Between July 1973 and January 1974, three time transfer experiments using the Timation 2 satellite were conducted to measure time differences between the U.S. Naval Observatory and Australia. Statistical tests showed that the results are unaffected by the satellite's position with respect to the sunrise/sunset line or by its closest approach azimuth at the Australian station. Further tests revealed that forward predictions of time scale differences, based on the measurements, can be made with high confidence

Luck, J. M.

Morgan, P.

English

NASA Technical Reports Server

STATISTICAL ANALYSIS OF TIME TRANSFER DATA 
FROM TIMATION I1 
J. McK. Luck and P. Morgan 
. Division of National Mapping 
A.C.T., Australia 
ABSTRACT 
Between July 1973 and January 1974, three time 
transfer experiments using the Timation I1 satel- 
lite were conducted by the Division of National 
Mapping, A.C.T., Australia and the US Naval 
Research ~aboratory, Washington, DIG,, to measure 
time differences between the US Naval Observatory 
and Australia. Statistical tests showed that the 
results are unaffected by the satellite's position 
with respect to the sunrise/sunset line or by its 
closest approach azimuth at the Australian station. 
Further teats revealed that forward predictions of 
time scale differences, based on the measurenents, 
can be made with high confidence. 
Measurements Against t%e Satellite Clock 
I The results of the first two-time transfer experiments 
between NRL and Australia have already been presented 
(Easton, Smith and Morgan, 19738 1974). 
The first series of statistical tests examined the residuals 
from a quadratic fit of the results TII-AUST, where TI1 re- 
presents the satellite on-board oscillator and AUST represents 
the local Australian time standard, in this case the National. 
Mapping portable cesium standard DNM590 whose performance was 
linear with respect to a four com2onent mean time scale. The 
measurements in July showed that the on-board oscillator had 
a constant aging rate during the experiment, and the residuals 
were normally distributed, while during the September run the 
residuals were not normally distributed. The oscillator was 
evidently much less stable in the January 1974 run, and a 
simple curve could not be fitted. 
The curves fitted were: 
2 July run: TII-AUST = 5.371 -t 4.7616(t-E) + 0.25214(t-E) 
microseconds, 
where t was the day of the year and w a s  the midpoint of the 
199.153). The standard deviation of the 38 residuals 
was run 6 -585 microseconds. 
513 
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September run: TII-AUST = 38.940 + 3.9779(t-E) - 
0.00956 (t-E) microseconds , where E was 
265.544. The standard deviation of the 42 
residuals was 1.653 microseconds. 
For each run, the residuals were analyzed by a two-way 
analysis of variance with unequal cell sizes (Hamilton, 
1964), the classifications being: 
(1) Effect of sunlight, to see if it affected either the 
satellite clock through temperature variation, or the signal 
travel time. 
This classification was sub-divided by the time of closest 
approach: 
(i) more than 2 hours before sunrise or more than 2 
(ii) 1 to 2 hours before sunrise or 1 to 2 hours after 
(iii) 0 to 1 hours before sunrise or 0 to 1 hours after 
(iv) 0 to 1 hours after sunrise or 0 to 1 hours before 
(v) 1 to 2 hours after sunrise or 1 to 2 hours before 
hours after sunset; 
sunset; 
sunset; 
sunset; 
sunset; 
hours before sunset. 
(vi) more than 2 hours after sunrise or more than 2 
(2) Effect of azimuth, to see if the local surroundings 
had any effect. This classification was sub-divided by the 
azimuth at closest approach: 
(i) 
(iii) 
in the quadrant North to East; 
(ii) in the quadrant East to South; 
in the quadrant South to West; 
(iv) in the quadrant West to North. 
The number of observations falling into each cell, and the 
row and column means, are given in Tables I and 11. The 
residuals from the fitted curves are shown in Figure 1. 
The analyses of variance presented show that there is no 
statistical evidence for effects due to the amount of SUA 
illumination on the satellite-station path, or on the quad- 
rant of observation; nor is there any significant interaction 
between these two factors. No data were available to the 
authors for testing more precisely the effects of temperature 
and proton bombardment on the satellite oscillator. 
514 
January 1974 Results 
I 
A third run was conducted in January 1974, principally to 
provide an interpolating line for calibrating the Timation 
system against a USNO flying clock which visited National 
Mapping on 7 December 1973. On this run, oscillator in- 
stability, the restriction to one transmission frequency, 
and turning the oscillator on and off during the run de- 
graded the results of TII-AUST to the extent that compari- 
sons with the satellite clock could not be statistically 
analyzed, The time transfer comparisons with USNO were also 
degraded, as can be seen from Figure 2, but the mean of the 
observations still proved useful. The results of fitting 
straight lines to the January measurements of USNO-AUST, 
together with combinations of all three runs, are given in 
Table 111. 
Inter-run Statistics 
Three series of tests were conducted to establish whether 
the runs, and their combinations, were statistically equiva- 
lent. 
In the following descriptions, the u2 are variances of re- 
siduals after fitting straight lines of form 
USNO - AUST = CI + P (t-E) , 
and the o2 are the variances of the observed rates P. 
number of observations in each data set is denoted by n. 
The B 
(1) The first series compared the results obtained, on 
the one hand, by subtracting direct Australian observations 
from points interpolated between NRL observations, and on 
the other hand, by subtracting interpolated Australian 
observations from direct NRL observations. The tests were: 
(i) Equivalence of Sample Populations, i.e, whether 
NRL-interpolated samples were drawn from the same 
population as AUST-interpolated samples. 
Null hypothesis Ho : Ol 2 2 = O2 
2 2 H1 : Gl # U2 Alternative 
: f = u2/02 (Fisher's F) 1 2  Test stat is tic 
Evaluation : Accept Ho if 0-60 < f < 1-67. 
515 
The f-column of T a b l e  I V  shows tha t  the populations w e r e  
s t a t i s t i c a l l y  equivalent a t  the 95% level  fo r  a l l  except the 
September runs, which were nearly equivalent. 
(ii) Non-zero Significance of Rates, i.e. whether the 
rates of each run w e r e  s t a t i s t i c a l l y  equal t o  zero. 
Null hypothesis Ho : p = 0 
Alternative H1:  @ # O  
T e s t  s tatist ic : t = [$1 / (0~) '  (Student's t) 
: Accept Ho i f  t e 1.96. 
The t-columns of Table I V  show t h a t  the r a t e s  were very 
d i f fe ren t  from zero i n  all runs except January. T h i s  i s  due 
i n  pa r t  t o  the large standard e r ro r  and s m a l l  data se t :  b u t  
reference t o  Table 111 shows t h a t  the r a t e  w a s  indeed small, 
which i s  possibly explainable by the vagaries of the satel- 
l i t e  o s c i l l a t o r  which m a d e  the interpolation scheme unstable. 
It will be shown i n  a later test that  the January rates w e r e  
d i f fe ren t  from the  r a t e s  determined from the other runs. 
(iii) Equivalence of R a t e s ,  i.e. whether r a t e s  obtained 
by interpolat ing NRL observations equalled AUST- 
interpolated rates. 
Null hypothesis Ho : p, = @, 
Alternative 
T e s t  s ta t is t ic  
H1 : 8, # p2 
: T = (p, - @,)/S (Student's t) 
Evaluation : Accept Ho i f  T < 1.96, 
The T-column of T a b l e  I V  shows t h a t  the two interpolation 
schemes gave the same rates. 
(2 )  The second series of tests evaluated whether the 
midpoint of one run (a a t  time E i n  Table 111) coincided 
with the value a t  e on the l i n e  f i t t e d  through another run, 
i.e. whether runs gave consistent values when extrapolated. 
516 
Null hypothesis: al =E a2 + P2(t-f) 
Alternative : a1 #a, + P2(t-E) 
Test statistic : 
d-2 degrees of 
t = al - [a,+@, (t-E) l/S with 
freedom, where s = CO1/nl 2 + ui/n23 4 8 
This statistic is approximately distributed as Student’s t - 
it is the incomplete Fisher-Behrens statistic (Welch, 1937: 
Hamilton 1964) . 
Evaluation : Accept Ho if t 1.96. 
Table V shows that forward extrapolation of the NRL-inter- 
polated samples i s  valid - even extrapolating from the July 
run into January is satisfactory at the 2% level. The fact 
that the AUST-interpolated samples do not give such good 
extrapolation characteristics is attributed to the sparser 
Australian data sets - otherwise it is a little puzzling. 
(3) The third series tested the hypotheses that, for the 
NRL-interpolated samples, the sample populations and rates 
from each run and combination were equivalent; and similarly 
for the AUST-interpolated samples. All single runs satisfied 
a ~2 goodness-of-fit test for normality. 
Equivalence of population variances. 
2 2 Ho : al = U2 Null hypothesis 
2 2 Alternative H1 : 01 z 0.2 
Test statistic 
Evaluation : Accept Ho if 0.60 f < 1.67. 
The f-column of Table VI shows clearly that the July run had 
significantly lower variance than any other run or combina- 
tion, but that the other runs were, by and large, front the 
same population. 
: f = G:/D; (Fisher’s F) 
517 
(ii) Equality of rates. 
Null hypothesis Ho : 8, = 8, 
Alternative H1 : 8, = 8, 
Test Statistic : T = [pl - p23/S (Student's t) 
where S = {[(nl-2)crl 2 + (n2-2)o2I[o  2 2  /a + u 2 2  /u21/[n1+n2-4]] 55 
$1 $2 
Evaluation : Accept Ho if t < 1.96. 
The T-column of Table VI shows that, for the NRL-interpolated 
samples, the rate determined from the January run was statis- 
tically different from the rates determined from all other 
runs and combinations, which were in turn statistically equal 
to each other. This confirms the result found in test (ii) 
of the first series. The poorer results obtained from the 
AUST-interpolated samples confirm the second series tests 
wherein extrapolation between some run combinations was not 
valid. 
Comparison of Time Scales 
The tests described above all used a single cesium standard, 
DNM590, for the time scale denoted AUST. To demonstrate 
that the out-of-character results of the January run were not 
due to a change of rate in this clock, a special Australian 
artificial time scale (AATS) was constructed, comprising the 
four cesium standards DNM590, the original Mount Stromlo 
standard DNM205, the newer standard NSL338 of the National 
Standards Laboratories, CSIRO, Sydney, and standard HP052 
maintained by Hewlett Packard (Australia) Limited, Melboarne, 
These clocks are all compared daily by ABC television com- 
parisons (Miller 1970) and were not stopped or adjusted in 
the period between 8 February 1973 and 23 May 1974. No other 
cesium standard in Australia satisfied both these conditions, 
The time scale was a simple unweighted mean of the four 
clocks, offset (in phase only) so that it agreed approximately 
with UTC (USNO) determined by flying clocks. 
An extrapolating ephemeris for AATS was constructed, using 
least squares straight line fits, in which: 
4 
i=l 
ECUSNO-AATSI = C E[USNO-CloCki}/4 
and 
i 
518 
E [ USNO-C locki 3 = E[ USNO-DNMS 90 1 (by Timation) +E[ DNMS 90-Clocki 1 
(by Television). 
Selected points on the graphs of the clocks against the arti- 
ficial time scale are shown in Figure 3. It can be seen that 
no significant rate change occurred in DNM590. Assuming that 
no such rate change occurred in UTC (USNO), the poor results 
in rate from the January run would reflect deficiencies in 
the Timation I1 technique. 
Table I11 includes the result of a visit by USNO flying clock 
PC572 in December 1973, and, in the column headed USNO-AUST, 
gives the values obtained by inserting t = 7 December 1973 
in the various formulae obtained for  different Timation I1 
runs and combinations. For each interpolating system, the 
July-September combination gave the best agreement, which was 
0.31 microseconds for NRL-interpolations and 0.17 microseconds 
for UAST-interpolations. The 95% confidence interval at 7 
December for the NRL-interpolated July-September combination 
was 20.27 microseconds, and for the AUST-interpolated combina- 
tion +0.20 microseconds. Thus, on the assumption that no 
error-was attached to the flying clock result, the former 
s e t  gave almost statistically’correct results, while the 
latter set showed excellent agreement. When the quoted error 
of +0.2 microseconds from the USNO certification was taken 
intz account, the NRL-interpolated result also became accept- 
able. 
Extrapolating the July-September combinations backwards to 
the date of the previous USNO flying clock visit on 8 February 
1973, the differences were 3.03 microseconds with $.05 micro- 
seconds 95% confidence interval for USNO-interpolates, and 
3.12 microseconds with 0.37 microseconds 95% confidence in- 
terval for AUST-interpo1ates. When the errors of the flying 
clock measurements are taken into account, these results are 
not unsatisfactory. On the other hand, the agreement between 
USNO-AATS by the extrapolating ephemeris and USNO-AATS by the 
flying clock measurement was 0.8 microseconds, which is regard- 
ed as very satisfactory over such an interval. It is unfortun- 
ate that no subsequent definitive flying clock trip has been 
made, as a further test of the predictive power of Thation 
would have been very beneficial, especially as time keeping 
in Australia during 1974 has been plagued with breakdowns 
both in a number of cesium standards and in the television 
network system. 
On the basis of the consistency of the NRL-interpolated 
July-September combination in both the extrapolative and 
interpolative senses, its agreement with USNO flying clock 
519 
measurements, and its consistency with a selected Australian 
time scale, the formula 4 in Table I11 was adopted as the 
definitive comparison between Australian clocks and UTC(USN0 
Mc), and was accordingly made the sole interpolating link 
between UTC(USN0 MC) and the regular television-compared 
Australian mean t h e  scale UTC(Aus) , 
Conclusions 
The foregoing statistical analysis shaws quite clearly the 
value of the Timation satellite for intercontinental time 
transfer at the sub-microsecond level. The major areas re- 
quiring particular attention are: 
Long runs are required to establish rates reliably. 
The durations of the three runs were thirteen, 
seventeen and twelve days in July, September and 
January respectively, and even then the sparser 
results in January produced anomalies. 
The stability of the satellite oscillator must be 
good enough to carry interpolations over several 
hours, The poor January results establish this 
point forcibly. 
The superior results from the July run show that _.. 
1 
> 2’ 
dual frequency transmissions (150 MHz and 400 MHz) 
do indeed reduce errors, 
Every effort should be made to have the data set 
as dense as possible. 
There are factors affecting the stability of the 
oscillator which we do not yet understand, since 
an analysis of variance failed to reveal two 
possible causative, or perhaps correlated, effects. 
It is significant here that the residuals AUST-TI1 
were not normally distributed yet the residuals 
from USNO-AUST = (USNO-TII) - (AUST-TII) were nom- 
ally distributed, thus indicating the presence of 
a perturbing influence in the region of the satel- 
lite. No data was available to test accurately 
the hypothesis that the temperature around the 
crystal caused it to fluctuate, 
The Timation I1 results presented here have been incorporated 
into UTC(AUS), so that predictions of the relationship be- 
tween Australian clocks and USNO can be made with confidence 
at the microsecond level. It is hoped that regular 
520 
observations of Timation I11 can be carried out - its im- 
proved clock should improve the statistics considerably and 
enable our geographically isolated clocks to contribute to 
International Atomic The. 
Acknowledgements 
The authors particulary thank Mr. R. Easton of the Naval 
Research Laboratory, Washington, D.C. for his enthusiastic 
collaboration in the project. They also thank Mr, R , J .  Bryant 
and Mr, R. D, Craven of the Division of National Mapping for 
their assistance with the calculations for this paper, and 
Miss M. Dowhy, Miss M. Schussig and Miss J. Thurling of the 
Division for preparing the manuscript. 
References 
Easton, R.L., Smith, H.M., Morgan, P.: "Submicrosecond Time 
Transfer Between the United States, United Kingdom and Aus- 
tralia Via Satellite," Proc. 5th Precise T h e  and T h e  In- 
terval Planning Meeting, Goddard Space Flight Center, Green- 
belt, Md., December 1973. NASA publication X-814-74-225. 
Easton, R.L., Morgan, Po, Smith, H.M.: "Satellite Time 
Transfer from the U.S. Naval observatory to the Royal Green- 
wich Observatory and to Australia," paper presented to 
conference on Precision Electromagnetic Measurements, London, 
U.K., July 1974. 
Hamilton, W.C.: "Statistics in Physical Sciences," The 
Ronald Press Co,, N.Y., 1964. 
Hoel, P,G,: "Introduction to Mathematical Statistics," 3rd 
ed,, John Wiley and Sons Inc., N.Y., 1966. 
Miller, M.J.: "Synchronization of Distant Clocks by Televi- 
sion Pulse Comparisons," Proc. Astronomical of Australia, 
Vol 1 No 7 p 352, April 1970. 
Welch, B.L.: "The Significance of the Difference Between 
Two Means when the Population Variances are Unequal," 
Biometrika, Vol 29 pp 350-362, 1937. 
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Statistical analysis of time transfer data from Timation 2

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