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Luck, John IVcKenneth

coNirraucnoN and comparison o f atomic tim e scale ALGORITHMS WITH A BRIEF REVIEW OF TIME AND ITS DISSEMINATION

The Ohio Siaie University PH.D. 1982

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CONSTRUCTION AND COMPARISON OF ATOMIC TIME SCALE ALGORITHMS WITH A BRIEF REVIEW OF TIME AND ITS DISSEMINATION

DISSERTATION

Presented In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy 1n the'Graduate School of The Ohio State University

by

John McKenneth Luck, B.Sc., B.A., M.S.

The Ohio State University 1982

Reading Committee: Approved By: Prof. I.I. Mueller, Chairman Prof. U.A. Uotila [ V O ( jl I Dr G.M.R. Winkler Adviser Department of Geodetic Science I dedicate this dissertation to the memory of n\y father. Rev. K.K. Luck, 3/24/05 - 9/14/81

(11) ACKNOWLEDGEMENTS

"The trouble with you, Johm, 1s that you have no Idea of time" [Luck, J.M. (Mrs.), personal communications, 1975-82].

Many people have contributed to make this work possible. First, I thank my adviser, Prof. 1.1. Mueller, for his encouragement, great patience, sound personal and academic advice, gentle bullying and stead­ fast support over the years. My reading committee also Included Prof. U.A. UotHa of the Department of Geodetic Science and Surveying and Dr. G.H.R. Winkler, Director of the Time Service Division, U.S. Naval Observatory. Their very pertinent comments have been deeply appreciated. My colleagues 1n the Division of National Mapping, Canberra, Australia, have given much help and understanding, fir s t by covering my absence 1n 1975-77 and subsequently by suffering patiently my preoccupa­ tions. Dr. P. Morgan 1n particular has supported, encouraged and driven me and found many ways to make my studies possible, as has Dr. 6. Greene to no lesser extent. Mr. A. Bomford, Director of National Mapping during most of this time, has been most sympathetic and helpful, and Mr. C. Veenstra has also provided constructive Impulses. I specially thank Mr. J. Woodger who has contributed greatly 1n the collection and editing of the Australian data used herein. I am most grateful to National Mapping for permitting me to use this data, to Mr. R, Bryant for assistance 1n transferring 1t between Canberra and Columbus, and to Mr. D. Barber for expertly drawing many of the figures. The faculty 1n the department of Geodetic Science have had a great Influence on my lif e , both through their courses and by providing an extraordinarily stimulating environment to. such an extent that people

(111) leave their kith and kin to work crazy hours at their behest. My fellow graduate students have also contributed to the pleasure of being part of the Department, and I particularly want to thank Alice Drew, John and Laura Hannah, Errlcos Pavlls, Brent Archlnal and Huseyln Iz for their personal help during 1981 Autumn Quarter. The s ta ff of the U.S. Navel Observatory, Washington, D.C. have provided data and many useful discussions. In addition to Dr. Winkler I thank Dr. W. Klepczynski for his continued Interest In my work. Simi­ la rly , Mr. J. Bulsson has been my principal point of a highly valued contact with U.S. Naval Research Laboratory, Washington, D.C. I am most grateful also to M M. Granveaud of the Bureau International de 1'Heure, Paris, France for providing an annotated copy of the program ALGOS. Figure 3 1s reproduced with permission of Academic Press. Rene Tesfal's typing and advice on layouts has been Instrumental 1n bringing this work to fruition; Margaret Bacon has carried on the good work 1n Canberra. To both these ladles I express my warmest apprecia­ tion. Finally, to my poor long-suffering family I express my heartfelt thanks. They have endured wonderfully during the last seven years, and have provided me with endless love and cheerfulness. To my son Ken I give special thanks for the manner 1n which he has accepted responsibi­ litie s occasioned by my absence from home; he and my daughters Katy and Georgina have brought me a ll the joys that children can bring. All that I dare say to my dear wife Joyce 1s untold thanks for everything, and I cannot wait to start leading a normal lif e again. This work has been supported by the Instruction and Research Com­ puter Centre at OSU and by the OSU Research Foundation, contract 711055.

(iv ) VITA

June 14, 1940 ...... Born - Ballarat, Victoria, Australia.

1^63 ...... B.Sc. - University of Melbourne, Victoria, Australia

1^63-1964 ...... Trainee Actuary, National Mutual L ife , Melbourne, Victoria, Australia

1965 ...... Programmer-1n-Train1ng, Department of Supply, Melbourne, Victoria, Australia

1966-1971 ...... Senior Technical O fficer, Mount Stromlo Observatory, The Australian National University, Canberra, A.C.T., Australia

1971-1977 ...... Physicist, Division of National Mapping, Canberra, A.C.T., Australia

1975 ...... B.A., The Australian National University, Canberra, A.C.T., Australia

1977 ...... M.S., The Ohio State University, Columbus, Ohio, U.S.A.

1977- ...... Surveyor, Division of National Mapping, Canberra, A.C.T., Australia.

PUBLICATIONS

Photographic Measures of Double Stars", Memoir of The Royal Astronomical Society, Vol.76, Part 3, pp.67-98, 1972.

'^Comparison and Coordination of Time Scales", Proc. Astronomical Society of Australia, Vol.3, No.5, pp.357-363, 1979.

(Several unrefereed articles 1n conference publications.)

FIELDS OF STUDY

Ffajor Field: Geodetic Science Studies 1n Geometric and Gravimetric Geodesy: Professor Richard H. Rapp Studies In Adjustment Computations: Professor Urho A. Uotila. Studies 1n Astronomy, Satellite Geodesy and Time: Professor Ivan I. Mueller (v) TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS...... 111 VITA...... v LIST OF TABLES...... x LIST OF FIGURES ...... x11

CHAPTER 1 INTRODUCTION...... 1

1.1 The Need for Time S cales ...... 1 1.2 Objectives of This Study ...... 6 1.3 Broad Description...... 8

2 REVIEW OF TIME AND ITS DISSEMINATION...... 12

2.1 The Nature of Tim e ...... 12

2.1.1 Time Perception ...... 12 2.1.2 Entropy and the Arrow of Tim e ...... 13 2.1.3 Time'Invariants and Geometrlzatlon ...... 14 2.1.4 Fundamental Constants and Units ...... 15 2.1.5 Time's Relationship to Length, Volt and Mass ...... 16

2.2 The Measurement of time ...... 19

2.2.1 Definitions ...... 19 2.2.2 History of Time Measurement ...... 22 2.2.3 Observational Time S cales ...... 24

2.3 Methods of Time Comparison ...... 27

2.3.1 Portable Clocks ...... 28 2.3.2 Television ...... 28 2.3.3 LORAN-C...... 29 2.3.4 Active Satellites ...... 30 2.3.5 One-Way Passive S atellites ...... 31 2.3.6 Two-Way Passive S atellites ...... 31 2.3.7 Advanced Systems ...... 31 2.3.8 Relat1v1st1c Effects ...... 32 2.3.9 International Coordination ...... 33

(v i) Page

REVIEW OF FREQUENCY STANDARDS ...... 35

3.1 Atomic Frequency Standards ...... 35

3.1.1 General Principles ...... 36 3.1.2 Cesium Beam Frequency Standards ...... 45 3.1.3 Hydrogen Masers ...... 47

3.2 Statistical Behaviour...... 50

3.2.1 Spectrum and Allan Variance ...... 50 3.2.2 ARIMA Representation ...... 53 3.2.3 Specific Processes...... 55

3.2.3.1 White Noise Phase Modulation . . . 55 3.2.3.2 White Noise Frequency Modulation . 60 3.2.3.3 Random Walk Frequency Modulation . 62 3.2.3.4 Flicker Noise Phase Modulation . . 63 3.2.3.5 Flicker Noise Frequency Modulation 64

STATISTICAL PROCEDURES...... 68

4.1 Simulation T e s ts...... 68

4.1.1 Simulations...... 68 4.1.2 Predictors...... 78

4.2 Unbiassed Variance Estimation...... 79

4.2.1 Inv1d1v1dual Clock Error Variance Estimates 79 4.2.2 Unbiassed Variance Estimates in Time S cales ...... 82 4.2.3 Detection of Correlations Among Clock Measurements ...... 89

CONSTRUCTION OF TIME SCALES...... 93

5.1 Characteristics of Time Scales ...... 93 5.2 Formulation...... 94

5.2.1 X0: Single C lock ...... 94 5.2.2 X}: Simple Mean of Two Clocks ...... 95 5.2.3 X2: Simple Mean of n Clocks, n > 2 ...... 95 5.2.4 X3: Weighted Mean of n Clocks...... 97 5.2.5 Xu: Corrected Mean of n Clocks ...... 97 5.2.6 X5: Iterated, Corrected Weighted Mean of n Clocks...... 100

(v11) Page

5.3 Prediction Formulae and Weighting ...... 101 5.4 Steering ...... 102

5.4.1 Reinitialization ...... 102 5.4.2 Current Observations of Rate ...... 103 5.4.3 Predictive Steering ...... 104 5.4.4 Least Squares Solution with External Rate Observations ...... 106 5.4.5 Steering TAI...... 109

5.5 Operational Time Scales ...... 109

5.5.1 Echelle Atomlque Libre (EAL) ...... 109 5.5.2 International Atomic Time (TAI) ...... 113 5.5.3 Time Scales at US Naval Observatory .... 113 5.5.4 Time Scales at US National Bureau of Standards...... 117 5.5.5 UTC(Austral 1a)...... 119 5.5.6 UTC(DNM), Division of National Mapping. . . 127 5.5.7 Other Time Scales ...... 129 5.5.8 Summary of Different Algorithms ...... 132

5.6 Comparison Between Algorithms ...... 132

5.6.1 Simulated Data S e ts ...... 132 5.6.2 Experiments on Simulated Data Sets ...... 135

5.6.2.1 Selection of Intervals and Criteria ...... 140

5.6.3 Results on Simulated Data Sets...... 142

5.6.3.1 Data Set 1 ...... 143 5.6.3.2 Data Set 1(a) ...... 144 5.6.3.3 Data Set 2 ...... 151 5.6.3.4 Data Set 3 ...... 162

5.6.4 Discussion...... 169

(v111) Page

5.7 Comparisons Using Real Data...... 176

5.7.1 Description of Data Sets...... 176 5.7.2 Results on Real Data S e ts ...... 177

5.7.2.1 UTC(DNM) ...... 177 5.7.2.2 UTC(AUS) ...... 184

5.7.3 Discussion...... 184

5.8 Algorithm Extensions ...... 186

5.8.1 Split Ensemble ...... 186 5.8.2 Extended Use of Historical Data ...... 187

SUMMARY AND CONCLUSIONS ...... 189

6.1 Summary...... * . . 189 6.2 Conclusions...... 195 6.3 Recommendations for Further Study ...... 196

LIST OF REFERENCES...... 199

(ix) LIST OF TABLES

Table Page

1. Characteristics of Observational Time Scales...... 25

2. Interpretation of Quantum Numbers ...... 39

3. Distribution of Electrons and Other Data, Selected Atoms ...... 40

4. Some Results of Primary Cesi urn Beam Evaluations (parts 1n 10ls) ...... 48

5. Clock Noise Processes ...... 51

6. Parameters of Autoregressive Integrated Moving Average Processes of Order (p,d,q) ...... 56

7. ARIMA Representation of Flicker Frequency Modulation...... 66

8. Results of Simulations of 50-Clock Ensemble ...... 71

9. Test of Unbiassed Covariance and Allan Variance Procedure ...... 86

10. Measurements of the EAL and TAI Frequency ...... 114*

11. Classification of Operational Free-Running Time Scale Algorithms ...... 131

12. Simulation Parameters 1n Data Sets 1, 1(a) and 3......

13. Behavior of Simulated Data, Data Set 3 ......

14. Description of Simulation Experiments ......

15. Results UTC-REF on Simulated Data Set 1: Prediction Errors (zR-z R) ......

16. Results UTC-REF on Simulated Data Set 1: Final Values (zR) ......

(x) Table Page

17. Results UTC-REF on Simulated Data Set 1(a): Prediction Errors (zR-z R) ......

18. Results UTC-REF on Simulated Data Set 1(a): Final Values (zR) ......

19. Results UTC-REF on Simulated Data Set 2: Prediction Errors (zR-z R) ...... 157 CM o

* Results UTC-REF on Simulated Data Set 2: Final Values (z R) ...... 158

21. Results UTC-REF on Simulated Data Set 3: Prediction Errors (zR-z R) ...... 163

22. Results UTC-REF on Simulated Data Set 3: Final Values (zR) ...... 164

23. Recovery of ARIMA Related Parameters from Time Scales* Data Set 1* Second Differences ...... 172

24. Parameters of Runs on DNM Data...... 178

25. Results UTC-REF on DNM^Data Set: Prediction Errors (zR-zR) ...... 180

26. Results UTC-REF on DNM Data Set: Final Values (z R) ...... 181

(x1) LIST OF FIGURES

Figure Page

1. Australian Broadcasting Commission television network used for clock comparisons...... 5

2. Analogies of quantised angular momenta of electrons and nucleus ...... 37

3. Magnetic hyperflne transitions In Hydrogen, Thallium, Rubidium and Cesium atoms - Dependence on external magnetic fie ld HQ...... 41

4. Dependence on atomic transition probability Pi2 on strength b and frequency w of applied microwave f i e l d ...... 43

5. Principal features of typical cesium beam frequency standard ...... 46

6. Principal features of typical hydrogen maser ...... 49

7. Log-1og graph of square root of two-sample Allan variance a ( t ) v s . sample time t for typical atomic clock noise. Units are arbitrary ...... 57

8. Log-log graph of frequency spectrum S (f) vs. Fourier frequency f for typical y atomic clock noise. Units are arbitrary ...... 57

9. Allan variances for attempted ARIMA simulations of flic k e r frequency modulation ...... 67

10. Time dispersion (R.S.S. of outcomes) from simulated 50-clock ensembles, normalised to 5 jjs at t=4096, showing slope y of log-log graphs...... 72

11. Allan variances from simulated 50-clock ensembles, showing slope B of log-log graph ...... 73

12. Evolution of selected clocks simulated with flic k e r of frequency modulation - short term. Scales are arbitrary ...... 74

(x11) FIgure Page

13. Evolution of selected clocks simulated with flic k e r of frequency modulation - long-term. Scales are arbitrary ...... 74 14. Allan variances of selected clocks simulated with flicker of frequency modulation ...... 75

15. Evolution of some simulated ARIMA processes, straight line trends removed. Scales are arbitrary .... 76

16. Allan variances of some simulated ARIMA processes ...... 76

17. Evolution of some more simulated ARIMA processes, straight line trends removed. Scales are arbitrary ...... 77

18. Allan variances of some more simulated ARIMA processes ...... 77

19. Evolution of 5 clocks simulated with white frequency modulation, showing sample cross- correlations...... 88

20. Time scale Xi: simple mean of two clocks, showing effect of jump 1n fir s t clock ...... 96

21. Time scale X*: simple mean of three clocks, showing reduced effect of jump 1n first clock...... 96

22. Estimate of X*,-xK when time scale Xi* 1s being steered to rate of UTC and current rate observations r» are a v a ila b le ...... 105

23. Estimate of Xn-xk as 1n Figure 28 when several past external rate observations are used, emphasizing prediction errors ...... 105

24. UTC(AUS)-Clock for selection of clocks 1n Canberra and Melbourne, 80 Jan 1 to 81 June 30 ...... 121

25. UTC(AUS)-Clock for selection of clocks 1n Canberra, Sydney and Melbourne, 80 Jan 1 to 81 June 30 ...... 121

tx 1 i1) Figure Page

26. UTC(AUS)-Clock residuals from straight line f it s , Melbourne, 1980 ...... 122

27. UTC(AUS)-Clock residuals from straight line f it s , Canberra, 1980 ...... 122

28. UTC(AUS)-Clock residuals from straight line f it s , Sydney and Alice Springs, 1980 ...... 123

29. UTC(AUS)-Clock residuals from straight line f it s , best clock 1n each of Canberra, Melbourne and Sydney, 1980 ...... 123

30. Allan variances of selected clocks, 1980 ...... 125

31. Autocorrelation function for UTC(AUS)-1109, 1980...... 125

32. Results of measurements yielding UTC(USN0 MC) - UTC(AUS) from mid 1979 to early 1981 ...... 126

33. UTC(DNM)-Clock residuals from straight line fits, 1980 ...... 130

34. Allan variances of UTC(DNM) ensemble clocks, 1980...... 130

35. structure of algorlth types tested showing parameter re-evaluation Intervals, f it Intervals and Improvement procedures ...... 139

36. Unbiassed Allan variances of prediction errors U R-zR) » Data Set 1 ...... 147

37. Unbiassed Allan variances of final values (zR), Data Set 1 ...... 147

38. Results on Data Set 1 ...... 148

39. Results on Data Set 1, normalized at t=0,96 ...... 149

40- Results on Data Set 1, normalized at t»96,192 ...... 150

41* Unbiassed Allan variances of prediction errors (zR-zR), Data Set 1(a) ...... 154

42. Unbiassed Allan variances of final values (zR), Data Set 1 ( a ) ...... 154-

43. Results on Data Set 1 (a ) ...... 155

44. Results on Data Set l ( a ) t normalized at t=96,192 ...... 156

45. Unbiassed Allan variances of prediction errors (zR -zR)» Data Set 2 ...... 159

46. Unbiassed Allan variances of final values (zR), Data Set 2 ...... ' ...... 159

47. Results on Data Set 2 ...... 160

48. Results on Data Set 2, normalized at t a96,192 ...... 161

49* Unbiassed Aj[lan variances of prediction errors (zR-zR), Data Set 3 ...... 165

50. Unbiassed Allan variances of final values (zR), Data Set 3 ...... 165

51. Results on Data Set 3 ...... 166

52. Results on Data Set 3, normalized at t=96,192 ...... 167

53. Ensemble mean of Data Set 3, normalized at t=96,192 ...... 168

54. Results of DNM algorithm on each Data Set, normalized at t-96,192 ...... 174

(xv) Figure Page

55. Results of AUS algorithm on each Oata Set, normalized at t=96,192 ...... 174

56. Results of ARIMA algorithm on each Data Set, normalized at t**96,192 ...... 175

57. Results of EAL algorithm on each Data Set, normalized at t a96,192 ...... 175

58. Unbiassed Allan variances of prediction errors (zR-z R), DNM Data Set ...... 182

59. Unbiassed Allan variances of final values (zR), DNM Data Set ...... 182

60. Results on DNM Data Set, normalized at t a96,192 hours ...... 183

61. Weighted predictions from results 1n several previous Intervals ...... 188

(xv1) 1. INTRODUCTION

1.1 The Need for Time Scales Time plays a central role 1n the lif e of modern man. Quite apart from Its regulation of his everyday affairs and Its fascination to his philosophical s p irit, time and Its twin brother, frequency, are of vital Importance to his communication, navigation, defense, metrology and science. Today, 1n certain applications 1t 1s necessary to have clocks synchronized worldwide to a millionth of a second or better, and frequency comparlslons to 1 part 1n 10 are becoming common. Some examples of time synchronization requirements are [Serene and Albert1nol1, 1979]:

Digital communications 10ms International telephone communication 1 ms Earth-based navigation 1 ms Deep-space navigation 20 ns Radio-astronomy 1 ns

Geodesy 1 j possible, d o s s Im !*3* 6 35 while an example of frequency s tab ility requirement 1s "better than l t f 1** a f /f for 1000 second averaging times" [Reinhardt and Rueger, 1979] for Very Long Baseline Interferometry. In order to give meaning to such specifications, Internationally accepted standards need to be established. For centuries, the rotation of the earth provided this standard and gave us the scale of time known as Greenwich Mean Time, or as 1t Is now called, Universal Time. When, earlier this century, 1t proved to be Inadequate the scale Epheneris Time was Introduced, based on the revolution of the earth around the sun. Though much more stable than Universal Time, Ephemerls Time also has proved to be Inadequate for practical purposes, so the timekeepers

1 of the world have turned to atomic frequency standards and clocks to provide the systems currently used for dating events with the highest accuracy and precision. These systems are called atomic time scales. The standard International unit of time Interval* the second* has been defined as 9,192,631,770 cycles of a certain transition of the cesium atom (see, for example, NBS Monograph 140: Time and Frequency: Theory and Fundamentals, [B la ir, 1974]); 1t Is generated by atomic clocks. But even though atomic clocks generate seconds with an accuracy and reproducibility unknown 1n any other physical quantity, yet the seconds so generated by different devices are 1dent1f1ably different, as are the seconds generated at different times by the same device. The 12 variations may be a few parts 1n 10 between otherwise identical cesium frequency standards built commercially, and on the order of a part 1n 13 10 In and betweeen cesium standards built and maintained 1n special laboratories which have the fa c ilitie s to estimate the sizes of system­ atic and random errors ("evaluable standards"). It 1s therefore desirable to average the seconds generated by Individual clocks 1n the hope that a better estimate of the second results. This 1s especially Important since no one specific device has been nominated that produces THE atomic second, unlike the astronomical time scales in which a single device (the earth 1n the case of Universal Time) was so nominated. Another reason for combining the results of the individual clocks 1s that, being made by man, they have fin ite lif e ­ times. When one "dies" there must be sufficient numbers remaining for the time scale to continue uninterrupted, otherwise the system for dating events would be broken. Again, until the last four or five years 1t was very d iffic u lt to operate the best devices (the evaluable standards) continuously, so i t was and 1s necessary to average a large number of the commercial variety to achieve a continuous time scale s ta tis tic a lly comparable with the evaluable standards. A number of proce dures for obtaining such averages having desirable properties have evolved. These procedures are known as atomic time scale algorithms. The Bureau International de 1'Heure (BIH) has been charged by the General Conference on Weights and Measures (CGPM) through the Consul­ tative Committee for the Definition of the Second (CCDS) with the responsibility to maintain the standard second. I t does this by calcu­ lating the average of over one hundred commercial cesium clocks and several of the evaluable standards which operate contlnously as clocks, 1n such a way that addition of clocks to the ensemble, or deletions therefrom, does not affect the continuity of the result. Further, the result of the procedure 1s compared 1n rate ("frequency calibrated") as often as possible with several evaluable frequency standards, and adjusted ("steered") to agree with their mean rate. This two-part procedure produces International Atomic Time (TAI) which 1s the agreed standard. Although the algorithms used to compute TAI give adequate results for current purposes, there are two major considerations which keep the subject of time scale algorithms alive. The firs t consideration Is that there w ill soon be a sufficient number of evaluable cesium standards operating contlnuosly to permit the construction of a time scale between them~at a recent meeting of the International Union of Radio Science [URSI, XXth General Assembly, Washington, D.C., August 1931] some fifteen were noted. To these may be added hydrogen masers which are rapidly having the property of long-term stab ility added to their well- known excellent short-term performances. When this promise becomes a reality, TAI will become obsolete. The essentials of the first part of the BIH procedure (ALGOS) w ill almost certainly remain, but the steering part w ill probably be redundant since the most accurate frequency stan­ dards will already be Incorporated 1n the time scale. A nomenclature: TAI (PC) has been suggested for 1t [Becker, 1981]. Optimum methods for Its calculation are yet to be determined. The second consideration concerns the current d iffic u lty experi­ enced by many establishments a ll over the world 1n obtaining ready and accurate access to TAI. The situation 1n Australia Illustrates this point rather well. That country contains a number of observatories 4 collaborating 1n International experiments (lunar and satellite laser ranging, VLBI) and a digital telecommunications industry which requires synchronization of clocks at the microsecond level or better, yet the only regular methods of obtaining that synchronization use very low frequency radio signals (VLF) which are Inadequate; in particular, LORAN-C cannot be reliably received. It 1s fortunate to receive visits by portable atomic clocks from USA at Intervals ranging from three months to a year, but must exist on Its own in the Intervening periods. Again, 1t 1s fortunate 1n having up to 30 cesium standards that can be intercompared dally through a television network vihlch extends through­ out the eastern half of the continent (see Figure 1). This enables the construction of a national time scale to serve the Australian time­ keeping community between clock trips. Several places have three or more good clocks close enough together that measuring errors are virtu­ a lly negligible; at those places local time scales are possible and desirable for giving excellent stability up to the point where many clocks are needed to average out long-term variations. The Australian situation 1s reflected 1n many countries in the southern hemisphere and 1n the Far East. Until the advent of routine, Inexpensive time transfer techniques having accuracies better than 100 nanoseconds, those countries and Institutions requiring precise and reliable time coordinated with others will have to rely on their own time scales. Conversely, the lack of acceptable time comparison media means that many perfectly good clocks 1n those regions are unable to be used 1n the computation of TAI. The problems associated with the calculation and comparison of time scales receive a great deal of attention from the International unions, and cause numerous meetings to be held. Two recent meetings high­ lighted the problems 1n the Far East and Oceania [IREE, 1980; Mathur, 1981] and complemented a call from the International Astronomical Union [IAU, 1980] for Improved time comparison methods. Satellite methods are being studied by the International Radio Consultative Committee (CCIR) of the International Telecommunication Union (ITU) [Beehler, 1980] and TV DISTRIBUTION

NORTH W ES T CARE AltC C SRMINCS

SID INC SR R IH O S

% CANBERRA

RECEPTION FROM A B C SYDNEY

RECEPTION FROM A B C MELBOURNE

uY U ? HOBART

FIGURE 1. Australian Broadcasting Conmission television network used for clock comparisons. [Woodger, 1981]. 6

Commission A of URSI as noted above. Finally, CCDS Recommendation SI (1980) states 1n fu ll [Giacomo, 1981]: "Algorithms for time scale computation

The Comite Consultatlf pour la Definition de la Seconde, Considering that TAI should be as stable and as accurate as possible, that the many clocks and frequency standards available have various degrees of s ta b ility and accuracy, that the uncertainties 1n the current time comparisons can 11m1t the quality of the computed time scales, that only a few primary standards are available to ensure the long-term stab ility of TAI and Its con­ formity with the definition of the SI second, and that the algorithm employed can significantly affect the quality of the resulting time scale, Recommends that the development of time scale algorithms to ensure optimum use of the available data be actively pursued."

1.2 Objectives of this Study

The prlnlcpal objective of this study Is to analyze systematically the effects of computation technique on the stab ility and uniformity of free-running atomic time scales. (In this context, "free-running" means self-calibrating, or not steered.) Put another way, the aim 1s to select one method from several, of filtering the autocorrelated noise Inherent 1n atomic clocks, 1n such a way that the average noise of several clocks after filte rin g 1s as nearly "white" (random uncor­ related) as possible, given that the only reference for determining the natures of the clock noise processes 1s the average Its e lf. The methods chosen have certain features In common. In order to maintain continuity of the time scale 1n time and rate 1n the face of addition or deletion of clocks, changes 1n weight, or sudden anomalous behavior, i t is necessary to p red ict the time error of each clock to the time being computed, and the predictor should be unbiased. A set of preliminary results 1s calculated for the time being computed, and corrected to minimize the total error between predictions and results; 7

In fact, because there 1s no external reference, the corrections are achieved by Imposing the condition that the weighted sum of the prediction errors 1s zero. Thus, weights are to be assigned on the basis of past behavior. I t 1s also useful to be able to detect anoma­ lous behavior, both 1n the clocks themselves and 1n the measurement system. The main variables 1n this study are the methods of prediction, weighting and anomaly detection. They are varied according to princi­ ples adopted 1n the following time scales that are 1n use today:

(a) IJTC (USNO, MEAN) • US Naval Observatory, unweighted;

(b) UTC (USNO, MC) - US Naval Observatory, weighted and using autoregressive Integrated moving average (ARIMA) methods:

(c) UTC (DNM) - Division of National Mapping, Australia, weighted 1n-house smal 1 -ensemble scale;

(d) UTC (AUS) - weighted Australian scale, Involving Iterations on current results (not merely current data);

(e) EAL - BIH, weighted Iterated fir s t part of TAI.

A method of Incorporating older data to take advantage of clocks having superior long-term stab ility w ill also be examined. Along the way to meeting the goal of choosing a "good" algorithm, c rite ria must be found for deciding upon the choice, or choices. The objective chosen for this study 1s to find a time scale whose long-term noise process 1n time error (rather than rate or frequency error) 1s white, for then the uniformity of the scale 1s assured. However, perfect results cannot be expected because fin ite data samples will lim it the accuracy of any parameter estimation, because the statistical models to be employed are only approximate, and above all because the problem of estimating the clock corrections at each step 1s Inherently rank-deficient. 8

1.3 Broad Description

Principally, this study 1s an Investigation of the effects of various averaging and correcting procedures on sets of data simulated to have the same statistical characteristics as real clocks. Simulated data 1s also used to study the evolution of separate clock noise processes, as well as the uncertainties and the effects of "blasslng" 1n sample variance estimates. Some of the procedures studied are applied to data from real clocks. Some fa irly confident conclusions are drawn concerning the principles of constructing atomic time scale algorithms, and some further studies are proposed which could lead to refinements 1n those conclusions. Firstly, though, a brief discourse on the role of time in thought, theoretical science and metrology 1s presented 1n Chapter 2, in order to provide a background appreciation of how the needs for time scales In general have arisen. In particular, the specific relationships between astronomical and atomic time scales are summarized In Section 2.2, while a rapid review -of those mthods of 1ntercompar1ng clocks which are currently, or soon lik e ly to be, used between places Involved 1n the calculation of time scales 1s given for the sake of completeness 1n Section 2.3. Since the prime objective of the study 1s a statistical comparison of time scale algorithms, some statistical tools are developed 1n Chapter 3. But fir s t, as an aid to understanding, the physical opera­ tion of cesium standards and hydrogen masers 1s reviewed, leading to a tabulation of the principal error sources In the evaluable cesium stan­ dards. Along the way, sane relevant elementary atomic physics 1s treated In a picturesque rather than a rigorous manner, drawing upon analogies with the solar system which should not be taken too far. In Section 3.2, the power law spectrum, commonly accepted as an appropriate model for the correlated noise processes that occur 1n atomic clocks, 1s Introduced, and Its relationships with the Allan variance and autocorrelation function are reviewed. The relatively recent concept of modelling the processes by ARIMA methods [Box and Jenkins, 1976] 1s then found to be a powerful way of understanding them. Advantage Is taken of this to derive simple means of Identifying separate models, especially white noises of phase, frequency and d r ift. Of particular Interest are derivations of the time dispersion character­ istics of the white processes, since time dispersion 1s what we're trying to minimize; the derivations are almost tr iv ia l. Explicit expres slons are also derived to demonstrate that the least squares method 1s superior to the "two endpoint" method for estimating the rate 1n a white phase noise process, and vice versa for white frequency noise. Attempts to model pure flicker noise by ARIMA were, however, unsuccessful. For Chapter 4, an extensive series of simulations was undertaken based on the numerical flicker model of Barnes and Jarvis [1974] and an evidently very effective random number generator. By estimating the time dispersion and Allan variance characteristics from an ensemble of 50 Independently simulated clocks, some Insight was gained 1n the behavior of (empirical) autocorrelation functions for both flicker of phase and flicker of frequency, and a simple way of filte rin g these processes to almost-whlte phase noise was verified. The white processes were also simulated, with excellent results. From Section 4.2, the assumption that there exists a perfect reference against which to measure and characterize clock behavior 1s discarded, and attention turns to the problem of estimating Individual clock statistics when only data from 1ntercompar1sons are available. Such data are necessarily correlated. The "three-cornered hat" tech­ nique 1s studied, with particular reference to the phenomena of negative variances which sometimes result when real data 1s used, and to the Impossibility of estimating the errors of measurement simultaneously. A similar problem exists with the accessible results of time scale calcu­ lations, which appear for each clock as a time comparison against the time scale "reference" which 1s biassed by each clock according to Its weight. The method of calculating unbiassed variance estimates given by Yoshlmura [1980] 1s examined and extended for the case where 1t can reasonably be expected that correlations appear to exist between some 1 0 (but not a ll) of the separate clocks. Simulations vividly verify that any noise process causing autocorrelation 1n a clock Independent of a ll other clocks, can also cause apparent cross-correlations with the other clocks because of fin ite data sampling; thus unbiassed variance estima­ tion, Including the three-cornered hat, should only be used with white Input. The construction of atomic time scales 1s developed systematically In Chapter 5, from a simple set of observations on one clock (which finds considerable application, both when no others are available and when one clock 1s demonstrably superior to the others) to fa irly sophisticated treatments where continuity of the scale Is virtu ally unaffected by changes In (a small number of) the contributing clocks, and schemes for weighting and prediction are based on knowledge of the noise processes that affect each clock. The notion of "steering" a time scale to conform 1n accuracy (though not necessarily 1n precision) with an external standard while maintaining essentially Its own statistical properties, 1s Introduced and developed 1n a manner rather simpler than the TAI method, 1n order to be useful 1n a less sophisticated envlroment. The comparison of different time scale algorithms really commences 1n Section 5.5 where several methods 1n actual use are described In sufficient detail to accentuate their differences. Those differences are classified, and summarized 1n Section 5.5.8. The remainder of Chapter 5 Is devoted to numerical exploration of the consequences of the differences. To this end, five data sets were made available—three by simulation, plus live data from the programs developed 1n the early stages of this work for the scales UTC (DNM) and UTC (AUS) which have become operational for separate purposes 1n Australia. Two of the simulated sets use the ARIMA parameters for 10 clocks as given by Perclval [1978], so they approximate real data sets 1n an Ideal environ­ ment. The third set simulated just flicker of frequency 1n the hope that i t would serve as a control set. Because some of the computer programs used 1n various places were not available, this study looks at the principal features by varying the weighting and prediction methods, length of calibration and prediction Intervals, Iterative techniques and criteria for rejecting uncertain readings. Not all parameters are varied 1n every case for the amount of data-processing would then be prohibitive; some selections were rejected quickly when their fallings became obvious. Oirlng the course of this study, 1t became clear that the only real way to create a time scale approaching the Ideal using real clocks, 1s constantly to re-evaluate the deterministic and stochastic parameter estimates made as new data become available. This would necessitate periodic recalculation of the scale, resulting 1n a multitude of versions—an Intolerable situation. An algorithm 1s therefore con­ sidered for creating a free-running time scale which retains relevant past Information without seriously affecting Its short term stability. The final chapter summarizes the results obtained and the points made that are considered Important, cautionary or Interesting. The general s p irit of the recommendations 1s that much more work remains to be done! 2. BACKGOUND

2.1 The Nature of Time

2.1.1 Time Perception

Man seems to be endowed with a natural sense of the passage of time. He has used the concept of time with ever-increasing sophistica­ tion for remembering events 1n his lif e , for placing those events 1n order, for measuring the duration of events and the duration betweeen them, for predicting events and for ascribing relationships among them, among other things. The external existence of time 1s a subject that has fascinated philosophers through the ages. Other physical quantities such as mass and length can be measured easily, assigned to physical objects enabling objective comparisons between them to be made, and recalled for later use when desired, whereas a moment 1n time 1s fleeting and apparently . can never be brought back, nor do humans have the capability to see Into the future. We can move around 1n space at w ill; motion 1n time 1s beyond our control. Our 1n-bu1lt notions of time have always been dependent upon astro­ nomy. I t would be Interesting to speculate upon what comparable notion would be devised by beings on a moonless, cloud-shrouded planet that always faced Its sun and whose rotation pole was parallel to Its orbital pole. There would be no day and night, no seasons, no change 1n star v is ib ility through the year, no regular periodic cosmic phenomena to Impress th$qselves on the consciousness of such beings. We are fortu­ nate, perhaps, 1n having these phenomena to regulate our lives and thoughts; 1t 1s no doubt fa ir to say that the great religions have evolved (at least 1n part) from our need to explain these motions and to elucidate their ultimate meaning. One great topic that has always been debated amongst theologians, philosophers and scientists 1s the creation

1 2 13 of the universe and Its ultimate demise, or whether 1t did begin and w ill end. The nature of time as conceived by the arguers 1s, of course, crucial 1n the debates: has time been flowing "like an ever-rolling stream" forever {— < t < »); or did 1t (t) have a beginning but our perception of 1t ( t) 1s somehow logarithmic:

0 < t < «

t ■ log t (2.1.1) therefore, -» < t < » ; does 1t even make sense to consider time as existing before the start of the universe or after Its end; 1s 1t merely a figment of the Imagination that w ill "stop, short, never to go again" when man's In te lle c t dies? To what extent does time depend upon matter 1n the universe and Its configuration—why would clocks appear to stop Inside a "black hole"? Does time really always travel forwards: 1s time travel logically possible? Accounts of these arguments can be found 1n the references [Whltrow, 1980; James Arthur Foundation 1936-49; Johnson, 1952].

2.1.2 Entropy and the Arrow of Time

Most processes occurring 1n physics are time-reversible 1n the sense that changing the sign of the time variable would result 1n another valid physical process. For example, reversing t 1n the equa­ tions of motion of planetary orbits would simply make them revolve 1n the opposite direction. I f photographs were taken of the (different) planetary configurations at Instants tj and tg but le ft unmarked by any timing Information, and re-examined later by a person with no prior knowledge of real orbits, he would find 1t impossible to deduce which photo had been taken fir s t, I . e . , he could not order the events recorded. Similarly, 1n a game of billiards employing only two or three balls, photos of different stages of the game would not later reveal the order of events since any competent professional player could re-create any configuration recorded starting from any other configuration of the balls. But 1n a game of snooker or pool, the balls initially are placed 1n a very regular configuration and then "broken"; 1t would be 14 next to Impossible for even a world champion to re-create the In itia l triangular configuration from the confused configuration after the break. In this case* the order of events could be deduced with high probability from unmarked "before and after" photographs. The property used to make the distinction here 1s the state of disorder of the balls—a great e ffo rt, a great amount of energy would be required to create (spatial) order out of the chaos. The state of disorder 1s called entropy, which according to the Second Law of Thermodynamics 1s always Increasing 1n the universe. I t 1s therefore possible, 1n the large, to place events 1n temporal sequence by comparing entropies—the forward flow ("Arrow") of time 1s a consequence of the Second Law of Thermodynamics (or perhaps vice versal)

2.1.3 Time Invariants and Geometr1zat1on

Much of physics can be construed as motivated by the desire to eliminate time and thereby to remove the discomfiture caused by time's Imprecise definition.. Thus Archimedes confined his studies to statics and geometry which did not a lte r with time. Aristotle was brave enough to address the questions of motion and dynamics, but formulated his law of motion Incorrectly (essentially, all motion requires a force) and s tu ltifie d progress accordingly for many centuries. His concept of time, though, was that time 1s the "numerable aspect of motion" [Whltrow, 1980], so In a way he anticipated Ephemerls Time. I t Is undoubtedly fortunate for the motivation of the study of mechanics that Newton considered that "absolute, true and mathematical time of Its e lf, and of Its own nature, flows equably without relation to anything external" [Whltrow, 1980], for he was then free to formulate his laws with beautiful consistency and fruitfulness 1n explaining the observations of his era. From Newton's laws, the laws of conservation of Energy, Momentum and Angular Momentum are derived, which became the fundamental laws of physics because they Imply invariance of these quantities in time. Likewise, his universal law of gravitation was appealing, not only because of its practical success but also because of Its Independence from time (as formulated). Later extensions of Newton's work, particularly by Hamilton [Goldstein, 1950] laid even greater stress on t1me-1nvar1ant quantities and have found great appli­ cation 1n quantum mechanics.

Despite the removal of time from the metaphysical foundations of physics, problems persisted until Einstein's cogitations concerning simultaneity produced his theory of re la tiv ity which placed time on an almost equal footing with space, and subsequently reduced gravitation to a consequence of the geometry of space-time. Yet an asymmetry remains, as the metric tensor employs opposite signs for the space-11ke and time-like coordinates—time has not been completely Identified with space.

2.1.4 Fundamental Constants and Units

Newton's law of gravitation depends on the gravitation constant G; Einstein's theory of re la tiv ity was deduced from the assumption that the speed of lig h t c 1s constant, while quantum mechanics relies on the postulate that energy comes 1n packets that depend on Planck's constant h. The electrostatic charge e and mass m. of an electron are also regarded as universal constants. Possibly even more fundamental constants exist, as speculated by Eddlngton, 1n a form such as the dimension!ess "Eddlngton numerals" [Johnson, 1952]:

( I) mp/me « 1836 ( 2.1.2)

(II) eVGmpme« 1039 (2.1.3)

( I I I ) hc/2re* « 137.036 (2.1.4)

(IV) NwlO78 (2.1.5) where mp 1s the mass of the proton and N 1s the number of primordial hydrogen atoms 1n the universe; oberve that (IV) Is the square of ( I I ) . Eddlngton had hoped that ( I I I ) would be an Integer. Again, the radius of an electron r0 Is [Besancon, 1974]:

re - *s(*rreo) e2/mec* a 2.8 x 10"15m (2.1.6) where %tte0 1s just a scaling factor. This 1s one of the smallest 16 distances known. I f we divide 1t by the greatest velocity* namely the speed of lig h t, the answer 1s a very small unit of time, about 10"23 s. This 1s a "natural” unit, being derived from the fundamental constants and has accordingly been given the name "chronon." I f time were quantized, 1t would be reasonable to expect the quantum of time to be the chronon. Dirac 1n 1937 calculated the age of the universe (as then estimated) 1n chronons, and 1t turned out to be close to 1039 which 1s Eddlngton's second numeral. Such a coincidence led to various cosmo­ logical speculations which allowed one or another of the constants to vary, expeclally 1n such a manner as to explain the observed redshlft 1n lig h t coming from distant galaxies. Prominent among these were the steady-state theory of Hoyle and Bondi and Its continuous creation of matter, and the logarithmic time explanation of de S itter (see equation (2 .1 .1 )). Suffice 1t to say here that the constancy of G has not been verified by lunar laser ranging and occultatlons to a part In 10" [van Flandern, 1981] - 1t does appear to be changing by -6.4x10"11 per year. An alternative value for the "natural" time unit could be the Planck time t p [Whltrow, 1980]:

t p - (Gh/c5)** - lO’ ^ s (2.1.7)

I t 1s surely one of the tasks of modern time observers to establish the actual relationship between gravitational time (depending on G) and atomic time (depending on h), and hence to establish the constancy or otherwise of the Planck time.

2.1.5 Time's Relationship to Length, Volt and Mass

As well as relationships between the fundamental constant of nature and time, 1t 1s Instructive to look at the relationship between time, Its Inverse: frequency, and the standard variables of physics: length, volt and mass from which a ll other variables can be derived. Let v be the frequency of oscillations of a clock used to define a scale of time (see Section 2 .2 .1 ), and T be the period of that clock. Then: v » 1/T (2.1.8) so that measurement of a frequency 1s equivalent to the measurement of a 17

time Interval. I f the clock 1s an electromagnetic radiation (as 1s the case with atomic clocks), the relationship between Its frequency and Its wavelsnth x 1s:

X - c/v (2.1.9)

Supposing c to be adopted as a universal constant, equation (2.1.9) shows that a measurement of frequency v 1s equivalent to a measurement of length X. At present, the definition of the meter 1s 1650763.73 wavelengths of an snlsslon line of krypton-86 whose wavelength 1s about 0.606 pm [Kartaschoff, 1978] and 1s reproducible to a part 1n 108. There are technological d iffic u ltie s 1n measuring the frequency of this emission (which Is 1n the visible lig h t region) 1n terms of the standard frequency produced by the cesium beam (which 1s 1n the microwave region of the spectrum), but these d iffic u ltie s are rapidly being overcome by the use of cascades of stabilized lasers. In particular, the frequency of a 3.39 pm methane stabilized laser has been measured to £3 x 10"11 [Knight et a l., 1980], which 1s a frequency measurement of 88 THz obtained by almost perfect multiplication (by a factor of nearly 9600) from cesium's 9.2 GHz. I t 1s confidently expected that this precision w ill be extended in the next few years to the visible region where useful precise length measurements are made Interferometrlcally; when that happens, 1t 1s very lik e ly that the meter w ill be defined by equation (2 .1 .9 )—1n terms of the standard of frequency [Giacomo, 1980]. A similar situation exists with the volt through the a.c. Josephson effect. When a direct current voltage V 1s applied across a Josephson junction [Young, 1979] composed of a superconducting material with a thin Insulation barrier at superconducting temperatures, an alternating current Is produced whose relationship to frequency v 1s [Harvey, 1976]:

V - (h/2e)v (2.1.10)

The value of 2e/h 1s 483.594 THz/uV [Besanpon, 1974]. By measuring the a.c. Josephson frequency against a cesium standard, Harvey has realized voltages with accuracy limited by the comparison standard cells, a few 18 parts in 10®. The technology 1s thus being developed whereby the volt could be usefully defined 1n terms of the standard of frequency. The case with mass 1s rather more hypothetical. From Einstein's energy equations:

E * hv (2.1.11)

E ■ me2 (2.1.12) the equivalent mass of a photon of given frequency 1s:

mp ■ (h/c2)v (2.1.13) the rea lity of which has been demonstrated 1n the famous measurement of the deflection of starlight by the sun when observed during eclipses. Again, the de Broglie wavelength of an electron \ Is related to Its mass mfi by:

me - h/(vxe) (2.1.14) where v 1s its velocity; or, with v ■ x. va: 6 6 me “ h/(X|ve) - (h/v*)ve (2.1.15)

To use this concept to define a practical macroscopic standard of mass would require the measurement of the frequency v against standard frequency, the measurement of X or v against standard frequency via the derived meter or the derived Josephson volt (since v depends on an applied voltage), and some means of accurately relating macroscopic mass to me. It does not seem likely that all these conditions will be fu lfille d 1n the near future. The same problem of relating mp to macroscopic mass rules out equation (2.1.13). To summarize, the other basic units of physics, namely the meter, volt and kilogram, can a ll be related to time's Inverse, frequency, by simple relationships Involving the fundamental constants c, h, e and mg. Theoretically, when sufficiently precise values of the constants are determined, all measurements could be reduced to the measurement of frequencies 1n terms of the standard frequency which, produced today by the cesium beam, 1s the most stable physical quantity known. Unfortu­ nately, practical considerations render the fu ll realization of this 19 goal unlikely. Further, the gravitation constant G did not appear 1n the discussion 1n this subsection; 1t and the other constants could conceiv­ ably be varying after a ll, hence gravitational and atomic time may not be equivalent.

2.2 The Measurement of Time

2.2.1 Definitions

H. Poincare proposed 1n 1929 "time should be so defined that the fundamental laws of physics, 1n particular the equations of mechanics, are as 'simple' as possible" [Whltrow, 1980, p.44]. Clemence made this concept more specific when, 1n the process of formulating Ephemeris Time, he wrote,

an 'Invariable measure of time' 1s a measure that leads to no contradictions between the observations of celestial bodies and the rlgourous theories of their motions. [Whltrow, 1980, p.44]

The Nautical Almanac Offices assert:

... measurement of time . . . requires two cardinal principles to be kept 1n mind: (a) In astronomy, we are concerned, not with defining time, but only with measuring 1t. To define a measure of time, 1t 1s not necessary to know the ultimate nature of time; we need only devise practicable means of realizing a unit of time and for comparing any interval of time with this unit. (b) A measure of time, like any physical measure, 1s entirely conventional. Any particular measure may be adopted on the basis of Its relative advantages for the specific purpose at hand; no restriction to a unique measure 1s Imposed by physical principles, and no ultimate standard of reference 1s physically attainable. [Explanatory Supplement, 1974, p .68]

In the remainder of this dissertation, the spirit of these concepts will be adopted without being restricted only to mechanical or solar theories, by means of the following operational definitions: 2 0 Frequency standard: an oscillating device having a theoretically uni­ form frequency (e.g., rotating planet, orbiting elctron);

Clock: a device for counting and displaying the number of oscillations (and fractions thereof where necessary) of a frequency standard that have occurred since a definable event*,

Time: that which 1s Indicated by clocks;

Time Scale: a system for assigning dates to events according to an agreed starting point (epoch) and unit of time Interval 1n terms of a prescribed frequency standard (e.g. Universal Time, Atomic Time). By "theoretically uniform frequency" we mean that there 1s no reason to suppose that successive oscillations of the frequency standard are not Identical In every respect. The unit of time Interval (TU) 1s then given by assigning a number (x) to a specified number (n) of such oscillations—the phase of the oscillation may need to be specified also, as 1n Ephemerls Time. Thus:

x(TU).-n. (2.2.1)

Actual adopted values for x and n are discussed 1n Section 2.2.3.

The period P of the oscillation 1s, clearly

P - x/n (time Interval units). (2.2.2)

To be useful, a time scale must be realizable 1n practice. Observ­ able devices may have Inherent Inaccuracies, as with real atomic clocks; they may need to be corrected for environmental effects, as with the reduction to sea level for relat1v1st1c gravitational effects due to altitude; or they may not even be the same type of devices as required by the time scale definition, for example, hydrogen masers realizing a cesium-defined second, or the moon being observed for Ephemerls Time which 1s defined 1n terms of the Sun's apparent motion. Hence, there must be defined a:

Time Scale Algorithm: a procedure for transforming observations on real devices Into an acceptable approximation to the defined time scale. 21

The conditions governing the acceptability of the approximation are:

Accuracy: freedom from systematic errors:

the accuracy of a primary frequency standard 1s a measure of Its a b ility to generate a frequency as close as possible to the Ideal level.

[Kartaschoff, 1978]

Precision: minimization of random errors, particularly measurement errors;

S ta b ility :minimization of cumulative errors, be they systematic or random;

Uniformity: agreement between successive realizations.

There 1s, of course, some Interplay between these factors. I f there 1s excellent uniformity over many realizations then the derived time scale w ill be stable, and vice versa. On the other hand, 1t may be wonder­ fu lly precise, stable and uniform yet Inaccurate because of nonpropa­ gated errors 1n the starting values, perhaps.

I t 1s probably worth emphasizing that the above factors w ill usually be applied to the algorithm relating observations to definition. However, they can be applied equally as well to the definitions them­ selves, as exemplified by Universal Time which 1s now known to be non- uniform and unstable 1n the sense of being unpredictable, yet the UT algorithm Its e lf 1s accurate (1 f precession, nutation and catalogue errors are Ignored!), precise within observational lim its, stable because the link between observation and definition Is simple and fa irly direct, and reasonably uniform because of smoothing. One other area needs to be mentioned:

Aoceaoibility: availab ility of the time scale's realization (observa­ tions reduced by algorithm) to all people who need 1t, when they need 1t. As w ill be seen la te r, this Introduces the concept of:

Time Transfer: the dissemination of tine Information to geographically 2 2 separated locations. In general, this will be Interpreted as the com­ parison of the times shown by distant clocks.

2.2.2 History of Time Measurement

The day has always been with us as a natural unit of time reckoning by virtue of the rising and setting of the sun. The year has also been relatively easy to determine, and was assigned a value of 365 days as early as 4236 BC [Breasted, 1936]. Since then there has been a continuing effort to define both the concepts and the units more precisely; the relationship between the day and the year has been Intimately connected with the development of the calendar [Mueller, 1969; Explanatory Supp., 1961; Jesperson and F1tz-Randolph, 1977; Breasted, 1936]. With the Improvement 1n techniques of astronomical observation from merely noting the heliacal rising of Sirius by the ancient Egyptians, through the use of graduated transit circles to modern transit telescopes, I t became clear that different years could be defined, e.g., tropical, Bessellan, Ephemerls, sidereal [Mueller, 1969] and that the length of day varied whether defined by the Interval between successive sunrises or by the Interval between successive transits of the sun over a given meridian. Hence, 1t has become necessary to state explicitly which type of day or year 1s being con­ sidered 1n precise work. The week and fortnight are units of time Interval that play Important roles 1n man's soda! lif e , but are scientifically unimportant so will not be treated further here, nor w ill the calendar month which 1s another arbitrary unit loosely based on the naturally occurring lunar cycle. Subdivisions of the day, on the other hand, have become Increas­ ingly Important. The hour was considered by the ancient Egyptians to be one-twelfth of the Interval between sunrise and sunset, so 1t varied with the seasons, and the water wheel clocks needed adjusting accord­ ingly. Until the rise of global exploration at the end of the Middle Ages, there was no real need for more precise knowledge of time than the 23 hour, so voter clocks, sand clocks (hour glasses) and sundials sufficed as the Interpolating devices between astronomical observations. Indeed, I t Is doubtful*If the minute and the second were defined, even 1n concept, before 1000 A.D. when they appeared 1n writings from the Orient [Breasted, 1936]. Nevertheless, they became defined sexagesimally from the hour and clocks to measure them Improved until Galileo, dissatisfied with using his heartbeat as the time base for his experiments In mechanics, designed the firs t pendulum clock, one being actually bu ilt fir s t by Huygens 1n 1656 [Jespersen and FItz-Randolph, 1977], Pendulum clocks culminating 1n the refined Shortt pendulum, and spring-based chronometers stemming from Harrison's famous navigation- inspired chronometer 1n 1761 [Smith, 1976] Improved as time Inter­ polators to the point where their accuracies and uniformities were better than those of the quantities they were Interpolating, and so the variations 1n the earth's rotation rate were measured. This non- uniformity In the device (rotating earth) which had served for so long as the frequency standard for a ll time calculations caused a major revision 1n thinking about time measurement, leading to the Introduction of two new concepts. One was the correction of earth-rotatlon observa­ tions for seasonal variations determinable by the Interpolators (and for polar motion) giving UT2; the other was the Introduction of Ephemerls Time (ET) based on the tropical year which, besides being philosophi­ cally d iffic u lt [Wilkins, 1974; Winkler and Van Flandern, 1977] places great demands on the observations and on the Interpolating devices. Fortunately excellent Interpolators In the form of quartz crystals and then atomic Instruments became available during the revision period. Once again, the Interpolators became superior to the underlying process, with the result that 1n 1967 the second was defined 1n terms of an atomic oscillation. This 1s the current situation for time Interval: the frequency standard and the Interpolator are now the same device. The developments that have occurred this century are well docu­ mented [B lair, 1974; Hewlett-Packard, 1974; Jespersen and F1tz-Randolph, 1977; Ramsey, 1972]. 24

2.2.3 Observational Time Scales

The principal characteristics of those time scales capable of being meaningfully observed and accepted or adopted by International agree­ ment* are shown 1n Table 1. The Information has been abstracted from the references given 1n the previous two subsections. The table attempts to highlight the evolution of time scales during this century. Universal Time (UT) Is not s tric tly related to the mean sun, but to a "fictitio u s mean sun" whose right ascension <*H 1s given by [Mueller, 1969]:

aM « 18h30f"45f836 + 8640184^542 t„ + 0?0929tfl (2.2.3) where t^ 1s the number of Julian centuries since 1900 Jan 0^5 UT. Greenwich Sidereal Time (GST) 1s then related to UT by the formula:

GST - UT + oM - 12h. (2.2.4)

The reference to Greenwich 1s to the zero meridian of longitude currently defined by the adopted longitudes of observatories contributing to the BIH. UT 1s not observed directly; rather sidereal time 1s Inferred from astronomical transit obervatlons and then transformed to UT using equation (2.2.4). Thus, sidereal time 1n practice 1s just an Intermediary 1n the realization of UT and could perhaps ju stifiab ly be deleted from the 11st of meaningful time scales. The usefulness of Universal Time as a true time scale 1s also questionable. Nowadays, the only Impact UT has on civil or scientific timekeeping 1s through the whole-second steps applied to UTC so as to maintain UTC within 0.7 of UT1. Even this relationship can be viewed as a buffer stage until the community-at-large 1s ready toaccept a purely atomic time scale. When that happens, the word "time"should be removed from the descriptor of earth rotation. Sim ilarly, the usefulness of Ephemerls Time as a time scale 1s under attack. The definitions of Its epoch and scale unit are perfect 1n the sense that they cannot change because they refer to specific TABLE 1 CHARACTERISTICS OF OBSERVATIONAL TIME SCALES

International Coordinated Sidereal (GST) Universal (UT) Ephemerls (ET) Atomic (TAI) Universal (UTC)

Epoch Greenwich hour angle GHA of fictitious 1900 Jan 0 & ET* 1958 Jan 1, oto TAI* TAI-UTC - n s of vernal equinox, mean sun, instant geometric 1958 Jan 1, 090 UT2 |UT1-UTC|<0.7 s B1H system BIH system mean; long, o f sun near 1900.0 was 279*41'48?04 Scale Unit 1 Day: successive 1 Day: successive 1 Ephemerls second 1 Atomic second 1 Atomic second (TU o f eq. 2 .2 .1 ) transits of vernal transit of flct. equinox a t Greenwich mean sun a t Green­ wich Length of Second 1/86400 of Sidereal 1/86400 of Univer­ 1/31556925.9747 of 9192631770 cycles o f As fo r TAI Day sal Day tropical year at (F*4, Bf*0) - (F-3, 1900 Jan 095 ET Bf*0) transition of Cs 133 atom Frequency Base Earth rotation Earth rotation Apparent orbit of Cs atom Cs atom sun Observation Base 1n Sidereal time Sidereal time Lunar ephemerls Atomic clock Atomic clock. Practice Universal Time Uniformity of Scale Irregular Irregular Perfect Excellent Excellent Limiting Factors In Precision of observa­ As for sidereal; ac­ lunar ephemerls un­ Environmental; As fo r TAI Realization Accuracy tions; star catalog curacy of conversion certainty; precision technological; accuracy formula of observations averaging algorithm Precision o f Obser­ •1 ms/night -1 ms/night -50 ms/9 years -0 .1 us/10 days As fo r TAI vation Accessibility of (by calculation) 5 days 9 years 60 days 60 days Definitive Results Derivatives UT (eq. 2.2.4) UTO: observed at giv­ UTC en location Proper Dynamical: UT1: UTO corrected fo r TDP-TAI ♦ 325184 polar motion Coordinate Dynamical: UT2: UT1 corrected for TDC*TBP reduced to seasonal variatio n solar system bary- center Implementation UT: 1895 1955 (IAU) 1967 1972 (GMT: 1884) 1956 (CGPH) 26 events which have already occurred. However, relating current observa­ tions back to those specific events depends on the exactness of the ephemerls used. The earth's orbit 1s known with sufficient rigor for this, but 1s not readily observable directly (by sun observations), while the orbit of the moon (which 1s the body observed) has to be related to the sun's apparent orbit and requires non-grav1tat1onal terms (tidal friction) for Its complete description. Hence realizations of the definition with any accuracy are very d iffic u lt, so ET-type observa­ tions will probably be used 1n future for examining physical processes and exploring the relationship between atomic and gravitational theories Instead of providing a time scale [Wilkins, 1974; Winkler and Van Randern, 1977; Van Flandern, 1981], The time scales based on atomic time have therefore become the only scales actually used for practical timekeeping. TAI 1s the continuation of the atomic time scale A.3 maintained by the BIH since about 1960; It was o ffic ia lly adopted by the General Conference on Weights and Measures (CGPM) 1n 1971 [BIH, 1974, p. C-2). Until July, 1973, 1t was a weighted mean of seven atomic scales maintained by separate laboratories; since then I t has been computed as theweighted mean of Individual clocks In many laboratories using the algorithm ALGOS. In addition, since 1 January 1977 the output from ALGOS has been corrected by a steering procedure based on calibrations against evaluable ("primary") cesium standards [BIH, 1980; Azoublb et a l., 1977; Granveaud, 1979]. Eight Independent laboratories also currently maintain their own scales of atomic time, designated TA(I). Coordinate Universal Time (UTC) 1s a time scale based on TAI and kept within certain bounds from UT to satisfy the requirements for civil usage, because by International agreements, radio time signals are broadcast 1n the UTC system. Until 1 January 1972, both the time and rate of UTC were offset from A.3 so that:

|UTC - UT2 | < 0?1 (2.2.5)

[Mueller, 1969], Since then, the criterion has been:

1UTC - UT1| < 0?9 (2.2.6)

[BIH Annual Report, 1980], with UTC being offset 1n time from TAI by a 27 whole number of seconds only, and no rate offset. Many International bodies are concerned with the definitions and limitations of UTC [Barnes and Hinkler, 1974; Fosque, 1976; Kartaschoff, 1978]. Dynamical Time (TD) was adopted by the IAU in 1976 to replace ET as the Independent argument 1n theories of celestial motion, because of the d iffic u ltie s with ET alluded to above. TOP 1s a "Proper" time scale, 1n that 1t 1s given by clocks on planet earth with only local corrections, such as to sea level. I t 1s given by:

TOP - TAI + 32?184 (2.2.7)

[Winkler and Van Flandern, 1977], and Is therefore continuous with ET 1n 1977. For theories expressed 1n a heliocentric coordinate system, the varying re la tlv ls tlc effects of gravitation and the earth's velocity must be taken Into account 1n reducing observations on real clocks to the barycenter of the solar system. The reduced time scale 1s called "Coordinate Dynamical Time" (TDC) and 1s related to TDP such that there are only periodic variations between them. The principal variation Is an annual one of amplitude 1658 us. However, the time scale used 1n the construction of solar system ephemerldes in the national almanacs was s till ET 1n the 1981 publications.

2.3 Methods of Time Comparison

Atomic time scale algorithms average the readings from several clocks. The readings must therefore be collected at a central computing fa c ility , such as the BIH. Because of the nature of time, this requires that the several clocks be compared against each other In as near real time as possible, and as often as possible. Current requirements are tending to 10 ns accuracy over Intercontinental distances on a routine basis. Similarly, time comparisons under the same or even more stringent conditions are often needed for specialized applications. A number of techniques have thus been developed, most of them using fa c ilitie s built for other purposes such as navigation or communications. Few realize 10 ns precision or accuracy at present, but great efforts are being made 28

to reach this goal. Status reports are available 1n the Proceedings of the Annual Precise Time and Time Interval (PTTI) Applications and Plannnlnq Meetings held each year 1n Washlnton, B.C., and can be Inferred from the BIH Annual Reports. Three journals that often describe time comparison methods are Metroloqla. Radio Science (see, for example, Vol. 14, No. 4, 1979) and IEEE Transactions on Instrumentation and Measurement. Books containing excellent descriptions are HBS Mono­ graph 140 [B lair, 1974], NBS Monograph 155 [Jespersen and FItz-Randolph, 1977], NBS Special Publication 559 [Kamas and Howe, 1979], and Frequency and Time [Kartaschoff, 1978]. Some of the techniques used today are now briefly Introduced.

2.3.1 Portable Clocks

Cesium and rubidium frequency standards with clocks have been equipped with rechargeable batteries and flown from country to country since 1958 [B lair, 1974]. This method Is s till considered to be the definitive method of time comparison, even though the mlsclosures after a good two-week trip may exceed 200 ns. Generally, commercial cesium clocks with supertubes are employed, although rubidiums have been developed which give 100 ns results for quick trips between continents, and 10 ns for trips lasting, say, an hour [Hellwlg et a l., 1978]. Smaller clocks are being developed to reduce the logistical and financial d iffic u ltie s of clock transportation [Putkovlch, 1979]; but methods of monitoring clock behavior while travelling through varying environments w ill have to be devised before 10 ns Intercontinental accuracy w ill be achieved [Hafele and Keating, 1972].

2.3.2 Television

TV 1s an excellent medium for time transfer because of the extra­ ordinary stab ility of the propagation delay time from transmitter to receiver (which is calibrated by portable clock trip every few months) and the c la rity of the signal (11ne-10 1s used in the USA.) A notable example 1s the Australian TV network 1n which, despite numerous repeater 29 stations and connecting links, standard errors of time comparisons are typically 100 ns between Melbourne, Canberra and Sydney, and only 250 ns at Alice Springs, 3000 km from those cities [Woodger, 1980; Harris, 1980]. I t 1s used extensively 1n Europe [Kaarls and de Jong, 1979; Mureddu, 1980; Kovacevlc, 1978, Kovacevlc et a l., 1979], and the USA [Kamas and Howe, 1979] and many other countries (e .g ., [Mathur, 1981]). I t 1s usually used 1n the passive mode, 1n which the I1ne-10 or synchro­ nizing pulse 1s merely a convenient "beacon" viewed simultaneously by several clocks; In the active mode the appropriate line or pulse 1s transmitted on time, for example, Channel 5 WTTG 1n Washlnton, D.C. [B lair, 1974; Putkovlch, 1975] which 1s synchronized to UTC (USNO) and thus looks like a clock. TV 1s also used for direct frequency compari­ sons [Kamas and Howe, 1979].

2.3.3 LORAN-C

The operating principles of LORAN-C are described 1n [Mueller, 1969] as well as In the genral references quoted above. The 1974 PTTI meeting contained a whole session devoted to this method. LORAN-C provides extensive coverage 1n the Northern Hemisphere, and 1s 1n fact the prinlcpal method of comparing clocks used 1n the computation of TAI. Standard errors of 100-200 ns .are routinely obtained over a two-month period [BIH Annual Reports]. Dally variations may be of the same order of magnitude, but can be reduced to less than 50 ns over a 1000 km path by the application of meteorological corrections [Dean, 1978; Klepczynskl, 1975]. Seasonal variations are also suspected, the evidence being annual fluctuations 1n frequency with amplitude about one part 1n 1013 between TAI, the evaluable cesium standards at NRC, NBS and PTB and several local atomic time scales [Becker, 1981]. These fluctu­ ations were discussed at the XXth General Assembly of the International Union of Radio Science (URSI), 1981, and are attributed variously to seasonal effects on the evaluable standards, seasonal effects correlated with humidity 1n commercial cesium clocks, and seasonal effects on LORAN-C propagation paths, even between the USA and Canada. 30

They directly affect the computation of TAI and therefore need to be c la rifie d , for example, by comparing the results of time transfer via LORAN-C against results via the Global Positioning System (GPS—see Section 2.3.4) and via two-way geostationary satellites (see Section 2 .3 .6 ).

2.3.4 Active Satellites

Earth orbiting satellites which contain clocks on board provide a convenient method of time transfer. Typical are the TRANSIT system of Navy Navigation Satellites [Leroy, 1979; F ell, 1980; Janlczek, 1980]. In these systems, the behavior of the clock, especially Its offset and rate with respect to some defined time scale, 1s evaluated 1n the orbit determination procedure. This Information 1s broadcast by the s a tellite along with orbital ephemerldes, pennlttlng real time correction of the received timing signal for propagation time between the s a tellite and a ground receiver at known location [Hunt and Cashlon, 1978; Raymond et a l., 1976]. Thus a direct real-time comparison between the ground clock and the onboard clock's time scale 1s possible. Results from the stan­ dard TRANSIT system show 10-20 us precision, while the NOVA update of TRANSIT should give about 3-5 us [Cashlon et a l., 1979]. GPS satellites contain redundant atomic clocks, Including rubidium, cesiums and hydrogen masers (planned). Pilot experiments are achieving 10 ns precision 1n measuring the onboard clock performance with respect to UTC (USNO) [Klepczynskl, W., private communication, 1981] using a commercially bu ilt receiver, while a NBS prototype receiver 1s giving as good as 3.5 ns receiver fluctuations and 5 ns time transfer 1n "common view" mode between NBS and USNO [Davis et a l., 1981]. I t 1s expected that worldwide accuracies on the order of 10-20 ns w ill be attained with GPS using relatively simple, "stand alone" receivers which make I t an attractive technique (see a ll PTTI Proceedings, 1972-80). A possible role for GPS 1n a worldwide high precision time transfer scheme 1s out­ lined 1n Section 2.3.9. 31

2.3.5 One-Way Passive Satellites

Geostationary satellites generallly retransmit ground-originated signals for communications, meteorology, environmental monitoring, etc. Typical 1s the Geostationary Operational Environmental S atellite (GOES) Which broadcasts an NBS time code and s a te llite coordinates, among other things [Kamas and Howe, 1979]. Inexpensive, commercially-ava1lable receiver/processors subtract the propagation delay time, and precisions of about 10 vs can be obtained as far away as New Zealand and Australia [Lohrey, 1980; Thorn, 1980].

2.3.6 Two-Way Passive Satellites

The most accurate regular time transfers occur at present between laboratories that have access to both receiving and transmitting equip­ ment. Each laboratory transmits Its signal to a geostationary s a te llite more or less simultaneously, and measures the time of arrival of the signal coming via the s a te llite from the other laboratory. The propaga­ tion paths of both signals are virtually Identical so satellite orbit errors have negligible effect on the results [B la ir, 1974; Brunet, 1979; Costain et a l,, 1978]. Standards errors of a signal at a receiver have been reported as small as 0.18 ns [Costain, 1980]. Intercontinental time transfers are achieving 15-30 ns [Costain et a l., 1979; Mathur et a l., 1980], while a closing error of 2 ns between three stations 1n North America has been obtained [Costain et a l., 1979], where the standard errors are calculated from runs of up to 20 minutes. Although fa irly Inexpensive transmit/receive terminals are being developed at NRC, Ottawa, I t 1s generally expensive to hire s a te llite transponder time and sometimes d iffic u lt to arrange suitable times, so this method 1s likely to remain restricted to time transfers between the major laboratories.

2.3.7 Advanced Systems

Even more accurate and/or u tilita ria n systems than GPS and Two-Way Passive are being proposed and developed. For example, NASA's Tracking 32 and Data Relay S atellite System (TDRSS) w ill have a number of geosta­ tionary satellites each having an accurate onboard clock, so that a combination of one-way and two-way techniques can be employed [Ch1, 1979]. The geostationary s a te llite SIRIO-1 has been used to Investigate the fe as ib ility of sequential two-way time transfer using only one channel, 1n which s a te llite motion between measurements 1s monitored by ranging and Doppler observations on the timing signals [Detoma and Leschlutta, 1979]. The next s a te llite 1n this series, SIRIO-2, w ill have an onboard clock and laser retro-reflectors to Implement the Laser Synchronization from Stationary Orbit (LASSO) experiment [Serene and Albert1nol1, 1979; Serene, 1980]. As well as normal laser ranging from two stations equipped with the atomic clocks to be compared, the system w ill Include a detector package on the sate llite to measure arrival tines of the laser pulses. Ulth this extra Information, the error budget gives a standard error of 3.5 ns. An augmented version of the LASSO project has been proposed as a Space Shuttle experiment [Decher et a l., 1980; Allan et a l., 1980]. The onboard clock w ill be a hydrogen maser, and laser ranging w ill be sup­ plemented by a three-channed microwave link for removing Doppler shifts and Ionospheric errors. Subnanosecond time transfer and frequency com­ parisons to one part 1n 10lu are anticipated. The potentialities of Very Long Baseline Interferometry (VLBI) are being explored by the NASA Deep Space Network, which 1s achieving 100 ns precision between California, Australia and Spain [Madrid et a l., 1980]. Three-station closing errors were from 10 to 33 ns. A 22 km baseline experiment 1n California gave results that agreed with portable clock measurements to 9 ns [Young, 1979], The accuracy potential of VLBI 1s 0.1 ns [Thomas, 1980], but 1t requires time on large, busy radio tele­ scopes and a great deal of post-processing.

2.3.8 Relat1v1st1c Effects

There Is a mutual 1nter-relat1onsh1p between time transfer experi­ ments and the special and general theories of re la tiv ity . Experiments 33 using moving clocks 1n aeroplanes [Hafele and Keating, 1972; Bertottl, 1979], rockets [Vessot, 1979; Vessot and Levine, 1976] and 1n orbiting spacecraft [Bulsson et a l., 1978], and using non-atomlc clocks (super- conducting cav1ty-stab1l1zed oscillators) compared against atomic clocks [W ill, 1977], have tested or can test various aspe ts of the theories— the ultra-high stability of atomic clocks, to five parts 1n 1016 over an hour 1n some cases, has been necessary 1n detecting the very small effects. Conversely, submicrosecond time comparisons a He affected sign ifi­ cantly by re la tiv ity . As examples, a portable clopk slowly circumnavi­ gating the equator eastwards w ill lose 207 ns; the corrections In a two-way geostationary time transfer can exceed 400 ns, and those affecting onboard sate llite clocks can exceed 200 jis; and the frequency of a clock fixed on the earth Increases by 1.09 x :.0“13 per km above sea level [Ashby and Allan, 1979; Shapiro, 1978; Reinhardt, 1974]. These effects must not be neglected.

2.3.9 International Coordination

In view of stated needs to Include as many decks as possible Into the TAI system [IAU, 1980] and Into local or natloral time scales, and to ensure the widespread availability of the results while not commit­ ting all users to very expensive or Inconvenient eti ulpment, a composite scheme along the following lines will probably be in ecessary [IREE, 1980]: (a) A very carefully maintained h1gh-qual1ty link between the major center on each continent, either by two-lv way geostationary (Section 2 .3 .6 ), minimally post-processed GPS (Section 2.3.4) or LASSO-type satellites (Section 2.3.7); (b) GPS satellites 1n "common-vlew" mode or at won: st In nearly common-vlew mode, or where applicable one-way passive satel- Htes 1n differential mode (Section 2.3.5), to transfer from the continental centers to the principal centers 1n each country; (c) Terrestrial TV (Section 2.3.2) or s a te llite TV (Section 2 .3 .5 )' supplemented by regular, quick portable clock trips (Section 2.3.1) within each country; (d) National and regional time scales computed rapidly, to Inter­ polate between the Intercontinental comparisons and to ensure the Integrity of the participating clocks and of the comparison links. 3. REVIEW OF FREQUENCY STANDARDS

3.1 Atomic Frequency Standards

The aim of this section Is to describe briefly the three types of cesium beam frequency standards* namely, commercial, high perform­ ance, and evaluable, and the two types of hydrogen maser, namely, active and passive, that currently form the basis of h1gh-prec1s1on time scales,with a view to highlighting the principal error sources. Readable introductions can be found 1nNBS Technical Notes 399 [R1s- ley, 1971] and 616 [Hellwlg, 1974a],NBS Monograph 140 [B lair, 1974], Hewlett Packard Application Note 52-1 [Hewlett Packard, 1974], and Frequency and Time [Kartaschoff, 1978]. For a deeper understanding, a knowledge of atomic physics 1s required (e .g ., [Born, 1944; B itter, 1956; Belser, 1968]) leading to a study of atomic and molecular spec­ troscopy and the classical work of Ramsey [1956]. The current situ­ ation 1s discussed at the Annual Frequency Control Symposia (AFCS) 1n Ft. Monmouth, N .J., the Annual Precise Time and Time Interval (PTTI) Applications and Planning Meetings 1n Washington, D.C., the triennial General Assemblies of the International Union of Radio Science (URSI) and the International Astronomical Union (IAU), and occasional symposia (see, for example, [Enslln and Proverblo, 1974; Hellwlg, 1976; Mathur, 1981]). Research articles are regularly pub­ lished 1n the journals, particularly Metro!ogla. Radio.Science and IEEE Transactions on Instrumentation and Measurement. Although other atomic devices such as rubidium standards and those that trap 1ons at zero velocity do and w ill have Important roles to play', they w ill not be considered here 1n any depth, nor w ill crystals or superconducting cavities. They are adequately de­ scribed 1n the references quoted above.

35 3.1.1 General Principles

I t Is convenient to view an atom as a nucleus surrounded by a given number of electrons In separate, well-defined orbits each hav­ ing a specific energy level governed by the Pauli exclusion principle that no two electrons may have the same total energy. Each electron 1s ascribed a spin which gives 1t a magnetic moment, as Is the nucle­ us. Strong analogy with a solar system of spinning planets can be drawn, with gravitational forces replaced by electrostatic forces and with nonspherldty precessions replaced by magnetic precessions. The various components of angular momentum associated with orbital motion, spin and precession and their directions are quantized, that 1s, they obey relationships of the form:

p * Xh/2* (3.1.1)

where p 1s the angular momentum concerned, h 1s Planck's constant, and x can take only Integral or half-integral values. They are Illu s ­ trated 1n Figure 2 and summarized 1n Table - 2 as an aid to further reading. The energy associated with each angular momentum component 1s therefore also quantized. The way 1n which electrons are added to build up successive elements 1n the Periodic Table 1s summarized In Table 3. When an electron changes Its state of quantization from one with energy Ex to another with energy E2, a photon Is emitted with frequency v given by

E2 - Ex = hv . (3.1.2)

Atomic frequency standards have to Induce such transitions and detect ,or assure the required frequency which must therefore be 1n the range accessible to current technology. Hyperflne transitions in the alka­ l i atoms hydrogen, rubidium, cesium and thallium have been found most suitable, as they emit radiation in the microwave region. The partic­ ular transitions chosen are quite Insensitive to weak external magnetic field s, so that the frequencies generated are Intrinsically constant. The dependence of these transitions upon magnetic field strength for these atoms is shown 1n Figure 3. Rlsley [1971, 1974] gives further 37

JL*Z

r\mi *n»2 7\»3 77*3

Principal (b) I a n t i Orbital a): px- £ (Morton a Ij major axese e ^ o a lj

H 4 Mor^nat'rC R eid

mx - 2

m z = l 7

m „ - 0 7 X 7 \ ___ j y x

Z e e m a n £ Electron spin (A) : — x f £ c o s e -

FIGURE 2. Analogies of quantised angular momenta of electrons and nucleus. 38

Total E lectron ^ + * 4 Nuclear Spin (I): j*i = I *5 t

H A Magnetic Field

~ T ^ ’ 2

m ^ l _ _ Z _ A " f

m F * 0 J T j r

m F » - l

\ / - t n F m ~ 2 _ _ _ \ / f

Total A tom £F ) 1 F *■ J+ £ J fbojecied Total Cm p) :

P s . " V ^

FIGURE 2. — Continued, TABLE 2 INTERPRETATION OF QUANTUM NUMBERS

Quantization Spectroscopic Quantum Name Range Number Analogy Interpretation

n Principal n i 1 Semi-major axis Lyman, Balmer, series £ Orbital 0<£

S Spin ±i Electron rotation Line splitting j Total electronic 1 =t+ s Rotation axis Anomalous Zeeman inclination I Nuclear spin 0 ,i, 1,3/2, Nucleus rotation (Fixed per atom)

F Total atomic |J-I|

1 H Hydrogen 1 13.5 1 2.79 1/2 2 He Helium 2 24.5 4 - 0

3 LI L ith lua 2 1 5.4 7 3.26 3 /2 4 Be Beryllium 2 2 9.3 9 -1 .1 8 3 /2 S B Boron 2 2 1 8 .3 11 2.69 3 /2 6 C Carbon 2 2 2 11.2 12 - 0 7 N Nitrogen 2 2 2 1 14.5 14 0.40 1 8 0 Oxygen 2 2 2 2 13.6 16 - 0 9 F Flourlne 2 2 2 2 1 18.6 19 2.63 1/2 10 He Neon 2 2 2 2 2 21.5 20 0.00 0

11 Na Sodium 2 2 2 2 2 1 5.1 23 2.22 3/2 12 Mg Magnesium 2 2 2 2 2 2 7 .6 24 0.00 0 13 Al Aluminium 2 2 2 2 2 2 1 6 .0 27 3.64 5/2 14 SI S ilicon 2 2 2 2 2 2 2 8.1 28 0 15 P Phosphorus 2 2 2 2 2 2 2 1 11.1 31 1.13 1/2 16 S Sulphur 2 2 2 2 2 2 2 2 10.3 32 _ 0 17 Cl Chlorine 2 2 2 2 2 2 2 2 1 13.0 35 0.82 3/2 18 Ar Argon 2 2 2 2 2 2 2 2 2 15.7 40 0.00 0

19 1C Potassium 2 2 2 2 2 2 2 2 2 1 4 .3 39 0.39 3 /2 20 Ca Calcium 2 2 2 2 2 2 2 2 2 - - - - _ 2 6.1 40 0.00 0 21 Sc Scandium 2 2 2 2 2 2 2 2 2 1 ------2 6 .7 45 4.76 7 /2

37 Rb Rubidium 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 6 - - 1 4 .2 87 2.75 3/2 55 Cs Cesium 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 6 10 - 2 6 - 1 3.9 133 2.58 7/2

56 Ba Barium 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 6 10 - 2 6 - 2 5.2 137 0.93 3/2

81 Th Thallium 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 6 10 14 2 6 10 2 1 6.1 205 1.63 1/2 41

W/htv «•* 'H % • |.97110* Ho F • I — »»n i • 4 391 10'* Ho

F«0~

W/h»o I-f 8 7 Rb i •4.|*I«T,Ho

W/h*.. I’T '"c. *•3-05 *1 0 '* Ho

FIGURE 3. Magnetic hyperfine transitions in Hydrogen, Thallium, Rubidium and Cesium atoms - Dependence on external magnetic fie ld H . [Kartaschoff, 1978. Reproduced by permission of the publisher.] 42 c rite ria for the selection of suitable atoms. Molecular transitions have also been considered [Beehler, 1974; VMneland, 1977; Vanier, 1978]. To Induce the required transitions, atoms In one energy state are Irradiated with electromagnetic energy of the correct frequency 1n a tuned microwave cavity, causing them to change to the other state. In paAtivz devices, the microwave radiation 1s synthesized (b u ilt up) from a 5 MHz crystal oscillator (which 1s, 1n fact, the actual oscillator that provides the working output) and fed Into the cavity. In active, devices, the radiation 1s Initiated by spontaneous transitions 1n some of the atoms themselves, the photons so produced Induce transitions 1n other atoms, and the process builds up by rnaser action [Beesley, 1971] to a steady state of emission. In both active and passive cases, 1t 1s necessary to ensure that there are sufficient atoms In the "before transition" state to provide an adequate number of transitions for good signal-to-no1se ratio and, 1n the case of active standards, to achieve the self- sustaining maser action. State selectors are provided to do this. They are generally magnets with strong magnetic fie ld gradients 1n the central hole through which the atoms pass on their way to the microwave cavity. Deflections of atoms in this fie ld depend on mag­ netic moments 1n the atoms, which 1n turn depend on the energy states —so atoms 1n unwanted states are deflected away, and atoms 1n the desired state are focused Into the cavity. (State selection 1n rubid­ ium standards Is performed by optical pumping, which may soon also be applied to cesium standards.) Figures 5 and 6 in the following section Illustrate these Ideas. After the atoms have been state selected, they pass through a region subjected to a steady weak magnetic fie ld H, the "C-f1eld." This region contains the microwave cavity. I f the effective oscil­ lating microwave fie ld strength 1s Hi, and the atom Is 1n the region where the two fields Interact for time t , the probability of transi­ tion occurring Pia 1s:

Pi* * C(2b)Vn*] sin* Or] (3.1.3) where 43

fl2 a (ioo-u)2 + (2b)2 uo a 2ir(E2-Ex)/h (from eq. (3.1.2)) b = U0Hx/2H

and u> Is the angular frequency of the fie ld 1n the cavity (which 1s synthesized from the crystal oscillator) [Rlsley, 1971; Ramsey, 1956]. The shape of the resonance curve of P22 against w 1s Gauss1an-I1ke near u * u0) so that transitions w ill occur when u f uo (see Figure 4). The half-width Au of this curve must thus be minimized, I.e ., the quality factor Q maximized:

Q * uo/Au) (3.1.4)

1n order that transitions occur for only a very small band of fre ­ quencies. Further, Pl2 Increases with t (up to a limit) so that the time of atomic Interaction with both fields must be maximized. The atoms have fin ite velocity, so the cavity should be very long or the atom velocities small. Further, the velocities have a velocity dis­ tribution which broadens the resonance curve. The distribution 1s

O.T 0-«

0-9

0 4

0 1

• 4

FIGURE 4. Theoretical Transition Probabilities. (From Rlsley [1971]) Maxwellian which applies to gases and which depends on temperature, so cold cavities could be expected to have sharper resonances. The velocity of the atoms with respect to the electromagnetic waves In the cavity produces a Doppler shift, in frequency. The second-order relatlvlstlc Doppler effect is very difficult to remove and varies according to the velocity distribution [Wlneland, 1979]. Other Im­ portant factors that broaden the transition resonance curve are col­ lisions between atoms, collisions with the container when Inside the cavity ("wall shift"), 1nhomogene1t1es 1n the C-fleld, Impurities 1n the applied frequency u, relative phase anomalies from one end of the cavity to the other ("cavity phase s h ift" ), overlapping with resonance curves from nearby transitions, and Interaction with the resonance curve of the microwave cavity itself ("cavity pulling"). The basic design of a frequency standard 1s directed towards eliminating or making compromises between a ll these effects, I. e . , minimizing the Unewldth, that 1s, reducing the region 1n which u can wander, and towards maximizing the number of atoms that undergo transition. The fact that atoms are making transitions 1n the desired manner needs to be detected. Active standards create radiation which is detected by a microwave loop behaving as a radio antenna. On the other hand, passive standards respond to radiation, and I t 1s the change 1n state 1n the atoms themselves which 1s detected. For exam­ ple, the gas 1n a rubidium cell Increases opacity when the applied frequency 1s correct, causing a minimum 1n a photocell detector's output. In practice, a low-frequency signal 1s modulated onto the signal entering a microwave cavity, and Ideally only Its second har­ monic should emerge 1n the detecting system; the phase and amplitude of the modulated signal's fundamental Indicates the direction and size of the error 1n the microwave frequency, and generates an error signal to drive the crystal's frequency back to the correct value, via a servo-loop [Kaufman, 1975]. Electronic noise 1s Inevitably Introduced at this stage. 45 3.1.2 Cesium Beam Frequency Standards

Figure 5 1s a diagram of a commercial cesium beam frequency standard [Kaufman, 1975]. Cesium atoms are heated to about 100° C 1n the oven, and pass through the dipole state selector magnets, after which the beam 1s focused Into the cavity area at about 100 m/s.

The Ramsey cavity 1s U-shaped, but the Interaction time t 1 s the same as I f the beam were Inside a cavity a ll the way between the two arms [Ramsey, 1956]; the 11new1dth Au 1s reduced by about 40% by this arrangement, but 1t 1s practically Impossible to keep the microwave frequency 1n phase at both ends, so cavity phase shift errors occur. The Interaction time 1s a few milliseconds, during which the atoms are raised from the F ■ 3, m^ » 0 state to the F ■ 4, mf ■ 0 state by resonant absorption [Hewlett Packard, 1974]. Those atoms so raised are selectively deflected by another dipole magnet to a hot-wire Ionizer, a strip of tungsten or similar material at 800-1000° C. For e ffic ie n t Ionization, the Ionization potential needs to be small— see Table 3. Thermal noise 1s added here. The 1on strength 1s measured by a mass spectrometer/electron m ultiplier assembly, and fluctuates with the modulation frequency or Its second harmonic (274 Hz). The error signal is shaped and fed to the voltage-controlled crystal oscillator. The time constant 1n this loop determines how rapidly the crystal responds to error signals, hence to what extent the short-term stab ility (up to a minute) depends on the crystal rather than on the atomic transition. HAQh.-pzfifatmanc.e. cesium standards employ two beams In the cavi­ ty [Seavey, 1976], which means there are more atoms Interacting which Improves the signal-to-no1se ratio and hence short-term stability. EvaZuabtz (primary, laboratory) cesium standards are so con­ structed that the factors affecting accuracy and stability can be measured. Their interaction regions are long, e .g ., 3.74 m 1n NBS-6 [Glaze et a l., 1977] which has dual beams 1n which transitions occur 1n opposing senses, and are now being b u ilt so that they can be eval­ uated without stopping the device which means they can be used con­ tinuously as clocks instead of Intermittently just as frequency 'A' MAGNET SHIELDS ' MAGNETMAGNETIC

*C' FIELD COIL HOT WIRE

OVEN m

RAMSEY CAVITY DETECTOR VACUUM CHAMBER

OVEN C-FIELD 1770 UHi CONTROL ADJUST PM 137 Hi 137/ 274 Hi X 102 HARMONIC GENERATOR AMPLIFIER

90M H i PM 137 Hi X 18 137 SYNTHESISER PHASE DETECTOR MULTIPLIER H i

VOLTAGE ERROR OP. AMPLIFIER OUTPUT FREQUENCY CONTROLLED SIGNAL CRYSTAL OSCILLATOR

FIGURE 5. Principal features of typical cesium beam frequency standard. 47 standards [Mungall, et a l., 1980; Becker, 1977]. The evaluable standards NBS-6, NRC-V and PTB-1 have been used to provide frequency corrections to TAI since 1978 (See Table 21, BIH Annual Report, 1980). Some recent evaluations of their errors are given 1n Table 4.

3.1.3 Hydrogen Masers

Figure 6 1s a diagram of an active hydrogen maser (courtesy P. Clements, OPL). Hydrogen molecules are dissociated Into atoms by electrical discharge. The atoms pass as a beam through a hexapole state selector magnet and enter a teflon-coated quartz bulb contained within the microwave cavity. The transition desired Is between the F"0, m^«0 and F«l, m^-0 energy levels. The atoms bounce around Inside the bowl for up to one second before leaving, which gives much longer interaction times than 1n the cesium beam. The 11new1dth 1s further decreased because the containing bulb dimensions, typically 0.15 m, are smaller than the wavelength of the radiation [Hellwlg, 1979; Wlneland, 1979]. However, collisions of the atoms with themselves, with foreign particles and with the bulb walls broaden the Unewldth again [Cheng et a l., 1980; Turner, 1978]. For maser action, the microwave cavity must be very accurately tuned, I.e ., have a large Q. I f Its resonance curve 1s not centered on the transition resonance curve (Figure 4), there will be conflict between the two curves, I . e . , cavity pulling with a resulting offset 1n frequency. Various methods of tuning the cavity automatically by altering Its effective dimensions have been Investigated [Morris and Nakag1r1, 1976; Hellwlg, 1974b; Hibbard, 1980]. Detection of the signal 1s by microwave antenna 1n the cavity; a crystal oscillator circuit 1s tuned to the received signal. This crystal 1n fact provides the actual outputs. In the passive hydrogen maser, the crystal frequency 1s multiplied up to the hydrogen resonance frequency and Injected Into the cavity, whence the servo-loop is completed just as 1n the cesium standard. In addition, the mlstunlng of the cavity can be detected and corrected as mentioned above. Stabilities approaching one part 1n TABLE 4 SOME RESULTS OF PRIMARY CESIUM BEAM EVALUATIONS (PARTS IN TO13)

NBS-6 1 NRC-V 2 NRC VIA, -B, --C3 PTB CS1 * Instability Bias Uncertainty Uncertainty Uncertainty Uncertainty (80 days)

Servo system amplifier 0 0.02 .020 .020 0.2 0.3 Servo system 2nd harmonic 0 0.15 .010 .010 Finite magnetic fie ld 536 0.03 .010 .010 0.4 0.5 Inhomogeneous magnetic 0.02 0.02 .001 .001 field

Majorana transitions 0 0.03 - - - -

Neighboring transitions 0.4 0.20 - - .010 .001

Cavity pulling 0 0.01 0.02 - .010 .001

Microwave impurities 0 0.02 -- .013 .010 2nd-order Doppler shift -3.1 0.10 0.2 0.2 .004 .001 Cavity phase shift 0.25 0.80 0.2 0.3 .050 .050

C-field reversal - - - 0.2 - -

Beam path -- -- .091 .031

Total error: RMS 0.85 0.53 0.71 .108 .064 Sum 1.38 1.02 1.5 .219 .135 Random 0.31 - - .065 .040

X[Wineland, 1977], 2[Mungall et a l., 1976], 3[Mungall et a l., 1980], ^[Becker, 1979] 49 lO1* over several days are being achieved, and an order of magnitude Improvement can be expected [Walls and Howe, 1980; White et a l., 1980; Vessot et a l., 1978; Vessot, 1979]. Long-term stab ility equalling that of evaluable cesium standards and working lives of many years can also be anticipated [Kartaschoff, 1978; Peters, 1978].

RECEIVE# MODULES

DEGAUSSING TERMINAL OVEN PLATE MICROWAVE COMPONENTS

VACUUM ENVELOPE RF CAVITY RF OUTPUT STORAGE BULB THERMAL OVEN MAGNETIC SHIELDS FOCUSSED ATOMIC HYDROGEN B E A M -

HEXAPOLE MAGNET

SOURCE COLLIMATOR

VACION PUMP HYDROGEN OISSOCIATOR

HYDROGEN ENTRY

FIGURE 6. Principal features of typical hydrogen maser. [Courtesy P. Clements, JPL] 50 3.2 Statistical Behavior

Despite the great care taken 1n constructing and, where possible, evaluating atomic frequency standards and clocks, they are not perfect. The fundamental physical causes of error were reviewed 1n Section 3.1, while Imperfections 1n mechanical construction and electronic circuitry add more noise. The natures of these errors are determined by compari­ sons between similar clocks—there 1s no more accurate accessible standard—so measurement errors can also Influence results. The errors that arise can, as usual, be classified as eyetematio and random. The systematic errors w ill henceforth be assumed constants 1n frequency or, equivalently, linear In time since there 1s no con­ clusive evidence to the contrary 1n atomic devices (quartz crystals need a quadratic systematic error model). The random errors, though generally correlated, are found to obey fa irly simple laws which are described 1n this section, and which are commonly referred to under the heading "frequency stab ility". A considerable literature now exists on this subject, starting with the work of Barnes and Allan [Allan, 1966] and reviewed, for example, 1n [Barnes, et a l., 1973; Allan et a l., 1974; Allan, 1974; Winkler, 1976, 1977; Barnes, 1978; Vanler and Tetu, 1978; Kartaschoff, 1978; Lesage and Audoln, 1979]. A summary 1s presented here, emphasizing certain aspects that are necessary as tools In characterizing, simulating and testing results later on.

3.2.1 Spectrum and Allan Variance

Let x (t) be the time error of a clock at time t , and le t yT(t) be the fractional frequency error determined over a time interval t from t. Clearly

yT(t) ■ [x{t + t ) - x(t)]/r. (3.2.1)

The power spectrum of fractional frequency fluctuations that occur 1n frequency standards has been found to have a simple form [Allan, 1966], namely: ^ S{f). r h f a a-d TABLE 5 CLOCK ttOISE PROCESSES

Spectra Allan Variance Modified Time AR1MA Process O Frequency Phase (N » 2, T - t ) Autocovariance Dispersion (P .d .q) Ux ( t )/ o*( t )U) * (a ) s«( f , tb) ”5 W c , <®x«t)>(^ h , 3f(| 2 t* White phase 2 M 1 -A r '. p ( 2 * ) i x* ~ r (0 .0 .0 ) hi h,[9/2-ln2 + 31n(afhx)] F lic k e r phase 1 2xl -A + ln ( t ) M (2»J ? ( 2 .) * i 2 White frequency n h i hjL h» t - t (0 ,1 ,0 ) (Random walk phase) ( 2 - ) ‘ f X h_i h*i 11> Flicker frequency -1 h_i 2 In 2 .*a f (2«) f ' o * -l 2-2 * • t Random walk Frequency h -i h.,(2.)2x x2 .9 - t 2 (White d rift) ( 2 » ) * f' 6 * T (0 ,2 ,0 )

(a) [Allan et at., 1974] (b) [Kartaschoff, 1978]

fj, Is high frequency cutoff: 2*fh t » 1 where f 1s'the Fourier frequency, not to be confused with clock frequency, and a 1s Integral [Kartaschoff, 1978]. A particular type of noise 1s Identified with each a; these are Identified In Table 5. The spectrum of the time errors x (t) corresponding to the spectrum of frequency fluctuations 1s denoted by S (f) and 1s: A Sx(f) » (1/2*0* Sy(f) 0 - (l/4»*) • E h . f° . (3.2.3) a=-4

The Allan variance Is an alternative means of characterizing the noise processes. The two-sample Allan variance of fractional frequency fluctuations y where each sample has been taken over an

Interval t 1s denoted by oy (2 , t ) or, more simply, by and has gained wide acceptance as a meaningful measure which 1s easy to calculate. I t 1s given by:

Oy(t) ■ <[y(tk + t ) - y(tk)]2/2> (3.2.4) where <> denotes an In fin ite time average over a ll sample times t k* y denotes an average frequency over Interval t , and there 1s no "dead time" between samples, I.e ., (3.2.5)

The characteristic Allan variance for each type of noise 1s given 1n Table 5. To compute an Allan variance from a fin ite but large number M of time errors, equation (3.2.1) 1s used 1n equation (3.2.4):

M*2 aj

Useful relationships can be derived between the Allan variance, the spectrum and the autocovariance function Ry(T) which Is defined [Kartaschoff, 1978]:

(3.2.7) T+~ 1 0 53 Then, m Sy(f) -4 / R ( t ) COS 2irfr dt (3.2.8) / 0 JT

00 U t) ■ / S (f) cos 2irfr df (3.2.9) Jr 0 /

00 Oy(x) 8 (2 /ttt) Jf $y(u/irr) sln^u/u2 du (3.2.10)

Also, Introducing a modified covariance function Ux ( t ) of the time fluctuations [Allan, 1966; Allan et al., 1974]:

Ux(t ) - <[x(t) - x(t+r)]2> ■ 2[Rx (0) - Rx(t)] (3.2.11) gives the relationship between Allan variance and covariance:

o2 ( t ) * ( l / 2 t 2) [4Ux (t ) - Ux (2t )]

- (1 /T a ) [3RX(0) - 4Rx ( t ) + Rx (2t )3 (3.2.12)

Formulae for U ( t ) for each type of clock noise are given 1n Table A 5.

3.2.2 AR1MA Representation

The description of clock noise in terms of autoregressive integrated moving average (ARIMA) processes [Box and Jenkins, 1976] has been explored at the U.S. Naval Observatory [Percival, 1978]. The basic Idea 1s that a correlated noise process

xk = x(tk), k ■ 1, 2, 3, ... (3.2.13) can be bu ilt up by adding uncorrelated white noise ("random shocks") ak at each step, together with a weighted sum of previous outcomes and previous random shocks. Thus: xk 8 ak + (

(3.2.15)

(3.2.16) which is known as a (p, d, q) process. Given a set of data, the appropriate value of d must be decided and the values of Sj calculated, as well as the variance of of the underlying white noise. a I t w ill be assumed that the mean of the process Is zero, which Implies that deterministic trends w ill have been Identified and removed prior to the application of statistical analyses. The nature of the process can mostly be deduced from examina­ tion of the autocorrelations and partial autocorrelations calculated from the data. For example. In an autoregressive process of order P (qB0):

rk = R(kx)/R(0) (3.2.16)

(3.2.17) which lead to the Yule-Walker equations:

which can be solved for the ^ when the r^ are known. The partial autocorrelation 1s the last coefficient of equation (3.2.17) when 1t 1s assumed that the process 1s of order j . I t 1s shown In [Box and Jenkins, 1976] that:

*jj " °* ^>p (3.2.19) where p Is the true orderj one calculates for j * 1, 2, ... until 1t is zero by an iterative method. The general ARIMA process gets computationally complicated as the orders p and q Increase. To ascertain the order of differencing d, calculate the autocorrelation coefficients rk for d » 1, 2, 3 ... u n til, for some d, only a fin ite number of the r k are non-zero, or until a recognizably exponential (possibly complex) decay pattern emerges. For clock noise processes, I t has been found that up to a (1, 2, 2) process Is generally adequate [Perclval, 1978]; Table 6 gives the equations for determining the parameters for such processes and Is the basis of Identification used 1n this study; o* Is the variance of the outcomes x^. The process parameters, once known, are used to make predic­ tions of future values from the present and past values. A partic­ ularly Important example Is the one-step-ahead forecast, which nay be denoted by 8k( l) which Implies 11 Is made with knowledge of the outcomes at t ** t k and previously. For the (2, 2, 2) process, 1t 1s;

*k(l) ■ t2+*i)xk - (l+24. 1-* 2 )x k _1 + (♦l -2*2)xk_2 + *2xk-3

-0^ - ®2®k—1 (3.2.20) where the a^ are 1n fact calculated as the one-step-ahead forecast errors:

a1 9 7*x1 " v2* i - i ^ ) » * * k» k-1 (3.2.21)

Equation (3.2.21) provides an alternative method to that given 1n Table 6 for calculating

3.2.3 Specific Processes

The characteristics of each of the five standard clock noise processes are described 1n the succeeding subsections, and some im­ portant results made explicit. Figure 7 shows the Allan variances, and Figure 8 the spectra, for these processes.

3.2.3.1 White Noise Phase Modulation. The errors In phase, I.e ., in time display, are random uncorrelated. Thus, i f r (k) is A the autocorrelation coefficient at lag k steps for the process x, TABLE 6 PARAMETERS OF AUTOREGRESSIVE INTEGRATED MOVING AVERAGE PROCESSES OF ORDER (p,d ,q ).

Parameters Process (p,d,q) ( i) (1.2.0) (0,2,1) (2,2,0) (0,2.2) (1.2,1)

♦l **i - ri(l-r2)/l-r?) - r 2/r i

4*2 - (r2-r?)/(l-r?) - -

g 1+0? - 1+0? +0^ l+O?- 2$i©

0i(from ri=) - 0i/g - - 0i ( l - 02)/g ( i - 4i 0i ) ( * i - 0i) / g

02(from r 2=) - - - 02/g - l-ri$ i- r2#2 aa^°x i/ g V g < i- * f) /g Behaviour r i Exp.decay Non-zero Exp.decay Non-zero Non-zero n r 2 " 0 Non-zero Exp.decay r k,k>2 0 n 0 ti

4>i i Non-zero Exp.decay Non-zero Exp.decay Non-zero

4*2 2 0 *1 Non-zero It Exp.decay «l 0 W tt 0

(a) Autocorrelation coefficients rk and process variances o£ are calculated after taking d-th difference of experimental values [Box and Jenkins,'Table 6.1]. 57

WHITE FLICKER WHITE FLICKER RAND. WALK PHASE PHASE FREQ FREQ FREO Lag ( Sample tlma T )

FIGURE 7. Log-1og graph of square root of two-sample Allan variance oy(T) vs. sample time t for typical atomic clock noise. Units are arbitrary.

RAND WALK ' FLICKER FREO Lag(Fou/lar fraquMcy f )

FIGURE 8. Log-log graph of frequency spectrum Sy(f) vs. Fourier frequency f fo r typical atomic clock noise. Units are arbitrary. 1n this case the phase or time x (t):

(3.2.22)

I t w ill be found useful for diagnostic purposes to realize that the autocorrelations of the firs t and second differences of white phase noise, which correspond to the rate and drift respectively, are:

k=0 r7x(k) k-1 (3.2.23) k>l 1 k=0 -2/3 k=l (3.2.24) rv2x(k) 1/6 k=2 0 k>2

The log-loa graph of the square root of Allan variance has a slope of -1 , which demonstrates that for this type of noise the error of a frequency calibration decreases linearly with the duration of the calibration. To see this directly, suppose that the frequency estl- A mate yhas been obtained by the least squares (L.S.) method over an odd number n = 2m+l of points equally spaced at apart:

t = (n-1) at (3.2.25)

Then: y_ a EtJXtJ - x] / • E (tr t): (3.2.26) 1 1-11 1 1=1 1

Now: n £ (t4-t)2 = (at)2 (n-l)n(n+l)/12 (3.2.27) 1=1 1 and, with n' (n+1 )/2:

n m st,[x(tf) - x] = at £ [j x(t_,+i) - (m-j+1) x jtj] 1=r 1 j=l n J J (3.2.28) Hence, 12 £ [j x(t ,.,) - (m-j+1) x (tj] j= l n J ll (3.2.29) at (n-l)n(n+l) Since x (t) 1s a white noise process, there 1s no correlation be­ tween x (tk) and x (t^ ), k f 1. Hence,

ECo^s. - 12 oJ/[{At)2.(n-l)n(n+l)] (3.2.30)

I.e., E[a|]L>s< - [12 (n-l)/n(n+l)] [oj/x2] . (3.2.31)

Thus, the error decreases linearly with x; i t also decreases as the number of data points Increases within the interval x. I f the frequency had been determined from the two end-point (E.P.) measurements, there would follow:

y - Cx(tn) - x t tjt t/x (3.2.32)

from which 1t Is readily calculated that

E[o|]E>p. « 2 oj/x2 . (3.2.33)

Again, the error decreases linearly with x. The ratio of the least squares to the end-po1nt methods 1s:

^ p L . S . 1 ^ E .P . “ ®(n-l)/n(n+l) (3.2.34)

so that, provided more than three points have been used In the least squares calculation, i t gives a superior estimate of the frequency than does the end-po1nt method. White noise phase modulation 1s an ARIMA (0, 0, 0) process. From equation (3.2.14):

xk ° ak (3.2.35) hence, oj = o2 (3.2.26) k a so that the error (time dispersion) 1s constant as k Increases, I.e ., as the process proceeds. Finally, the phase spectrum S (f) of this process 1s constant while the frequency spectrum s^(f) Increases quadratlcally with f , so that the noise power 1s greatest at very short periods. 60 3.2.3.2 White Noise Frequency Modulation. The errors In fre­ quency, I.e ., In rate, are random uncorrelated. Thus, i f r Vx(k) 1s the autocorrelation coefficient at lag k steps for the fir s t d iffe r­ ences of process x (t), then

(3.2.37)

The autocorrelation coefficients r^ 2x(k) of the second difference of the process are:

(3.2.3B)

The log-log graph of the square root of Allan variance has a slope of - i , which demonstrates that, for this type of noise, the variance of a frequency calibration decreases linearly with the cali­ bration's duration. By calculations similar to those 1n the previous subsection, but rather more complicated because the least squares estimated rate must be expressed 1n terms of the phase differences of the process 1n order to sum variances of uncorrelated variables, 1t is found that:

ECc^ l . s. " C(6 At)(n2+l)/5n(n+l)][<^/t] (3.2.39) where 1s the variance of the uncorrelated normalized variables:

(3.2.40)

Thus, for constant data spacing At, the error decreases as the square root of the total calibration Interval t , especially 1f n 1s large. Again, the end-po1nt calibration Is

y * Cx(tn) - x(tx)]/T (3.2.41) n * E bi/(n-l) j»2 J Therefore,

E[o|]^ ■ AtCo^/x] . (3.2.42) The ra tio o f the least squares to the end-po1nt calculation 1s:

ECoF l .S./ECo) ] E.P. “ 6(n2+U/5n(n+l) (3.2.43) which tends to the value 6/5 as n Increases Indefinitely, I.e ., as the calibration interval Increases (since At 1s here held fixed). Thus, the end-point method gives a superior estimate of the rate than does the least squares method when the noise process Is white modulation of frequency. This type of noise Is an ARIMA (0, 1, 0) process. From equation (3.2.16), 1t 1s seen that:

xk a xk-1 + ak . (3.2.44)

To describe the time dispersion, suppose without loss of generality that x0 B 0. Then:

Xi a x0 + ai = ai x2 « Xi + a2 = ax + a2 : k x,, “ E a. (3.2.45) K j . j J

Hence, since the a^ are random uncorrelated:

E[c* ] B kal (3.2.46) k a

With an obvious change of notation using the fact that t = kAt:

E[aJ{t)] = t [oJ/At] (3.2.47) which shows that the process tends to grow according to the square root of elapsed time. I t 1s noteworthy that this process Is also named natidom mZk o

r y 2x(k) 1s the autocorrelation coefficient at lag k steps of the second difference of the process x (t),

Vx 0 (3.2.48)

The log-1og graph of the square root of Allan variance has a slope of + i, which demonstrates that, for this type of noise, the error of a frequency calibration Increases with the duration of the calibration, 1n contrast to the two processes previously con­ sidered. This occurs simply because the random walk 1s a divergent process. Random walk frequency modulation Is an ARIMA (0, 2,"0) pro­ cess. From equation (3.2.16) I t Is seen that:

*k " 2xk-l - xk-2 + ak <3'2-49> or vxk ■ vx^ + ak ( 3 .2 .50 )

The optimum predictor for the next rate 1s thus just the last rate obtained. To describe the time dispersion, I.e ., the growth of x(tk), suppose without loss of generality that x<> ■ x_i. ■ 0. Then

Xi » 2xo - xi + ai a ai x2 * 2xi - x0i + a2 = 2ax + a2

xs * 2x2 - Xi + a3 = 3ai + 2a2 + a3 •

xk * k a2 + (k-1) a2 + (k-2) a3 +...+ 2ak-1 + ak (3.2.51) and, because the a^ are random uncorrelated k E[o* ] - • Z I2 aJ - [k(k+l)(2k+l)/6] a2 •(3.2.51) xk is l a a which tends to (k3/3)o2 as k Increases indefinitely. With the 63

change 1n notation due to t * k at:

E[oJ( t ) ] * t3[o*/3(At)s] (3.2.52) which shows that the process tends to grow according to the 1.5th power of elapsed time. The frequency spectrum Sy(f) of random walk frequency modu­ lation decreases quadratlcally with Fourier frequency, while the phase spectrum S (f) decreases as the fourth power. Hence the A noise power 1s concentrated at very long periods.

3.2.3.4 Flicker Noise Phase Modulation. The errors 1n phase are correlated to a certain degree. From simulations using an adaptation of the method of Barnes ahd Jarvis [1971] as explain­ ed 1n Section 4.1.1 1t has been found empirically that the auto­ correlation coefficients r (k) are approximately: A rx(k) =(0.75)(.94)t k > 0 (3.2.53) while the autocorrelations of the first and second differences are -0.42 k = 1 ■WO ■ I 0 n'42 kI > 1 (3.2.54) -0.64 k » 1 V* 0.14 k « 2 0 k > 2

The log-1og graph of the square root of Allan variance has a- slope of nearly - l i from Table 5 i t can be seen that the re­ lationship has the form:

ln[

< o^(t) > ~ (constant + In t ) . (3.2.57)

From the autocorrelation values given In equation (3.2.54), It may be inferred that the ARIMA representation of the flicker of phase process 1s (0, 1, 1) with 9i » 0.55. ARIMA simulations with these parameters are discussed 1n Section 4.1.1. The frequency spectrum Sy(f) of flicker noise of phase modu­ lation Increases linearly with f so that the frequency noise power predominates over short durations,'whereas the phase noise power predominates at longer periods since the phase spectrum Sx( f ) . varies Inversely as f.

3.2.3.5 Flicker Noise Frequency Modulation. The errors In frequency are highly correlated. From simulation using the method of Barnes and Jarvis [1971] as explained 1n Section 4.1.1 i t has been found empirically that the autocorrelations In phase and Its firs t two differences are:

rx(k) - {.} I ; I (3.2.58)

r7x “ i(0.75)(0.94)k k > 0 (3.2.59)

1 k « 0 ■0.42 k = 1 (3.2.60) v * (k> 0 k > 1

The sim ilarity In the values to those obtained 1n subsection 3.2.3.4 for flicker phase modulation autocorrelations at lower differences Is a consequence of using the same flicker noise generator in both cases and summing ("Integrating") successive values to obtain phase when the flicker is of frequency modulation. The log-log graph of the square root of Allan variance for flic k e r noise frequency modulation has a theoretical slope of zero; the simulations gave a slightly negative slope on average, due

mainly to a fluctuation at the high frequency (small t ) end. The precision of a frequency calibration will therefore not improve as the calibration Interval Increases when this type of noise 1s operating. Attempts to derive the time dispersion characteristics high­ ligh t the well-known analytical d iffic u ltie s associated with flic k ­ er noise [F e ll, 1980; Wu, 1980]. The auto-correlations found 1n equation (3.2.60) would suggest that the ARIMA representation 1s (0, 2, 1) with 9i ■ 0.55. Equation (3.2.16) would then be written:

xk ■ 2xk -l - Xk-E - V k - 1 + ak <3-2-611

which, by a procedure similar to that leading to equation (3.2.51) gives for the time dispersion:

E[aJ ] « [k3(l+%)2/3 + k2(l-eJ)/2 + k(l-4^+02)/6] o2 (3.2.62) k

which has the same characteristics as the time dispersion for ran- don walk of frequency (unless 9 B *1 in which case a white frequen­ cy noise process results.) The desired dispersion characteristic 1s, from Table 5:

E[o5 ] - k2 (3.2.63) k which cannot be achieved by an economical ARIMA representation. Results of time dispersions and Allan variances for several seem­ ingly promising ARIMA representations of flicker of frequency are shown in Table 7. Some of these are plotted in Figure 9 which shows that the ARIMA representations are not successful. Both the phase spectrum Sx(f) and the frequency spectrum Sx(f) decrease as Fourier frequency f increases. Hence the noise power of each predominates at low frequencies, i. e . , over long time intervals. 66 TABLE 7 ARIMA REPRESENTATION OF FLICKER FREQUENCY MODULATION

Representation B(T1me Dispersion) v{Allan Variance) (P*d»q) Ccx(t ) * tBj toy (t) * t Yj

Theoretical 1.0 0.0

Barnes & Jarvis Simulation 0.96 -0.04

(1.2.1) .035 0 i“ .61 1.30 0.34 (0,2,1) » 0.0 0i" .55 1.42 0.38 (1,2,1) « .15 0 i“ .85 1.21 0.19 (1.2.1) * .15 0 i“ .90 1.14 0,11 (1,2,1) ■ .15 0i" .95 1.03 -0.03 (1,2.1) 4>x ■ .15 0i" .98 0.90 -0.21 67

ARIMA ( 0 ,2 , I ) 0 i 0 ‘S0 10

O I T ) BARNES a JARVIS y

ARIMA (1 ,2 ,1 ) 0,*O*I5 ^*0*B5

i - ARIMA (1,2,1) 0*0*15 V*0*98

i j o Too ' i o o o t FIGURE 9. Allan variances for attempted ARIMA simulations of flic k e r frequency modulation.

* 4. STATISTICAL PROCEDURES

4.1 Simulation Tests

4.1.1 Simulations

The simulations performed 1n this study are based on a random number generator written for 16-bit word computers by R. Brent of The Australian National University, Canberra. I t generates uniformly distributed random numbers on the Interval (0, 1) and does not repeat for at least two million numbers. It 1s not known 1f correlations exist 1n 1t, but results obtained give confidence that any correlations are minor. When necessary, normally distributed random numbers aL are derived u from these uniformly dlstrlubted numbers a^ by 12 a!J - z(ai1 - 0.5) (4.1.1) K 1-1 1 so that

ajj * N (0,1).

The three white noise processes were simulated by the following variants of equation (3.2.16):

xk ■ ak (a-2; xo“0) (4.1.2)

xk " xk-l + ak (“B°; x°“°) (4.1.3) xk “ 2*k_i “ xk-2 + ak (“"“Zixo-x^-O) (4.1.4) while the flicker noise processes used the recursive generator of Barnes and Oarvls [1971] with the uniformly dlstrlubted random numbers a*j* as Input. ARIMA processes were simulated using equation (3.2.16).

6 8 69

To gain Insight Into the nature of the processes, and to tdst the In teg rity of the simulators, fiv e sets of simulations were performed. Each set was of one of the basic processes (a«-2, -1 , 0 , 1, 2 ), and con­ tained 50 clocks extending to 4096 points each (k ■ 1, 2, . . . , 4096). Initial rates and offsets were zero. If xk(1) denotes the outcome at time t ■ t k for clock 1, the following quantities were calculated:

Ensemble average: 50 - Xi, a E x,,(1)/50, k ■ 1, 2, 4, 8, 16, ..., 4096 k 1=1 K Root sum square: 50 . s., ■ [ 2 x'?{1)/50] , k = 1, 2, 4, 8, 16 ...... 4096 K 1=1 * Mean Allan variance: 50 o ( t ) = E o (t ; 1)/50, t 6 1, 2, 4, 8, 16...... 512 y 1=1■ -

Standard error of ov (t ):

50 , S[o ( t ) ] = [ Z (a (t;1) - o

50 rx (k) =• E r x (k;1)/50, k = 1, 2...... 5

50 7oy(k) - E r (k;1)/50, k = 1, 2, 5 v x VX

50 r72x(k) = ■ £ r72x(k;1)/50, k = 1, 2, 5

Standard errors of

50 , S[r,(k)] ■ [ E (r (k;1) - r (k))2/49]*, k = 1, 2, .... 5; 1**1 ** 4 I - X, vx, Dispersion slope 6: S = {In (sk)/ ln(tk)> by least squares over tk; 70 Allan variance slope y: Y ■ d n (o ( t ) ) / 1n(x)> by least squares over x. J

The results are summarized In Table 8. The units 1n these simulations were arbitrary, so no real significance 1s to be attached to the magni­ tudes of the results (except for the autocorrelations and slopes). Figure 10 shows the root-sum-square (R.S.S.) of the outcomes as each process proceeds, where the R.S.S. 1s taken over the ensemble of 50 clocks. (It 1s tacitly assumed that the processes are ergodlc, I.e ., that time-evolutionary characteristics can be obtained by ensemble aver aging.) The R.S.S. Is an appropriate measure of time dispersion since the process means are zero by hypothesis In this case. The simulations are evidently very satisfactory. Figure 11 shows the ensemble-averaged square roots of Allan variances. Again, the simulations provide the results desired. The standard errors of square roots of Allan variances (note: not of Allan variances) reveal Interesting patterns which depend on x. There has been discussion recently on the computation methods for, and significance of, such estimates [Yoshlmura, 1978; Lesage and Audoln, 1979]. They are affected by correlations between samples 1n the esti­ mates, and by the number of samples, so the values given In Table 8 are not necessarily rigorous. The same comments apply to the standard errors quoted for the autocorrelation coefficients. Figure 12 shows the outcomes of some of the clocks simulated as flicker of frequency, up to the 512th time point, and Figure 13 extends the graphs of some of these up to the 4096th point at reduced scale. Figure 14 shows square roots of Allan variances for some Individual clocks, giving an appreciation of how they vary. Figures 15 to 18 show some results of ARIMA simulations, with straight-11ne trends removed so as to emphasize graphically the nature of their fluctuations. TABLE 8 RESULTS OF SIMULATIONS OF 50-CLOCK EHSCMLES (Units a rt arbitrary)

a 2 1 0 -1 •2 Process White Phase F licke r Phase White Freq. Flicker Frcq. R.W. Freq.

Enseabl* A n . k-1 -.136 2.96 -.389 1.97 .001 .142 .000 .003 .000 .003 (xk) and Root 2 -.784 2.86 -.975 2.09 -.035 .187 -.001 .006 -.001 .006 Sul Square .301 2.18 -.940 2.73 -.067 .272 -.002 .010 -.003 .014 a .488 2.98 -.560 2.51 -.092 .369 -.006 .019 -.010 .036 1(SJ k 16 -.496 2.95 -.564 2.44 -.014 .506 -.005 .038 -.020 .096 32 -.508 2.86 -.385 2.7B -.015 .807 -.004 .082 -.024 .286 64 -.798 3.14 -.031 2.56 .020 .857 -.003 .127 -.031 .701 12B .430 2.70 -.232 2.65 -.030 1.42 -.014 .293 -.096 2.086 256 -.026 2.55 -.829 2.91 .141 1.73 .012 .507 -.050 5.262 512 -.142 2.92 .336 2.89 .313 2.76 .053 1.010 .513 13.96 1024 -.118 2.89 -.221 3.07 -.810 4.28 -.198 2.323 -.835 44.26 204B .017 2.82 .352 3.24 .456 7.00 .038 5.037 -2.277 124,6 4096 .167 2.97 .559 3.33 .933 9.62 .277 10.01 6.019 224.7 *(sfc* tk") .005 .052 .503 .965 1.405 Theoretical t 0.0 *0 0.5 1.0 1.5 (*100) (*100) (*100) (xtOO) Allan T«1 4.99 .049 2.82 .029 .144 .0011 .298 .0022 .204 .002 Varfanca 2 2.50 .031 1.50 .019 .102 .0012 .240 .002B .251 .004 (5„{t>) and 4 1.25 .013 .622 .010 ,0723 .0014 .221 .0046 .340 .008 ItX -standard 8 .624 .0088 .458 .0079 .0513 .0014 .223 .0072 .476 .016 arror 16 .313 .0032 .251 .0042 .0361 .0013 .223 .0090 .572 .031 (S[0y (T )]) 32 .156 .0020 .133 .0029 .0257 .0012 .221 .012 .954 .063 64 .0783 .0010 .0700 .0020 ,0182 .0014 .220 .022 1.35 .12 128 .0390 .0005 .0369 .0017 .0127 .0017 .219 .029 1.92 .21 256 .0196 .0003 .0196 .0011 .0088 .0011 .211 .028 2,77 .51 512 .0098 .0001 .0099 .0005 .0061 .0013 .209 .057 4.06 1.3 y(5v(t) ' tv) -.9997 -.90 -.5056 -.0364 .4886 Theoretical y •1.0 -1.0 -0.5 0.0 0.5 Autocorre- k-1 .0035 .015 .694 .033 .998 .0011 .999 .0003 .999 .0002 la tlo n 2 -.0017 .017 .647 .038 .996 .0021 .999 .0006 .999 .0005 o f phase 3 .0016 .013 .609 .041 .995 .0032 .999 .0009 .999 .0007 (?*(k)) and 4 .0002 .012 .575 .045 .993 .0043 .998 .0012 .999 .0010 Its standard 5 .0008 .016 .545 .048 .992 .0054 .998 .0015 .998 .0012 arro r

Autocorre- k - l -.665 .008 -.643 .010 -.498 .011 -.423 .012 .003 .015 la tlo n of 2 .162 .017 .140 .021 -.002 .022 -.013 .021 .000 .017 drift (P-.Jk)) 3 .003 .017 .002 .022 .000 .020 -.009 .019 .001 .017 and it s " 4 -.001 .017 .002 .023 .002 . 018 -.006 .017 .002 .014 standard arror 5 .000 .020 -.003 .023 -.002 .016 -.009 .016 -.001 .012 (S[r-T, xlk ) ]) • t of flicker .00001 .00163 . filte re d Dispersion y of flIcker • * •1.011 -1.011 filtered Allan var.

F licke r f i l ­ k-1 -.170 .014 -.169 .013 tered auto­ 2 -.088 .017 -.086 .017 correlation 3 -.047 .016 -.049 .017 o f phase, and 4 -.030 .015 -.031 .016 standard error 5 -.027 .017 -.027 .014 I U E 0 Tm dseso (... f ucms fo smltd 0cok nebe, omlzd to ensembles, normalized 50-clock outcomes) simulated from of (R.S.S. dispersion Time FIGURE 10. 001

0-1 RSS £x(t)] D O - JVH PH^ JVH - O D H R L FR FL FR^ WH y a t49, hwn soe o lg1g graph. log-1og of showing Y slope t=4096, at ys 5 FL PHFL T X >0 - O 8 9 - 0 100 ^ 8 0 - 0 W FR RW f S 1000 j t

73

1-00

090' WH PH

0 04

0-1

BW BB

10 100 T

FIGURE H . Allan variances from simulated 50-clock ensembles, showing slope 8 of log-log graph. 74

SIMULATED CLOCKS - BARNES & JARVIS;^

sc u 0 _j u 1 X u. u oc

III TIME FIGURE 12. Evolution of selected clocks simulated with flic k e r of frequency modulation - short term. Scales are arbitrary.

4* SIMULATED CLOCKS BARNES & JARVIS MODEL OF FLICKER OF FRM. * scout Mtltmv. . —

oSC o _J o

u. t il Q£

* TIME FIGURE 13. Evolution of selected clocks simulated with flic k e r of frequency modulation - long-term. Scales are arbitrary. I U E 4 Aln aine o slce cok smltd ih lcker f rqec modulation. frequency of r e k flic with simulated clocks selected of variances Allan FIGURE 14.

LOG SIGMPI (2,TRU) -3 -E -6 0 LA VARIANCES ALLAN I 2 O TAULOC 3 CL fk TRARY. fikW SCRLL F LCE O FREQUENCY OF FLICKER -MODELOF Sc JRRVIS BARNES •4 S «

r 76

ft# SIMULATED CLOCK NOISE PROCESSES (RESIDUALS) I...... ?

<1

•it

i l l M l

FIGURE 15. Evolution of some simulated ARIMA processes, straight line trends removed. Scales are arbitrary.

I ALLAN .. VARIANCES (R.M LOG-LOG)

4

1

2 4 f9 4 f I

FIGURE 16. Allan variances of some simulated ARIMA processes. SIMULATED* CLOCK NOISE PROCESSES (RESID'S)

t ® 12 , 31 ' '. ' - V..' 1

■

FIGURE 17. Evolution of some more simulated ARIMA processes, straight line trends removed. Scales are arbitrary.

i i ALLAN VARIANCES CR .11 .S . ,. LOC-LOC)

4

9

3 3 4 t f I

FIGURE 18. Allan variances of some more simulated ARIMA processes. 4.1.2 Predictors

I t w ill be seen 1n Chapter 5 that a necessary part of con­

structing a time scale is the availability of a predictor 2 ^(t) for the value of (UTC - x^) at the computation time t (see Section 5.2.5). It 1s most desirable that the prediction errors e^t) con­ stitute a zero mean white noise process for each clock 1

e^t) = zi (t) - ^ (t) (4.1.5)

for then the basic time scale equation (5.2.16) 1s Imposed with minimum error. To conform to the notation of Section 3.2.2, le t xR ■ Zj(tk) (4.1.6)

*k " *k-l*1J ° M V (4.1.7)

akaxk “ V e1(V (4.1*8)

As usual, 1t w ill be assumed that deterministic trends have been removed. I t 1s easy to show [Box and Jenkins, 1976] that the mini­ mum variance unbiased one-step-ahead predictors for non-fI1cker noise processes are, for the Indicated processes:

White phase (a = 2): xk+1 = 0 (4.1.9)

White frequency (a = 0): xk+1 = xk (4.1.10)

Random walk frequency (a ° -2): xk+1 = 2xk - xk-1 (4.1.11)

ARIMA (1, 2, 2): xk+1 3 (2+$i-0i) xk - (l+2$i+e*) xk-

+

yk a [yk + m yk-1]/(nH-l) (4.1.13)

where yk 1n this context 1s vxk, and m » 0.6. This equation 1s essen­ tially a recursive filte r rather than a predictor, yet an approximate predictor can be obtained from 1t by assuming that

E[yk+i - W " 0 (4.1.14) from which It transpires that

* k+l “ ^k + F(xk) “ F(xk-1) (4.1.15) where

F(xr ) - [xk + m F(xk_1)3/(«H-l)i F(*0) a 0 (4.1.16) Similarly, an approximate predictor 1n the case of flicker noise of phase modulation may be derived:

xk+1 - F(xk) (4.1.17) These predictors were tested on the simulated data sets summarized 1n Table 8, by calculating the Allan variances and auotcorrelatlon coeffi­ cients of the prediction errors ak as 1n equation (4.1.8). The results are presented as the final three sets 1n Table 8. The slopes of the log-log graphs of root-sum-square time dispersion (e) and square root of Allan variance (y) are completely consistent with white noise, while the .autocorrelations of phase, though not a ll zero, are considerably smaller than their unfiltered counterparts and Indicate a residual ARIMA

(1, 0, 1) process with 9i » 0.16, 4^ " 0.52 which might be worth Investigating further,

4.2 Unbiassed Variance Estimation

4.2.1 Individual Clock Error Variance Estimates

So far, 1t has been tacitly assumed that clock error processes are characterized with respect to an Ideal or errorless clock. In practice, of course, 1t 1s generallly only possible to acquire the necessary data by comparisons between two or more Imperfect clocks. 80 If only two similar clocks are available there 1s only one compari­ son possible so that, 1n the absence of any other external Information, 1t Is necessary to assume that the error processes 1n each are statist­

ically Identical. Thus, 1f o £(t ;1) I s the Allan variance of clock 1,

and o *(t ; 1, J) 1s the Allan variance computed from (perfect) comparisons between clock 1 and clock j , there results:

oJ(t ! 1) - aJ(T; j) = 1, J) (4.2.1) and similarly for covariances. When three Independent clocks are available, three sets of compari­ sons are possible. Writing o jj for the measured variance (or covariance or Allan variance) and correspondingly a^ for the Individual clock, 1t 1s clear that:

° h " °? + °3 (1 » J) * 2 )' (2 ’ 3 )»(3» (4*2*2) This set of three equations can be solved for the o^:

a\ - Jj[o^ + o|i - ojk], 1 » 1, 2, 3; (4.2.3)

This solution has come to be known as the three-cornered hat method [Perclval, 1973]. An extension to take account of measurement errors o^(m) 1s simply [Allan et a l., 1974]:

ah " + + (4.2.04) Therefore,

°? “ + °^1 “ °3k^ “ + °lc1^ “ °3k^m^ (4.2.5) or, simplifying 1n case of equal measurement errors,

+ aj^ - - o2(m)] (4.2.6)

I t has been found that the application of this technique using real data sometimes yields negative variances o| [Perclval, 1973]. The fallacy of the three-cornered hat suite of equations evidently resides 1n the use of Imprecise estimates of the measured variance and in the d iffic u lty 1n estimating the measurement errors o?j(m) which are probably correlated among themselves, especially when the same time Interval counter 1s used in rapid succession to make the three measurements. The matter 1s discussed further 1n Section 4.2 If the Individual clock errors and the measurements are all truly Independent of each other, the measuring error a2(m) can be estimated from the closing errors, Let be an actual measure- ment, and m^ be Its error. Then

£1 j “ X1 " xj + m1J (4.2.7) and the closing error c 1s the sum of these, taken in order:

c * * U - + *Jk + *k1. <4-2-8) Therefore,

c ' m1j + mjk + mk1.

Hence, since m ^, mjk, m^ are assumed mutually independent,

o2(m) » o2(c)/3 (4.2.9) and this value Is then substituted in equation (4.2.6) above. I f the estimation of o2(c) 1s Inadequate, there 1s an excellent chance that negative clock variances w ill result. Now suppose there are p clocks, p > 3. I t 1s possible to make p (p -I)/2 comparisons among them 1n which case the system becomes overdetermlned and a least squares approach may be employed. The model equations when p = 4, for example, are (sim ilarly to equation (4 .2.2)):

°12 " l 1 0 o’ 1 0 1 0 ’o f _2 Oil, 1 0 • 0 1 oi s (4.2.10) °£s 0 1 I 0 oi o|„ 0 1 0 1 o* 2 Ojl,_ 0 0 1 1_ or, In condensed matrix notation 1n the general case, 82

P(P-1)/2L1 “ P(P-1)/2AP*PX1 (4.2.11) 4 whose (unweighted) solution Is, by the standard method:

X = (A^)"1 ATL . (4.2.12)

It 1s fairly easy to show that this solution 1s, for p = 4,

oj ■ C2(ofj + ofk + 0}e) -

where 1 * 1, 2, 3, 4 In succession, and 1, j , k, i are all different. Note that the determinant of A^A 1s 48 for p “ 4; i t 1s 648 for p = T -1 5; so (A A) does 1n fact exist. The measuring error o2(m), 1f known, should be subtracted from the raw estimates of each before processing. Since the system 1s overdetermined, i t 1s tempting to deter­ mine the measuring error as an extra parameter and write In place of equation (4.2.10):

0?2 "l 1 0 0 1 al _ 2 Oil I 0 1 0 1 02

Oil, S 1 0 0 1 I oi (4.2.11)

ois 0 1 1 0 1 o5

o|«, 0 1 0 1 1 o*(m)

0 0 1 1 1 I but the determinant of A A In such cases 1s zero. Hence the mea­ suring error must be determined separately.

4.2.2 Unbiassed Variance Estimates in Time Scales

The three-cornered hat method described above, and Its exten­ sions, are examples of the estimation of the error characteristics of individual clocks when only data from clock combinations are available. A similar situation exists In the computation of atomic time scales. In Section 5.4.4, the solution for a free-running time scale Xv, sometimes called UTC, Is derived. Combining equations (5.4.31) and (5.4.32) yields the essential output z^ of the system

1 * pj [V * j ] / J 1 pj ' xi 1,1 n <4-2-,2>

where 1s the weight given to clock 1 1n the algorithm. The variances, covariances and Allan variances of the several z^ can estimated directly 1n the usual manner, but the weighting and pre­ diction formulae generally require the characteristics of each clock Xj Individually to be known, so these must be derived. Suppose the weights p^ have been normalized. Then from equation (4.2.12)

Z1 * ^ pj h * jJ, PJ * i ' 1 * 1...... n (4.2.13)

To c la rify the exposition, i t w ill be considered that the predictors z are perfectly known, and that the clocks are truly Independent. Let the variance (or covariance) of z^ be denoted o|^; this Is an accessible or "measured" quantity. Propagating variances 1n equa­ tion (4.2.13) gives, assuming that the clocks are Independent,

(4.2.14)

or, 1n matrix form

- 1 -o|- ' ° z r - ( l - P i ) * pi ( 1-P 2) o| °z2 a pi (4.2.15) • • • * • • • ------— i i Q » 1 M znM * 1 ( - Pi Pi Lon J i . e . , S * B S (4.2.16) z x

1n which the matrix of weight coefficients B 1s square (cf. matrix A in equation (4.2.11) The solution 1s simply: 84

B' (4.2.17) x z Yoshlmura [1980] specifies the inverse when the weights are equal:

1/n, i « 1, n (4.2.18)

(n2-n-l) -1 -1 -1 (n2-n-l) -1 B' x l/(n-l)(n-2) « -1 -1 ... (n2-n-l) (4.2.19) and makes the point that oj -► o2^ as n » provided that the weights and variances are approximately equal. By summing over 1 in equation (4.2.12), one obtains n z 2 Pi (Xi - X.) (4.2.20) 1=1 1=1 1 1 1 which,with the assumption that the predictors are perfect, and that the clocks are truly independent of each other, gives n • £ p2 o2 1 (4.2.21) 1=1 1

This enables the resulting time scale Itself to be statistically characterized. This w ill be a very useful technique for testing various algorithms against each other; the variance relationships for the "best" algorithm should Indicate minimum variance white phase noise. A test of this procedure was performed, 1n which a set of five clocks was simulated using the techniques described 1n Sec­ tion 4.1.1. The simulations provided the raw Individual clock data Xj, 1 = 1 , ..., 5 for 4096 time points, from which the vari­ ances, covariances, Allan variances and cross-correlations between clocks were calculated. The procedure should recover these values. A time scale X was then constructed simply, by putting 5 5 I Pj Xj £ p, = 1 (4.2.22) 1 = 1 1 1 1=1 1

The outputs z.j = X - x^ (cf. equation (4.2.12)) were generated at each time point, and used to calculate "observed" variances, e tc ., oz2r Equation (4 . 2 . 1 7 )was then solved, and the results compared against the originals. Table 9 sumnarizes the results. I t was found that white phase noise simulations gave excel­ lent recovery, whereas white frequency noise variances were not recovered at a ll w ell. The reason for this 1s related to the actual Independence of the clock values x^. The clocks simula­ ted by white phase noise were, 1n fact, Independent as shown by the cross-correlation coefficients (which gives further confidence In the Integrity of the random number generator). The set of clocks having white frequency noise was b u ilt up using the same sets of random numbers as 1n the white phase noise case, yet the cross-correlations between clocks were fa r from zero. The effect can be seen In F1g. 1 9 ,.and Is due to the random walk nature of the process resulting 1n a finite probability that Independent processes will appear to track together (or 1n opposition) for a certain length of time. Although the effect noted here 1s a property of the simulations and could undoubtedly be reduced by appropriate choice of seeds 1n the random number generator, 1t does highlight behavior that 1s often observed 1n real clocks. Such behavior should not be Interpreted as Indicating genuine cor­ relation between clocks until after the appropriate filte r has been applied to reduce the process to white noise. Note that the Allan variances were recovered w ell, because the filte r appropriate 1n this case (differencing) 1s applied as a matter of course during their computation. TABLE 9 TEST OF UNBIASSED COVARIANCE AND ALLAN VARIANCE PROCEDURE

Lag CLOCK (1) 1 2 3 4 5 O rig 'l Recov'd O rig 'l Recov'd O rig 'l Recov'd O rig 'l Recov'd O rig 'l Recov'd

Generating o ,M ) 1.0 2.0 1 .5 2.0 1 .0 Weight pi 0 .3 0.1 0 .2 0.1 0 .3 Random No. Seed 6132 8055 1173 7685 6789 WHITE PHASE NOISE Cross Correlations: Clock 1 vs. Clock 1 • -0.004 -0.0158 0.0178 0.0257 2 -- -0.0072 0.0155 -0.0018 3 --- -0.0130 0.0082 4 ---- -0.0151 UTC vs. Clock 1 0.5251 0.3355 0.4873 0.3321 0.5212 Variance 0 1.028 1.007 3.884 3.889 2.171 2.209 3.858 3.864 1.010 .987 Autocorrelation 1 -.017 -.037 -.002 -.003 -.001 -.016 -.001 -.015 .001 .023 2 .001 .010 .029 .031 -.0 1 2 .004 -.0 0 3 -.0 0 3 .019 .011 3 -.006 -.019 .003 .004 .008 .021 .007 -.0 0 4 -.006 -.020 4 -.035 -.042 -.024 -.011 -.014 -.012 .012 .016 .019 .027 5 -.031 -.030 .010 .014 .040 .051 .005 -.008 .003 -.016 Allan Variance 1 1.78 1.78 3.43 3.44 2.55 2.60 3.45 3.44 1.74 1.70 (square root) 2 .872 .856 1.67 1.67 1.28 1.28 1.71 1.71 .862 .857 4 .450 .445 .867 .862 .645 .652 .843 .84- .432 .425 8 .217 .219 .426 .425 .315 .315 .427 .430 .214 .210 16 .111 .110 .212 .212 .160 .161 .212 .210 .108 .105 32 .054 .054 .107 .107 .080 .081 .105 .106 .054 .054 64 .028 .028 .054 .054 .040 .040 .053 .053 .027 .027 128 .014 .014 .026 .026 .020 .020 .027 .027 .014 .013

Continued TABLE 9 (Continued)

Lag CLOCK (1 ) j g - 4 l Orig'l Recov'd Orig'l Recov'd Orig’l Recov'd Orig'l Recov'd Orig'l Recov'd

WHITE FREQUENCY Cross Correlations: Clock 1 vs. Clock 1 - 0.1726 -0 . 5361 -0 . 7167 0.5806 2 - - -0 . 3934 0. 1848 0.3556 3 - -- 0.5391 -0.4775 4 ---- -0.5056 UTC vs. Clock 1 0.0729 0.6105 0. 2280 0. 4412 0.4508 Variance 0 260.0 333.9 1462. 1041. 741.9 835.7 7688. 8390. 677.4 779.8 Autocorrelation 1 .998 .998 .998 .998 .998 .999 .999 .999 .999 .999 2 .996 .995 .997 .996 .997 .997 .999 .999 .998 .999 3 .994 .993 .995 .994 .996 .996 .998 .998 .998 .998 4 .992 .991 .994 .992 .994 .995 .997 .998 .997 .997 5 .990 .989 .992 .991 .993 .994 .997 .997 .996 .997 Allan Variance 1 1.02 1.02 1.97 1.97 1.47 1.50 1.99 1.98 1.00 .982 (square root) 2 .714 .702 1.37 1.37 1.05 1.04 1.38 1.39 .705 .706 4 .513 .505 1.01 .999 .725 .720 .960 .971 .499 .504 8 .342 .345 .695 .697 .508 .503 .679 .679 .352 .351 16 .251 .232 .491 .499 .384 .403 .507 .496 .254 .249 32 .176 .165 .357 .367 .271 .284 .350 .346 .188 .192 64 .120 .122 .277 .284 .190 .202 .226 .223 .141 .120 128 .062 .077 .175 .178 .116 .115 .169 .183 .097 .086

co IU E 9 Eouin f cok smltd ih ht feuny ouain showing modulation, frequency white with simulated clocks 5 of Evolution FIGURE 19. 300" X i (PHASE) 200 no ■•X sample cross-correlations. oo 1024 oo 2048 o O O o • • • - • • 3072

TIME •• •• 4.2.3 Detection of Correlations Among Clock Measurements

The discussion 1n the previous section raises the question of what Is the effect of genuine correlations on time scales, and whe­ ther the unbiassed variances can be determined 1n their presence. This 1s a question of some Importance, because the Inputs to (and therefore the outputs from) a time scale algorithm contain not only the true clock errors but also the measurement errors. In two of the time scales considered 1n section 5.5, namely EAL and UTC(AUS), there may be several clocks in each laboratory while there 1s only one actual time transfer medium to each laboratory. To Illustrate, denote (fo r the time being) the TV comparisons of clock 1 in labora­ tory i by xj^|. The Input from that clock, now denoted by g^» 1s in practice

*1 xil + e£ (4.2.23) where e. is the error 1n the TV measurement, which may be comparable to, or larger than, the clock errors. (The same applies to LORAN-C and most other inter-laboratory comparison techniques.) The time scale equation corresponding to (4.2.13) 1s then n (4.2.24) ■ A PJ zi + A PJ qi * (P rl) 91 or, 1n matrix form:

z = Z + BG (4.2.25) where “P l-l P2 Ps . . . p„ Pi P2-1 Pa (4.2.26) • * • pn [{b lj} ] • « • • ■ • * • • • Pi Pz p. . . . p n "1 Assuming as before that Z 1s non-stochast1c, the relationship be­ tween the covariance matrix of observed quantities, z^, and of the covariances to be determined, Eg, 1 s by the law of propagation of variances: Ez ■ B Zq BT (4.2.27) 90 The matrix B 1s square but singular because of the condition that n E p< - 1, so equation (4.2.27) cannot be solved directly 1n order to 1-1 1 find Eq. However* solutions can be found 1n certain cases, as the following example shows. Suppose that a ll but two of the clocks, and their measurement methods, are Independent. Let clocks k and % share the same measurement method *. Then the covariance matrix zG has the structure:

0 '91

9 Z

*

0 (4.2.28) v 9k 9kga

0 *«• | 0 9k9* h

0 9n

1n which

°91 " °*1 + ° V 1 " U ’**’ " (4 ,2 *29)

°9t,9 - o2 “ a2 K A E|c

The correlation coefficient between gu and g 1s 1 i f 02 » 02 .o2 K 4 e t xk xa (Inter-laboratory time scale) and 1s 0 1f a2 « 05 » o5 (Ideal Intra- e A x k X A laboratroy time scale). Equation (4.2.27) may be written explicitly 1n terms of elements of ^ 1n the form shown 1n equation (4.2.30), by expanding the matrix products and equating the desired terms: 91

‘Si bIi Ma Ma • • • Mk anbu bU - M« Mi ‘Sz Mi Ma Ma • ♦ • Mk aZkbZi bh ••• Mn M®z : t > : : I • • t : : t : • i t « :

♦ • • M °K Mi Mz bta Mk akkbkt Mr Mn ®k

m bklbtl MzMz bk3bl3 • * bkkbtk *bkkblt * bklbtk) bkibn bknbUi Vi O b . 2 * ♦ * b*3 b'k atkbu Mi - b b »

• * « • 1 t : % t fr : ■ s 1 : t * S *

• • Mz M i M k » • Mi ankb« Mi Mn Mvn

(4.2.30) In which P< # J > 1 bu prl , j - 1. (4.2.31)

Note that equation (4.2.30) 1s Identical to equation (4.2.15) except for the augmentation of the (la) row and columns. It may be written as

Sz - Q 3q (4.2.32) where Q has dimension (n+l)x(n+l) because c^g “ gfc* and o_ _ for 1 i j t k i %provides no new Information. z1zj The unbiased variances c? and the covariance a„ „ are found by g1 9kg* solving equation (3.2.32) provided that Q exists. An example was tested, using n ■ 5 and ■ 1/5 for all 1; the resulting 6x6 matrix had a determinant of 0.0726 and a crude condition number | ^ j | max . | q1j Imax **00 which Indicated that a very good solution 1s obtained. By contrast, an example was tested 1n which a ll variables were supposed correlated, and the resulting n(n+l)/2 square matrix corre­ sponding to Q constructed with p^ as given 1n Table 9. Q failed to Invert, obviously because of the fu ll set of correlations. It 1s sup- spected that a limit exists at which Q will fall to Invert, such as when clock measurements carrying more than half the total weight are all correlated among themselves. A situation like this exists with EAL and TAI, In which the large number of contributing clocks 1n North America are a ll Intercompared with European clocks by a (more or less) common LORAN-C link. It 1s therefore difficult to estimate the exact effect of LORAN-C variations on EAL without using additional external Information (such as knowledge of from Intra­ laboratory time scale calculations). The application of equation (4.2.32) time scales contain­ ing a large number of apparently correlated clocks would Involve a great deal of computation 1n estimating a ll the cross-covariances

0 _( _ * while the Q matrix would have dimension as large as n(n+6)/8 K SL 1f half the clocks were Intercorrelated. Perhaps the solution of that equation can be streamlined. 5. CONSTRUCTION OF TIME SCALES

5.1 Characteristics of time Scales

Basically, AT and UTC scales are calculated as weighted averages of a number of atomic clocks, with the objective of obtain­ ing a system for dating that 1s more uniform and stable than any of the contributing clocks. For the purposes of this discussion, uniformity Implies continuity 1n time (date; epoch; offset) and 1n rate (frequency), while s ta b ility 1s a measure of departure from an Ideal time scale usually expressed either as a frequency variance or as a time dispersion (see Section 2.2.1). Procedures to maintain uniformity are necessary since contributing clocks may f a l l , may be given Intentional or unintentional junps 1n time or frequency, may be added as a result of purchase or refurbishment, or may be deteriorating. Stability procedures involve the selection of appropriate weighting and prediction methods to account for the random noise processes that are known to occur 1n atomic clocks, and where possible to Incorporate the results of calibration of the time scale against other, superior time scales or frequency standards. In many laboratories or countries maintaining a time scale, 1t 1s not yet practical to incorporate calibrations against external standards. Hence the behaviour of Individual clocks can only be determined with respect to the time scale towards which they contri­ bute. Time scales formed this way are essentially self-calibrating, and are called free-running. When external calibrations form an Integral part of the computational procedure and affect the resultant time or rate, the time scale 1s eteered. A third type Is conceivable, In which uniformity Is maintained by self-calibration while stability 93 94

1s controlled by external calibration of the random noise processes. Such a time scale could be called stabilised. A further distinction may be drawn between operational and s c ie n tific time scales. For example, to keep operationally the physical realization of the second of the International System (SI) of Units, TAI as maintained by the BIH 1s steered so that its frequency conforms to that of the several laboratory ("primary") frequency standards 1n Europe and North America [Azoublb et al., 1977]. The BIH also calculates a Free-running Atomic Scale (EAL) which, besides Interpolating between calibrations, provides scientific data on the long-term behavior of time scales. In general, operational time scales strive for accuracy In maintaining synchronisation with TAI or UTC, while scientific time scales concentrate on stab ility and uniformity. Again, operational time scales are sometimes used to drive a physical clock which displays the computed average In real­ time, whereas scientific time scales are often computed weeks, or even months, 1n arrear [Cochran, 1980; Perclval, 1978].

5.2 Formulation

Suppose a signal S 1s generated at some time tQ 1n an (unobser­ vable) Ideal time scale. S may be a pulse from a reference clock, a TV equalising pulse, or Indeed any arbitrary signal. Suppose 1t travels with negligible delay to each of n clocks C^, 1 ■ 1, ...,n . Denote the time of arrival of S (as measured by ) by x.j = x^(t). A time scale 1s some average of the n clocks. Let the time scale's estimate of the time of arrival of S be (t) where the subscripts k a 0 ,1 ,2 ... denote different levels of sophistication.

5.2.1 XQ; Single Clock

The simplest time scale of a ll 1s just the reading on a single clock, say the fir s t clock:

X0(t) - X j(t) (5.2.1)

For consistency with subsequent formulations, write this as a condition Imposed on the time scale: x0(t) - Xx (t) = 0 (5.2.2)

Clearly, the time scale XQ 1s no more stable, uniform or long- lived than the one clock on which 1t 1s based.

5.2.2 X^: Simple Mean of Two Clocks

This time scale 1s defined by:

Xi(t) = i[xx(t) + x2(t)] (5.2.3) or, expressed as a condition:

2 E CXi(t) - x,(t)] = 0. (5.2.4) 1=1 1 Such a scale 1s Illustrated 1n Figure 20, which also shows what happens 1f one clock jumps. A d iffic u lty arises 1f the jump 1s unintentional, since the only results of the calculations are, 1n fact, the differences Xi(t) - Xi(t) and Xx(t) - x2(t). Suppose clock 1 has jumped by an amount &xx. Then the time scale X i(t) 1s affected by an amount AXi/2 which 1s revealed as equal and opposite jumps 1n the differences. With only two clocks and no additional external Information, 1t 1s therefore Impossible to te ll * which clock jumped. The same argument applies for changes 1n rate and other anomalous behaviour.

5.2.3: X2: Simple Mean of n Clocks, n > 2

This time scale 1s defined by:

X2( t ) = i E xj(t) (5.2.5) n 1=1 1 or, expressed as a condition:

n E [X2(t) - x.(t)] = 0. (5.2.6) 1=1 1

Figure 21 Illustrates this time scale in the case n = 3. A jump axx 1n clock 1 affects X2 by Axx/n, so that: 96

TIME FIGURE 20. Time scale Xi: simple mean of two clocks, showing effect of jump in first clock.

- 2

IDEAL CLOCK

TIME FIGURE 21. Time scale X2: simple mean of three clocks,l -: showi reduced effect of jump in firs t clock. a [X2 - Xi] = -[(n-l)/n]axx (5.2.7)

a [X2 - v j * [l/n]AX!, 1 « 2, 3..., n. (5.2.8)

Anomalous behavior can thus be detected with confidence Increas Ing as n Increases. Note that:

n 2 [X2 - x J = 0 (5.2.9) 1=1 1

5.2.4 X3: Weighted Mean of n Clocks

Let a weight be associated with clock C^. The time scale X3 is defined by: n n X3(t) = E p, x ,(t) / I Pj (5.2.10) 1=1 1 1 1=1 1 or expressed as a condition:

!p,[X3(t) -x,(t)] =0. (5.2.11) 1=1

The same argument for jumps 1s used as for X2, with obvious modifications for weighting.

5.2.5 Xi*: Corrected Mean of n Clocks

So fa r, the time scales considered suffer from the disadvantage that, 1f one of the contributors stops altogether or a new one is Introduced, a step and a rate change w ill generally occur 1n the scale. Further, the time and rate follow those of the contributing clocks and cannot be offset to follow UTC. These violations to uniformity can be overcome at least to firs t order, by requiring the weighted sum of the errors of the clock results to be zero, rather than just the weighted sum of the results as 1n equation (5.2.11). A Let the time scale so formed be designated X*, and define by z^(t) the expected, or predicted, value of X ^ t) - x^(t) at time t :

Zj(t) = E[z1(t)3 (5.2.12) where z^t) = Xi*(t) - x,j(t), 98

( I f r.j Is the rate of X«, - x^ estimated from results at previous times, one possible form of z^(t) 1s:

Z j(t) = [Xi»(t- T) - x ^ (t-r)] + r^r = z^(t-x) + r^x (5.2.13)

where x 1s the Interval since the previous calculation.) The condition defining the time scale Is then:

= P^XUt) - Xj(t) - z1(t)] = 0 (5.2.14)

which corresponds to:

X j t ) = E^p^Cx^(t) + Z j( t ) ] / E p.,. (5.2.15)

A Writing a^(t) = z^(t) - z^(t) for the prediction errors, equation (5.2.14) 1 s seen to be simply: n E Pj a*(t) ® 0 . (5.2.16) 1-1 1 1 I f one clock, say Cn, has stopped or 1s otherwise a rb itra rily removed from the ensemble, the revised time scale xj 1s calculated as: n_i ^

xi(t) - E Pj [X j (t) + Zj(t)] / E Pj (5.2.17) i-1 1 1 1 1=1 1 from which 1s readily obtained by subtraction from equation (5,2,15): , n-1 E[X*(t) - X,(t)] = [pn/ E p j E [a (t)] (5.2.18) n 1=1 1 n which 1s zero 1f the estimator 1s unbiassed. Thus the expected value of the time scale 1s unchanged when a clock 1s removed. A similar argument shows that the expected value of Xi,(t) is unchanged upon Introduction of a new clock Cn+^, provided that an unbiassed estimate zn+^(t) 1s available. I t can also be shown that 1f the weight of clock Cn, say, 1s changed from pn to p^, the resulting change 1n the time scale 1s:

xi(t) - X *(t) = (pn/p^ - 1) [(X „(t) - xn) - ; n] (5.2.19) the expected value of which Is again zero I f the estimator zn 1s unbiassed. A jump of Axp In clock Cn w ill cause a jump of - (pn/ 2 P j)fixn i =1 in Xt, unless i t is detected and allowed for by adjusting the estimate A 2n* In the possible form given in equation(5.2.13) for 2^(t)» suppose that the rates r^ have been calculated from two previous sets of results: ^ - [z^t-r) - z^t^iM/T. (5.2.20)

The defining condition for X*(t) can then be written as:

n ^p.| [z1 (t) - 2z1(t-t) + (t-2x)] ■ 0 (5.2.21) which is a second-order difference equation requiring 2n arbitrary In itia l conditions for Its solution. These In itia l conditions can be chosen so that:

z^O) » UTC - Xj(0) (5.2.22)

r^O ) - (d/dt) [UTC-x1 (0)3 (5.2.23) which are obtained from measurements of the individual clocks against the external time scale to be emulated (eg UTC(BIH)). I t is worth noting that the internal relationships between the clocks:

z1(0) - Zj(0) = Xj(0) - x1 (0) (5.2.24)

ri(0) - rj(0) - (d/dOCxjfO-x^O)] (5.2.25) must be maintained, otherwise future estimators w ill be biassed. The same argument applies i f different forms of the estimators are employed. This formulation X j t ) produces a time scale whose expected values are continuous 1n time and frequency 1n spite of changes in the number and weight of the contributing clocks and 1n spite of known jumps in time or rate. It can be initially set to follow another time scale I f so desired. With the assumption that the basic behavior of atomic clocks Is linear In time (constant frequency) for certain durations, Xi»(t) is the basis of most practical time scale algorithms - different establishments use different methods of 100

weighting, prediction and treatment of anomalous behavior. The procedure for calculating the time scale when observations are time Interval comparisons, rather than notional "readings", 1s given 1n section 5.4.4, especially equations (5.4.31-32).

5.2.6 X$: Iterated, Corrected Weighted Mean of n Clocks

Small jumps and changes 1n frequency often escape notice 1n preliminary data editing, and 1f not allowed for w ill degrade the time scale's performance. Such defects can be caused by divider noise, propagation anomalies, counter errors and many other things. As hinted at 1n the discussion on X2 1n Section 5.2.3, 1t may be possible to detect such jumps from within the procedure. In itia lly , calculate Xi, using equation(5.2.14) summing over all n clocks, and determine a^. If:

|ak| - max1en|a1| > hk (5.2.26) where Is some tolerance applied to clock C^, delete this clock (for this time t) and repeat the process without 1t. I f a second clock exceeds Its prescribed tolerance, delete 1t also and repeat again; and so on until all clocks' errors are within limits, or until the total weight 1s Insufficient. I f q clocks are deleted at a given time t , the time scale X5(t) 1s defined by: n-q(t) s P^Xstt) - x^t) - Zi(t)] = 0. (5.2.27)

Although, as seen above, the expected values 1n offset and rate are continuous at time t , this procedure requires the exercise of considerable judgment, especially 1n the choice of h^ which must be sufficiently large that normal random clock processes are not excluded, yet small enough that genuine anomalies are detected. This method 1s probably necessary when calculations are done 1n real-time to drive a physical device, whereas 1n the case of a pure "paper clock" calculated after-the-event, the opportunity exists for checking anomalous data so this method could be used 1n a preliminary run for deciding which clocks to accept during the computation period. A statistical procedure for detecting anomalies was given by Ganter [1972], 101

5.3 Prediction Formulae and Weighting

The time scales X* and Xs w ill 1n practise fluctuate with the weighted mean a of the prediction errors: n n a(t,x) * I pj a.(t,x)/ z p, (5.3.1) 1-1 1 1 1-1 1 where we write a^(t,x) rather than a^(t) to emphasise that we are dealing with predictions over a period x since the last "known" value, at time t . The requirement for stab ility generally leads to a choice A of the weights and predictors to minimise the expected variance o2(t,x) of a(t,x) over as great a range of t and x as possible. Denote 2 the variance of prediction errors by a ^ (t,x ), and suppose that they can be estimated from past results. I f the stochastic processes generating the aj(t,T) are stationary,then:

oj(t.x) - oj(x). (5.3.2)

I f also the processes are random uncorrelated Normal ("white noise"), 2 2 ^ ( t ) - and i t 1s well known that the optimum weights are given by: p.| ° 1/a^. (5.3.3)

* In such a case, allowing a linear trend 1n the behavior of each clock, the optimum (minimum variance unbiassed) predictor 1s obtained by evaluating the straight line fitte d through previous results by the 2 method of least squares. (Of course, with fin ite sample size w ill 1n fact vary with the prediction Interval x). Unfortunately, noise processes occurring in atomic clocks are, 1n general, neither stationary nor white, as explained 1n Section 3.2. More complications occur because the previous results z^(t) are contaminated by measurement noise, and because only the variances of Zj(t) are directly available whereas (see equation(5.3,15)) 1t 1s the variances of the x^(t) which are required. Much effort has therefore gone into devising linear filters (linear combinations of past results) which reduce the prediction errors to white noise with minimum variance over a range of prediction Intervals. 102

Commonly, the f ilt e r chosen 1s appropriate to the dominant process over the prediction Interval. For example, over very long periods random walk FM dominates. In this case, the second difference of clock phase (time) 1s a white noise process, so the f ilt e r 1s obtained from:

ECv^ft+t)] * 0 (5.2.4)

1e Zj (t+ t) « ZzAt) - Zj (t-T ). (5.2.5)

(Note the use of here rather than x^, which Implies that the difference between the clock Cj and the time scale 1s random walk FM. Clearly, this 1s the required quantity 1n a free-running time scale). The weight would be Inversely proportional to the unbiassed two-sample Allan variance for lag t . Filters for the several processes were described 1n Section 4.1.2, and I t 1s assumed that deterministic trends have been removed. They are to be Interpreted as one-step-ahead predictors, and the step size 1s chosen to suit the measurements pro­ cedure - typically one day. More complex procedures are possible, to retain one-step-ahead optimality while allowing other processes to be accounted for over longer periods. Details of some of these will be given 1n section 5.5

5.4 Steering

A free-running time scale can be supposed to be set in itia lly , perhaps by wel1-separated portable clock visits, after which It adjusts to its own constituent clocks. Its accuracy therefore depends both on the accuracy of the Initialisation and on the stability of the algorithm used subsequently; the tlme-error dispersion w ill increase with time. When external Information 1s used to correct the time scale for measured departures, 1t 1s said to be s t e e r e d .

5.4.1 Re-1n1t1al1zat1on

The simplest form of steering occurs when the time scale 1s just given a jump at the time of a portable clock measurement. Suppose that the portable clock carries UTC, and that 1t 1s compared 103

against Xt, via a clock Ck which contributes to X*. The error in X* is then: AXM s X* - UTC - (X., - xk) - (UTC - xk). (5.4.1)

After re-setting so that Xi a UTC, it is clear that:

Xi » X„ - AXU; (5.4.2)

thus a jump has been introduced. Sim ilarly, the time-scale's rate may be adjusted as the result of two or more measurements against portable clocks. I f both the rate and the offset are adjusted after the second portable clock measure­ ment, the time scale has effectively been re-in itialised to coincide with UTC. It is still possible, however, to retain the prediction and weighting schemes In use just prior to the adjustment, provided that the same modifications are made to each of the predictions ^ ( t ) : z^(after) » z^(before) - [ aXi* aXi*.(t-ta)3 (5.4.3)

where t a 1s the date at which the adjustment was made. Application

of equation ( 5 . 2 . 1 5 )ensures that the re-1n1t1al1sed time scale X i(t) results.

5.4.2 Current Observations of Rate

For the re in itia lis a tio n case just considered, 1t was assumed that the relationship between the time scale (Xi,) and UTC obtained by portable clock was error-free. In many cases this assumption 1s desirable, especially when no other external Information 1s available for long periods. However, many organisations receive radio signals of one type or another each day, which can be Incorporated in the time scale calculation with appropriate weighting. Suppose that the rate r k of clock Ck has been measured against UTC using the dally relative phase comparisons of a VLF transmission. The variance of r k calculated from the measurements is denoted by

a*. I f t is the interval between successive computations of Xt», then:

rk = [(UTC(t)-xk(t)) - (UTC(t-x)-xk(t-T))]/T. (5.4.4) 104

Since 1t 1s desired to use this rate to steer X* to UTC, replace UTC by Xt, 1n this equation and use the notation

zk{t) » XH(t)-xk(t) (5.4.5) to arrive at the relative external estimate (see Figure 22):

zk(t) = 2k(t-x) + tyr. (5.4.6)

This estimate could also arise directly from a time transfer via satellite, say. For simplicity, assume that zk(t-r) 1s error-free (1t w ill have been adopted after the previous calculation of X*.) A The variance of zk 1sthenT2o2. A least squares method of Incorpor­ ating this estimate and Its variance 1n to the solution for the z^(t) 1s given 1n Section 5.4.4. That method can be extended 1n an obvious way to Include several direct external rate observations. Obviously, the measured rate could be applied directly to the time scale, which would then suffer a rate discontinuity.

5.4.3 Predictive Steering

Observations of rate may be available from several sources, such as Omega, LORAN-C and GBR which to fir s t approximation could be considered Independent keepers of UTC, the more so when tables of corrections are published. For the sake of convenience, suppose that each of these signals 1s compared against one clock Ck of the time scale's ensemble, and denote the rates measured over the Interval x by r ^ , where, for example:

rj[1} ° v [UTC(OMEGA) - xk3/T

r£2) =v[UTC(LORAN-C) - xk] /r (5.4.7)

r£3) =vCUTC(GBR) - xk]/T .

Equation (5.4.6)must then be modified to read:

zkq){t) “ zk(t“x) + rkq)T* q " 1,2,3 (5,4*8) var[zjq)(t)] = t 2o2 (q) (5.4.9) k and can be treated as explained 1n Section 5.4.4. Note that a (q) index has not been attached to zk(t-x ), which emphasises that these 105

(Mtaiurad)

*k Z k f t - r )

^Kk.

I-4 t «-3t » - 2 t t- T t TIME

FIGURE 22. Estimate of X^-Xi, when time scale X»» is being steered to rate of UTC and current rate observations r^ are available.

UTC

K k .q .n

l( k ,q , m )

t - t TIME FIGURE 23. Estimate of Xn-x^ as in Figure 28 when several past external rate observations are used, emphasizing prediction errors. 106

observations are only used to correct the Increment In the time scale. This has the advantage that the time scale carries on free- running whenever these observations are not made, and 1s not distorted unduly when these observations are resumed. However, there 1s 1n fact an error associated with z ^ t-x ) and, of course, with a ll similar and previous results, arising from the accumulation of prediction errors a^(t) (equation 5.2.16), measurement errors s^, and the rate errors o ^ fq ). Because the errors are added at each step of the calculation, the overall dispersion of the results w ill Increase - even I f the added noise 1s white, the resulting process w ill be a random walk whose dispersion can be decreased by taking numerous observations over a long period of time, as discussed 1n Section 3.2. In the present case, past observations on the rates may give strength to the current solution by the follow­ ing procedure. Let r(k,q,m) s rj^(t-mx) (5.4.10)

be the rate of clock measured against time scale q (eg UTC(OMEGA)), m steps ago. Construct the "pseudo-observation" £(k,q,m) as depicted 1n Figure 23:

£(k,q,m) ® ZjJt-mr) + mi. r(k,q,m) (5.4.11)

and calculate Its variance oz (k,q,m) having due regard to the prediction Interval mx. Augment the vector L and the matrices A, S and Q (see Section 5.4.4) and perform the solution as Indicated - G and H remain unaltered, while Z, N and U retain their original dimensions. The solution 1s given by equation (5.4.27).

5.4.4 Least Squares Solution with External Rate Observations

At time t , each clock 1s compared against a local reference signal R, which may be another clock pulse, a TV pulse or even the output of a device following the time scale. The measurements are written as: *1 B Xr - x, (5.4.12) 107

with measurement variance s|. From equation (5.2.12):

zi « X* - x1 . (5.4.13)

whence equation (5.4.12)may be written 1n the form of an observation equation: » Zj - zR, 1 » l,2,...,n (5.4.14)

where the n + 1 quantities zR are to be determined. From equation (5.2.16): A Ep^Zj * Ep^z^ (5.4.15)

which enables an exact solution to be found. The weights p^ are assigned so do not necessarily bear a close relationship to the s*. The predictions z^ are here assumed exact. The extra observation on z ^ t ) given 1n equat1on(5.4.6)is modified to observation equation form: III *kit- t ) * T r k ■ *k(t) (5.4.16)

with variance t 2o 2. V* The following vectors and matrices are defined:

Observation vector: L «* Cti £2 ... (n+l) x 1 (5.4.17)

Parameter vector : 1 ■ Czi z2 ... x 1 (5.4.18) zn ;zr ] t (n+1) Condition vector : G * (n+l) x 1 (5.4.19) Cpi P2 ... pn i° r W 1 • • • — > >

c Residual vector : V » Cvx v2 ... 1 (n+l) x 1 (5.4.20)

Design matrix : A ■ 'l 0 . . . 0 :-i' (n+l) x (n+l) (5.4.21)

0 1 . . . o :-i

* * • • • • • • • 0 0 . . . i :-i 1 1 0 .1 0 ... 1 .0 ... 1n which k *» 1 has been chosen for convenience; 108 Covariance matrix S Si 0 ...0:0 (n+1) x (n+l) (5.4.22) of observations. 0 s2 . . . 0 : 0

t •

0 0 . . . sjj: 0 n.

0 0 ... 0 I t 2a2 r Weight matrix Q * S“l (5.4.23) of observations. Perfect observations would be modelled by the equation L - AZ (5.4.24) but, because of measurement error i t is necessary to introduce the residual vector V: L ** AZ + V (5.4.25) The weighted sum of squares of the residuals, 1e VTQV, 1s minimised subject to the constraint condition:

GtZ - GtZ (5.4.26) where GTZ » Ep,jZj is assumed known and is henceforth denoted by H - 1t is in fact a scalar. The solution is: [UotHa, 1967]: Z - N“1U - N_IG (GtN"1G)“1 (GTff lU + H) (5.4.27) where N = ATQA 1s the "normal matrix", (5.4.28) U = AtQL, (5.4.29) and the required Inverses do exist. Calculation of the Inverses is facilitated by observing the partitioned structure of N:

1 I 1 + - J - r 0 .. 0 sT7 l 0 .. 0 1 si7 s i7 • * • « • 8 (5.4.30) 1 1 0 0

1 1 1 *' " s2 's T * ■ 1 7 n 109

I f the extra observation on rate, is not used, there are just n observations and one condition for determining n + 1 unknowns. The solution (which does not Involve Q) 1s then: n . n 2r " P^ 1 " / i^iPl (5.4.31)

Zj * zR - l y 1*1, 2.., n (5.4.32) which 1s thus the result for a free-running, unsteered time scale.

5.4.5 Steering TAI

A similar concept 1s used by the BIH 1n steering TAI, which 1s generated mainly from commercial cesium standards, towards the rates generated by laboratory standards at NBS, NRC and PTB. In analogy to the time scale Xi,, the BIH calculates EAL (Echelle Atomlque Libre - free-running), while "external" observations are made from time to time on the rates of the laboratory standards using LORAN-C as the transfer medium. The weights corresponding to o|(k,q,m) are based not merely on the errors of measurement but also on knowledge of the correlated noise processes occurring 1n the primary standards and 1n the time scale Its e lf; they are optimised to produce minimum dispersion over an estimation Interval as described fu lly 1n [Azoublb et a l. 1977].

5.5 Operational Time Scales

5.5.1 Echelle Atomlque Libre (EAL)

EAL 1s the free-running reference time scale computed every two months by the BIH. I t uses data from many clocks In Europe, North America and North Africa - for example, 1n 1980 some 133 commercial cesium standards, 6 prototype cesium standards and 6 evaluable cesium standards contributed with non-zero weight at one time or another. As further examples, for the two-month Interval ending on Modified Julian Date (MJD) 44289, there were 88 commercial, 2 prototype and 1 evaluable cesium standard contributing, while for MJD 44599 the numbers were 88, 6 and 6 respectively. (The 88 commercial standards 110 were not a ll the same 1n the two Intervals.) [BIH Annual Report* 1980]. The principles of computation of EAL are set out in the BIH Annual Report* 1974; the algorithm 1s called ALGOS. The notation TAI/UTC 1n that report should be replaced by EAL since 1977 (when TAI started to be steered from EAL.) Converting to the notation of this study, the method is as follows. Within a laboratory x, measurements between each clock 1 and the laboratory's reference clock Rx will be denoted by £xj. Thus:

4X1 “ Rx " x1* *"*»•••» nx» (5.5.1) When the laboratory calculates Its own atomic time scale TA(x), the difference z^ between Xx B TA(x) and Rx 1s

^XA ** RX " V (5.5.2) The measurement of the local reference Rx against a time signal s (such as LORAN-C or networked TV) is reduced prior to a computation of an effective measurement relative to the time of emission of the signal, by subtracting the propagation delay time tXs. This delay is calibrated as often as possible by portable clocks. Denoting the actual measure­ ment of the signal as received by Rx - xs, and the reduced measurement by t xs, the signal measurement equation Is:

B (R^ * Kj) - S“1 , . . . ,S. (5.5.3)

The signal measurements are averaged over non-overlapping ten-day Intervals to f ilt e r out noise 1n the signals; thus the data points occur every tenth day, for which the MJD ends in a 9. The adopted measurement values for Intra-laboratory comparisons are simply the results obtained on the standard days directly or by Interpolation. When known time or frequency steps have occurred 1n a clock, the results prior to the step are corrected for continuity. The effects of bad measurements are also removed prior to computation, so that the iterated correcting procedure described in section 5.2.6 1s not Invoked. The measurements *x^» * xA and JtXs are expressed 1n the form of observation equations, 1n which the unknowns are: I l l

z.j ■ EAL - x^ for each clock 1«*1,...,n (5.5.4)

z^ = EAL - for each laboratory Xal,...,L (5.5.5)

zxA“ EAL - for each Independent TA(x) (5.5.6)

zg a EAL - x$ for each signal s»l S (5.5.7)

The observation equations are found by combining equations (5.5.1) - (5.5.7): £X1 " 21 ‘ ZX (5.5.8) zxA - ZXA * ZX <5 - 5 - 9 ’ z xs " zxs ' zx (5.5.10) and are constructed for each standard day (t) of the two-month Interval. It 1s to be noted that the full range of Indices does not necessarily apply 1n a ll equations; for example only appears for those few laboratories which actually have their own Independent time scales, and different laboratories X do not all receive the same signals s 1n forming £^s.

The defining condition equation is as given 1n equation (5.2.14) which can now be written as: n n E Pj Z i 8 E P i Zj . (5.5.11) 1*1 11 i»l 1 1 where z^ » z^(t°t0) + r^(t-tQ) (5.5.12)

The In itia l value z^ (t°t0) 1s the resultant (EAL - x^) at the end of some previous two-month Interval, while the rate r^ 1s obtained by a least squares straight line fit through the results of the previous two-month Interval. The method of assigning the weights to each clock 1 1s an Important feature of EAL. I t 1s performed Iteratively. F irst, the sample variance of the rates r^ obtained1n the previous six two- month Intervals 1s calculated for each clock, and the weight assigned 1s proportional to the inverse of that variance. A preliminary run of ALGOS 1s performed, from which preliminary rates during the current Interval are obtained. A new sample variance 1s computed for each clock from Its preliminary current rate plus Its five previous rates, 112 and Its weights reassigned for thenext (intermediate) run [Granveaud, M., private communication, 1980]. The resulting Intermediate rates yield the final weights to be used 1n the third (final) Iteration. An upper lim it Is placed on the weight, so that about half the contributing clocks receive maximum weight. I f no such lim it were enforced, one clock would eventually carry a ll the weight which 1s a most undesirable state of affairs when a ll the clocks are statis­ tic a lly comparable. Now that the evaluable standards are Increasingly contributing as clocks to EAL, the upper lim it may need to be re­ appraised [Becker, private communication, 1981]. In fact, the BIH has raised the lim it from 100 to 200 starting from 1980 December 26 which, since

p1 = lOOOO/o* (5.5.13) means that the 6-sample standard deviation lim it of the 2-month rates has been reduced from 10 ns/day to about 7 ns/day. Another lim it to the weighting scheme 1s designed to nu llify the effects of sudden changes 1n clocks, which might not show up very well 1n the 1nverse-var1ance weights. I f the change 1n rate ar.j from the last two-month interval to the current interval exceeds 47 ns/day, that clock 1s eliminated (given zero weight), whereas 1f the rate change 1s smaller than 7 ns/day that clock is given maximum weight. This criterion 1s combined with the Inverse-varlance criterion as follows:

200 1f or.j 5 7 and |Ar^( s 7 ns/day; M1n [lOOOO/o^; 2 0 0 -5 (1 ^ 1 -7 )] (5.5.14) P-i s j 1^ > 7 or |ar^| > 7 ns/day; [ 0 i f |Ar-j | is 47 ns/day.

Interestingly, of 102 clocks with non-zero weight during the Interval ending on MJD 44659 (1981 Feb 24), 20 were assigned weight 200, of which only 2 were evaluable standards. Having assigned weights according to the stage of the Iterations, the set of equations (5.5,8) - (5.5.11) 1s solved by least squares, the redundancy coming from the fact that some of the participating 113

laboratories measure several signals (which 1s the only way to ensure that a ll laboratories are connected to the system, given that not all of them can receive the same signal.) The solution follows essentially the method given 1n Section 5.4.4 with the "external rate observations" removed and multiple observations on several signals added along with their signal parameters z£ which behave as extra zR parameters (c .f. equations (5.4.31) - (5.4.32)). Thus, the relationships defined by the observations and (equations (5.5.1), (5.5.2)) are strictly preserved as is the condition (5.5.11): the free parameters are really only the zx and zg.

5.5.2 International Atomic Time (TAI)

As mentioned in Section 5.4.5, TAI is obtained from EAL by steer­ ing the results of EAL towards the rate of the three evaluable cesium standards PTB Cs 1, NRC Cs V and NBS-6 [Gulnot, 1974; Azoublb et a l., 1977; Gulnot and Azoublb, 1980]. In practise, the rate correction between EAL and TAI 1s estimated for the "estimation Interval" being the current two-month Interval and Included in the ALGOS algorithm as a series of 10-day time corrections, so that the outputs for both EAL and TAI are produced simultaneously. Table 10 reproduces Table 21 of the 1980 BIH Annual Report; from 1t can be Inferred the frequency corrections actually applied between 1978 and 1980. The accumulated time difference between EAL and TAI over the period MJD 43479 to MJD 44629 (1250 days) amounted to 87 ys, which very largely reflects the frequency change 1n TAI of 1 part 1n 1012 Introduced on 1 January 1977 to bring 1t Into line with the primary standards. It should not necessarily be used as a measure of the instability of ALGOS.

5.5.3 Time Scales at US Naval Observatory

The US Naval Observatory calculates several time scales which are comprehensively described by Perclval [1978]. The basic one 1s A.1 (USNO, MEAN) which has been running as an Independent atomic time scale since 1 January, 1958 [Winkler et a l., 1970]; from 1t 1s derived UTC (USNO, MEAN) by the addition of a constant which changes 114

TABLE 10 - MEASUREMENTS OF THE EAL AND TAI FREQUENCY

GRAVITATIONAL FREQUENCY CORRECTIONS ARE APPLIED . THE FREQUENCIES ARE EXPRESSED AT SEA LEVEL . INTERVAL CENTRAL F(EAL)-F(NBS6) F(EAL)-F(NRC CSV) F(EAL) -FCPTB ( MJD DATE IN 10**-13 IN 10**-13 IN 10**-13

40479.43599 1978 JAN31 9.00 43509-43309 1978 FEB10 9 .2 3 43338-43618 1978MARI 1 8.97 43389-43669 1978MAY 1 8.32 43626-43706 1978JUN 7 8 .0 9 43669-43749 1978 JUL20 8 . 12 43749-43829 1978OCT 0 7.49 7.68(2) 43769-43849 19780CT2B 8.48 7.93(1) 4 3829-43909 1978 DEC27 8 . 10 8 .6 1 43909-43989 1979 MAR17 8 .2 4 8 .8 0 43989-44069 1979 JUN 5 7 .7 2 8 .2 4 44064-44116 1979 AUC 5 8 .3 44069-44149 1979 AUC24 7 .7 9 7 .6 4 44149-44229 1979 N0V12 7 . 0 5 ( 1) 8 .0 7 44229-44309 1980 JAN31 8 .8 4 9 .3 1 44249-44329 1980 FEB20 8 .6 0 44309-44389 1980APR20 9.57 8.99 44309-44469 1980JUL 9 8.9B 8.43 44469-44549 1980 SEP27 8.33(1) 8.04 44049-44629 1980 DEC 16 8 .3 2 8 .6 7

INTERVAL CENTRAL F( TAI)-F(NBS6) F(TAI)-F(NRC CSV) FCTAI)-F(PTB I MJD DATE IN 10**-13 IN 10**-13 IN 10**-13

43479-43399 1978 JAN31 -0.20 43309-43389 1978 FEB10 0 .0 3 43338-43618 1978 MARU -0.23 43389-43669 1978 MAY 1 - 0 .6 8 43626-43706 1978 JUN 7 - 1 .1 1 43669-43749 1970 JUL20 - 1 . 08 43749-43029 1978 OCT 8 - 1 .6 6 - 1 .4 7 ( 2 ) 43769-43849 1978 0CT20 - 0 .6 2 - I . 17( I) 43829-43909 1978 DEC27 - 0 .9 1 - 0 .3 9 43909-43989 1979 MAR17 - 0 .7 6 - 0 .2 0 43989-44069 1979 JUN 3 - 1 .2 3 - 0 .7 1 44064-44116 1979 AUC 3 -0.3 44069-44149 1979 AUC24 -0.91 -1.06 44149-44229 1979 N0V12 — 1.4 0 ( 1) - 0 .3 8 44229-44309 1980 JAN3t 0 .4 4 0 .9 1 44249-44329 1980 FED20 0 .2 0 4 4309-44389 1900 APR20 1. 17 0 .3 9 44389-44469 1980 JUL 9 0.58 0.03 44469-44349 1980 SEP27 -0.03(1) -0.36 44349-44629 1980 DEC 16 0 .1 2 0 ,2 7

( 1) COMPUTED JUST AFTER A FULL EVALUATION OF NRC-CSV (2 ) THE THREE LAST CALIBRATION RESULTS VIA PTB-CSt PUBLISHED TABLE 22 (ANNUAL REPORT 1978) ARE REPLACED BY THE VALUES SHOWN IN THIS TABLE 115

by one second at leap second dates. This constant was 20.0343817s during the second half of 1981. The time scale UTC (USNO) 1s coordinated, I.e . steered to be within a few microseconds of UTC (BIH). A physical device based on a phase microstepper attempts to provide UTC (USNO) 1n real time; this device 1s called the Master Clock (MC) and Its output designated UTC(USN0 MC). However, UTC(USN0, MEAN) and UTC(USNO) are subject to reprocessing when rate changes have been detected (which may take several weeks), hence the master clock 1s driven by a provisional version of UTC(USNO) which 1s run twice weekly with the latest available data on rate changes. The algorithms which generate A.1(USN0, MEAN) and provisional UTC(USNO) are now briefly described [Winkler et a l., 1970; Perclval 1978]. The basic equation of A.1(USNO, MEAN) 1s equation (5.2.27) with a ll weights equal - i t 1s an Iterated, corrected unweighted mean of n clocks. The predictions Z j(t) are as given in equation (5.2.13) where the rates r.j were obtained from the two previous readings, according to the earlier paper, with the result that

z^(t) * 2 z(t-x) - z(t-2r) (5.5.15) where t 1s the Interval between data points, namely one day. From equation (4.1.11) of Section 4.1.2 1t 1s seen that this predictor Is appropriate to random walk of frequency. (Since this type of noise is rarely predominant over such short Intervals, 1t may have been better to predict by

z^t) = z^t-x) + [z^t-x) - z1(t-kx)3/(k-l) (5.5.16) where the second term on the right 1s x times the mean rate over the previous k days; i f the dominant noise process over the Interval x and over the longest Interval kx were white frequency modulation, A this z^(t) would be the minimum variance optimal predictor - see sub-section 3.2.3.2). This objection 1s largely overcome 1n the later paper, where the frequency, or rate, 1s modelled as a sequence of discrete frequency steps plus white noise (of frequency) a ^ (t), viz: 116

r.j(t) » p.,(t) + a^(t) (5.5.17)

(5.5.18)where

and t 0

using the ARIMA method rather than those previously considered 1s that, provided that the prediction model 1s accurate, the prediction errors w ill be white noise of phase rather than of frequency (or worse), I.e . 2 f(t) “ Zj‘ (t) a ,(t) (5.5.18) where, as usual, a^(t) represents a random uncorrelated process. It does not, however, guarantee that the predictions themselves form a white noise process. Thus

^(t) - z4(t-l) sa^t) (5.5.19) 1 and, since equation (5.2.15) shows that the time scale depends on both the predictions and the prediction errors, 1t 1s not to be expected that the time scale noise wll be white of phase. Nevertheless, ARIMA solves half the problem simply and elegantly. I t 1s also well adapted to real-time computation, and the parameters need re-evaluat1on only every 100-200 points. The weight p^ of each c ock (1) 1n this algorithm 1s Inversely proportional to the variance j)f of the random shocks a^, normalised so that i:pi 1. The option to assign zero weight 1s maintained. A rejection scheme as described 1n Section 5.2.6 operates, 1n which the tolerance h^ of equation 5.2.26) 1s set at The choice of random shocks to character se clock performance, together with the efforts made at USNO to ensure that the clocks are physically Independent, suggest that the Yoshimura method of calculating unbiassed variances as discussed 1n Sec1ons 4.2.2 and 4.23 could be used to improve the weighting s till fitrther.

5.5.4 Time Scales at US National Bureau of Standards

The US National Bureau cf Standards (NBS) maintains three basic time scales, ATo(NBS) which 1 a free-running Independent scale of atomic time based on about el dht commercial and one primary frequency standards; AT(NBS) which 1s peered from ATo(NBS) by an accuracy algorithm using the results of calibrations against the NBS primary evaluable laboratory cesium s tjandards; and UTC(NBS) which 1s 118 coordinated from AT(NBS) to be within a few microseconds of UTC(BIH) and hence also within a few microseconds of UTC(USNO). The algorithm used for AT0(NBS) 1s called a "first-order11 algorithm at NBS, and 1s described 1n NBS Monograph 140 [Allan et a l., 1974]. I t Is a weighted, corrected mean of n clocks (see section 5.2.5) 1n which time and frequency steps are calibrated from the time scale Its e lf. Predictions 1n AT0(NBS) are similar to equation (5.2.13), namely: z^ft) » z^to) + r^(t-to) (5.5.20) where the rates r^ are estimated using the exponential prediction method discussed at equation (4.1.13) 1n Section 4.1.2 since a flicker frequency modulation noise model 1s assumed for the prediction Interval t ■ 1 day. Weighting Is Inversely proportional to the variances of the Individual clocks' time errors, which are determined from tlme- Interval comparisons between clocks using the "three-cornered hat" method (Section 4.2.1) and which correspond closely to unbiassed random shock estimates obtained from the time scale calculations. The "second-order" algorithm which steers AT(NBS) has been described by Allan et a l. [1975], Its principles are summarised here as they shed some light on the TAI steering algorithm as well. Steering 1s accomplished using both past and current calibration data on the fractional frequency differences ys(k) between the contin­ uous reference time scale AT0(NBS) and an Intermittently operating primary cesium standard; t k 1s the midpoint of the k-th calibration. It 1s supposed that the best estimate of the frequency difference after the current U-th) calibration, yU ), 1s a linear combination of the current measurement y .U ) and an estimate y (jt) based on past results: yU) yeU) + 0 -b a) ysU) (5.5.21) where B Is a weighting parameter to be estimated, and

yeU) B b4_i yeU-i) + ) ysU-D + Ae(*-i,t). ( 5 .5 .23) Thus yeU ) Includes previous results by recursion. Ae( t - l , i ) 1s a term that must be included to account for frequency fluctuations 119

1n the reference time scale since the previous (U -l)-th ) calibration, because the only way to compare calibrations Is through some medium which operates continuously between calibrations. The optimum value of y (t) 1s found by minimising Its variance with respect to the para­ meter 0, with the results:

h “ + cs(£) “ 2C(*)3 (5.5.23) al » [o?U).fff(0 - C*(t)]/[o!(£) + al(l) - 2C(l)] (5.5.24) y(t) s e e where ®SU ) is the variance of the t-th calibration ysU)» and Includes the effects of possible correlations between successive evaluations 2 * of the primary standard's frequency; o U) 1s a combination of the variances of the previous best estimate y (£ -l) and of the reference scale's probable fluctuations between the previous and the current calibration; and C(t) 1s the cross-covariance between the a priori estimate ye(t) and the current calibration ysU ). C(a) reduces to a combination of a ll the previous and the cross-covariances between the a-th and the (t-k )-th frequency calibrations. The estimation of Og(i)t °s^) and based on Previ°us results and certain assumptions about the noise processes operating. The NBS algorithm thus obtains an optimal estimate of the fre­ quency of the reference time scale at the time of calibration against co-located primary standards. This differs slightly from the BIH approach to steering TAI, in which the variance of the time errors 1s minimised for estimation Intervals which may come after the latest calibration. A The resulting corrections yU) are applied to the reference time scale either as a frequency d rift or as a series of small frequency steps applied 1n such a way that the noise Introduced by the steps 1s no worse than the intrinsic noise of the reference time scale.

5.5.5 UTC (Australia)

Some 30 clocks 1n Eastern and Central Australia are Intercompared dally by TV, as described 1n [Woodger, 1980], and used to form a time scale designated UTC(AUS). Their distribution was shown 1n Figure 1. 120

UTC(AUS) 1s a free-running time scale of type Xs. I t 1s computed monthly In batch mode, yielding results separately for each day 1n the month. A preliminary run 1s performed 1n which a least-squares straight line is fitted through all the dally results from the previous three months (or from the last known Jump 1n that period) for each clock. Predictions z^(t) are obtained by simply evaluating the straight line formulae, and weights p^ are assigned Inversely proportional to the variance of the mean point of the f i t , subject to a fairly low maximum weight. The rejection criterion hk of Section 5.2.6 Is the same for a ll clocks, being Inversely proportional to the square root of the mean weight; this was found empirically to give a reasonable balance between giving a ll the weight on any particular day to very few clocks on the one hand, and Including very bad performers on the other. With these parameters, preliminary results are obtained. The predictions, weights and rejection criteria are then re­ evaluated using the previous data plus the preliminary results for the current month, and the procedure 1s repeated. On each run, clocks with obviously bad behaviour are excluded altogether, as are Individual measurements that show serious TV anomalies. Smaller anomalies are accounted for 1n the daily Iterative rejection tests. Some results of computations of UTC(AUS) using real data kindly supplied by the Division of National Mapping, Canberra are displayed 1n Figures 24 - 25. The d iffic u ltie s Imposed by jumps, additions, deletions and rate changes are evident. Results for selected clocks 1n major centres are shown 1n Figures 26 - 29 at greatly Increased vertical scale obtained by subtracting a linear trend from each clock. Figure 26 clearly shows the effect of a bad TV link to Melbourne near MJD 44424, and small short-term correlations attributed to dally TV fluctuations; 1t also shows the Independence of long-term clock behavior. Similar comnents apply to the Canberra clocks shown 1n Figure 27 - note that DNM590 was used as a portable clock during 1980. Figure 29 1n which trend-removed results of the best clock 1n each city are shown, tends to confirm the long-term independence from TV effects, so that the apparent correlations 1n Figure 28 can be 121

* MOt

FIGURE 24. UTC(AUS)-Clock fo r selection of clocks in Canberra and Melbourne, 80 Jan 1 to 81 June 30.

• I t W n a oz o IJ o z o H z

FIGURE 25. UTC(AUS)-Clock for selection of clocks in Canberra, Sydney and Melbourne, 80 Jan 1 to 81 June 30. I U E 7 UCAS-lc rsdas rm tagt i ts, , s it f e lin straight UTC(AUS)-Clock from FIGURE 27. residuals IU E 6 UCAS-lc rsdas rm tagt i fi , s it f e lin straight from UTC(AUS)-Clock residuals FIGURE 26.

MICROSECONDS 4 • tmitao') I ) ' o a t i m t “ T(lS - CLOCK - UTC(flUS) ebun, 1980. Melbourne, abra 1980.Canberra, CLOCK MELBOURNE J/DNM1109 lC 902 flTC 153

I U E 9 UCAS-lc rsdas rm tagt ie ts, s it f line straight from UTC(AUS)-Clock residuals 29. FIGURE

MICROSECONDS » MICROSECONDS M 8 UCAS-lc rsdas rm tagt ie is» fits line straight from UTC(AUS)-Clock residuals 28. yny 1980. Sydney, et lc 1 ec o Cner, Melbourne and Canberra, of each 1n clock best yny n Aie pig. 1980. Springs., Sydney Alice and EIUL : UTC(AUS) :RESIDUALS CLOCK CLOCJ! YNYAIE SPR SYDNEY.ALICE M 201 NML H NAVY1195

124

attributed to fo rtu ito u sly similar behavior of the clocks themsleves over the interval displayed. This provides a concrete example of the d iffic u ltie s enocuntered with simulations in Sections 4.2.2 - 4.2.3. The Allan variances of the results UTC (AUS)-clock are shown in Figure 30 for a selection of the clocks. The slopes on these log-log

plots are approximately -1 in these cases, indicating that the dominant noise process from 1 day ( T B 0.64 x lO^s) to 64 days 1s white phase modulation (cf Section 3.2 .3 .1 ). This conclusion 1s given qualified support by the autocorrelation function r_ (k) for z4 = UTC(AUS)- Z1 1 DNM1109 depicted in Figure 31 (note - this 1s the autocorrelation of the time results, not of their fir s t or second differences), for r , (k) is effectively zero for k>2. The anomalous result for NML201 Z1 Illustrates that clock’s non-linear behaviour which can be seen 1n Figure 28. A conservative explanation for the near-whiteness of the UTC(AUS) results out to 64 days 1s that they are dominated by TV propagation

noise, which 1s not surprising when such great distances are Involved and that the TV phase noise 1s white. The standard deviation of the phase noise, derived from the Allan variance, 1s 0.10 ys. The principle of using a least-squares straight line fit for filte rin g and predicting over periods on the order of 30 days Is thus

ju s tifie d , since such a f i t 1s optimal for this noise process as was demonstrated 1n Section 3.2.3.1. However, the practice of revising an

In itia l estimate and then fittin g through about 120 days Introduces

correlations and biases 1n the estimators which undoubtedly affect this time scale's stability for Intervals exceeding 60-100 days, and may account for the rather disappointing results achieved In comparisons made against UTC(USN0 MC). These results are depicted 1n Figure 32, and cover about three years; the difference UTC(USN0 MC) - UTC(AUS) was set to zero for the first half of this Interval (see Section 5.4.1). (A new

line fitted through all the results would d iffe r by about 6 parts 1n

1 0 1t 1s believed that this 1s not an unusual result when comparing computed time scales.) ALLAH VARIANCES UTC(AUS)-CLOCK •

• a * . * f .. • ♦ ( • • • ♦ «

: . UtClMCI ..... '• ‘ " J ...... * : • h i • * l i t • • i i 4 « 4 T 10 TAU 10 (SEC) 10

FIGURE 30. Allan variances of selected clocks, 1980.

AUTOCORRELATION FUNCTION UTC(AUS) 1109

FIGURE 31. Autocorrelation function for UTC(AUS)-1109, 1980, 1 1 1 1 ► 43800 44100 444 00 4 4700

MODIFIED JULIAN DATE

FIGURE 32. Results of measurements yielding UTC(USNO MC) - UTC(AUS) from mid 1979 to early 1981. 127

5.5.6 UTC(DNM), Division of National Map ping

A small, in-house time scale design ited UTC(DNM) is maintained by the Division of National Mapping, Australia at its Lunar Laser Ranger observatory on an isolated h llls ld i outside Canberra. Its 4-5 commercial cesium standards are suppl smented via a microwave link by another one at the nearby Orroral Vail »y NASA S1DN fa c ility . At any given time, several of the clocks may be out of action, or the automatic data acquisition and processing system may halt for an extended period due to power failures. F jrther, I t has not been possible to provide the clocks with preci ;e temperature control or any control over humidity: their operating environment may indeed be described as non-optimum! The aim of UTC(DNM) 1s to provide ah accessible, real-tim e, reliable time scale for the observatory's Time comparisons are acquired every hour automatically, and processed Immediately, The results automatically correct both the tltjie offset and rate of a physical device, driven by one of the ces urn standards and called "Station Clock" [Cochran, 1980]. The out iut from Station Clock is therefore similar 1n concept to UTC(USN0 He). UTC(DNM) 1s an iterated, weighted, Corrected mean of n clocks, i.e . is of type Xs as described in Section 5.2.6. Predictions are made by least square straight line fits through the results from the previous 25 successful hourly readings, wlv ch are not necessarily consecutive because of the possibility of data loss. In any event, the predictions and weights are recalculal ed every hour. The weights are proportional to the inverse variances of residuals from the 25- point f it s , subject to a fa irly low maximim. The use of a low weight lim it tends to de-we 1ght Inferior clocks rather than to place a greater weight on an excellent clock. This 1s particularly Important 1n a small, nominally homogeneous ensemble: or one hand, I t 1s desirable to keep poorer clocks as low-we 1ght contri butors in case some of the other clocks fa ll and these poorer clocks then have to "carry" the time scale, whatever their faults; on the) other hand, fortuitous smallness of one clock's prediction errors during a fittin g Interval could easily give 1t over half the total vjelght 1f no lim it were 128

applied, from which vantage point 1t would quickly and artificially proceed to carry all the weight. This latter point was abundantly verified during early tests of UTC(DNM) to determine what a realistic weight 11m1t would be. A value corresponding to a standard error of 5 ns was fin a lly chosen, since most fits yielded 2*3 ns standard errors. One solution to the "carry a ll the weight" problem would be to evaluate weights weekly, and Impose them without alteration 1n the forthcoming week; another would to be f i t through much longer past Intervals, thereby minimising the chance of a fortuitous but spurious excellence's occurence 1f computer resources were available. A related problem 1s the establishment of a realistic rejection criterion h^ (see Section 5.2.6). It 1s essential to have some such method of detecting errors 1n an unattended system because occasional bad readings, as well as systematic jumps, do unfortunately occur, and must be eliminated. In line with the philosophy of treating most clocks equally, a single rejection criterion 1s used for a ll clocks at any given hour: n s i Iv » 2.39 [n/ l (1/of)]* (5.2.25) K 1»1 1 where 2.39 1s student's t for 97.5% confidence with 25 degrees of freedom, and n Is the number of clocks actually contributing to the time scale. This choice, an Inverse mean weight, 1s representative of the ensemble as a whole, and 1s relatively Insensitive to either one large or one small o^. Again, a lim it (this time a lower lim it of 12 ns) 1s applied which 1n practise operates nearly a ll the time. I f three successive results for a clock exceed the criterion but are otherwise consistent, a jump 1s deemed to have occurred and that clock

1s assigned zero weight for at least the next six hours. UTC(DNM) 1s not continuously steered to an external time scale; rather, 1t 1s reinitialised to UTC(AUS) whenever Its deviation from UTC(AUS) exceeds about two microseconds. I t may also be re in itia lis e d to UTC(USN0 MC) after a flying clock v is it. 129

A residual plot of UTC(DNM)-x^ for two contributing clocks 1s shown 1n Figure 33 along with the residuals of direct measurements between the clocks whose serial numbers are 1109 and 153. The effects of re-1n1t1al1sat1ons 1n rate at MJD 44303 and 44440 are clearly visible* as 1s a temporary failu re of the algorithm around MJD 44290. A preliminary Allan variance graph 1s given 1n Figure 34 1n which the scales are labelled arbitrarily. Any Interpretation other than that the data contains many more bad points than realised* seems fu tile .

5.5.7 Other Time Scales

The time scale UTC(NRC) of the National Research Council of Canada, Ottawa, 1s derived directly from NRC Cs V which operates continuously as a clock [Costain, C., private communication, 1981].

I t 1s therefore of type X 0 (Section 5 .2.1), but 1s, of course, backed up by the three 1-metre primary cesium clocks and a number ( 2) of commercial standards. The proper time generated by Cs V 1s designated PT(NRC Cs V), from which are obtained:

UTC(NRC) « PT(NRC Cs V) - (MJD-43144) x 0.00097 + 52.041 us TA(NRC) * UTC(NRC) - 31.069 us [BIN Annual Report, 1980]. The same reference discloses that TA(PTB) and UTC(PTB) at the Phys1kal1sch-Techn1sche Bundesanstalt, Braunschweig, West Germany "are derived directly from a local oscillator monitored by the primary clock Cs 1", so these time scales are type Xo, backed up 1n this case by 11 commercial cesium clocks and a hydrogen maser. Other time scales are listed 1n Tables 12 and 13 of the BIH Annual Reports, and their results are given 1n Tables 15 (TAI-TA(i)) and 17 (UTC-UTC(i)) therein. ( I t 1s noted here that UTC(AUS) is

1n fact a continuous free-running time scale, despite the apparent jump at the end of July 1980 in Table 17 of the 1980 report. That "jump" reflected a change In the external estimate of the relation­ ship between UTC(AUS) and UTC(USNO).) 130

1 1 0 9 -1 5 3

*o». M.J.D. (I960)

FIGURE 33. UTC(DNM)-Clock residuals from straight lin e f it s , 1980.

C LOCKALLAH VARIANCES : UIC(DNM) CLOCKALLAH

^ i ocs _J 153

LOG 1AU

FIGURE 34. Allan variances of UTC(DNM) ensemble clocks, 1980. TABLE 11 CLASSIFICATION OF OPERATIONAL FREE-RUNNING TIME SCALE ALGORITWS

EAL UTC(USN0. UTC(USN0, AT, (KBS) UTC(AUS) UTC(DNM) UTC(NRC) UTC(PTB) MEAN) K) Type X.. x . x , x* x , X , x . Xg Data In terval 10 days 1 day 1 hour 1 day 1 day. 1 hour Continuous Contlnuot Predictions:

Class D irect Corrected D irect Recursive Iterated Direct -- Method Least Last value, ARJHA Exponential Least Least Squares rate Squares Squares

Interval 60 days 1 day 1 day 1 day 1 month 1 hour - - Weights: Class Iterated 0,1 D ire c t, Ite ra te d Direct 2-oonthly updated monthly updated periodically periodically Proportional 1/o ff,b u t I/O* - - to 1/o!| v ' ! i see e q .5.5.14 U -S F it) (L .S .F It) Upper U n i t o, ,afi<7ns/d - - - basis - oZj<.12us oZf<5ns - f 1 Contributing Clocks Ntsnber 120 16-25 16-25 8 -9 10-16 4-6 i 1 Type Lab. ,P r o t., Cona.Cs Cocnm.Cs Lab., Coom.Cs Cona.Cs Lab.Cs Lab.Cs Coen. Cs Cona.Cs Distribution Europe, N. USNO USNO MBS East Onroral NRC PTB Africa, USA, A ustralia Canada

Comparison 10RAN-C, Interna! Internal Internal TV In te rn a l. -- Techniques TV u wave Derived/Steered TAI UTC(USNO) - TA(NBS) ---- Scales UTC(NBS) 'E xtern al' NBS-6, UTC(BIH) - NBS-4 o r Reference NRC-V, NBS-6 & PTB Csl BIH 132

5.5.8 Summary of Different Algorithms

Table 11 is a summary of the main differences between the algorithms discussed above, culled as far as possible from the available literatu re. In the table, the "direct" class of predic­ tions Implies that the previous adopted results are used without further modification; the "iterated" class means that the predictions are modified by preliminary current calculations, but the preliminary results are not published, while the "corrected" class involves recalculation after preliminary publication. The same words apply to the weighting classifications. NBS uses an exponential recursive f ilt e r in Its predictions, and Individual unbiassed variances 1n Its weights. The difference between types Xi* and Xs 1s principally that clocks in type X5 scales are examined for rejection at each computa­ tion point, whereas in type Xt, scales they are accepted or rejected for the whole computation interval.

5.6 Comparisons Between Algorithms

5.6.1 Simulated Data Sets

Three data sets were created by simulation. Each contained ten clocks having expected values of zero 1n both offset and rate. No intentional jumps in offset or rate were introduced, as they would merely have complicated the data processing unnecessarily. Data Set 1 was simulated using the ARIMA parameters of the ten real clocks discussed by Percival [1978]; these parameters are given 1n Table 12 along with the starting values usually adopted during the tests. A sub-set of this data set, designated Data Set 1(a), was used 1n a number of tests. This subset used only clocks 1 ,56,8 , and 10 because they have relatively large e 0 terms, so that the clocks exhibit non-Hnear deterministic behavior to some degree. Data Set 2 consisted of ten clocks modelled as flic k e r noise of frequency modulation by the method of Barnes and Jarvis [1971]. The standard error of the white noise input to the generator was 999 ys for each clock, which resulted in effective random shocks in TABLE 12 SIMULATION PARAMETERS IN DATA SETS 1, 1(a) and 3

C lock^ Degree of ARIMA PARAMETERS INITIAL L.S. VALUES INITIAL RATE ^ Differencing^ a b I °o era r (PS) (VS) (us) (us/day) (day) (us) (u s /d a y )

1*(#) 2 0 . 0 0 0 . 8 6 0 . 0 0 9.5 .0124 0 .1 2 2 0.00793 16.5 0 .0 2 0 0 0.00875 (#3) 3 1.00 0.00 0.00 0.0 .00005 0 .0 0 0 0 .0 0 0 0 0 16.5 .0 2 0 0 2 2 0 . 0 0 0.84 0 . 0 0 0 . 0 .0104 0.013 0.00497 16.5 .0329 0.00886 3 2 0 . 0 0 0.75 0.05 0.0 .0107 - 0 .2 0 1 -0.01347 16.5 0 .0 2 0 0 0.00740 4 2 0.30 0.90 0 . 0 0 2.6 .0042 - 0 .0 1 1 -0.00117 16.5 0 .0 2 0 0 0.00314 5* 2 0 .0 0 0.92. 0 . 0 0 11.2 .0139 0.059 0.00373 16.5 0 .0 2 0 0 0.00349 6 * 2 0 .0 0 0.60 0.15 -10.4 .0060 -0.049 -0.00520 16.5 0.0219 0.00385 7 2 0.45 0.80 0 . 0 0 0.0 .0070 0.016 0.00160 16.5 0 .0 2 0 0 0.00683 8 * 2 0 . 0 0 0.73 0 . 0 0 - 8 . 6 .0048 - 0 .0 1 0 -0.00309 16.5 0 .0 2 0 0 0.00451 9 2 0.45 0.90 0 . 0 0 6.0 .0047 0.043 0.00423 16.5 0 .0 2 0 1 0.00724 1 0 * 2 0 . 0 0 0.78 0 . 0 0 -19.9 .0080 -0.064 -0.01097 16.5 0.0474 0.00571

(a) *CIocks included in ensemble in Data Set 1(a).

(b) Obtained from linear least squares f i t through fir s t 32 points.

(c) Obtained from standard error of rate in fir s t 6 16-day intervals.

CO CO 134

the second differences of about 14 ns. This data set was therefore comparable with Data Set 1, at least 1n the short term. Data Set 3 was simulated using the same ARIMA parameters as Data Set 1, except that clock 1 was modelled as a (0,3,0) process (white noise of d r ift) with a generating standard error of 0.05 ns to make 1t look rather like a rubidium standard. Further, the fir s t 2400 points were Ignored and the origins re-set to zero at the 2401-th

point; 1 n addition an arbitrary constant rate was subtracted from the fir s t clock to make the numbers more convenient without affecting Its statistical properties. Moreover, different seeds were used 1n the random number generators between the data sets, so a ll the simulated

clocks 1n both data sets should be truly Independent of each other. Independent white phase noise with standard error 1 ns was

super-imposed on each clock 1n each data set, to simulate measurement' noise appropriate to modern standard equipment. Each simulated data point & j(tj) for clock 1 at time t j 1s to be Interpreted as a t 1me- 1nterval comparison against the reference, clock REF (abbreviated to R) 1n the sense:

*j(tj) * R(tj) - Xjftj). (5.6.1)

The results of an algorithm giving time scale X appear as:

zR(tj) » X(tj) - R(tj) (5.6.2)

*l(tj) - X(tj) - x^tj). (5.6.3)

I t 1s always assumed that REF does not contribute to the time scale

( 1 t 1 s a simple matter to modify the algorithms in practical cases 1n which REF 1s actually a member of the ensemble). The essential results z ^ (tj) are thus Independent of REF which, as stated ea rlie r, may be quite erratic. However, the intermediate results zR(tj) are automatically produced as part of an algorithm. This Is used to advantage 1n the simulations by specifying REF to be Ideal, i.e .

R(tj) = 0, j « 1,2,3.... (5.6.4) so that the results Zp(tj) give the behavior of the time scale X directly, and provide a standard against which the effectiveness 135

of methods used to evaluate the commonly accessible results Z j(tj) can be judged. I t 1s Interesting to look at the actual behavior of the simu­ lations. Figures 12-17 displayed some typical examples. In Table 13 the results of fittin g straight lines by least squares through various segments of Data Set 3 up to the 256-th point of each clock are given. They show quite noticeable changes 1n frequency and 1n standard error from segment to segment. This behavior 1s emphasized when 1t 1s attempted to extrapolate each segment back to t “0 (cf Figure 23) for the variation 1 s much greater than would be expected from normal least squares prediction. Of course» this occurs because the clock errors are highly correlated, but 1 t highlights the care needed in adopting starting values In each process. I t 1s primarily for this reason that a ll test runs on a given data set used the same starting values, around t B0, and created a time scale using a commoi? ARIMA algorithm up to t**96 so that more realistic starting values for each algorithm could be calculated - 1 n fact, each test effectively started at t=96 (days). This behavior also highlights the difficu lties encountered in trying to find more stable algorithms.

5.6.2 Experiments on Simulated Data Sets

Several runs were performed on each data set, using different algorithms or variations thereof 1n each case. The details are summarised 1n Table 14. In general, a run with a given number uses the same algorithm and compatible parameters 1n each data set, but some minor variations do occur. Notable exceptions are Run 1 on Data Set 3, Run 3 on Data Set 2 and Run 5 on Data Set 1, 1n which actually mistakes were made when keying 1n the parameters; but i t was con­ sidered worthwhile to retain their results as illustrations of what can happen In such circumstances. In Table 14 the "number of passes" 1s closely related to the algorithm type. Thus, a DNM algorithm will predict one day ahead using the least-squares straight line fitte d through the previous "Fit Interval" number of points (usually 32), and there 1s no further 136

TABLE 13 BEHAVIOR OF SWULATED DATA, DATA SET 3

Clock F it Number Are. a o f F irs t Six Seoments Httn Simple (p*d.q) tn te rv il o f SegmentsRetei Extr. to s.d. o f (Diys) Segment! (VI) e o f F itt t, Extr. Extr. to 1 2 3 4 5 6 1

(4) ( b ) ' 1 4 64 .0031 R 3.2911 3.2776 3.2630 3.2518 3.2373 3.2253 (0,3.0) .0040 .0040 .0040 .0040 .0020 .0042 E .003 .040 .154 .281 .506 .736 0.288 0.285 B 32 .00B4 R 3.2840 3.2569 3.2307 3.2047 3.1786 3.1532 .0087 .0066 .0082 .0094 .0087 .0079 E .014 .212 .621 1.236 2.062 3.065 1.202 1.178 16 16 .0321 R 3.2700 3.2162 3.1660 3.1141 3.0617 3.0096 .0332 .0317 .0313 .0318 .0320 .0308 E .062 .665 2.510 4.977 8.297 I12.442 4.859 4.771 32 B .1276 R 3.2441 3.1400 3.0359 2.9317 2.8275 2.7233 .1272 .1273 .1275 .1275 .1278 .1275 E .266 3.538 10.148 20.099 33.362 49.9B9 19.569 9.158 2 4 64 .0064 R .0611 .0511 .0428 .0508 .0299 .0265 (0.2.1) .0106 .0092 .0052 .0062 .0059 .0034 E -.006 .001 .037 -.048 .261 .312 0.093 0.154 B 32 .0080 R .0526 .0481 .0252 .0265 .0369 .0269 .00 B3 .0072 .0G64 .0060 .0059 .0090 E -.008 -.013 .342 .318 -.005 .306 0.157 0.182 16 16 .0122 R .0467 .0265 .0326 .0322 .0243 .0191 .0135 .0063 .0121 .0121 .0198 .0053 E .000 .317 .147 .166 .658 1.053 0.392 0.391 32 8 .0246 R .0364 .0323 .0208 .0137 .0130 .0078 .0491 .0119 .0195 .0186 .0163 .0303 E .084 .15B .906 1.570 1.729 2.647 1.182 0.992 4 4 64 .0026 R .0942 .0926 .0916 .0940 .0930 .0852 (1.2.1) .0054 .0040 .0025 .0021 .0042 .0012 E .004 .000 .004 -.017 .001 .148 0.023 0.062 B 32 .0033 R .0915 .0941 .0882 .0893 .0927 .0945 .0049 .0031 .0058 .0019 .0020 .0051 E ,007 -.019 .085 .051 -.053 -.119 -0.008 0.071 16 16 .0050 R .0922 .0831 .0940 .0943 .0974 .1011 .0052 .0046 .0043 .0035 .0054 .0060 E .004 .0B5 -0.97 -.105 -.288 -.745 •0.191 0.299 32 6 .0100 R .0910 .0943 .1001 .1016 .0994 .0971 .0117 .0040 .0147 .0062 .0053 .0137 E .015 -.110 -.481 -.612 -.344 .003 -0,255 0.263 5 4 64 .0000 R .2161 .2054 .2160 .2158 .2054 .2056 (0,2,1) .0074 .0144 .0032 .0039 .0085 .0104 E .002 .039 -.035 -.039 .126 .120 0.035 0.074 B 32 .0093 R .2107 .2147 .2051 .2079 .2104 .2046 .0116 .0034 .0078 .0118 .0054 .0083 E .010 -.024 .131 .097 .011 .225 0.075 0.094 16 16 .0129 ' R .2123 .2095 .2065 .2050 .2229 .2292 .0094 ,0130 .0099 .0115 .0100 .0153 E .004 .048 .148 .235 -.926 • 1.430 •0.320 0.688 12 B .0233 R .2110 .2065 .2259 .2320 .2331 .2329 .012B .0118 .0195 .0198 .0273 .0195 E .014 .147 - 1.140 - 1.727 - 1.877 - 1.799 •1.064 0.925 Explxmtioni (•} Root mm of tho vtrisnces of fit residusls In esch segment. (b) R ii rits during segment inrf/d»y; q i t simple stm d ird d e v litio n of re sid u ils from fitt E ft extripolition of fitted strtlght line beck to t ■ 1. 137

TABLE I I DESCRIPTION OF SIMULATION EXPERIMENTS

tun Type o f Number Parana ter Weight Past Predictions ( t .) WtlBhtt (p,) Rejection Algorithm Pastes Re>eva1uat1on Bias C riterion No. Data Interval Method F it Method Max Sett Interval fo r : (I) (4) (2) (3) (4) (4) (S) (u*)(6) (w*l (7) 1 1.1(4) Din 1 1 U 1 LS 32 LS .010 KLS, .030 2 DM1 1 1 U 1 LS 32 LS .025 MLS, .100 3* D irect 1 32 U 1 LS 32 LS .020 HLS, .025 1 1.1(a) DM1 1 1 B 1 LS 32 LS .010 MLS, .030 3 AUS 2 8 U 1 LS 24 LS .016 HLS, .050 2 LS 32 LS .020 MLS, .060 3 1 .1 (a ),3 AUS 2 32 U 1 LS Prev 96 LS .100 MLS, .300 2 LS 128 LS .132 MLS, .400 2* AUS 2 32 U 1 LS Prev 96 LS .750 MLS, .225 2 LS 128 LS .100 KLS. .300 4 1 AUS 2 32 U 1 LVaLSr Prev 95 LS .too MLS. .300 2 LV+LSr 12B LS .132 HLS, .400 1(a) AUS 2 32 B 1 LS Prev 95 LS .100 HLS, .300 2 LS 12B LS .132 HLS, .400 2 AUS 2 32 U 1 LS Prev 96 LS .100 MLS. .225 2 LS 128 LS .125 HLS, .375 5 1* HYBRID 2 32 U 1 ARIMA «6 LS .100 HLS, .300 2 LV+LSr 32 RS .008 KLS, .024 6 1.2 ARIHA 1 32 U 1 ARIHA (64 LS .020 .020 3 ARIHA 1 32 U 1 ARIHA (64 LS .050 .040 7 1 ARIHA 1 32 U 1 ARIHA (32 RS .008 HRS, .024 1(a) ARIHA 1 32 U 1 ARIHA (32 RS .008 HRS, .024 2 ARIHA 1 32 U 1 ARIHA (32 RS .010 HRS, .040 3 ARIHA 1 32 U 1 ARIHA (32 RS .008 HRS, .024 B 1 ARIHA 1 32 B 1 ARIHA P* LS .003 HRS, .024 1(a) ARIHA 1 32 B 1 ARIHA 32 RS .008 HRS, .024 2 ARIHA 1 32 B 1 ARIHA 132 RS .020 HRS, .080 3 ARIHA 1 B U 1 ARIHA (B RS .004 HRS, .015 9 1 .1 (a ),3 EAL 3 16 B 1.2.3 LS Prev 16 BIH 7 ns/d • 2 EAL 3 16 B 1,2,3 LS Prev 16 BIH B ns/d - 10 1 EAL 3 32 B 1,2,3 LS Prev 32 BIH 7 ns/d - 3 EAL 3 16 B 1 LS Prev 16 BIH 3.5ns/d 2.3 LS Prev 16 BIH 7 nt/d - 11 1.1(a) USNO OLD 3 16 1,2,3 LV*LSr 16 Equal (lOns/d) .020 2 USNO OLD 3 16 1,2,3 LVatSr 16 Equal (lE ni/d) .030

Explanations (1) O ltt S«t l i 10 clocks simulated uilng P e rclva l't parameter. Data Set 1(a)i Clocks 1,5.6,9,10 o f Data Sat l j tana value*. Data Sat 2: 10 clock* simulated a* flic k e r of frequency. Data Sat 3i 1 clock random M ilk o f d r i f t , 9 a* In Data Sat 1 but simulated Independently. * : Pathological ease*. (2) U : Weight* calculated from unbiased variances (Section 3.2 .7 ). B : Weights calculated from raw variance estimates. (3) IS t Least squares stra ig h t Una f i t through number of paints given In next column. Formula altered only at end o f each ’ Parameter Re>evaluat1on Interval* (PRI). LV+LSr : f . ( t j ) ■ t i ( t , .) + r , ( t , * t 4 , ) where the rate r . Is computed by LS each PRIt thus *LV* - Last Value adopted. 1 J 1 1 J J * 1 (4) Number of points used to determine LS values. For AUS Pass 2, these Include preliminary estimates In current PRI. For ARIMA, numbers In parentheses give the number o f points used for assessing weights. For EAL, nmber of points of previous PR! used for prediction, and number of points In EACH PRI used for weighting. (5) LS ! p. • n./ol where n, 1s no. of points In PRI used for clock 1, and o f Is corresponding variance of fit residuils. 1 RS : P, ■ n,/el where 9i (RS) 1s the variance of prediction errors (Random Shocks) i . BIH : See equation (S.5.14), Section 5.5.1. (6) p. ■ p.(nax) I f 0. < o_. For USNO OLD, the values In parentheses are the cutoffs below which chinges In rate a rt decned Instgnlfleant. (7) KLS, a 1 Hu[3«|(LS)|m]| US, a : Hax[3 ci|(RS)io ] . 138

refinement. AUS algorithms employ two passes; on the fir s t pass, a straight line 1s fitte d through the previous three "Parameter Re- evaluation Intervals" (PRIs) to provide preliminary predictions and corresponding preliminary results during a ll of the current Interval. A new straight lin e 1s then fitte d through four PRIs Including the current one, to provide the predictions and weights for the second pass. Figure 35 helps to explain this concept - the PRI therein 1s

8 days, although the tests usually use 32 days. ARIMA techniques generally Involve just one pass - a second pass

cannot refine the predictions, while the change 1n weight obtained by

Incorporating one new result 1s miniscule (unless the weighting scheme 1s unstable!). Because only the previous three or four results are used for prediction, the Fit Interval column of Table 14 shows Instead how many points are used 1n reevaluating the weights. No attempts were made to re-assess the ARIMA $, e parameters within the algorithms, since they were known a priori 1n Data Sets 1, 1(a) and 3. A (0,2,1) model was adopted for Data Set 2, with 0 i ** 0.5. With EAL type algorithms, three passes are employed to refine the weights 1n the current PRI, as described In Section 5.5.1. Predictions were obtained only from the final (third pass) results in the previous PRI, and used unchanged thereafter In a ll three passes of the current PRI. The value of the standard error of rate, or change of rate, giving rise to maximum weight 1s given 1n the column of Table 14 labelled "Max for a:". The techniques of the "OLD" USNO algorithm were approximated by a three-pass method, In which a simpler method of detecting rate changes than that described by Percival [1978] was used (because of data processing d iffic u ltie s encountered later on with real data.) A rate change was ascribed 1f the slope of a straight line f i t through the results of a pass 1n the current PRI exceeded the final slope of the previous PRI by more than a tolerance specified (not quite appropriately) 1n the "MAX for a:" column of Table 14. When a rate change was detected 1n a contributing clock 1 , a correction ar^:

Ar1 = [p^/d-p^JlCr^(current PRI) - ^(previous PRI)] (5.6.5) T OHM DAY I 1 | PREDICTIONS D N M ~ DAY T ^ l JAMO WEIGHTS

______■*-rrt« r*T I a u s PASS 1 ^PREDICTIONS JANO WEIGHTS » 1 * AUS PASS 2

, x ARIMA DAY t j PREDICTIONS *VW

ARIMA WEIGHTS

EA L PASS 1,2,3 PREDICTIONS

•••*• EAL M SS I

EAL PASS 2 WEKHTS

E A L PASS 3

USNO PASS I

USNO PASS 2 RATES PREDICTIONS RESULT USNO PASS 3 PARAMETER REEVALUATION INTERVAL (PR I ) FIT INTERVAL WEIGHT INTERVAL

FIGURE 35. Structure of algorithm types tested showing parameter re-evaluation intervals, f i t intervals and improvement procedures. 140

was applied to Its prediction prior to the next pass; 1f any such changes were detected after the third pass, the corresponding correction was applied to the rates of a ll clocks ready for the next set of data. Further, after any given pass, the largest rate change was dealt with first and Its correction was applied to the current rates of a ll clocks to eliminate the time scale bias; the next-largest (corrected) rate change was then similarly treated; and so on until no corrected rate changes exceeded the criterion or until only three clocks had not yet been ascribed a change. (This procedure 1s similar to that described in Section 5.2.6 for aberrant readings.) The method of predicting used 1n this algorithm 1s to add the Increment due to (least-squares) rate to the value calculated by the algorithm at the previous point. All contributing clocks not temporarily deleted for excessive rate changes received equal weight.

5.6 .2.1 Selection of Intervals and Criteria

Factors that can critically affect the success of an algorithm are the lengths of the Parameter Reevaluation Interval and the Fit Interval, the magnitudes of the maximum weight permitted any clock and of the rejection criteria, and to some extent the number of passes. It 1s necessary to strike a balance between responsiveness to changes and stab ility. It became evident over maqy preliminary trials that, when any

Inverse variance weighting scheme 1 s used, 1t 1 s essential to assign a maximum weight. Further, this maximum must be chosen so that any one clock w ill never have as much as half the total weight, otherwise 1t w ill rapidly come to dominate the ensemble a rtific ia lly . ( It 1s assumed here that all contributing clocks are nominally similar, as 1n the simulations; 1f one clock 1 s demonstrably superior, i t may be desirable to le t I t dominate.) The method of choosing the maximum weight 1 n this study was largely tria l and error based on the variations such as those displayed 1n Table 13; the final choices were made so that most of the time, at least half the clocks had maximum weight. The emphasis was therefore very definitely on 141 down-weighting clocks that were performing poorly, rather than on up- weighting momentarily superior clocks. A possible alternative to the Imposition of a maximum weight was considered, 1n which the variance used 1 s not merely the sample variance over one PRI, but the mean variance from a number of non­ overlapping PRIs. This would have given a much more stable estimate of the true variance, especially 1n the simulated data sets where each clock's progress 1n time was statis tic a lly homogeneous. However, few algorithms 1 n practice used this method which takes much too long to recognise a genuine deterioration in a real clock, so 1 t was not employed 1n this study. (This 1s a good example of the conflict between stab ility and response.) Rejection criteria (h^ of equation (5.2.26)) likewise need careful consideration. The motivation for Including this facility was again largely dictated by experience with real data, because . safeguards against faulty equipment operation are necessary. The rejection criteria used with the simulated data sets tended to be generous so as not to upset unduly the controlled evolution of the noise processes; nevertheless 1t was observed to operate from time to time. Different forms of criteria used were: 3 times sample standard deviation 1n previous PRI; fixed value throughout a run; the maximum of these; or In fin ity . The choice of Parameter Reevaluation Interval was generally 32 days, as experience showed that 1n multi-pass algorithms this was sufficient without invoking too much computation, and i t 1 s close to the interval used for UTC(AUS). In applicable one-pass algorithms, however, i t was found necessary to re-evaluate each day to maintain short-term s tab ility ; in DNM-type algorithms this required fittin g a new line each day, while In ARIMA algorithms,this daily "reevalua­ tion" occurred as a matter of course - the number 1n the PRI column of Table 14 shows how often the standard errors were recomputed for weighting purposes. The "Fit Interval" 1s the number of points used In evaluating the least squares parameters and/or the weights. Thus, for example, Pass 1 of an AUS-type algorithm would f i t a straight line through 142 the 96 points adopted 1n the three previous PRIs. -The numbers given 1n this column of Table 14 for ARIMA algorithms show the number of points used 1n assigning weights to each clock. In many runs, the weights were calculated using unbiassed variances (designated "U" 1n Table 14) as explained 1n Section 4.2.2 and contrasted against similar runs using direct variances (designated "B"). The distinction was not necessary with the USNO OLD algorithm, since the clocks are equally weighted. The results for the fir s t 96 points In a ll runs on a given data set are the same, since a common ARIMA technique and parameters were used. This was done to provide a uniform, realistic basis for Initial values, especially for AUS and EAL algorithms.

5.6.3 Results on Simulated Data Sets

The principal output of each run consisted of: (1) Results zR and z^, 1al , ..,1 0 at selected points, and the weights assigned to each clock at those, points; (11) Variances and autocorrelations of zR and z^, 1=1,..,10 residuals from a straight line fit 1n the range 96stsl024, and corresponding unbiassed estimates of X and x^, 1=1,...,10 1n which the mean weights 1n the stated range were used as elements 1n the B matrix of equation (4.2.15). The autocorrelations were taken out to lag k«16. (111) Variances and autocorrelations of the fir s t and second differences of Zg.XiZ^Xj, 1=1 , . . . , 1 0 as in (11). (1v) Allan variances (and logarithms of their square roots) of zr i X,Zj ,Xj ( 1=1,...,10 for t = 1,2,4,8,..,128 as In (11). (v) Variances and autocorrelations of the random shocks A Zj - Zj, 1=1 , . . . , 1 0 and of their fir s t and second differences and of the corresponding unbiassed estimates x^ - x^ and X - X; and the direct and unbiassed Allan variances of the undifferenced quantities, as In (1 1 ) - ( 1v). (vi) A plot file , showing the results zR = X-R, 143

The results are summarized for each data set in the ensuing tables and figures. In Tables 15-22 the "Allan slopes" are the

slopes of the graphs of log O y ( 2 , x ) against log x ; values of -1 are desirable over the whole range of x. Standard errors of straight lines fitte d through points t»96 to t=1024 are quoted as o(LS F it) and are not terrib ly meaningful except as broad Indicators of behavior. However, those lines were subtracted from the Final Results (zR)

before computing correlation coefficients (p) 1n order to reduce the apparent calculated autocorrelations to the point where dissimilari­ ties between runs would become evident. Since the quantities o(zR,Direct) are computed after subtraction, they are Identical to o(LS F it) whereas the standard errors of the prediction errors o(RS,Direct) are calculated with respect to an expected mean of zero so differ substantially. With these provisos, the quantities o(LS F it), o(RS.Direct) and c(zR,Direct) were calculated from the results zR « X-R. On the other hand, o(RS,Unbias), a(zR,Unbias), Allan slope and autocorrelation coefficients were calculated using unbiassed values Inferred from the clock results z.| ** X-x^ and their weights (obtained by averaging the weights over the interval 96stsl024.) I t 1s noted that the unbiassed estimates for Allan variances and autocorrelation coefficients 1n the Final Results zR were very similar to the values calculated directly, which verified that the technique for obtaining unbiassed variances and covariances 1s valid. Whenever the autocorrelations indicate that a given difference

1s a white noise process, p (l) for the next difference should be -0.5, while p(l) and p{2) for the following difference should be -0.67 and +0.17 respectively. This pattern 1s readily visible in the tables, and was used to assess the likely ARIMA character of the residual prediction errors.

.5.6 .3.1 Data Set 1

For data set 1, Table 15 gives the statistics of the predic­ tion errors 1n X-R (Time Scale-REF), and Table 16 gives the 144 statistics of the actual results X-R (rather than of the random shocks). Unbiassed estimates are shown unless stated otherwise. Figure 36 shows Allan variances of prediction errors of each run, while Figure 37 shows Allan variances of the results. Figure 38 shows the results X-R of selected runs, Figure 39 shows them normalised to X-R «* 0 at t=0 and t»96, while Figure 40 shows a selection normalised at t “96 and t«192 to emphasize the evolution of each different time scale a fte r 1t has truly settled down. Outstanding features 1n this data set Include near-equality of runs 1 and 2 (DNM unbiassed and direct respectively) 1n a ll respects; the discontinuities every PRI (32 days) 1n run 3 (AUS, least square; Figure 38); the distinctly superior nature (white phase noise) of the prediction errors 1n algorithms whose predictions are based on the last adopted value (ARIMA, LV+LSr; runs 4-8,11) rather than on least squares; and the tendency on the part of all algorithms to display long-term behavior similar to that of the ensemble average (Figure 37), although run 10 (EAL, 32) does rather better than the others 1n the longer term.

5.6.3.2 Data Set 1(a)

The results for data set 1(a) are summarised 1n Tables 17 and 18 with Allan variances displayed In Figures 41 and 42 for predic­ tion errors and actual results respectively. Figure 43 shows the raw results of X-REF for each algorithm, with Figure 44 showing them normalised at t°96 and 192. Generally, the variances are slightly higher than found In data set 1 , as would be expected from the smaller number of contributing clocks, otherwise thecharacteristics are very similar. The principle exception 1s the occasional failure of the UTC(AUS) algorithms both unbiassed and biassed (runs 3 and 4 ), which show up most clearly 1n Figure 41. This 1s caused by the long prediction Interval, Inadequately spanned by Insufficient clocks whose parameters are only corrected once. Also, the Fit Intervals chosen were evidently too long at 96 and 128 days for straight line fittin g to be appropriate, yet recovery after gaps was good and the TABLE 15 RESULTS UTC-REF ON SIMULATED DATA SET 1: PREDICTION ERRORS (2R - zR)

UTC-REF RUN

l , 2 taJ 3 4 5 6 7 8 9 10 11 DNH AUS AUS HYB ARIHA ARIHA ARIHA EAL 16 EAL 32 USNO (LS) (B) (0 ) PREDICTION ERRORS Allan slope -0 .6 5 -0 .5 7 -0.88 -0.94 -0 .9 9 -0 .9 9 -1 .0 0 -0 .6 4 -0 .5 5 -0.98 Standard errors: o (LS F i t ) tl>: , , 0.20 1.21 1.27 1.44 0.45 0.33 0.54 1.12 1.03 0.92 c RS, D ir e c t ) ^ ' 2.08 1.29 3.51 6.67 0.63 3.56 '4 .0 6 ' 3.07 5.08 3.90 o (RS. Ikibias) 0.010 0.045 0.004 0.003 0.003 0.002 0.002 0.020 0.047 0.003 Autocorrelations: P ( l) 0.96 0.93 0.52 0.28 -0.02 -0.04 -0.04 0.90 0.95 0.07 P(2) 0.91 0.86 0.49 0.19 -0 .0 2 0.00 0.00 0.81 0.90 0.01 p(k>2) Exp.dec Exp.dec Exp.dec Ofrom 0 0 0 0 from Exp.dec *0 fc*6 k-14 First Dlffs: o(vRS.Unbias) 0.003 0.017 0.004 0.003 0.004 0.003 0.004 0.009 0.014 0.004 p( n 0.10 -0.02 -0.47 -0.44 -0.50 -0.52 -0.52 0.00 0.00 -0.47 p 2) 0.06 -0 .0 2 -0 .0 2 -0 .0 3 -0.01 0.01 0.01 0.00 0.00 -0.02 P(k>2) 0 0 0 0 0 0 0 0 0 0 Second D lffs : °Iv*RS,linblas) 0.004 0.024 0.007 0.005 0.007 0.006 0.006 0.012 0.020 0.006 P i ) -0 .4 7 -0 .5 0 -0.65 -0.64 -0.66 -0 .6 7 -0.67 -0.50 -0.50 -0.65 p (2) -0 .0 3 -0.01 0.15 0.13 0.16 0.18 0.18 0.00 0.00 0.14 p(k>2) 0 0 0 0 0 0 0 0 0 0 ARIHA character (0,1,1) (0,1,0) (0,1,1) (0,1,1) (0,0,0) (0,0,0) (0,0.0) (0,1,0) (0.1,0) (0,0,0) (qualitative) Notes: (a) Results for runs 1 and 2 are nearly Identical. (b) RMS of residuals from least squares straight line fit - for Information only. (c) Values on results as computed, before application of unblasslng. (d) All autocorrelation coefficients are computed by the unbiassed method. They generally differ greatly from biassed values. 146

TABLE 16 RESULTS UTC-REF ON SIMULATED DATA SET 1: FINAL VALUES (zR)

UTC-REF RUN

l . f W 3 4 S 6 7 8 9 10 11 DNM AUS AUS MTB ARIMA ARIMA ARIHA EAL 16 EAL 32 USNO a * ) (B) (0) RESULTS Alim si Opt t "lt.tB •0.38 -0.30 -0.2S -0.2S -0.33 -0.23 -0.24 -0.24 -0.31 -0.31 r “16... 0.46 0.44 0.63 0.49 0.41 0.48 0.48 0.54 0.47 0.4S Standard t r r o r t : °(LS F1t)*b7 T 0.20 1.21 1.27 1.44 0.45 0.33 0.54 1.12 1.03 0.92 ezfi,Direct)1" 0.20 1.21 1.27 1.44 0.45 0.33 0.54 1.12 1.03 0.92 otzp.Unblat) 1.06 1.02 1.03 1.28 1.00 1.22 1.22 1.0S 0.74 1.21 Autocorrelation*:^ p{rr ------.99 .99 .99 .99 .99 .99 .99 .99 .99 .99 .99 .99 .99 .99 .99 .99 .99 .99 .99 .99 SM I1** vsed sed ted ted tad vsed vsed vted vted vted F1r*t Dlffs: o VZp.Unblat) 0.010 0.010 0.010 0.012 0.010 0.011 0.011 0.010 0.008 0.011 p .93 .83 .92 .96 .92 .96 .96 .92 .85 .94 e .92 .81 .91 .95 .92 .95 .95 .92 .84 .93 a ed ed ted sed vsed ted ted vted ted vted Second D 1 fft: cfc'zp.Unblai) 0,004 0.006 0.004 0.003 0.004 0.003 0.003 0.004 0.004 0.004 op) -.48 -.46 -.47 -.45 -.46 -.44 -.44 -.47 -.49 -.47 p 2) -.03 -.02 -.02 -.02 -.03 -.03 -.03 -.02 -.01 -.02 p(k»2) 0 0 0 0 0 0 0 0 0 0 Value*: , M tnlm irn -8.59 -3.29 -13,30 -23.76 -1.14 •13.23 -15.02 -12,00 -18.96 -2.16 Maximum 0,01 1.(2 -0.39 •0.39 0.75 0.01 0.01 •0.56 •1.52 10.55 No. of valid point* 1n 96*t*1024 929 911 929 929 929 929 929 929 929 929

Note*: (c) See Table 15. IS Unbiassed correlation co e fficie n ts. They are generally d o te to the values computed directly (cf. prediction errors). "ed* • exponential decrease: “vs* • very slow. H2- PREDICTION ERRORS TIME -SCALE RESULTS (RANDOM SHOCKS)

-I2-J 10

RUNS Py(T)

-IS RUNS 10 5 .7 .8 .1 1 ^1,2,4,6.9 10 1.2 -14- 10

7 .8

-J L X X J L X I 2 4 B 16 32 64 128 t 8 16 32 64 128 T

FIGURE 36. Unbiassed Allan variances of prediction FIGURE 37. Unbiassed Allan variances of final errors (zR-zR), Data Set 1. values (zR), Data Set 1. FIGURE UTC - RCr (rucro**£ondi) 20 lb e - le e — 8 Rsls n aa Set Data on Results 38. l s ' zse fl r ------04 DAYS 1024 -l- i . . . 1 i EL32) 2 EAL(3 i 18 . i AUS132) i 3 i RI1 3| ) (32|U It1R AR i I16) L A E 7 l 9 : RM( 7) ,7 2 ARIMA(3 : 8 l DNJK32) l 1 IU E 9 Rsls n aa e 1 nraie a t=0,96. at normalized 1, Set Data on Results 39. FIGURE

UTC - RCF (111 e 0 512 10 l RRIHA(32,2) l 0 ? i RRIMf»(64,UI • 6 i EflL(16) i 9 3 1 nusozi z o s u n i J flRtrinos.u) l)Hn(32)

I U E 0 Rsls n aa e 1 nraie a t=96,192. at normalized 1, Set Data on Results 40. FIGURE

UTC ' REF inUro**eand*l 0- DCIYS 102-1 6 ffl ,tl 2 3 C A flfflH i 6 ( UitlUIULB) B L U I U l t i U 1 1 1 0 EAL132) t 10 ISHL ncmt CIISCHBLC i nfl U) ,U 2 fll3 in R A i V EALCl6) ) l 2 9 AUS(3 l 3 ) |r)ft(64,U R A t 6 HCIE HEAH EHSCNILE i SHHI32) i 1

151 long-term accuracy of these algorithms was no worse than any others, as can be seen 1n Figures 43 and 44. This suggests that fittin g over­ lapping quadratic curves might produce improved results. The relative success of the EAL and ONM approaches 1n this data set lend weight to this suggestion, since their straight lines are fitte d over much shorter Intervals and updated more frequently.

5.6.3.3 Oata Set 2

Data Set 2, 1t w ill be recalled, was simulated from a flicker of frequency model. The summaries of results are given 1n Tables 19 and 20, Allan variances 1n Figures 45 and 46, and results raw and normalised 1n Figures 47 and 48. A The results for prediction errors zR-zR are very similar to those for Data Set 1, with the decays in autocorrelation coefficients being rather slower, although not so much as to prevent ARIMA predic­ tion errors from being considered white of phase. EAL and AUS algorithms gave essentially white frequency prediction noise. Run 3 on this data had an unusually low maximum weight, so a ll weights . throughout this run were virtually equal; the prediction error results were Indistinguishable from those for run 4 1n which the standard AUS weight lim it was employed. Incidentally, run 4 In Data Set 2 was the counterpart of run 3 1n Data Set 1 and gave very comparable results. Run 11 (USNO OLD) gave anomalous results 1n both prediction errors and final values, the prediction errors looking like flic k e r of phase even though the "last value plus rate" prediction method was used. The flic k e r f ilt e r developed 1n Section 4 .1.2 would undoubtedly have given better results. In this context, the results on runs 6, 7 and 8 show that, 1n the ensemble, an ARIMA (0,2,1) approximation to flic k e r frequency modulation Is acceptable. The anomalies 1n run 11 show up well 1n Figures 46 to 48 - I t was perhaps fortunate that such an example was encountered, although the final values yielded by this algorithm on Data Set 1 also run against the general trend. They are probably due to overly tight rejection crite ria for both phase and rate changes used here. Allan variances indicate that the final values from run 11 of Data TABLE 17 RESULTS UTC-REF ON SIMULATED DATA SET 1( a ) : PREDICTION ERRORS (z R - ZR)

UTC-REF1al RUN

1 2 3 4 7 8 9 11 DNH(U) DNH(B) AUS(U) AUS(B) ARIHA(U) ARIHA(B) EAL 16 USNO

PREDICTION ERRORS

Allan slope -0 .6 8 -0 .7 0 -0 .5 7 -0 .5 8 -1 .0 2 -1 .0 3 -0.65 -1 .0 4 Standard errors 0 iLS F it ) 1.01 0.70 0.53 0.83 0.14 0.47 0.29 0.35 a[RS. D ire c t) 4.53 6.82 3.85 2.70 0.56 1.80 3.96 1.64 o(RS, Unbias) 0.017 0.017 0.120 0.118 0.004 0.004 0.029 0.004 Autocorrelations: P 1 .95 .95 .92 .92 -.0 2 -.0 3 .90 -.01 p z> ih l .91 .91 .84 .84 -.01 -.01 .80 -.0 3 P k>2)(b l Id Id ed ed 0 0 0from k*14 0 First Dlffs: 0 vRS, unbias) 0.005 0.005 0.048 0.048 0.005 0.005 0.013 0.006 P 1) .01 -.01 -.0 3 -.0 7 -.51 -.51 -.01 -.4 9 P 2) .07 .08 -.01 -.0 4 -.01 .00 -.01 -.0 1 P k>2) 0 0 ' 0 0 0 0 ' 0 0 Second D lffs : o 7*RS, unbias) 0.007 0.007 0.069 0.070 0.009 0.009 0.018 0.010 P -.5 3 -.5 5 -.51 -.51 -.66 -.6 7 -.5 0 -.6 6 P 2) .03 .06 .00 -.0 2 .16 .16 .00 .15 P >>2) 0 0 0 0 0 0 0 0 ARIHA character (0 ,1 ,1 ) (0,1,1) (0,1.0) (0 .1 ,0 ) (0,0,0) (0,0,0) (0 ,1 ,0 ) (0 .0 .0 ) (qualitative) Notes: (a) See notes for Table 15. (b) "id* - linear decrease; “ed" * exponential decay. TABLE is RESULTS UTC-REF ON SIMULATED DATA SET 1 (a ): FINAL VALUES (z R)

UTC-REF^ RUN

1 2 3 4 7 8 9 11 DNH(U) DNH(B) AUS(U) AUS(B) ARIMA(U) ARIKA(B) EAL 16 USNO

RESULTS • Allan slope -0 .4 2 -0 .4 5 -0 .3 9 -0 .3 8 -0 .2 8 -0 .2 9 -0 .4 0 -0 .3 8 t *1 6 ,. .. 0.38 0.39 0.08 0.05 0.52 0.53 0.43 0.44 Standard Errors: 0 JLS Fit) 1.01 0.70 0.53 0.83 0.14 0.47 0.29 0.35 a 2 r , D ire c t) 1.01 0 .7 0 0.53 0.83 0.14 0.47 0.29 0.35 o( zr , Unbias) 1.52 1.53 1.91 1.93 2.16 2.32 1.82 1.88 Autocorrelations: p 1 .99 .99 .98 .98 .99 .99 .99 .99 p 2) .99 .99 .97 .97 .99 .99 .99 .99 p k>2) vsed vsed ed sed vsed vsed vsed vsed F Irst Otffs: o(vzp, unbias) 0.015 0.014 0.018 0.018 0.020 0.021 0.017 0.017 .89 .88 .89 .88 .97 .97 p i1 .93 .93 p 2) .89 .89 .87 .86 .96 .97 .92 .93 p(k>2) vsed vsed Sid sed vsld vsld vsld vsld Second D lffs : a >*Z0. unbias) 0.007 0.007 0.008 0.008 0.005 0.005 0.006 0.006 P (1) R -.5 3 -.5 5 -.4 8 -.4 5 -.46 -.47 -.49 -.4 9 P 2) .03 .06 .00 -.0 4 -.0 4 -.0 3 -.0 1 -.01 p(k>2) 0 0 0 0 0 0 0 0 Values: Hinlnun -0.11 -0 .0 9 0.04 0.04 -1.94 -5 .9 7 -0 .0 3 -0 .7 8 Haxlmm 15.24 22.99 12.09 8.41 0.02 0.02 12.77 3.80 No. of v a lid points 929 927 789 802 929 929 929 929 In 96sts1024

Note: (a) See notes for Table 16. PREDICTION ERRORS TIME SCALE RESULTS (RANDOM SHOCKS)

° r (t> N RUNS 14. -IS RUNS 5 . 7 , 8 . 1 1 'L i . 4 . 6 , 9

1.2 -15

7 .8

I 2 4 8 16 3 2 6 4 128 t I 2 4 8 16 32 64 128 t

FIGURE 41. Unbiassed Allan variances of prediction FIGURE 42. Unbiassed Allan variances of final errors (zR-zR), Data Set 1(a). values (zR), Data Set 1(a). I U E 3 Rsls n aa e 1(a). Set Data on Results 43. FIGURE UTC - Rtr (hi 20 0 0 S12 768 I ARinA(32,U) I 7 t 3, > (32,B A M R R t 8 I U E 4 Rsls n aa e 1a, omlzd t=96,192. t a normalized 1(a), Set Data on Results 44. FIGURE

utc - REr i n i IB e S12 , : AUS(UIB) : 3,H HEL HEBH CHSEMLE USHOtOUD i t I i Dlini32,ls> i 2 i CALI16) i 9 t M1HRI ,» 2 I3 R H 1 M t S UtM>»2, ) ,U 2 » > ttM U t 7 i MNI U) .U 2 I3 N M i 1

TABLE 19 RESULTS UTC-REF ON SIMULATED DATA SET 2 : PREDICTION ERRORS (z R - zR)

UTC-REF 1 3 4 6 7 8 9 11 ONH(U) AUS AUS ARIHA(LS) ARtHA(U) ARIKA(B) EAL 16 USNO ______(loose) (tight) ______PREDICTION ERRORS A llan Slope -0 .5 2 -0.54 -0.55 -1.02 -1.02 -1.01 -0.58 -0.93 Standard Errors: o ( lX F U ) 1.11 0.65 0.73 0.27 0.65 0.39 0.44 2.55 a(RS, Direct) 1.39 0.67 1.55 1.24 0.67 0.95 2.48 6.70 o(RS, Unbias) 0.022 0.042 0.042 0.004 0.004 0.004 0.040 0.004 Autocorrelations: p(1 ) .97 .96 .96 -.0 6 -.0 7 -.0 7 .91 0.28 p(2 ) .92 .91 .91 -.0 8 -.0 8 -.0 8 .82 0.15 P(k>2) O .rtl 2 Id Id 0 0 0 0,1*11 -.0 5 First Diffs: o(vRS, Unbias) 0.005 0.012 0.012 0.006 0.006 0.006 0.017 0.005 p (1 ) .36 .05 .04 -.4 9 -.4 9 -.4 9 .03 -.41 p(2 ) .28 .03 .03 -.0 2 -.0 2 -.0 2 .02 -.0 3 p (Ic>2) ed 0 0 0 0 0 *0 0 Second D iffs : 2) 0 0 0 0 0 0 "0 0 ARIMA character (0 ,1 .1 ) (0 ,1 ,0 ) (0,1,0) (0.0,0) (0 ,0 ,0 ) (0 ,0 ,0 ) (0,1.0) (0,0.2) (qualitative) TABLE 20 RESULTS UTC-REF ON SIMULATED DATA SET 2 : FINAL VALUES (z R)

RUN UTC-REF 1 3 4 6 7 8 9 11 DW(U) AUS AUS ARIMA(LS) AR1MA{U) ARIMA(B) EAL 16 USNO (loose) (tiq h t) RESULTS

Allan Slope t*1 ..4 -0 .3 0 -0.26 -0.25 -0.22 -0 .0 9 -0.10 -0.10 -0.20 t = 8 . . . 0 -0.26 -0.25 0 0 0 0 0.18 SItandard Errors: 0 LS F it) 1.11 0.65 0.73 0.27 0.65 0.39 0.44 2.55 0 Zp, D irect) 1.11 0.65 0.73 0.27 0.65 0.39 0.44 2.55 0 zR, Unbias) 0.3S 0.26 0.28 0.33 0.27 0.26 0.25 0.79 Autocorrelations: p ( l) 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 P 2) .99 .99 .99 .99 .99 .99 .99 .99 p(k>2) vsld vsld vsld vsld vsld vsld vsld vsld F Irst Diffs: ovzD, Unblas) 0.006 0.008 0.007 0.006 0.006 0.005 0.006 0.009 P 1 r .61 .27 .46 .67 .65 .64 .63 .86 P 2) .56 .25 .42 .61 .59 .58 .57 .83 p k>2) sed sed sed ed sed sed sed sed Second D iffs : p(v*zD, Unbias) 0.005 0.010 0.007 0.005 0.005 0.005 0.005 0.005 P I ) R -.44 -.48 .-•45 -.41 -.41 -.41 -.41 -.41 0(2) -.0 2 .00 -0.1 -.03 -.03 -.03 -.0 3 -.0 3 p(k>2) 0 0 0 0 0 0 0 0 Values: Minimum -1.10 -1.13 0.08 -.2 7 0 -1.61 -2.65 -7.51 -19.49 Maximum 4.25 1.59 5.12 1.52 0.98 0.11 0.08 0.11 No. o f valid points 929 929 929 929 929 929 929 929 In 96sts1024 PREDICTION ERRORS -«T TIME SCALE RESULTS (RANDOM SHOCKS)

-1 2 ‘

O I t ) RUNS i RUNS

10

»' 14'

I 2 4 6 J* 3 £ 64 128 T I 2 4 8 16 32 84 128 T

FIGURE 45. Unbiassed Allan variances of prediction FIGURE 46. Unbiassed Allan variances of final errors (zR-zR), Data Set 2. values (zR)» Data Set 2. 10 T

4 l PUS(LOOSE) 1 i PHH

3 i AUSIT1CHS) 0 7 i BBIHAQ2.U) « ■ PRlnAI32|l) o i ARIHA 1 (4 ,U) (POOR START)

9 » C A LtlO yu oc -10 i u 3*- 11 : USNOtOLD)

-20 - » 256 512 1024 onvs FIGURE 47* Results on Data Set 2* I U E 8 Rsls n aa e 2 nraie at t=96,192. t a normalized 2, Set Data on Results 48. FIGURE

UTC - fctr t h l c r o f t e o n d d 18 e 512 04 s r r w 1024 USKOCOLD) Tl ■ H ITIC S nU t I I .71 ) / 132 M U 3 M t - » { _ — HCfiH ’feCNSCHll.C l tU) ) tt.U C llM k A i 7 t ( SL SlI O iSILO (H t 4 WItl64, » ,U 4 IItfl(6 fW 162

Set 2 exhibit random walk of frequency, whereas other algorithms show long-term flic k e r of frequency (zero "Allan slope") consistent * with the ensemble mean. One other feature which shows up clearly from Data Set 2 Is the Inferior short term s tab ility of final values from run 3 (loose AUS). The jumps at the end of each 32-day Parameter Reevaluatlon Interval are quite evident In Figure 58, as are their effects on the Allan variance 1n Figure 46.

5.6.3.4 Data Set 3

Results for Data Set 3f, 1n which the fir s t clock was given random walk of d r ift noise, are summarised 1n Tables 21 and 22, with their Allan variances displayed In Figures 49 and 50. The raw and normalised final values are plotted 1n Figures 51 and 52 respectively, while the normalised values of the ensemble mean are plotted 1n Figure 53 at much reduced scale. The average weight assigned clock 1 1s Included 1n Table 22 to Indicate Its effect on each algorithm. The most striking feature 1s the failure of run 1 1n which a prediction based on one 32-day PRI was used for the next PRI without further correction. The failure has little to do with the characters tics of the fir s t clock, but 1s due to the prediction method which proved quite unrealistic, and nothing further need be said about 1t. A comparison between runs 2 and 3 (AUS with PRIs of 8 and 32 days respectively) 1s Interesting. Run 2 succeeds better at rejecting the influence of clock 1, and Its prediction errors are smaller and whiter; In fact run 3 falls at one point despite Its fairly generous rejection c rite ria , yet i t gives the best-looking set of final values

1n Figure 5 2 . Their long-term stab ilities are about equal. Again, 1t may be inferred from Table 21 that the prediction errors In ARIMA algorithms (runs 6, 7, 8) are not quite white, due to the influence of clock 1 on these time scales. This also affects their long-term s tab ilities which become characterised by random walk of d rift much sooner than do the other algorithms (see Figure 50). TABLE 21 RESULTS UTC-REF ON SIMULATED DATA SET 3 : PREDICTION ERRORS (z R - zR)

RUN UTC-REF* 2 3 6 7 8 9 10 DNHtb) AUS(8,U) AUS(32,U) ARIHA(LS) ARIHA(U) ARIMA(D) EAL 16 EAL 32

PREDICTION ERRORS Allan Slone -0.40 -0.75 -0.53 -1.01 -0 .9 7 -1 .0 4 -0.61 • -0.61 Standard Errors: o(LS F it ) 2.48 0.87 0.50 0.18 0.20 Q.58 0.37 0.34 °l RS. Direct) 6.11 4.03 1.72 0.53 0.80 2.03 2.27 2.20 ° l RS. Unbias) 0.116 0.006 0.044 0.003 0.002 0.002 0.021 0.021 Autocorrelations: *>b) .94 .75 .89 -.05 -.14 -.11 .90 .90 P 2 )* .88 .53 .80 -.01 .03 .03 .80 .80 p(k>2) ed dec ed 0 0 0 ed ed FI rs t D iffs : 0 VRS, Unbias) 0.041 0.004 0.021 0.004 0.004 0.003 0.009 0.009 P 1 ) -.0 4 -.0 8 -.1 4 -.52 -.57 -.56 .01 .01 P 2) -.0 5 -.1 0 -.0 2 .03 .09 .08 -.01 -.01 P [k»2) 0 0 exc.k-8,16 0 0 0 0 -o '0 Second D iffs : °(v*RS, Unbias) 0.058 0.006 0.032 0.007 0.006 0.006 0.013 0.013 p (1 ) -.4 9 -.4 9 -.5 5 -.68 -.71 -.70 -.4 9 -.4 9 R(2) -.0 3 .00 .05 .20 .25 .24 -.01 -.0 1 p(k>2) 0 0 exc.k*8,16 0 0 0 0 0 0 ARIHA character (0,1,0) (0 .1 .1 ) (0 ,1 .1 ) (0.0.0) (0.0,1) (0 .0 .1 ) (0 .1 .0 ) (0 ,1 .0 ) (qualitative)

Notes: (*) For general explanations* see Table >5. ( b j 3 2 -day fits at 32-day Parameter Reevaluation Interval. TABLE 22 RESULTS UTC-REF ON SIMULATED DATA SET 3 : FINAL VALUES (z R)

RUN UTC-REF 1 2 3 6 7 8 9 10 DNM AUS(8,U) AUS(32,U) ARIMA(LS) ARIMA(U) ARIMA(U,8) EAL 16 EAL 32

RESULTS

Allan Slope t “1 ...8 -0 .6 4 -0 .2 9 -0 .3 5 -0 .3 0 0.83 -0 .2 2 -0 .3 0 -0 .3 0 t *1 6, • • 0 0.49 0.33 0.55 0.83 0.71 0.48 0.48 Standard Errors: o(LS F it ) 2.4B 0.87 0.50 0.18 0.20 . 0.58 0.37 0.34 o(zD, Direct) 2.48 0.87 0.50 0.18 0.20 0.58 0.37 0.34 o(zRI Uhblas) 2.01 1.82 1.79 2.10 10.13 5.68 1.75 1.75 Autocorrelations: p (1> 1.00 .99 .99 .99 .99 .99 .99 .99 p(2) .99 .99 .99 .99 .99 .99 .99 .99 p(k>2) vsed vsed vsed vsed vsed vsed vsed vsed First Diffs: o(vzD, Unbias) 0.024 0.015 0.016 0.018 0.085 0.048 0.015 0.015 n il r .45 .97 .86 .98 1.00 .99 .97 .97 p (2 ) .42 .96 .85 .97 .99 .99 .96 .96 p(k>2) ' . 5 vsed S Id vsed vsed vsed vsed vsed Second D iffs : o(vzzn, Unbias) 0.025 0.004 0.008 0.004 0.003 0.003 0.004 0.004 P ( 1 P ‘ -.4 8 -.4 8 -.4 9 -.4 7 -.4 5 -.4 3 -.4 7 -.4 7 p (2 ) -.1 6 .00 .03 -.01 -.0 1 -.0 2 -.01 -.01 p (k>2) *0 0 0 0 0 0 0 0 Values: Nintnun -14.88 0.18 0 .29 -0.01 -0.01 -0.01 0.16 0.16 Maximal 1.01 12.57 6.21 2.04 2.57 6.25 6.94 6.73 No. o f v a lid points 806 929 928 929 929 929 929 929 in 96sts1024 Av.weight of first 0.0025 0.0035 0.0061 0.0122 0.0939 0.0531 0.0004 0.0004 clock (0.3,0) PREDICTION ON ERROR TIME SCALE RESULTS (RANDOM SHOCKS)

-13 - 1 2 . RUNS RUNS

o,

-14-

9 ,1 0 Si 10,2

-15 -14-

I 2 4 8 16 32 64 I2S 7 f 2 4 8 16 32 64 120 t

FIGURE 49. Unbiassed Allan variances of prediction FIGURE 50. Unbiassed Allan variances of final errors (zR-zR), Data Set 3. values (zR), Data Set 3. I U E 1 eut o Ot St 3. Set Oata on Results 51FIGURE U1C - Rcr imeroi*eond*t to r T T>ay a > -T 2 3 « n M P i i — * — 04 DAYS 1024 H — ■ * ■ I— fillS IB) : 2 t RRiMAt t e BUS 132) t e»ui EFIL1161 3 t « l nRIM«I32,U) l 6 ------PREDICTION) . -----

h

IU E 2 eut o Dt St , omlzd t=96,192. t a normalized 3, Set Data on Results 52 FIGURE UTC - RCr tniero**tofid*l 20 0 256 766 64 DP.YS 1624 I AUS(01 I £ PREDICTION) 3- Y R 32-D M N D

I U E 3 nebe en f aa e 3 nraie a t=96,192. at normalized 3, Ensemble Set Data mean of 53 FIGURE a > * m c» * tRCr - Clock) mierof»cendti 150 -50 60 266 0 768 1024 03 o> 169 The results for EAL algorithms are very similar to those obtained 1n previous data sets - their long-term stab ilities are as good as the best of the other alrotlthms, because of their ability to down-welght clocks that do not perform well 1n the long term. They made practically no use of clock 1. (The Inadvertent omission of a test on the "old" USNO algorithm 1n Data Set 3, corresponding to runs 11 on the previous data sets, was perhaps unfortunate, but did not affect unduly the general conclusions drawn.) Another point that emerges from the final results on Data Set 3 Is the effect of different weighting schemes 1n ARIMA algorithms. The weights 1n run 6 were obtained from the unbiassed variances of residuals from (overlapping) 64-day fits reevaluated each 32 days, whereas runs 7 and B used unbiassed variances of random shocks over successive (non­ overlapping) Intervals of 32 and 8 days respectively. Run 6 showed much superior stab ility 1n Its long-term final results, much better rejection of clock 1, and marginally "whiter" prediction errors than runs 7 and 8. This result was masked 1n the other data sets because of the greater homogeneity of their ensembles.

5.6.4 Discussion

In a ll, 35 runs were performed on 4 data sets, three of which were Independent. Ideally, more simulated data sets should have been used with greater disparity 1n statistical characteristics between ensemble constituent clocks without necessarily having them as radically dif­ ferent as clock 1 1n Data Set 3; and a more comprehensive variation of the parameters Involved 1n weighting and predicting could have been devised 1n such a way that the comparisons could have been amenable to rlgourous statistical testing. Nevertheless, It 1s felt that, having due regard to the non-tr1v1al time and cost of each run, sufficient results have been obtained to draw some highly suggestive conclusions. The aim has been to find an algorithm whose final output 1s a white phase noise process. Obviously, this has not been achieved, nor has the production of a pure white frequency process, from the 170 selection of algorithms tried. However, the following results have emerged:

(a) Prediction errors (zR-z R) In the time scale can be made white noise of phase by basing the one-step-ahead prediction for each clock on the final value of the last point considered plus an optimal rate. ARIMA methods accomplish this automatically provided that the model adopted 1s correct, while rates calculated by least squares over Intervals that are not too long will produce substantially the same effect. The only justification for using direct evaluation of a straight line f i t 1s 1n the case when white phase noise 1s the dominant process over the fittin g Interval, which may occur when there 1s substantial measurement error.

(b) Superior long-term stab ility occurs when the weighting Is determined from the variances of straight line fits through the final results (rather than the random shocks) over extended periods, as exemplified 1n run 6 on each data set. This seems reasonable and does not conflict with point (a); Indeed, since point (a) discourages the use of long lines for prediction, the only place available for Incorporating long term Information concerning linearity 1n this time- scale formulation, 1s 1n the weights. (Alternative formulations w ill be discussed briefly 1n Section 5.8).

(c) The final results a ll tend to follow the ensemble mean, with the possible exception of USNO OLD (runs 11 ). This Is Inevitable to greater or lesser degree, because a t each step the weighted mean prediction error 1s set to zero. The practice of testing the errors against a rejection criterion alleviates the tendency somewhat, as can be seen contra by the fact that EAL algorithms (runs 9 and 10) generally follow the ensemble means closely, and they were not subject to rejections.

(d) The evidence presented does not strongly Indicate that the use of unbiassed variances in weighting 1s superior to the use of biassed variances. This 1s believed to be due to the nature of the data 171 sets used 1n the runs where Massing was tested, as a ll the clocks therein were more-or-less equally weighted anyway. {No bias-contrast­ ing runs were performed on Data Set 3, because I t was evident that the Influence of clock 1 was too small to provide any sensitivity to such a contrast. A more heterogeneous data set would be required, with perhaps the statistical parameters of each clock being slowly varied as i t evolves 1n such a way that 1t 1s among the "best” clocks for a significant proportion of the data span.) On the basis, therefore, of the theoretical arguments given previously, i t 1s considered desirable to weight the clocks according to the unbiassed variances, subject as always to a maximum weight lim it.

(e) Table 23 gives the standard errors of the random shocks of each clock 1n Data Set 1, and the autocorrelation coefficients of second differences of the final results (as opposed to random shocks) obtained by the unbiassed method on a selection of runs encompassing the types of algorithms being compared. The expected values shown are those used as Inputs to the simulations (see Table 12;, transformed In the case of the autocorrelations by means of the relations In Table 6. The degree of agreement between the recovered values and the expectations 1s a measure of how well the process parameters of each clock individu­ a lly may be estimated from results that w ill actually be available when using real data. The recovered values quoted 1n the last lines (REF) are taken from Tables 15-16 and therefore reflect primarily the departure of each time scale from Ideal (since REF was simulated to be almost perfect). It can be readily seen that the random shock standard errors are only properly estimated when predictions are based on the previous final value as 1n ARIMA (run 6) and USNO OLD (run 11) algorithm. This 1s entirely consistent with the discussions above. Assessment of the recovery potential for ARIMA weights ((l)] a 0.02 and ctCp (2)3 a 0.04, 172

TABLE 2 3 RECOVERY OF ARIMA RELATED PARAMETERS FROM TIME SCALES, OATA SET 1, SECOND DIFFERENCES

Clock Quantity Expected Unbiassed Value Recovered on Run: p j Value ( '-V ) 1 3 6 10 11 ______(a ) (b)

0.0124 0.0372 0.1298 0.0133 0.1346 0.0123 -.494 -.464 -.501 -.470 -.476 -.469 & 0. -.054 .029 -.042 -.041 -.048 oR 0.0104 0.036B 0.1600 0.0110 0.1724 0.0098 0 -.492 -.508 -.465 -.469 -.473 -.473 nS’ 0. .026 -.022 -.038 -.016 -.018 0.0107 0.0505 0.1844 0.0109 0.2300 0.0101 -.455 -.521 -.475 -.492 -.481 -.482 £ -.032 .018 -.030 -.024 -.023 -.040 0.0042 0.0150 0.0851 0.0046 0.0568 0.0041 -.345 -.509 -.420 -.317 -.348 -.360 £ -.103 .030 -.055 -.199 -.219 -.143 0.0139 0.0352 0.1492 0.0152 0.1459 0.0142 -.498 -.527 -.459 -.517 -.518 -.513 p(2 0. .039 -.002 .023 .022 .022 0.0060 0.0399 0.1466 0.0065 0.1808 0.0061 -.369 -.505 -.417 -.316 -.367 -.351 6 £ -.108 .034 -.089 -.183 -.154 -.189 0.0070 0.0528 0.1750 0.0076 0.2374 0.0080 -.243 -.416 -.391 -.283 -.301 -.342 7 £ -.110 .024 -.067 -.045 -.038 -.011 0.0048 0.0296 0.1141 0.005B 0.1445 0.0049 -.476 -.540 -.476 -.538 -.533 -.506 8 £ 0. .083 .007 .115 .109 .070 0.0047 0.0238 0.1444 0.0054 0.0857 0.0052 -.268 -.423 -.434 -.202 -.278 -.259 9 £ -.120 -.066 -.040 -.186 -.129 -.108 0.0080 0.0377 0.1602 0.00B3 0.1727 0.0075 10 to -.485 -.532 -.4B8 -.503 -.529 -.514 p ( 2 0. .034 .053 .000 .051 .029 REF 0.0010 0.0104 0.0454 0,0028 0,0467 0.0027 -.667 -.501 -.458 -.461 -.486 -.469 £ .167 .019 -.016 -.031 -.011 -.020

Notes: (a )o [p (k )] - 0.04 I f E[p(k)] ■ 0 for a ll kkq.

(b ) Expected values calculated directly from at, alt e, used In simulations. 173

the latter being consistent with Bartlett's test [Box and Jenkins, 19761 quoted 1n Table 23. Examination of recovered values of p(2) for clocks 7 and 8, for example, show that only four of the ten values lie within a ±2o confidence Interval about the expected values. It therefore seems that even unbiassed autocorrelation coefficients are only marginally useful in Identifying the (ARIMA) processes operating, le t alone 1n estimating the parameter values themselves; the extensive estimation procedures described by Box and Jenkins would need to be Invoked. (Time did not permit this In this study). The reason for such poor estimation of autocorrelations 1s that even second differen­ cing has not reduced the processes to pure white noise, as discussed 1n Section 4.2.2; similarly, only those prediction errors which are white are properly unravelled Into truly unbiassed random shocks.

(f) Figures 54-57 show the final results (zR) over appropriate data sets, for algorithm runs l(DNM), 3/4(AUS), 7(ARIMA) and 9(EAL) which have the same parameters. These are attempts to Illu strate the overall statistical characteristics of each type of algorithm, using the very limited number of samples available 1n each case. It would be Interes­ ting to plot on each graph the results on at least six Independently generated data sets extending out to t°4096, equivalent by ergodoclty to 24 1024-point runs, but that 1s not possible at present. Accordingly, no comment 1s offered here on these graphs. The overall conclusion arrived at 1n this discussion, then, 1s that for time scales bu ilt on the formulations of equation (5.2.14) or equation (5.2.27), the best results are obtained 1n terms of long­ term stab ility (lin e a rity ), short-term accuracy and computational tractablHty when predictions are true one-step-ahead predictions yielding white phase noise random shocks, and the weights are not necessarily equal but found by the method of unbiassed variances of residuals from straight lines fitted over extended intervals. The weights need to be restricted to prevent the occurrence of some un­ desirable effects (which actually become evident as ill-conditioning 1n the variance bias matrix B of equation (4.2.16)). The cautious application of po1nt-by-po1nt rejection criteria appears to improve I U E 5 Rsls f U agrtm n ah aa e, normalized Set, Data each on AUS algorithm of Results 55. FIGURE

UTC - RCT ( n ic r o « r <0 '.d;i gj 07C - *Er (llicroitcorxlt I -5 20 10 e 0 4 Rsls f N agrtm n ah aa e, normalized Set, Data each DNM on of algorithm Results 54. 0 0 ------t =96,192. t at t t=96,192. at ■ r ■ 256 512 ) 4 ( 1 buys 1024

174 175

BATA SET

* V o u • om

t

u I

DAYS -S e S I2 FIGURE 56. Results of ARIMA algorithm on each Data Set, normalized at t°96,192.

Vw s uft * co r

l . au I u

-S 0 25* 7*e

FIGURE 57. Results of EAL algorithm on each Data Set, normalized at t=96,192. 176 the results. The techniques and parameters chosen for run 6 1n Data Set 3 exemplify the preferred approach, although the practical evaluation of Its ARIMA parameters would not be simple. Despite the checks and precautions Included 1n the tests, each time scale tends to follow the mean d rift of Its constituent clocks. The simulated data sets have consisted of a sufficiently small number of clocks (5-10) for this tend to be apparent; clearly a much larger ensemble w ill give much better results 1

5.7 Comparisons using Real Data

Two sets of data were made available by the Division of National Mapping for testing. The data were obtained as part of the Division's operational timekeeping activ itie s, and have been processed by the algorithms described In Sections 5.5.5-6 which were developed 1n the early phases of this study.

5.7.1 Description of Data Sets

Both data sets used here were, 1n fact, the rem its of processing by the operational algorithms, so each Input datum had the form UTC(AUS)-Xj or UTC(DNM)-x^. Thus the operational time scales UTC(AUS) and UTC(DNM) acted as the counterparts of the reference clock REF 1n the simulated data sets; this 1s of no consequence for the tests, since 1n the formulations of Section 5.2 REF was always assumed to be arbitrary, while these choices for REF provided the possibility of assessing the performances of the real time scales. The UTC(DNM) data were the results of hourly readings taken over nearly a year from the five available commercial cesium standards; a period of 1024 hours was selected for testing, to accommodate the limitations of the testing program. It 1s remarked that the selection was not a simple matter, 1n view of the practical d iffic u ltie s associated with maintaining an automated timing system in an unfavorable environment - there 1s one gap of over 50 hours 1n the data set chosen, and several smaller gaps. Furthermore, much effort was expended in detecting and correcting for jumps and bad readings prior to testing 177

because, though they were excluded from the original time scale calcula­ tions, they were s till recorded 1n the data. Figure 33 showed some typical examples. The available UTC(AUS) data consisted of the dally results over a period of nearly three years, Involving a selection of 13 clocks. This data, obtained from the regular TV comparisons, was well edited since Its Input was manual and subject to close scrutiny each month; neverthe­ less many real jumps and rate anomalies persisted as well as births and deaths, as was Illustrated In Figures 24-29. The average measurement error was at least 0.1 us.

5.7.2 Results on Real Data Sets 5.7.2.1 UTC(DNM)

Table 24 shows the parameters allotted to experimental runs on this data set (cf Table 14 for simulation runs). The parmeters chosen reflect a good deal of tria l and error just to get the algorithms to cope with data deficiencies. In addition, 1t was found essential to modify ARIMA algorithms to accommodate gaps by using least squares pre­ dictors over (but only over) the gaps. Otherwise, the choice of para­ meter values was based on the experiences gained from the simulations. Results are presented In Tables 25 and 26, Flrueres 58 and 59 (Allan variances) and Figure 60 (plots). In the tables and Allan variance graphs the results for prediction errors are followed by the results for acutal final values. Unbiassed estimates are presented unless stated otherwise. The most striking outcome 1s the general failure of the technique of estimating unbiassed variances. This 1s revealed 1n the pattern of autocorrelation coefficients of prediction errors (Table 25) for successive differences 1n runs 4, 6, 8 and 11 - In run 11, for example, the coefficients for the zero-th difference would suggest that the firs t two lags 1n the fir s t difference should be approximately -0.5 and 0, and for the second difference should be -0.67 and +0.16. Examination of the raw computer printout showed frequent occurences 1n which the crude condition number of the matrix Inversion exceeded 2, which TABLE 2 4 . PARAMETERS OF RUNS ON DNM DATA

Run Type of No. of PRI Weight Pass Predictions(z<) Weights (p*) Rejection No Algorithm Passes Bias No Method Fit Method Max for Criterion In t'l(h ) o(ps) Cps)

1 ONM 1 1 U 1 LS 32 LS .003 MLS, .015 3 AUS 2 32 U 1 LS Prev 96 LS .010 MLS, .100 2 LS 128 LS .013 MLS, .132 4 AUS 2 32 U 1 LV+LSr Prev 96 LS .010 MLS, .100 2 LV+LSr 128 LS .013 MLS, .132 6 ARIMA 1 32 U 1 ARIMA (64) LS .004 MLS, .015 7 ARIMA 1 32 U 1 ARIMA (32) RS .005 MLS, .015 (LS over gaps) 8 ARIMA 1 32 U 1 ARIMA (32) RS .003 MLS, .015

10 EAL 3 32 B 1,2,3 LS Prev 32 BIH (.009 ns/h) - 11 USNO OLD 3 32 B 1,2,3 LV+LSr 32 Equal (.0028 ns/h) .040

Explanations As for Table 14. Note that data interval is one hour, rather than one day. ARIMA model for each clock is (0,2,1), ej = 0.75. 179

Indicated either insufficient clocks or Inadequate distribution of weights (cf equation 4.2.19 ). These occurrences happened both 1n the computation of weights each Parameter Reevaluation Interval during the evolution of a run, and 1n the estimation of the run's overall statistics as presented. The sim ilarity to the results for simulation run 1(a) based on five clocks 1s pronounced. It Is therefore clear that five clocks are insufficient for a time-scale ensemble using automatic unblasslng. In Figure 58, 1t 1s seen that, overall, run 7 has the most desirable Allan variance of prediction errors. It's fla t region

t * 2 - 8 , Is due to the change 1n prediction method over gaps, otherwise the prediction noise 1s white of phase and has the lowest variance. Run 1 also has low variance although Its process 1s more random walk of phase. In both these runs, the prediction formulae are updated each hour 1n a robust manner. The other true one-step-ahead prediction methods (ARIMA, runs 6 and 8) are badly affected by the data vagaries. In runs 4, 10 and 11 the predictions use the previous adopted value corrected by a rate which 1s only updated at the PRI; the poor perfor­ mance of run 10 1s caused by the fact that the rate Is not filte re d (corrected Iteratively from current observations) which makes 1t much more d iffic u lt for the algorithm to recover after a data loss. The predictions 1n run 3 are less white than In any other runs because they are purely least squares over a long Interval, but have smaller errors than run 4 (1n particular) because poor data are more effectively filte re d . The concept of "algorithm recovery after data loss" Is Illustrated 1n Figure 60 1n which results after loss are what would be expected 1f there were no such loss, 1n a ll runs except 10. The apparent recovery 1n run 10 can be considered fortuitous since there are obvious discontinuities 1n both offset and rate. (The results of run 8 are not plotted as they showed similar tendencies.) When the results of final values (z^) as presented 1n Table 26 and Figures 59-60 are examined, 1t 1s Immediately seen that runs 1 and 3 are superior to the others in a ll respects; run 3 1n particular has TABLE 25 RESULTS UTC-REF OH DNH DATA SET: PREDICTIOH ERRORS ( z R- i R)

UTC-REF*a* RUN

1 3 4 6 7 8 10 11 DNH AUS AUS ARINA ARIHA ARINA EAL US NO (32) (LS) (LS+LSr) (64) (32,LS )tbJ (32) (32)

PREDICTION ERRORS Allan Slope -0 .5 9 -0.61 -0 .9 0 -0 .8 4 -0 .6 8 -0 .3 9 -0.61 -0.91 Standard Errors: o(L5 F k i 0.060 0.022 0.103 0.586 0.386 0.495 1.229 0.116 o (R S , Direct) 0.834 0.654 1.207 0.595 1.810 1.294 3.880 1.538 o(RS, Unbias) 0.006 0.010 0.024 0.021 0.007 0.017 0.390 0.025

Autocorrelations: p (D 0.79 0.88 -0 .2 9 -0 .0 9 0.81 -0 .0 9 0.95 0.00 P12) 0.61 0.80 0.00 -0.07. 0.66 -0.06 0.90 0.00 p k>2) Id ed 0 0 ed 0 Id 0

First Diffs: o( RS, llnbias) 0.003 0.004 0.032 0.025 . 0.003 0.020 0.120 0.026 p | 1) -0 .0 5 -0 .1 0 -0 .2 9 -0.09 -0 .2 8 -0 .0 5 0.00 0.00 p ( 2 ) 0.02 -0 .0 3 0.00 -0 .0 3 0.13 -0.03 0.01 0.00 P(k>2) 0 0 0 0 0 fo r k>3 0 0 0 Si>cond D iffs : 0 V* RS, Unbias) 0.004 0.007 0.043 0.028 0.005 0.022 0.172 0.026 P -0 .4 3 -0.52 -0.29 -0.18 -0 .6 5 -0 .0 5 -0 .5 0 -0.02 p 2) 0.02 0.02 0.02 0.03 0.17 0.01 0.00 0.00 p k>2) 0 0 0 0 0 1 0 * 0 No. o f v a lid p o in ts '6* 834 837 837 809 793 750 655 836 ARINA character (0 .1 .1 ) (0 ,1 .0 ) ? 7 < 0 ,1 .2 ) ? (0 ,1 ,0 ) ? (qualitative)

Notes: (a) REF In this case Is operational UTC(DKH). lb ) Least squares predictors used over gaps In date. (c, Number of points used to evaluate unbiassed estlnates. Also see notes fo r Table IS . TABLE 26 RESULTS UTC-REF ON DNH DATA SET: FINAL VALUES (z R)

UTC-REF1a* RUN

1 3 4 7 10 . . 11 DNH AUS AUS A R IH A ^ ARIHA ARIHA*C* EAL,C* USNO (32) (LS) (LV+LSr) (64) (3 2 .LS) (32) (32) (32) RESULTS

A llan Slope t *1 . . . . 8 -0 .7 5 -0 .7 9 -0 .4 9 -0 .3 2 -0 .2 3 -0 .2 9 -0.49 40.63 -0 .0 7 -0 .2 4 -0 .5 6 0.09 0.24 0.48 •0.36 -0 .5 3 Standard Errors: oJLS Fit) 0.060 0.022 0.103 0.586 0.386 0.495 1.229 0.116 o(zR, Unbias) 0.049 0.025 0.064 0.298 0.219 0.323 0.780 0.145 Autocorrelations: p i ) 0.9B 0.96 0.96 0.98 0.97 0.98 0.98 0.98 p Z> 0.96 0.94 0.94 0.96 0.96 0.96 0.97 0.96 p(k>2) sed sed sed sed sed sed sed sed to k-12 First Diffs: otvzo. Unbias) 0.002 0.004 0.007 0.007 0.003 0.004 0.057 0.002 p(l) -0 .0 9 -0 .0 8 0.00 0.39 0.69 0.74 0.00 0.01 P(Z) 0.04 -0 .0 3 0.00 0.37 0.71 0.76 0.00 0.14 p(k>2) *0 ‘ 0 0 *0 *0 vsed 0 0 from k»14 Second D iffs : c (v * zr , Unbias) 0.003 0.005 0.009 0.007 0.002 0.002 0.082 0.002 p(1) -0 .3 9 -0 .5 0 -0 .5 0 -0 .4 9 -0 .5 4 -0 .5 4 -0 .5 0 -0 .5 5 p(2) 0.05 0.01 0.00 -0.04 0.04 0.03 -0.01 0.04 p(k>Z) -o *0 0 >0 at k-7 0 0 0 0 Values: M inim a 0.000 0.162 0.168 0.000 0.000 0.000 0.182 -5.511 Hajciwa 3.073 2.376 4.176 2.393 6.845 3.344 12.054 -0.149 Ave weight, clock 4 0.027 0.054 0.052 0.109 0.055 0.076 0.031 0.073

Notes: (a ) See Tables 16 and 23 fo r explanations. (b) Autocorrelations In differences non-zero at k«16. (c) Unblasslng not really successful. PREDICTION ERRORS ( RANDOM SHOCKS)

*13

10 E A L

ARIMA ' 3 AUS >* 1 DNM v 7 ARIMA — i— i— i— i— i— i____!__► 1 2 4 8 1 6 32 64 128 t I 2 4 8 16 32 *4 128 t

FIGURE 58. Unbiassed Allan variances of prediction FIGURE 59. Unbiassed Allan variances of final errors (zR-zR), DNM Data Set. values (zR), DNM Data Set. FIGURE

UTC - RCrUHrti (hierofttondi> 0 Rsls n N Dt St nraie a t9,9 hours. t=96,192 at normalized DNM on Results Set, Data 60. 26 1 768 04 HOURS 1024 8 6 7 512 256 B t H 31 ■ PHH <321 t I j USNOfOLD) i I I 0 ) 2 O L 0 E i 10 7 i US3,VR) S(32,LV*R RU i 4 fRMRl U) ,U 4 l6 R M flR t 6 3 fRl 2LS» S 32.L » lW fiR : fUSt2L ) t32,LS S flU i

184

a very low scatter about a straight line fit which indicates that 1t provides the best long-term f il t e r against noise processes. Further,

runs 1, 7, 8 and 11 have better short-term stab ility than the others which is a consequence of their superior prediction methods. These results tend to confirm the conclusions arrived at from the simulations, namely that predictions should be based on s tric t one-step-ahead methods while weights should be assigned from straight line fits through final results over extended Intervals.

5.7.2.2 UTC(AUS)

Preliminary runs on the UTC(AUS) data set gave such poor results that the tests were abandoned. This 1s attributed to the regularity

of gaps 1n the data which occurred every weekend, and to the measure­ ment noise occasioned by the TV method which masked the clock noise processes. I t proved unexpectedly d iffic u lt to find the correct range of rejection criteria, weight maxima and starting values to allow each algorithm to catch on. (An extended data set with up to 14 contributing clocks, few weekend gaps and Improved TV reception 1s now available, but time has not permitted Its testing.)

5.7.3 Discussion

Within the lim its Imposed by real data, the tests using the UTC(DNM) data set verified the simulation results that short-term

stab ility of a time scale 1s enhanced by using rigorous one-step-ahead

prediction methods, whereas long-term s tab ility 1s best served by assigning weights based on the long-term linearity of the contributing clocks. Note that the weights should be calculated from the final results and not from the prediction errors. The tribulations caused by real data have been emphasised. Loss of data to the extent that Insufficient clocks are available to the algorithm, 1s of particular concern. Such a state of affairs can be envisaged 1n even the best-run operational environments, for failures

1n data acquisition and clock comparison equipment are Inevitable, not to mention failures and changes in the clocks themselves. Multiple 185

redundancy of the equipment 1s, of course, necessary (for example, measurements should be made against a ll available GPS s a te llite s, not just one) but a properly designed algorithm w ill accommodate loss of data over a certain period of time. Pure ARIMA techniques are not satisfactory here, for the variance of predictions more than a few steps ahead Increases too rapidly. By substituting when necessary a technique with smaller prediction variance, I.e . with a smaller confidence Interval around the prediction, I t has been shown that the algorithm can recover satisfactorily. Least squares was used as the bridge 1n this study but

1s not necessarily the best method - i f the gap extended 1n to the white of frequency region of a clock's relationship to the time scale, for example, the two-end-po1nt predictor would be appropriate. There 1s, naturally, some loss of statistical purity when a bridge Is used, but this 1s a small price to pay for maintaining the time scale. The purity can be retained to some extent by tailoring the rejection criteria to the specific noise processes known to be occurring 1n the clocks. The problem of measurement error has only received passing attention 1n the preceding chapters, but Its Importance has been revealed 1n this section. In the operational UTC(ONM) program the rejection criterion 1s designed primarily to reject spurious observa­ tions rather than to weaken the link between ensemble mean and time scale, while the least squares predictor's principal function 1s to smooth the remaining measurement errors. Clearly 1n the tests described here the more successful runs have been those which employ least squares as these perform the required data smoothing functions; again, ARIMA Is not satisfactory 1n this respect. In the operational UTC(AUS) algorithm. Isolated bad readings are removed prior to processing, while those clocks suspected of being subject to large measuring errors are simply not Included In the ensemble. Unbiassed variance estimates were employed throughout where appropriate, but were not really effective because of the smallness of the ensemble and because determination of the effects of measure­ ment errors would have required several Iterations of whole runs. Also, they are only applicable to white noise, so the usefulness of the 186

results (except for most prediction errors) 1s questionable. In

particular, I t was not possible 1n practice to estimate the characteris­ tics of the operational UTC(DNM) time scale as suggested at the start of Section 5.7.1.

5.8 Algorithm Extensions

A large amount of time was spent 1n trying to find ways other than those described previously, of building a time scale with excellent short-term and long-term stab ility without the need for substantial revision. Some of these are quickly described.

5.8.1 Split Ensemble

An Intuitively appealing way of enhancing short-term stab ility Is to base the predictions on only the clocks with known excellent short-term performance. In Its barest essentials, then, suppose that a hydrogen maser 1s available but not wanted as an actual ensemblemember because of Its long-term variations 1n frequency. Let this clock be the reference clock R (cf equations (5.4.31-32)). Then, Instead of using the standard prediction method for each clock separately, e.g.

^ ( t ) ■ Zjf t - l) + »yr (5.8.1) as 1n equation (5.2.13), base the predictions on R its e lf:

Zj(t) a zR(t) + tj(t) (5.8.2) where ^ ( t ) 1s the actual measurement R(t) - x^(t) at time t , and zR(t) is the optimal predictor for X-R. It Is easy to show that the solution of the basic time scale equation (5.2.16) 1s, in this modified case:

Zj(t) » zR(t) + *t(t) (5.8.3) whereas the standard result was

Zj(t) * zR(t) + tjU) (5.8.4)

1n which zR(t) Includes any fluctuations 1n ensemble members, as shown by equation (5.4.31). Since the only errors 1n the *^ (t) 187 are those of measurement, the modified solution (5.8.3) w ill evolve 1n

conformity with the reference clock alone, which 1s not what was wanted. The standard solution (5.8.4) w ill evolve as the weighted mean of ensemble members, as desired. Thus the suggested modification, while undoubtedly Improving the time scale's short term s tab ility , sacrifices altogether the desired long-term stab ility. The same result 1s obtained 1f the predictions are based on a

supposedly superior subset split off the main ensemble, or 1f the reference clock 1n fact belongs to the ensemble. It would appear that

the only use for such a subset 1s for creating a preliminary but separate time scale for driving physical devices or for detecting anomalous data.

5.8.2 Extended Use of Historical Data

Another approach wqs considered 1n an attempt to Improve long­ term stab ility 1n a manner analogous to that employed by the BIH when steering EAL to TAI. It 1s Illustrated 1n Figure 61. For each clock,

a straight line 1s fitte d through the final results In several preceding Parameter Reevaluation Intervals. A prediction z jk^(t) and a confidence

Interval sj^(t) 1s calculated for the computation time t from the k-th preceding PRI. The confidence Interval Increases with the distance of t from the mean data of Its PRI. The final prediction z^(t) 1s the mean of m such predictions, weighted according to the confidence Intervals:

zAt) = a ? zjk)(t)/[s!k)(t) ] 2 (5.8.5) i k„i i where a 1s just a weight normalising factor; the weight assigned to * Z j(t) 1n the time scale solution 1s derived from a combination of the confidence Intervals sj ; (t) with the variance of the scatter of the Individual predictions about their mean. The weight should Increase

either 1f the sjk^(t) are all small, or If there 1s good agreement

between the z j^ ( t) , thus achieving good stability 1n both short and long terms. Assuming that this weight 1s Inversely proportional to a variance S |(t), two expressions were obtained from slightly different standpoints; 188 sj(t) - I [s( k ) ] 2 + m Z [z {k) - Z j]2/(m -l) (5.8.6) i k=l 1 k»l 1 sUt) - { " Cz\k) - Z j]2/(m -l)> { ! Cl/sSk>] %>/( I [l/s i(k)]*)*.{5.8.7) 1 k-1 1 1 kBl 1 kal 1

The former was tested on Data Set 3 of Section 5.6. As could be anticipated from the extrapolations quoted In Table 1 3 , the short term s tab ility was so atrocious that no further tests or refinements were attempted. Clearly, the use of least-squares predictors from results 1n the distant past 1s Inappropriate. It 1s thus evident that there has to be a trade-off between short-term and long-term s tab ility . From a ll the previous discussions, the most satisfactory solution Is to base the predictions on a re­ latively short t 1me-span so that they are truly unbiassed, while weights should be based on variances over the longest realistic t 1me-span desired.

PARAMETER RE-EVALUATION INTERVAL

JL

FIGURE 61. Weighted predictions from results 1n several previous Intervals. 6. SUMMARY AND CONCLUSIONS

6.1 Summary

This study has basically been an examination of solutions to the

basic time scale equation, given 1n Its simplest form by equation (5.2.16), 1n the presence of clock random noise processes assumed to follow the standard stochastic models, summarised 1n Table 5, with par­ ticular attention being paid to Yoshlmura's method of estimating clock variances In a manner not biassed by the Influence of the clock on the time scale, described 1n Section 4.2.2. In order to obtain an understanding of the nature and scope of the work, 1t was found desirable to explore the relationship between time and clocks. After a brief review of the philosophical nature of time, a series of definitions was adopted 1n Section 2.2.1 leading to a defini­ tion of time as "that which 1s Indicated by clocks". I t could be argued that this definition 1s recursive, but 1t 1s certainly useful and consistent for all subsequent discussions. Armed with this foundation,

1t was then possible to describe the formation of time scales, I.e . systems for dating events, In terms of clocks as they have evolved throughout history (Section 2.2), and to describe modern methods of transferring time from place to place by comparing clocks (Section 2.3). From Chapter 3 onwards, only atomic clocks based on transitions In cesium, hydrogen and rubidium atoms were considered. Section 3.1 described their general operating principles 1n some detail, albeit In an lllumlnatory rather than a physically rigorous style, to show where the main sources of error H e. That the errors thus arising follow certain statistical laws Is accepted without question; these laws are summarised 1n the "power law spectrum" of equation (3.2.2) and are described Individually 1n subsections of Section 3.2.3. The operation of a particular error law can often be Identified

189 190 from the shape of Its Allan variance graph (except the flicker of phase modulation law) as Illustrated 1n Figure 7, but 1n real atomic clocks several laws operate simultaneously which blurs the graph. An alterna­

tive method of representing the errors 1n terms of autoregressive Inte­ grated moving average (ARIMA) processes, as Initiated 1n the time scale context by Perdval [1978], was Introduced 1n Section 3.2.2 and found to be an excellent tool. ARIMA describes the white noise processes opera­ ting on phase, frequency and d r ift very simply, and was used to prove the time dispersion characteristics of these processes 1n Section 3.2.3. The power of ARIMA was demonstrated 1n subsection 3.2.3.1 where the

superiority of the least squares method over the two-end-po 1nt method 1n determining clock rates 1n the presence of white phase noise was rigor­ ously and easily proved, as was the contrary case for white frequency noise. For flicker noise processes, 1t was found 1n subsection 3.2.3.5 that ARIMA could not represent them economically. The evidence was obtained empirically by simulations, 1n which 1t was attempted to f i t ARIMA models to data simulated by the very effective flicker noise generator of Barnes and Jarvis [1971], and 1t was confirmed from time dispersion arguments (see equation (3.2.62)) that a likely looking ARIMA model, (0 ,2 ,1 ), was Inadequate. These explorations, and those 1n Chapter 4, were done primarily to understand the methods of weighting, prediction and evaluation used 1n the various formulations of solutions to the basic time scale equation which are discussed 1n Chapter 5. To provide a predictor for flicker noise, the NBS recursive filte r [Allan et a l., 1974] was modified and tested 1n Section 4.8.2, with reasonably encouraging results. The well-known time dispersion and Allan variance functions for flicker processes were reproduced numeri­ cally by simulating 50 such processes Independently to obtain strong estimates of the variances concerned, and of their uncertainties. Together with similar experiments for white noise processes, these provided (1n Section 4.1.1) a high degree of confidence 1n the simula­ tion and analysis techniques used later on. 191

The remainder of Chapter 4 dealt with the problem of estimating the variances of Individual clocks when the only data available Involves such combinations of clocks as time-1nterval comparisons or accessible time scale results. A fallacy In the application of the "three- cornered-hat" method was exposed 1n Section 4.2.1 by showing that 1t only works when there are no sample cross-correlations between clocks, which generally w ill only occur 1n fin ite samples when there are no autocorrelations within the clocks. In Section 4.2.2, this observation was extended to the method of Yoshlmura [1980] for extracting the "unbiassed" estimates of clock variances (Including covariances and Allan variances) from the "direct" estimates obtained 1n time scale algorithms, since the direct estimates for a given clock Include contri­ butions from all the clocks (equations (4.2.13), (5.4.31-32)). A suggestion vras made 1n Section 4.2.3 for the application of unblasslng to determination of correlations between measurement errors because these are known to exist 1n practise. I t was found that unblasslng could work provided that not too many measurements used for a time scale calculation are cross-correlated. The main work 1n the study was contained 1n Chapter 5, the objec­ tive being to Identify those procedures which lead to the best free- running time scale out of a selection of those available, where "best" Is a combination of superior short-term and long-term s ta b ility , and I f possible to Improve upon 1t. The restriction to free-running scales was not Inappropriate, for 1t 1s almost certain that the next generation of TAI w ill be calculated from laboratory cesium standards (and perhaps hydrogen masers) operating as clocks for which there w ill be no superior,

1f intermittent, reference. After some Introductory definitions, the formulation of the basic time scale equation (5.2.16) was built up step-by-step 1n Section 5.2 to

Illu strate and demonstrate how 1t averages a number of clocks, and how

1t has the b u ilt-in fa c ility to cope with changes 1n the number and quality of the clocks being so used. I t does this by imposing the basic condition that the weighted sum of prediction errors be zero at any 192 given time, where the predictions are estimates of the final results for each clock relative to the time scale based on previous performance. If the predictions are unbiassed estimators, then the expected value’of the

time scale 1s unaffected by the changes just referred to at the times of

occurrence of those changes*, in other words, continuity 1s ensured at each step. The basic time scale equation 1s the heart of nearly every

algorithm In current usage - 1t 1s a condition Imposed on an otherwise

rank-deficient set of observation equations, and 1s chosen for Its continuity property. The differences between time scale algorithms are 1n the methods actually used for prediction, weighting, and acceptance or rejection of the ensemble clocks. The c rite ria tor temporary rejection of clocks were discussed In Section 5.2.6 1n the context of editing bad data, but 1t was subsequently found that they could be used to advantage 1n smoothing the scale even when a ll the data Is free of measurement error. The

simpler methods of prediction and weighting were Introduced 1n Section 5.3. A temporary excursion was made 1n to the fie ld of steered time scales 1n Section 5.4. "Steering" 1s the correction of time scale results so that they agree at least partially with some external scale or reference with which comparisons may be sporadic or error-prone. This section was Included primarily to Introduce the concept 1n prepara­ tion for susequent descriptions of steering procedures with potential application to local time scales struggling to maintain contact with the International standards. It was comforting to see that the general solution of the free-running time scale equation emerged out of the solution for a rate-steered system (Section 5.4.4) 1n the special case when there were, after a ll, no external observations. Continuing on the way to actually comparing algorithms, Section 5.5 provided a compendium of methods In current usage at the BIH, US Naval Observatory, US National Bureau of Standards, Australian Division of National Happing, and others, summarised 1n Table U . Particular atten­ tion was given to the different ways 1n which they calculate predictions 193 and weights (which are also Illustrated In Figure 35) and to how they

select or reject particular clocks at any given step 1n the calcula­

tions* as 1t 1s Important to recognise when the prediction and weighting methods become Inappropriate. The practice of assigning an upper weight

lim it was also highlighted, for without such a lim it 1t 1s easy for one clock fortuitously to receive all the weight, especially with small ensembles, therby negating the benefits of averaging. Steering proce­ dures Implemented at the BIH and NBS were Included, because 1t was hoped

that those methods could be Incorporated 1n to the free-running algo­ rithms to Improve long-term s tab ility . (This hope was not realised.) Comparisons between algorithms were effected 1n Section 5.6. It should perhaps be emphasised that the comparisons were between algo­

rithms rather than between time scales, because a time scale s tric tly 1s the result of a particular algorithm applied to particular real clocks, whereas Section 5.6 compared variations of several algorithms applied to common data sets. Four data sets were generated, three of which were Independent simulations of ten clocks, the other being a five clock subset of the fir s t. Clock noise models were either ARIMA or flicker of frequency, and the only arbitrary assumptions made were that their In itia l offsets and rates were zero. I t 1s fe lt that ten was a good choice for ensemble

size, as 1t allowed fu ll scope for the features of each algorithm to be displayed without being swamped by the averaging effects that larger ensembles would have produced, I.e . 1t was not too large, nor was 1t

too small as some breakdowns 1n the five clock ensemble demonstrated In

subsection 5 .6 .3.2. I t Is also fe lt that taking each ensemble out to 1024 points was a satisfactory compromise between statistical validity as judged from the results that had been obtained 1n Section 4.1.1, and processing costs. Full descriptions of the data sets were given 1n Section 5.6.1. Numerical experiments were performed on the four data sets, as described In Section 5.6.2, especially Table 12. A certain amount of subjective judgement was called upon 1n deciding which parameter values 194 (fit Intervals, weight maxima, rejection criteria, etc.) to choose for any run, to keep the processing within reasonable lim its. This was justified by the emergence of unmistakable trends 1n the results. The only procedure not adequately tested was that for unbiassed weighting because the data sets were too homogeneous. The results of a ll these tests were given 1n detail 1n the subsec­ tions of Section 5.6.3, and summarised 1n Section 5.6.4. I t 1s from these that the prlnlcpal conclusions are drawn 1n the next section. The basis chosen for the conclusions was the relative phase whiteness of the prediction errors and final results, as judged from Allan variances (which proved quite adequate for the task, although a test for random­ ness such as Kolmogorov-Smlrnov's would have Improved discrimination.) Using the experience gained on the simulated data sets, a selection of algorithm variations was tested on some real data described, with a ll Its faults, 1n Section 5.7.1. Table 20 set out their details. On only one data set of the two made available was 1t possible to get meaningful results, and with considerable d iffic u lty at that due to large gaps 1n the data and to non-negl1g1ble measurement errors. Within these lim ita­ tions, the conclusions drawn from the simulations were confirmed, although with some reservations about the usefulness of ARIMA in a nosly, loss-prone environment. Unbiassed variances were not well esti­ mated, either, because only five clocks at most were In the successful ensemble. The detailed results and their consequent admonitions were discussed 1n Section 5.7.3. Finally, 1n Section 5.8 i t was shown that, for the basic time scale formulations considered 1n this study, 1t Is not feasible to improve both short-term and long-term stab ility simultaneously within the one algorithm by the apparently attractive technique of predicting on only the best short-term clocks (Section 5.8.1) or on only the best long-term clocks (Section 5.8.2) respectively. This 1s ultimately a consequence of the rank-deficiency of these formulations. 195 6.2 Conclusions

The principal results and conclusions to emerge from this study are:

(a) The study of free-running time scales Is s t ill necessary 1n the light of likely future trends.

(b) Autoregressive Integrated moving average stochastic processes are highly applicable 1n describing most clock noise processes, and are very convenient for studying and modeling them.

(c) The estimation of variances of Individual clocks, whether by the "three-cornered-hat" method for clock comparisons or by Yoshlmura's method for time scale results, 1s only valid when there are no sample cross-correlations between the clocks. In the time scale case, at least five good clocks are required 1n practise to apply unblasslng with confidence.

(d) Variances of measuring errors cannot be estimated directly from "three-cornered-hat" types of observations, but may be by modifying Yoshlmura's method 1n which case correlations between measurements may be detected 1f they are not too severe.

(e) In the construction of a time scale from clean data, predictions must be completely unbiassed, which Implies that they update results of the previous step using rates that are determined optimally from a relatively short preceding Interval. ARIMA 1s recommended, but others such as least squares are satisfactory 1f used judiciously. The quality of the predictions 1s the principal determinant of short term stability when long term stability 1s also desired.

(f) Long-term stab ility 1s best ensured by assigning weights on the basis of final results, not of prediction errors or random shocks, over as long a period as possible. The value of using unblaseed variances was not established. A maximum weight lim it needs to be Imposed to safeguard the benefits of averaging. 196

(g) Polnt-by-polnt application of rejection criteria helps to break the connection between time scale results and simple ensemble means, but should not be made too tig ht. I t can be Implemented by manual selection of clocks deemed desirable at any stage.

(h) I t does not appear possible to fkilly optimise both long and short term s tab ility 1n a single algortlthm under the formulation considered; rather, a compromise must be reached within the desired operational goals. Nor does 1t appear possible to produce a time scale whose results are pure white phase noise.

(1) Real data w ill 1nvev1tably require modification to the predictions. In parlcular, ARIMA does not handle gaps adequately. Also, vari­ ance unblasslng requires correction for measurement error.

(1) Finally, 1t must be remarked that the best results of all will be obtained from a very large ensemble of superb, truly Independent clocks, Ideally Intercompared I

6.3 Recommendations for Further Study

(a) No conclusive answer on the effectiveness of variance unblasslng emerged. Further simulations of more heterogeneous data sets may provide an Indication on whether the unblasslng equations can be modified to account for clock auto-correlations, and on what are the appropriate weights to Insert in the B-matrlx of equation (4.2.16) when the weights actually used 1n an algorithm change from time to time.

(b) There were no cross-correlations between the random shocks used to generate the various clock processes with each data set In Chapter 5. Since 1t 1s known that real clocks do exhibit such cross-correlations, their effects on a given algortlthm could be studied using the following procedure: Let x^(t), Xg(t) be Independent but identically distributed zero-mean outcomes of a white noise generator. A transformation to outcomes y ^ (t), y2(t) can be obtained via: 197

x ,(t) (6.3.1) Lx2ct)

whereby the new outcomes have vdiatever cross-correlation coefficientp Is specified [Winkler, G., private communication, 1982]. I t may be that the inverse of equation (6.3.1) could be used to break the sample cross-correlations that spoil attempts to remove blasses from variance estimates.

(c) A much better appreciation of the evolution of algorithms than that hinted at In Figures 54-57, would be obtained by creating, say, 10 data sets each having 10 clocks extending out to 4096 points or further, simulated 1n such a way that there would be no correla­ tions between the data set means. Application of the same algo­ rithm to each data set would then provide results for numerical evaluation of time dispersion and other statistical properties Introduced by the algorithm. By combining a ll 100 clocks Into one ensemble and running the chosen algorithm again, 1t should then be possible to establish exactly what effect ensemble means have on the results 1n the long term.

(d) General conclusions were made on the basis of relative whiteness as judged by Allan variances. More specific conclusions, such as the optimum prediction Interval for a given algorithm, would require more sensitive tests such as the Kolmogorov-Smlrnov test for ran­ domness.

(e) There are strong sim ilarities between the methods of statistical modeling chosen In this study and the methods of collocation theory which, In turn, resemble those of the Kalman f ilt e r . I t would be very Interesting to study the relationships between these methods, and the circumstances under which they could or should be applied to time scale calculations. They may be particularly appropriate 198

when near real-time results are required and the Input data are corrupted by significant measurement errors.

(f) The basic time scale equation (5.2.16) Is the (single) condition chosen to enable the otherwise rank-deficient set of observation equations (5.4*14) to be solved uniquely. It states that, 1f certain safeguarding properties are desired, then clock error predictions need to be unbiassed. I t Is not Inconceivable that a different formulation would lead to properties that are even more desirable. LIST OF REFERENCES

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