [22] by introducing the quadratic model in phase data for the atomic clocks. This method is based on a quadratic model for describing the frequency drift of the H-masers or the ageing of the caesium clocks. Considering two successive intervals of UTC calculation, $I_{k-1}(t_{k-1},t_{k})$ and $I_{k}(t_{k},t_{k+1})$ we impose several conditions on the prediction term $h'_{i}(t)$ at time t_k to avoid or minimize time and frequency jumps in the resulting time scale. The prediction term $h'_{i}(t)$ can be expressed as the following quadratic form:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{pred_new} h'_{i}(t)=a_{i,I_{k}}(t_{k})+y_{ip,I_{k}}(t)(t-t_{k})+\frac{1}{2}C_{ip,I_{k}}(t)(t-t_{k})^{2}. \nonumber \end{align} \tag{ 9 }$$

To evaluate the parameters in (9), that are the phase, the frequency and the frequency drift of the atomic clocks, we assume that at time t_k the following conditions exist on $h'_{i}$:

• 1.  
  no time steps, by imposing the continuity to EAL;
• 2.  
  no frequency steps, by imposing the continuity to the first derivative of EAL;
• 3.  
  no change in frequency drift, by imposing the continuity to the second derivative of EAL.

We obtain a system of three equations with three unknowns and by solving this system we find that the relation (9) can be expressed as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{pred_new_fin} \widehat{h}'_{i}(t) &=\widehat{a}_{i,I_{k}}(t_{k})+\widehat{y}_{ip,I_{k}}(t-t_{k})\nonumber \\ &+\frac{1}{2}\widehat{C}_{i,I_{k-1}}(t_{k}-t_{k-1})(t-t_{k})+\frac{1}{2}\widehat{C}_{ip,I_{k}}(t-t_{k})^{2}.\nonumber \end{align} \tag{ 10 }$$

The physical meanings of the terms present in (10) are:

• $\widehat{a}_{i,I_{k}}$ is the estimation of the time correction relative to EAL of clock H_i at date t_k
• $\widehat{y}_{i,I_{k}}$ is the estimation of the frequency of clock H_i, relative to EAL, predicted for the period [t_k, t]
• $\widehat{C}_{ip,I_{k}}$ is the estimation of the frequency drift of the clock H_i, relative to a frequency reference, predicted for the period [t_k, t]
• $\widehat{C}_{i,I_{k-1}}$ is the estimation of the frequency drift of the clock H_i, relative to a frequency reference, for the period $[t_{k-1},t_{k}]$.

Considering that $\widehat{C}_{i,I_{k-1}}$ is equal to $\widehat{C}_{ip,I_{k}}$ the prediction takes a more simple form:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{pred_new_fin1} \widehat{h}'_{i}(t) &=\widehat{a}_{i,I_{k}}(t_{k})+\widehat{y}_{ip,I_{k}}(t-t_{k})\nonumber \\ &+\frac{1}{2}\widehat{C}_{ip,I_{k}}(t_{k}-t_{k-1})(t-t_{k})+\frac{1}{2}\widehat{C}_{ip,I_{k}}(t-t_{k})^{2}.\nonumber \end{align} \tag{ 11 }$$

The physical assumptions described in the relation are the constant drift and the variability of the frequency of each clock in the ensemble during the calculation interval.

Parameter estimation.

The relationship (11) describing the correction applied to the atomic clocks includes three parameters, the phase, the frequency and the frequency drift. A major issue is the optimization of the estimation procedure. In particular the estimation of the frequency drift requires special attention. The estimation of the time correction $a_{i}(t_{k})$, added to avoid time jumps, is given by the last known point of the difference between EAL and each clock and is expressed as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{pred_cor2} \widehat{a}_{i,I_{k}}(t_{k})={\rm EAL}(t_{k})-h_{i}(t_{k})=x_{i}(t_{k}). \nonumber \end{align} \tag{ 12 }$$

From a theoretical point of view the best estimation of the mean frequency $y_{ip,I_{k}}$ is the difference between the last and the first point on the interval duration of the clock data with respect to EAL as expressed by the following relationship:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{est_freq} \widehat{y}_{ip,I_{k}}=\frac{x_{i}(t_{k+1})-x_{i}(t_{k})}{t_{k+1}-t_{k}}. \nonumber \end{align} \tag{ 13 }$$

To estimate the frequency drift of the ensemble of the clocks under the hypothesis that this drift remains constant on a one month interval, the choice of the frequency reference is very important. While the phase data $({\rm EAL}-h_{i})$ have been used to estimate the time and frequency corrections, the frequency data of the clock with respect to terrestrial time (TT) as realized by the BIPM TT(BIPM) [27, 28] (shortly reported as TT in the relationships), $y_{{\rm TT}-h_{i}}$ are used to estimate the drift. In details, the calculation of the drift of the ensemble of atomic clocks is obtained by applying the linear least square technique to the values of $y_{{\rm TT}-h_{i}}$ over three months. In the procedures described the biggest impact on the estimation of the drift is given by the choice of the frequency reference (TT(BIPM)) and the choice of the length of the evaluation interval:

• In [22] the use of EAL as a reference for frequency drift estimation has been tested showing that an external, very stable time scale is essential for having a satisfactory estimate that is not affected by the instability of the reference. It has been decided to use TT(BIPM) (its algorithm will be presented below) calculated with the data of PFS/SFS provided by the time laboratories to the BIPM as frequency reference for drift estimation.
• Considering the length of the interval for the drift evaluation, three months are used since the last modification of the algorithm. Before this, the best estimation interval (used during two years) was considered of six months but a shorter period resulted a better choice to cope with the variations of the H-masers drift over time. Very probably an equal interval for the estimation of the frequency drift of all clocks is not the optimum solution, and some other strategy should be designed.

In this section the weighting algorithm as currently used in UTC calculation is presented [24]. The weighting algorithm was updated in 2014 to take in to account the frequency prediction model of the atomic clocks. The objective is to obtain a weighted average that is more stable in the long term than any of the contributing elements. 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). The weighting strategy applied in UTC takes into account the prediction used in the calculation of EAL based on the principle that a good clock is a predictable/stable clock as used in other time scale algorithms [29, 30]. By using the EAL prediction the deterministic signatures affecting atomic clocks, such as the frequency drift or ageing, can be minimized or eliminated. H-masers are characterized by a very significant frequency drift that can usually be predicted with a very low uncertainty [31]. By taking into account this prediction the H-masers can contribute to the timescale ensemble with a realistic (significant) weight without degrading the long-term stability of EAL. This can be achieved by analyzing the difference between the frequency of the clocks $y_{i,I_{k}}$ and their prediction ${\hat{y}}_{i,I_{k}}$, obtained by using the relationship (11). Clearly the frequency drift of the H-masers must not be allowed to degrade the long-term stability of EAL. In UTC calculation an upper limit of weight is set to limit individual clock contributions to prevent domination of the scale by a small number of very stable clocks. 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 has been tested by the efficiency with which the stability of the scale is improved. The value is fixed equal to $w_{\max}=4/N$ where N is the number of the clocks used in the calculation.

The principle behind the new weighting algorithm is that a good clock is a predictable/stable clock, by considering that in a time scale ensemble only the difference between the clock and its predicted value is used. During studies of the new prediction algorithm we observed that deterministic signatures such as frequency drift and ageing do not affect the EAL stability if well predicted. In the current procedure a four-iteration process is used where the weights are obtained by differencing the frequency with the predicted values. The differences between the predicted and the real frequencies are evaluated for each one-month interval over one year and these values are used to define the weight. By using one year's worth of data we maintain the long-term stability of EAL and UTC. In particular each iteration runs as follows:

• 1.  
  The values $[{\rm EAL}-h_{i}]$ are found using a given set of relative weights. In the first iteration, the weights are those obtained in the previous computation interval after normalization. In the following iterations, they are those obtained from the previous iteration;
• 2.  
  The absolute value of the difference between the real frequency $y_{i,I_{k}}$ and the predicted frequency $\hat{y}_{i,I_{k}}$ obtained by the relationship (13) corresponds to:

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{epsilon} \epsilon_{i,I_{k}}=|y_{i,I_{k}}-\hat{y}_{i,I_{k}}| \nonumber \end{align} \tag{ 14 }$$

  where the index i identifies the clock and I_k the time interval;
• 3.  
  The square of (14) is evaluated for each clock;
• 4.  
  One year of $\newcommand{\e}{{\rm e}} \epsilon_{i,I_{k}}$ is considered to ensure long-term stability of EAL and UTC.
• 5.  
  A filter has been implemented to give a more predominant role to more recent measurements with respect to older ones, considering that new measurements have most reliable statistics:

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{filter22} \sigma_{i}^{2}=\frac{\sum_{j=1}^{M}(\frac{M+1-j}{M})\epsilon_{i,j}^{2}}{\sum_{j=1}^{M}(\frac{M+1-j}{M})} \nonumber \end{align} \tag{ 15 }$$

  where i identifies the clock, j  the calculation interval and M the number of available measurements (which can vary from 5 to 12 given that 5 is the minimum number of months requested to observe the behaviour of a clock prior to its introduction into the UTC calculation and one year is the standard period of observation).
• 6.  
  The relative weight of clock H_i is computed theoretically using a temporary value given by

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{weight_temp1} \omega_{i,{\rm temp}}=\frac{1/\sigma^{2}_{i}}{\sum^{N}_{i=1}1/\sigma^{2}_{i}}. \nonumber \end{align} \tag{ 16 }$$

The new weight $\omega_{i}$ of clock H_i is equal to $\omega_{i,{\rm temp}}$ except in two cases:

• 1.  
  Clock H_i satisfies the requirement set for the limitation of weight as for the current algorithm.
• 2.  
  Clock H_i shows abnormal behaviour during the interval of computation so it cannot contribute. In this case the current value of the difference between the real frequency and the predicted one is checked. If the value is greater than a fixed threshold (that means 5 ns d⁻¹ of difference between the prediction and the real data corresponding to about 150 ns at the end of calculation interval) the clock is temporarily excluded from the ensemble. In such a way we eliminate about 1% of the total number of clocks participating in UTC that is a good compromise between maintaining the largest possible number of clocks without negatively affecting the quality UTC.

The maximum weight put equal to $w_{\max}=4/N$ is a good compromise between the stability of EAL, the distribution of the weight among the laboratories and the conditions stated in [32] that are:

• that a minimum of, for example, 5%–7% of the clocks be at maximum weight and
• that a combined weight of clocks at maximum weight be at least 40% of the total weight.

In figure 2 the weighting algorithms is schematized.

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In the design of the new weighting algorithm an important part is dedicated to the choice of the filter presented in (15). The simplest form by optimizing the long- and short-term stability of EAL was chosen. The use of different filters, as for example the exponential filter [29, 30] and the use of a different value for M (number of available measurements) were tested. The results did not show a significant improvement in EAL stability by changing these parameters in the filter.

Another algorithm used for maintaining the frequency of UTC close to the SI second frequency is the steering algorithm. TAI is a realization of TT, a coordinate time of a geocentric reference system. TAI gets its stability from some 420 atomic clocks kept in some 80 laboratories world-wide and its accuracy from a small number of PFS and SFS developed by a few metrology laboratories. The frequency of EAL is compared with that of the PFS/SFS 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 BIPM Circular T. The accuracy of TAI therefore depends on PFS/SFS measurements, which are reported more or less regularly to the BIPM. Data from several PSF/SFS are combined to estimate the duration of the scale unit of TAI [27, 28]. The algorithm used at the BIPM to estimate the duration of the scale unit of TAI [27, 28] combines the individual calibrations of PSF/SFS and calculates the frequency of the time scale during a given interval T (usually the month of calculation of Circular T). The features of TT as realized by the BIPM (TT(BIPM)) will be presented in section 7.

The calibrations of the PFS/SFS are usually referred to a local independent time scale. When using them to improve the accuracy of TAI, we should account for the transfer resulting 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. This is due to the fact that we calculate the frequency of EAL for the calculation interval T (corresponding to the month of calculation of Circular T) but the evaluation of PFS/SFS can have a different duration (starting from 5 d) and the measurements be in a different interval. For example, when a new PFS/SFS starts contributing to the Circular T, all the available past calibrations are taken into account even if their evaluation period does not correspond to the current calculation month of Circular T.

A frequency standard j  carries out n_j calibrations. If N is the number of standards considered, the number of available calibrations will be $\sum^{N}_{j=1}n_{j}$. We calculate the rate of EAL over an interval T as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{sol_TT} y=\sum^{N}_{j=1}\sum^{n_{j}}_{i=1}a_{j,i}W_{j,i} \nonumber \end{align} \tag{ 17 }$$

where W_j ,i is the rate difference between EAL and the PFS/SFS j  for a given interval T_ji, and a_ji are the filter coefficients. The filter coefficients a_ji, are normalized and depend on:

• 1.  
  the uncertainty of the evaluation i of the standard j  (i is the index used for counting the number of evaluations of the standard j ).
• 2.  
  the distance between T_ji and T
• 3.  
  the instability of EAL, which transfers the evaluation from T_ji to T.

In figure 3 the frequency of EAL is compared to PFS/SFS with the algorithm previously presented.

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The uncertainties of the differences between UTC and UTC(k) are published in section 1 of Circular T [33, 34]. They are affected by three major elements: clock variations, the means of comparisons of remote clocks (time transfer) and the time-scale algorithm. The analytical solution for the evaluation of the uncertainty of [UTC–UTC(k)] is based on the relation (8). In fact, as explained in section 4, the difference between EAL and the clock h_k (8) may also represent the difference between EAL and UTC(k) because H_k may also represent a UTC(k) and finally the difference between UTC and UTC(k). The three elements that could be source of uncertainty are the prediction, the weights and the time link differences. Considering the predictions and the weights fixed by appropriate algorithms and considered as time-varying deterministic parameters, the measures x_i,j in (7) are thus the only contributors to the uncertainties in x_j (8) and are the only source of the uncertainty of [UTC–UTC(k)]. The current algorithm [33, 34] applies the law of uncertainty propagation to (8) for obtaining the following solution for the uncertainty:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{sol_unc} u^{2}_{x_{k}}=\sum^{N}_{i=1}w^{2}_{i}u^{2}_{x_{i,k}}+2\sum^{N-1}_{i=1}\sum^{N}_{j=i+1}w_{i}w_{j}u_{(x_{i,k},x_{j,k})} \nonumber \end{align} \tag{ 18 }$$

where $u^{2}_{x_{k}}=u^{2}_{{\rm EAL}-h_{k}}$. The weights of the clocks are available from the BIPM website, and the uncertainties of links between the clocks are published in Circular T [15]. The second part of the equation (18), representing the correlation between clocks in different laboratories, is currently put equal to zero and consequently the clocks are considered uncorrelated. A revision of the algorithm is being studied and will be officially applied in the near future [35]. One of the most important difference, with respect to the current method used for the calculation of uncertainties is the introduction of the correlations. The evaluation of the correlations is generally a complex procedure and actions could be envisaged for their evaluation. The most important consideration concerns the distinction between the two main techniques of remote clock comparison (GNSS and TWSTFT) used for UTC generation. In fact the result of a GNSS calibration can be associated to the laboratory's receiver while the calibration of TWSTFT measurements characterizes the links. Based on this physical assumption the GNSS time links will be considered with respect to an auxiliary time scale (ATS) instead of UTC(PTB). Being ATS external to UTC calculation (this is not the case of UTC(PTB) in the current UTC structure) the evaluation of correlations is possible. For the TWSTFT time links, the PTB laboratory will remain the pivot, even if in the case of redundant time links all the possible TWSTFT measurements should be considered. The formalism used and considered in [35] will be finalized for describing this physical reality.

In order to assess the progress with the current algorithm for UTC we have started studies on individual clocks. In particular, the interval for the evaluation of the frequency drift of each clock, that is currently three months, could be considered depending on the stability property of the atomic clocks. This could lead to the use of different intervals of estimation for each clock. A test has been done by considering three years of atomic clock data with respect to TT(BIPM). The Hadamard Variance is calculated for the data (TT-Hi) (3 years), its minimum value corresponds to the optimized interval for the ith clock (the longer predictable period for the drift estimation). The results show that 15% of the clocks are very predictable and the best drift evaluation interval is 6 months. For the rest of the clocks, 3 months is the longer predictable period. In the future the characterization of each single clock with respect to its drift predictability will be developed to rapidly detect clock anomalies and prevent their negative impact on UTC.

In figure 4 the frequency of EAL with respect to PFS/SFS calculated by evaluating the frequency drift with 6 months (black line), 3 months (red line) and optimized period (blue line) are compared. In figure 5 the Allan deviation of the previous data is reported.

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A Kalman filter time scale has been tested to improve the performance of UTC [36]. The Kalman filter is used in many fields because of its ability to clean data with white phase noise. In the time and frequency domain, use of the Kalman filter is especially important because of its use in building time scales. The short-term stability of UTC is dominated by the white noise of time transfer used to compare clocks; the idea is to improve UTC stability by means of the Kalman filter. A typical example of real-time time scale build with the Kalman filter algorithm is IGST (international GPS service time scale) [37]. In this first stage of the test an ensemble of 139 atomic clocks having run continuously without time and frequency steps for 3 years has been selected. UTC is calculated with this ensemble of clocks. The Kalman-based time scale is calculated by using the weights automatically obtained by the Kalman routine and the results are compared to those acquired by using the weights obtained from the UTC algorithm; the main difference between the ensemble of weights is the maximum weight fixed in UTC algorithm and not implemented in the classical Kalman filter routine.

EAL₁₃₉ (calculated by using 139 atomic clocks) is less stable than the current time scale but the quality is sufficiently good to be used for the test. We should observe that the parameters (frequency of the clocks, frequency drift etc) are optimized for having a long-term time scale in the case of 450 atomic clocks. Different choices should be made in the case of about 150 atomic clocks. However the results are good enough for the scope of our first analysis. The input data for our computation was therefore of the form EAL$_{139}-h_i$ with $i=1,...,139$. From that, we estimated the clock parameters (initial state and noise coefficients) attributing the whole noise signature to the h_i. The noise coefficients are calculated by comparing the clocks to TT(BIPM) and by performing the Hadamard variance. Before analysing this comparison there is another aspect on which we need to focus our attention: the clock weights. The x-reduction procedure in the case of measurement noise delivers, together with Kalman's estimates, a weight for each laboratory, based solely on the clock stability. However this kind of weight does not suit our purposes as it does not introduce a maximum for each clock weight (thus affecting the reliability of the scale). In this case the weights are completely dominated by the Rubidium fountains, as their stability is much better then those of the other clocks. Therefore we use the EAL weights for our computation, based, and previously described, on the principle that a good clock is a predictable clock. Figure 6 shows the stability comparison between EAL, the KF scale with EAL weights and the KF Scale with Kalman weights all with respect to a Rubidium fountain. We can observe that the KF Scale shows an improved stability with respect to EAL, particularly in the case of Kalman weights. This is not surprising as not having a maximum weight constraint implies that the scale can rely only on the better performing clocks. This fact, even if it allows a better stability, makes the scale much less robust to possible problems that can affect these high performance clocks.

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Time transfer is possible using the signal broadcast by GNSS satellites, which contains timing and positioning information. It is a one-way method, the signal being emitted by a satellite and received by specific equipment operated in a laboratory. For this purpose, GNSS receivers have been developed and commercialized to be used specifically for time transfer. This first generation of receivers evolved from single-frequency, single-channel to single-frequency, multi-channel. In the early 2000s the use of geodetic-type receivers had been studied and validated; it become possible implementing externally to the receiver the time comparison with the laboratory clock. They had the advantage of performing dual frequency measurements, thus correcting the ionospheric delays. The interaction of time metrology experts with GNSS receivers developers resulted in the conception of receivers specific for time comparisons which incorporate the geodetic-type receivers advantages. The text below explains the progress achieved with this equipment [38]. The common-view (CV) method [39] relies on the reception by several receivers of the same emitted signal. GNSS time transfer is intrinsically affected by the position errors of the satellite and by the instability of the satellite clock. These effects are eliminated or minimized when the two stations involved in the clock comparison observe the same satellite simultaneously. The CV method requires simultaneous observation of satellites at the two stations, with the drawback that with increasing distance the number of simultaneously observed satellites decreases, and the number of high elevation satellites becomes poor. Common-views of satellites of the US global positioning system (GPS) were used for the calculation of UTC at the BIPM until 2006.

The observation equation [40] for the GPS common view method between receivers i and j  is given as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{gps} P^{k}_{i}-P^{k}_{j}=\rho^{k}_{i}-\rho^{k}_{j}+I^{k}_{i}-I^{k}_{j}+T^{k}_{i}-T^{k}_{j}+c({\rm d}t_{i}-{\rm d}t_{j})+e_{i}-e_{j} \nonumber \end{align} \tag{ 19 }$$

where k is the satellite pseudo-random noise (PRN) number and $\rho$ is the geometric distance between the satellite antenna and receiver antenna. Here, I represents ionospheric refraction delay and T is tropospheric refraction delay, ${\rm d}t$ is the difference between the satellite time and the local clock, and e is measurement error including multi-path effects and the delay in the receiver. Because P is the observation value, and I and T are derived from the GPS navigation message and measurement error, $(e_{i}-e_{j})$ is assumed to be 0; so we can obtain the time difference (d$t_{i}-{\rm d}t_{j}$).

Then advances in obtaining precise satellite orbits and clock parameters allowed the introduction of another technique, named All in View (AV) [41] that eliminates the constraint of having simultaneous observations, thus becoming independent of the length of the baseline for having suitable observed satellites. In AV, data from all satellites in view at a station are used. The International GNSS Service (IGS) has provided since 2004 high precision GPS satellite clock products referred to the timescale of the IGS (IGST) [37], which relative frequency instability is of order 10⁻¹⁵ for a one-day averaging time, two orders of magnitude better than that of the GPS time. Since this minimizes the impact of the error coming from satellite orbits and clocks, it has been possible to use the AV method instead of the GPS CV with the benefit of adding data from satellites at high elevations and thus improving the statistical uncertainty of the time links, particularly the very long ones [41]. The GPS links obtained using dual-frequency receivers, termed GPS P3 [38, 42], provide ionosphere-free data and allow clock comparisons with nanosecond statistical uncertainty or better. The delay imposed by the ionosphere to the electromagnetic signals is one of most difficult to model accurately. Dual-frequency receivers perform simultaneous measurements in two frequencies (f _a and f _b) and eliminate almost completely first order ionospheric effects by a linear combination of the measurements M_a and M_b (code or phase) at the two frequencies, as in the simple equation:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{ionofree} M_{I-{\rm free}}=\frac{f^2_{a}M_{a}-f^2_{b}M_{b}}{f^2_{a}-f^2_{b}}. \nonumber \end{align} \tag{ 20 }$$

In a GPS P3 link only code measurements are used. Most remaining error sources are under 0.1 ns, the precision with which TAI is reported. Code-multipath effects can reach 1 ns on a short-term basis and higher values in the long term, representing the ultimate limit to code-only time transfer either with CV or AV. Tropospheric delay is still present in the data, introducing short-term noise and bias of a few 0.1 ns, with slow variations depending on weather conditions. The addition of phase measurements from geodetic-type receivers minimizes the effects of these two error sources. The precise positioning technique (PPP) [43, 44] in which dual-frequency phase and code measurements are used for comparing via GPS the reference clock in a station to a reference timescale has been implemented for use in the computation of time links for UTC and has been used on a routine basis since September 2009. By this technique we obtain the smallest statistical uncertainty of clock comparison, at present about 0.3 ns.

Two time transfer data protocols are accepted at the BIPM for data submission. For the code measurements, a special format and procedure for data tracking and averaging had been developed in 1994 for GPS [45] by the CCTF Group on GPS Time Transfer Standards, commonly known as 'CGGTTS format' and extended later to support GLONASS observations. These directives evolved with the income of new satellite systems, and its last version, CGGTTS version 2E [46] supports data from all GNSS fully and partially operational today. For the time comparisons using the phase of the signal combined with the code (presently the GPS PPP solutions) is necessary to make use of the 'Receiver Independent Exchange Format' RINEX, originally developed for geodetic applications. Its latest version, RINEX 3.03 [47] supports multi constellation data, including global (GPS, GLONASS, Galileo and BeiDou) and regional (Japanese QZSS and Indian IRNSS) satellite systems.

Thanks to new hardware and to improvements in data treatment and modelling, the uncertainty of clock synchronization via GPS has been reduced from a few hundred nanoseconds at the beginning of the 1980s to below 1 ns today. Old single-channel, single-frequency C/A code receivers have been replaced in time laboratories by multi-channel receivers, which allow the simultaneous observation 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. GPS observations with single-frequency receivers used in regular UTC calculations are corrected for ionospheric delays by making use of ionospheric maps produced by the IGS. Dual-frequency receivers installed in most participating laboratories allow the removal of the delay introduced by the ionosphere, thus increasing the accuracy of time transfer. All GPS links are corrected for satellite positions using IGS post processed precise satellite ephemerides.

Studies on the use of the Russian satellite system GLONASS for clock comparisons in UTC started at the BIPM since the early 1990s, but had slow progress into real application due to the time elapsed before fully displaying the system satellites. The first GLONASS time link between the PTB and the Russian Institute of Metrology for Time and Space (VNIIFTRI) in Moscow (represented by the acronym SU in BIPM Circular T) was introduced in the computation of UTC by the end of 2009 [48]. After many years of limited operations with a small number of satellites, GLONASS constellation is now fully operational. In UTC, GLONASS observations have been processed in common-view only, and the resulting link is combined with that of GPS. Different from GPS, the GLONASS satellites do not all transmit the same frequency, and the signal delay in the receiver is different for each satellite group transmitting the same frequency, provoking the unresolved technical issue of having multiple frequency biases affecting the code measurements. Studies on the possible causes and effects of these biases [49] did not reach a clear conclusion, but showed that using the L1C code with the common-view technique they are no larger than the measurement noise. Defraigne et al [50] showed that combining GPS and GLONASS code and phase observations the PPP solution is comparable to that with GPS observations only. Furthermore, the combined solution improves when only GLONASS phase observations are included.

The TWSTFT [51, 52] technique utilizes a telecommunications geostationary satellite to compare clocks located in two 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 two-way method has the advantage over the one-way method of eliminating 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 two clocks placed in the two stations are directly computed.

The observation equation [40] for the time transfer by TWSTFT is written as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{txw} T_{i}-T_{j}=\frac{1}{2}({\rm d}t_{i}-{\rm d}t_{j})+(e_{i}-e_{j})+t_{s} \nonumber \end{align} \tag{ 21 }$$

where t_s is a correction term for the path difference due to the Earth's rotation (the Sagnac effect), T_i and T_j are the compared time scales, $(e_{i}-e_{j})$ the measurement error and ${\rm d}t_{i}-{\rm d}t_{j}$ the time difference measured for stations i and j .

The first TWSTFT link was introduced into TAI in 1999. Since then, the number of laboratories operating two-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 three days per week, which were not sufficient to give significant statistics for their use in TAI calculation. With the installation of automated stations in most laboratories, however, the TWSTFT link observations in TAI are now made every day at two-hour intervals with an uncertainty below 1 ns. However, this statistical uncertainty has been degraded by additional noise with a daily signature, the origin of which is not yet clear. We will discuss in the section 10.3 on the possibilities for improvement. The TWSTFT time links are characterized by an excellent accuracy, coming from the long-term stability of the hardware and represented by a 1 ns calibration uncertainty.

The protocol for TWSTFT data exchange has been developed at the International Telecommunication Union (ITU) for use in multiple applications [53]. This protocol is used for the regular submission of TWSTFT data to the BIPM.

The number of institutes operating two-way stations for time comparisons remains small for a number of reasons. First, the high cost of the TWSTFT equipment, several times that of a GNSS receiver; second the complexity of its operation, and finally the fact that a satellite service provider is necessary for the transmission of the signals, involving additional cost and coordination with the other participating institutes. It has, however, the advantage of the easy treatment of the observations, since time comparisons are already computed in the ITU protocol, and the final solution is purely geometrical, and safe from corrections.

For two decades, GPS C/A-code observations have provided a unique tool for clock comparisons in UTC, rendering impossible any test of its performance with respect to other independent methods. The present situation is quite different; the introduction of the TWSTFT technique and the GLONASS observations have resulted in the opportunity of comparing the time links obtained with GPS techniques with those coming from the independent GLONASS and TWSTFT techniques, and making the system more reliable. In parallel, new GNSS receivers had been developed, and refined methods for treating their observations had been implemented. With the dual frequency receiver data and the provision of RINEX files, iono-free P3 links and code/phase measures solutions (PPP) are computed whenever possible [44]. This gives the BIPM the possibility of computing and comparing a redundant number of time links for some baselines, the best being used in the calculation of UTC and the others kept as backup. The time links and the results of time link comparisons are available on the BIPM ftp server [54].

Redundant links over a baseline can be combined to enhance the comparison of the clocks involved [55]. This approach has been in use at the BIPM since January 2011, with the introduction of combined GPS/GLONASS and TWSTFT/GPSPPP time links. The combination of TWSTFT and GPSPPP results in a link characterized by the accuracy of TWSTFT (u_B 1 ns) and the short-term stability of GPS PPP, at the sub-ns level [56].

Data from single channel receivers stoped in mid-2015. As of January 2018, 22% of the links in UTC are obtained using GPS single-frequency data, 70% of time transfer data is from dual frequency GPS receivers, from which 58% is used in GPS PPP solutions. The TWSTFT links represent only 12% of the time links, they are all combined with GPSPPP. The few time links computed with a combination of GPS and GLONASS have been temporarily interrupted.

A new technique that has the potential to revolutionize clock frequency comparisons using GPS signals has been developed by the BIPM time department in collaboration with colleagues from the Centre National d'Etudes Spatiales (CNES) and the Collecte Localisation Satellites (CLS). Signals of the global positioning system (GPS) have been used to compare clocks at a distance for decades. The best present technique of choice is precise point positioning (PPP), which provides the majority of the links used for UTC. Its performance for frequency comparisons is of the order a few parts in 10¹⁶ in 10 d, which is adequate for all commercial clocks, but insufficient to compare the best caesium or rubidium fountains, which claim an accuracy below $2\times10^{-16}$, not to mention the forthcoming optical frequency standards. One way to overcome the current limitations of GPS for clock comparisons is to base the results only on the phase of the transmitted signals and to explicitly account for the integer-cycle nature of the phase ambiguities that need to be resolved between the different arcs and satellites that are successively observed during the comparison. Developing this technique, called IPPP (PPP with integer ambiguity resolution), has been a topic of collaboration with colleagues at the CNES and the CLS over the last few years. The CNES and CLS teams constitute one analysis centre of the International GNSS Service (IGS), and they are developing specific IGS products needed for IPPP. In the framework of a post-doctoral position during 2013–2014, jointly funded by the BIPM and the CNES, the operational procedures for IPPP were developped that allowed several long-term tests to be carried out (comparisons over weeks to months). A joint publication with colleagues at the BIPM, CNES and CLS describes the work [68]. The most significant result compares IPPP to a 420 km optical-fibre time link in Poland, which is in continuous operation and which reports to the BIPM. As a result of the fibre link's better accuracy, it was possible to demonstrate that the IPPP technique reaches, in about 5 d, a performance of $1\times10^{-16}$ in comparing the frequency of two clocks and that the achieved accuracy continues to improve with longer averaging times [54]. This represents a significant improvement over classical PPP and is the best published performance for a GPS frequency comparison. Because IPPP can compare two clocks whatever their locations on Earth and can be operated immediately with existing equipment, it will be a significant step towards the comparison of ultra-accurate clocks. Optical fibre links will progressively build up into networks that cover more extended continental areas, particularly in Europe. For world-wide frequency comparisons at the sub-10⁻¹⁶ level we could refer to [69] and to atomic clock ensemble in space (ACES) mission [70, 71], which is expected to fly on board the International Space Station in 2020.

The optical fibre link (OFL) techniques have grown fast. Over thousands of kilometres, a coherent fibre link can compare clocks with an uncertainty of few parts in $1\times10^{18}$ in 1000–10 000 s. Also time transfer over fibre reports large advances, and today it can offer a sub-ns inaccuracy. Nonetheless, only two time links in Europe (AOS-GUM and BEV-TP) regularly report data to the BIPM connecting remote atomic clocks. The implementation of fibre links reporting data to the BIPM shall be encouraged and pursued to take advantage of a widespread optical fibre network.

Fibre links comparisons demonstrated to be the currently most suitable means to compare remote optical frequency standards. More comparisons shall now be realized in order to achieve the target identified in the roadmap for the possible redefinition of the SI second, collecting more and more comparisons of optical frequency standards with an uncertainty of parts in 10¹⁸. Nonetheless, there are not projects planned for an intercontinental fibre link. In the next years, the remote intercontinental comparison of optical frequency standards will become an issue. Intercontinental fibre links could be a possibility, but probably the use of advanced satellite techniques will be investigated and pursued to compare optical frequency standards at the right level of accuracy and stability.

Europe has, in 2018, the most extended network of fibre connections relying metrology institutes and research institutions where optical clocks are operated or in development. Microwave standards (Cs and Rb standards) operated at the national metrology institutes in France and Germany had been compared with a few 10⁻¹⁶ uncertainty [72]. With the support of EURAMET, projects have been put in place for optical-clock comparison with the target of operating long-distance optical links continuosly with measurement uncertainty better than 10⁻¹⁷ over one day. Progress has also been made in Japan, relying institutes in the Tokyo area for remote comparisons of optical and microwave frequency standards [73].

The software-defined radio (SDR) receiver [74–76] has been developed at the Telecommunications Laboratory (TL, Chinese Taipei) for implementation in TW Earth stations. It is aimed at improving the stability of TW time transfer, with particular impact on the diurnal signature present in almost all the links, which is the major uncertainty source in TW time links. The BIPM and the Consultative Committee for Time and Frequency (CCTF) Working Group on TWSTFT launched a pilot study in February 2016 for validating the SDR receiver in view of its implementation for use in UTC time links. Participants to this pilot study were the laboratories contributing to UTC which operate the SDR receiver and the BIPM time department. Goals of the pilot study were first, to validate the efficacy of the SDR receiver for improving the TWSTFT uncertainty and significantly reduce the diurnal effects; second to implement routine TW code measurement data through SDR receiver to improve the UTC time comparisons. In the framework of the pilot study, SDR receivers have been installed and are operational since the end of 2016 at TL, NICT, KRIS, NTSC in Asia, PTB, OP and SU in Europe and NIST in the US. The project to install the SDR receiver in all the TW laboratories is ongoing. The analysis of the data obtained from the pilot study shows that in some TW links there is significant improvement, while in others very small or no noise reduction is visible. This result needs more investigation, together with other open points such as the calibration etc. At this step, the results of of this pilot project are positive. The development of a SDR achieving complete independence from the currently used SATRE modems remains necessary, since the present software uses the SDR for the reception of the signal, and keeps the SATRE modem for the signal transmission. Other studies concerning TW development can be found here [77, 78].