2015 SIM TFWG Workshop and Planning Meeting January 27 – 29 Panamá City, Panamá.

1. 2015 SIM TFWG Workshop and Planning Meeting January 27 – 29 Panamá City, Panamá. Building an Ensemble Time Scale from Multiple Atomic Clocks Ricardo José de Carvalho National Observatory Time Service Division January 28, 2015

2. Introduction Time Scale from one clock Time Scale System The Clock Ensemble Automated Data Acquisition The Clock Ensemble Automated Data Acquisition Time Scale Algorithm Time Scale Formulation Time Scale Clock Model NIST Time Scale Algorithm A Case Study with Real Data Generating UTC(lab) from TA(lab) References Outline

3. Initial Condition: ► We have an ensemble of atomic clocks that are running at the same time. Objective: ► We want to build an independent atomic time scale [TA(laboratory)] from an ensemble of atomic clocks. ► The time scale [TA(laboratory)] will be used as an internal reference for the generation of a local UTC(laboratory) time scale. Introduction

4. ►What are the possibilities and the difficulties we are facing? ►What are the ingredients necessary to build TA(lab) from ensemble of atomic clocks? ►How can we evaluated the stability of TA(lab)? ►How can we generated UTC(lab) from TA(lab)? Introduction

5. Simple Solution: ►We can select one of the clocks and we can decide that our atomic time scale is identified with the output of this clock. Is it Valid? ►This solution is valid if the clock exhibits qualities superior to the others. ►All the time! Time Scale from one clock

6. Better Solution: ►We can build a time scale system. Advantages: ►Enable a time laboratory to keep time with stability, accuracy, and reliability beyond the performance level of the best physical clock. How? ►Combining all the clocks resulting in a Virtual Clock. Time Scale System

7. Time Scale System

8. A time scale system can be divided into four parts: ►The first is a clock ensemble that will be used as a reference for the generation of a local time scale TA(lab). ►The second, an automated data acquisition system that measures the time differences between the clocks. ►The third is a time scale algorithm that computes the time scale TA(lab). ►The fourth is the equipment to generate UTC(lab) from TA(lab) and how to do that. Time Scale System

9. The Atomic Clock: ►An atomic clock delivers a series of physical electric pulses separeted from one another by a duration of 1 second. Each pulse is an event that can be associated a number. This number is the time of the clock: for example, it may read as 2015 January 28 13h45min00s. ►The time of an individual clock can not be measured! ►What we can do is to measure the time differences between clocks. The Clock Ensemble

10. The atomic clocks are very stable and accurate but: ►They may be sensitive to the environmental conditions. ►They are to be maintained in the most stable environment to avoid injecting instabilities. The atomic clocks are independent but: ►Some correlations between clock frequencies may be detected. ►These correspond mainly to responses to changes in the environmental conditions that it affects all the clocks at the same time. The Clock Ensemble

11. ►Inside the laboratory the time differences are obtained with a high degree of accuracy using electronic counter. ►The measurement process is affected by intrinsic noise. ►The noise of the measurement process has to be negligible when compared with the clock noise. Automated Data Acquisition

12. Function: ►Combining the clock difference measurements from clock ensemble to produce an average time scale. Objective: ►Generate time and frequency with more reliability, stability and frequency accuracy than one of the individual clocks in the ensemble. Time Scale Algorithm

13. Time Scale Formulation • The Simplest Time Scale: • ►It is just the reading of a single clock, that is: • ►The condition imposed on the time scale is:

14. Time Scale Formulation • The Simple Mean of Two Clocks: • ►The time scale is defined by: • ►The condition imposed on the time scale is: • ►The time scale defined will be affected by any anomalus behavior that one of the clocks present. For example, suppose that the clock 1 has jumped by an amount • ►Then the time scale is affected by an amount

15. Time Scale Formulation • The Simple Mean of n Clocks, n > 2 : • ►The time scale is defined by: • ►The condition imposed on the time scale is: • ►Anomalous behavior can thus be detected with confidence incresing as n increases.

16. Time Scale Formulation • The Weighted Mean of n Clocks: • ►The time scale is defined by: • ►The condition imposed on the time scale is: • ►Anomalous behavior can thus be detected with confidence incresing as n increases.

17. Time Scale Formulation • ►The time scales considered previously possess a disadvantage that, if one the clocks stops or a new clock is introduced, a step and a rate change will generally occur in the time scale. • ►To solve this problem the definition of the time scale should be modified, in such a way that, the considered average is on the time offset of clock with the clock ensemble as reference.

18. Time Scale Clock Model • ►The general model of clock i behavior may be write as: • is the time deviation of the clock i at time t + Ƭ; • is the synchronization error of the clock i at time t; • is the frequency offset (syntonization error) of the clock i at time t, • which produces a linear ramp in the time deviations; • is the frequency drift term, which produces a quadratic time deviation; • is the all random fluctuations. It is this term which is typically characterized by one or more of the five power law processes.

19. NIST Time Scale Algorithm • ►A first prediction of the time offset for each clock against the ensemble is given by: • ►The best estimate of the time offset of each clock at time given the measurements is: • ►Once the are known the average frequency of each clock over the last interval can be estimate by:

20. NIST Time Scale Algorithm • ►An exponentially filtered estimate of the current average frequency of clock i that will be used in the next prediction interval is given by: • where miis an exponential time constant determined from the relative levels of white noise and random walk FM, that is: • ►The clock weights wi are calculated from:

21. NIST Time Scale Algorithm • ►The prediction error estimate is given by: • because ensemble time is a weighted average ofeach clock times, the prediction error estimate is biased, because each clock is a member of the ensemble, so it is necessary to correct this biasing by he condition imposed on the time scale is: • ►Since the noise characteristics of a cesium clock may not be stationary, the current prediction error of each clock is exponentially filter where the past prediction error are deweighted in the process, that is • the time constant for the filter is typically chosen to be 20 days and the initial value of is estimated as:

22. A Case Study with Real Data • ►The art of building a time scale from an ensemble of clocks requires various choices. We will show some of them by the following example. • ►We used in this example time differences of six commercial cesium clocks for the period of August 08, 2008 to September 30, 2009 with 1 hour of interval between measures. • ►The time difference measurement data are coming from an automated acquisition system that measures the time differences between the clocks with comercial electronic counter.

23. A Case Study with Real Data

24. Figure 1. Time Difference between clocks.

25. Stability of the Clocks • ►To evaluate the individual stability of the clocks we use the N-cornered hat technique by:

26. Stability of the Clocks • Figure 2. Comparison of Allan deviations of the clocks.

27. Initializing the Algorithm • ►To initialize the algorithm the UTC(BIPM) was used as reference to calculate the following parameters • that is • T130 – UTC(BIPM) = -14.738960µs at 54704 00h00min • T129 – UTC(BIPM) = -13.138160µs “ • T103 – UTC(BIPM) = -10.633442µs “ • T123 – UTC(BIPM) = -60.150347µs “ • T125 – UTC(BIPM) = +5.664442µs “ • T102 – UTC(BIPM) = -31.095ns “

28. Initializing the Algorithm • to • we calculated using Excel a linear regression of the time differences [clock - UTC(BIPM)], to all clocks we have • yT130 – UTC(BIPM) = -9.532ns/day (R2= 0.990) • yT129 – UTC(BIPM) = -1.731ns/day (R2= 0.964) • yT103 – UTC(BIPM) = -7.874ns/day (R2= 0.997) • yT123 – UTC(BIPM) = -3.092ns/day (R2= 0.999) • yT125 – UTC(BIPM) = +4.283ns/day(R2= 0.999) • yT102 – UTC(BIPM) = +5.945ns/day (R2= 0.976) • period 54704 00h00min to 54794 00h00min

29. Initializing the Algorithm • Figure 3. Linear Fit of T130 – UTC(BIPM)

30. Initializing the Algorithm • Figure 4. Linear Fit of T129 – UTC(BIPM)

31. Initializing the Algorithm • Figure 5. Linear Fit of T103 – UTC(BIPM)

32. Initializing the Algorithm • Figure 6. Linear Fit of T123 – UTC(BIPM)

33. Initializing the Algorithm • Figure 7. Linear Fit of T125 – UTC(BIPM)

34. Initializing the Algorithm • Figure 8. Linear Fit of T102 – UTC(BIPM)

35. Initializing the Algorithm • ►To initialize • the prediction error estimate in the algorithm we used the individual stability of the clocks estimated by the N-cornered hat technique, reminding that Ƭ = 1 hour we have: • (εT130)2= (3600)2x(1.978x10)-26 • (εT129)2= (3600)2x(1.222x10)-26 • (εT103)2= (3600)2x(1.243x10)-26 • (εT123)2= (3600)2x(3.340x10)-26 • (εT130)2= (3600)2x(3.781x10)-26 • (εT130)2= (3600)2x(8.260x10)-26

36. Initializing the Algorithm • ► To initialize • Ƭmini is the value of Ƭ at minimum σy(Ƭ)on Allan variance curve for • clock i. It was used the specification of the 5071A supplied by the • manual the value is Ƭmini = 5 days so: • mi= 68.78263370667524

37. Results • Figure 9. The time deviation of the clocks and scale (normalized).

38. Results • Figure 10. The time deviation of the clocks and scale (normalized).

39. Results • Figure 11. The time deviation of the TA(lab) (normalized).

40. Results Table 1. Allan Deviation with UTC(BIPM) as reference

41. Results • Figure 12. The Change of Clock’s Weights with Time.

42. Results • Figure 13. The Change of Clock’s Weights with Time.

43. Results • Figure 14. The Change of Clock’s Weights with Time.

44. Generating UTC(lab) from TA(lab) • How to generate UTC(laboratory) : • ►First, after each publication of the circular T we calculated the rate of TA(k) with reference UTC(BIPM), that is, we obtain the following value: • {TA(k) – UTC(BIPM)}ns/day (One sees a month) • ►Second, we chose one of the clocks as reference and we calculated de rate of the clock with TA(k) as reference, that is, • {Clock – TA(k)}ns/day (every hour) • finally we will have for every hour • {TA(k) – UTC(BIPM)}ns/day (One sees a month) • + • {Clock – TA(k)}ns/day (every hour) • ____________________________________________________ • {Clock – UTC(BIPM)}ns/day (every hour)

45. Generating UTC(lab) from TA(lab) • ►Third, with an equipment denominated phase stepper that have as input the frequency of the chosen clock, we applied the correction: • {Clock – UTC(BIPM)}ns/day (every hour) • == == UTC(k) • Phase Stepper • -0.27ns/day clock frequency

46. Results • Figure 15. The time deviation of the UTC(lab) (normalized).

47. Time Scale System ►More Robust that one clock. ►Provide Evaluation of the clocks. Some Difficulties ►Real clock behavior may be change. ►Breakdowns in the measurementsystem. Conclusion

48. References • Weiss, M. A., Allan D. W., Peppler T. K., 1989, “A Study of the NBS Time Scale Algorithm,” IEEE Transactions on Instrumentation and Measurement,vol.38, n. 2, pp. 631-635. • The Statistical Model of Atomic Clocks and the Design of Time Scales • Review of Scientific Instruments, Feb. 2012

49. THE END THANKS BY YOUR ATTENTION Ricardo José de Carvalho – carvalho@on.br Observatório Nacional / Divisão Serviço da Hora ON/DSHO – BRASIL Phone: +55 21 2580-7781 Fax: +55 21 2580-6071 http://www.horalegalbrasil.mct.on.br/