the u. s. naval observatory clock time scales

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-27, NO. 4, DECEMBER 1978

Each check of the galvanometer zero for IIN = 0 is madeby opening the measuring circuit; in correspondence a valueIOUTo (relative zero) can be found for IOUT, due to thermaland mechanical drifts of the galvanometer moving coil.Because of (1), in this condition every recovery of the valueIIN = 0, whatever is the source, always means 'OUT = IOUToBut the current 'OUT = IOUTO determines a reflected voltageVG at the galvanometer in the measuring circuit. Its value is,as RO > RG,

VG = RGHIOUTO.

As the additional correction term VG is usually small, itdoes not need a very precise calculation in practical cases. Infact, in actual operation of the 13-bit A/D converter (about2 x 4000 leyels), the 1-bit resolution corresponds to anequivalent input current IIN = 0.25 pA. Therefore, the bestpossible dynamics (and also the maximum tolerable devia-tion for the relative zero), expressed as an equivalent inputcurrent HIOUT, is + 1 nA. Then the absolute.value ofHIOUTois always less than 1 nA, i.e., for RG = 10 Q, VG is less than 10nV.

ACKNOWLEDGMENT

The authors wish to thank M. Angelino for suggestionsbased on his experience in cell comparisons and M. Negrofor help with the construction.

REFERENCES[1] T. M. Dauphinee, "Principles, current practice and future prospects

for high precision neutral potentiometers," IEEE Intern. Conv. Rec.,pt. 11, pp. 184-192, 1965.

[2] L. Julie, "A universal potentiometer for the range from one nanovoltto ten volts," IEEE Trans. Instrum. Meas., vol. IM-16, pp. 187-191,Sept. 1967.

[3] M. P. MacMartin and N. L. Kusters, "The application of the directcurrent comparator to a seven-decade potentiometer," IEEE Trans.Instrum. Meas., vol. IM-17, pp. 263-268, Dec. 1968.

[4] H. Hirayama and Y. Hurayama, "Automatic measuring system for acontrol of standard colls," IEEE Trans. Instrum. Meas., vol. IM-21, pp.379-384, Nov. 1972.

[5] A. F. Dunn, "Automatic intercomparison of standard cells," IEEETrans. Instrum. Meas., vol. IM-23, pp. 278 282, Dec. 1974.

[6] D. W. Braudaway and R. E. Kleimann, "A high-resolution prototypesystem for automatic measurement of standard cell voltage," IEEETrans. Instrum. Meas., vol. IM-23, pp. 282-286, Dec. 1974.

[7] S. Harkness and C. H. Dix, "A new NPL voltage standards withcomputer controlled measurement," in Proc. EUROMEAS 77 (Brigh-ton, England), 1977, pp. 158-160.

The U. S. Naval ObservatoryClock Time Scales

DONALD B. PERCIVAL

Abstract Several changes have recently been made to improvethe real time physical approximation of UTC(USNO), the coordi-nated time scale generated by the U.S. Naval Observatory. Aprovisional UTC(USNO) is generated in real time, and a physicalclock is steered to this time scale by a phase microstepper. When finalvalues for UTC(USNO) are available, a method has been devised tosteer the real time UTC(USNO) to match the final UTC(USNO).Further improvement will involve the adoption of an algorithm togenerate UTC(USNO) on a real-time basis. Details of this newalgorithm are presented.

I. INTRODUCTION

T HE BASIC atomic clock time scale generated by theU.S. Naval Observatory is A.1(USNO, MEAN). Cur-

rently it is based on 16 to 20 commercial cesium beamfrequency standards selected from a collection of approxi-mately 30 standards located at the Observatory. As shown inFig. 1, three other time scales are mathematically related toA.1(USNO, MEAN). The most important of these three

Manuscript received June 8, 1978; revised August 21, 1978.The author is with the U.S. Naval Observatory, Washington, DC.

UTC (USNO,M)

Fig. 1. Relationship of various U.S. Naval Observatory clock timescales.

time scales is UTC(USNO), whose physical realization isUTC(USNO, MC), the time scale generated by the NavalObservatory master clock. By means of the coordinationoffset in frequency, UTC(USNO) is steered to be within afew microseconds of UTC(BIH), the time scale generated bythe Bureau International de l'Heure.

In the past, the stability of the UTC(USNO, MC) clock

U.S. Government work not protected by U.S. copyright

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PERCIVAL: THIE U S. N\\AL OB8SERVATORY CLOCK TIME SCACEES

time scale has not been as good as the UTC(USNO) papertime scale, due to three factors: the iterative nature of thetime scale algorithm, the manual procedure used to steer themaster clock to UTC(USNO), and the instability, oversampling times of less than a few days, of the frequencystandard which drives the master clock via a phase micro-stepper. A new time scale algorithm has been devised toeliminate iterations as much as possible. The relationshipbetween the old and new algorithms is described in SectionII of this report. The new algorithm is given in the Appendix.An improved steering procedure was implemented in late1977 and is discussed in Section III.

II. OLD AND NEW TIME SCALE ALGORITHMSThis section first considers the heuristic derivation of the

basic equation needed to form a reasonable time scale. Thisequation is not dependent upon a specific model for thestatistical charactenrstics of atomic clocks. Two noisemodels are then examined, one of which leads to the oldA.l(USNO, MEAN) time scale algorithm, and the other, toa new algorithm. The advantages and disadvantages ofbothalgorithms are discussed.

A. Heuristic Derivationi of Basic Time Scale EquationConsider n clocks, indexed by i = 0, 1, t, 1. Clock 0

will be called the master clock (or reference clock). Let x,(i)be the measured time difference between the master clockand clock i at time t, where t = 0, 1, 2, - , and is expressed inunits of some arbitrary sampling time (usually days). Wemay consider these time comparisons to be made with a onepulse per second input from the master clock on the startchannel of a low noise counter and with a pulse from one ofthe other ni - 1 clocks on the stop channel. For brevity wesay that x&(i) represents the master clock minus clock i attime t. A positive value for x,(i) indicates that the masterclock is ahead of clock i in time. By definition, x,(O) = 0.

Let z,(i) be the estimated time difference at time t betweenclock i and the mean time scale. Contributors to this meantime scale are usually either all } clocks or all clocks exceptthe master clock. In either case, zt(°) is the estimated value ofthe master clock minus the mean time scale at time t. Thisquantity is calculated by the relation

nn-l

zt(0) =- - E (i) (1)

where m = I if the master clock is to be excluded from themean time scale, and m = 0 if it is to be included.We may justify (1) by the following argument. Let z'(i)

represent the true time difference at time t between clock iand "perfect" time. The quantity

1n t

rt E t(i)

represents the time difference at time t between the mean of nclocks and "perfect" time. From the definition of x,(i), itfollows that

xt(i) = t(0) - Zt(i) + £t(i)

where v;(i) represents the measurement noise. Summingboth sides of the above equation over all n clocks anddividing by n yields

1 -1n- 1

Z xt(i) = z(O) - zt(i) + Z ()1j=0 n j=O nj=

The first term on the right-hand side of this equationrepresents the master clock minus "perfect" time, and thesecond term, the mean of n clocks minus "'perfect" time.Combining these two terms yields the true difference be-tween the master clock and the mean of ai clocks. Theleft-hand side of this equation thus represents z,(O),the estimated value of the master clock minus the mean of nclocks, since it equals the true time difference plus the termdue to measurement noise. In a similar fashion, one mayshow that m = 1 in (1) when the master clock is not to beincluded in the mean time scale.

Equation (1) would be enough to form a reasonable timescale, if new clocks were never added and old clocks werenever deleted from the time scale. There are two problems inadding or deleting a clock: one due to the time differencebetween the clock and the mean time scale, and the other dueto the difference in frequency between the clock and themean time scale.The time difference problem is seen easily from (1). If we

delete the jth clock immediately following time t - 1, at timet there will be a time step in the mean time scale ofapproximately X.- U(j)/(n - m), assuming that the frequencydifference between clock j and the mean time scale is close to0. Likewise, if a clock is added to the mean time scale, ingeneral it will cause a time step in the mean time scale. Tohandle this problem, we modify (1) to use increments oftimedifferences instead of time differences directly:

Zt(°) = zt- 1(0) +1 n-l

n- i (Xt(i)-X_l,(i)). (2)

Equation (2) may be derived from (1) by evaluating it attimes t - 1 and t and subtracting. Since we are now dealingwith time increments, (x1(i) - x, 1(i)), the time differencebetween clock j and the mean time scale does not affect themean time scale when clock j is added or deleted.The second problem, due to the frequency difference

between a clock being added or deleted and the mean timescale, arises from the following considerations. Suppose onehas four clocks which keep "perfect" time, except that two ofthe clocks are high in frequency by 3 x 10 12 and two arelow in frequency by 3 x 10- 12 with respect to "perfect" time.A mean of these four clocks formed by using (2) would be a"6perfect" time scale. If one of these clocks is removed, themean of the three remaining ideal clocks will differ infrequency with respect to "perfect" time by I x 10 12 inmagnitude. A similar situation exists in general when a clockis added to a time scale. To solve this problem, correctionsfor clock rates are introduced into (2). The rate of the jthclock at time t, r,(j), is defined to be the estimated change oftime difference per unit time of clock j minus the mean timescale from time t - I to t. If r,(j) is positive, clockj is gainingtime with respect to the mean time scale. By definition ofthe

377


mean time scale, the initial set of clock rates has thefollowing constraint:

n-1

E r,(i)=O.i-m

As clocks are added and deleted later on, this condition willno longer be true. Equation (2) is now modified in thefollowing fashion:

1 n-i n-I

ZtO)= t 1(0) n-+ E (Xt(i)-Xt- I(i)) + T Ert(i

1 n-I= Z1.i(0) + -_m (xt(i) - Xt-1(i) + rtr(i)). (3)

If the r,(i)'s are good estimates of the clock rates, then

(Xt(j) - Xt-1(j) + Tr,(j)) (x1(k) - x,_ l(k) + Trr(k))m < j, k < n - 1. If the above approximation is valid, addi-tion or deletion of a clock will have little effect on thefrequency of the mean time scale generated by (3). To add anew clock to a time scale, its rate must first be determinedwith respect to the existing time scale. Likewise, before thejth clock is deleted, the system of rates should be checked toinsure that

(Xt(j) - Xt l(j) + Trt(j))1 ni

~~S ~(Xt(i)-Xt- l(i) +TIr(i)).n-m-1 i=m,i:tj

From (3), the mean time scale will not be adversely affectedas clocks are added and deleted, provided the rt(i)'s areproperly adjusted.For some applications, (3) will be adequate for generating

a mean time scale with the desired stability. The time scaleformed by (3) will be referred to as a simple average insubsequent discussion. It is the simplest procedure forgenerating a time scale that is unaffected by the addition anddeletion of clocks.

Equation (3) implicitly assigns an equal weight of1/(n - m) to all clocks included in the mean time scale. Fortime scales composed of clocks with significantly differentstability characteristics, various weighting schemes may bedesirable. The addition of the possibility of unequal weightsleads us to the basic time scale equation:

n - I

zt(O) = Zt- 1(O) + E w,(i)(xt(i) - Xt 1(i) + Trt(i)) (4)i=M

where

Zw,(i)= 1 and O < wt(i) < 1.

To justify (4), the rate estimate for a clock was defined tobe measured in the sense of the clock minus the mean timescale. It is also possible to define the rate of a clock as beingwith respect to some time scale other than the mean timescale. Suppose, for example, that one has estimated the rateof the mean time scale minus TAI, the atomic time scale

generated by the BIH. Denoting this rate by rc, one couldreplace the original set of rates (the r1(i)'s) by

tr,(i) = r,(i) + r

and use the r'(i)'s instead of the r,(i)'s in (3). As long as r, is agood estimate of the rate of the mean time scale minus TAI,the resulting time scale should be approximately equal, infrequency, to TAI.The basic time scale equation was devised to handle the

bookkeeping problems of adding and deleting clocks. Manyother schemes could also handle these problems. Equation(4) is particularly useful in connection with the two algor-ithms discussed in this paper. These two algorithms arebased on different models for atomic clocks. Both algor-ithms attempt to filter out clock noise by manipulating theestimated rates in (4). The time scales generated by bothalgorithms are critically dependent upon specifying theweights and properly estimating the rates for all clocks inthe mean time scale.

B. The Old Time-Scale AlgorithmThe old A.1(USNO, MEAN) algorithm is based on the

following model for clock noise. Let

y,(i) = (z,(i) -Z-I(i))ITbe the fractional frequency averaged from time t - I to timet of clock i minus the mean time scale, where T is thesampling time. We model y,(i) by

Yt(i) =-p(i) + at(i) (5)where

At(i) = E(yt(i))= co(i), 0 < t < tl(i)-c1( t(i < t < t2(i

= Ch(i), th(i) < t < th+i(i)

and at(i) is a zero mean white noise process, i.e., E(a7aj) = 0,for r * s. The tk(i)'s represent the times at which the meanfractional frequency changes from c,_ (i) to ck(i). Thismodel states that the fractional frequency of atomic clocksmay be approximated by a sequence of frequency steps pluswhite noise. The motivation for this model comes fromvisual inspection of time and frequency plots ofatomic clockintercomparison data.The form in which this model is presented above raises

many questions. To complete the model, one would need tospecify one process to govern the times of occurrence of thesteps in frequency and another process for selecting theCk(i)'s. The reader is referred to [1] for an attempt to answerthese questions. Fortunately, these questions do not have tobe answered explicitly in order to utilize this model in atime-scale algorithm.

If one assumes that this is a good model, it is desirable tocorrect for the effect of these frequency steps on a mean time

378

PERCIVAL: THE U.S. NAVAL OBSERVATORY CLOCK TIME SCALES

scale. If one knew both the Ck(i)'s and the tk(i)s, the effect ofthese steps could be removed by using (4) and setting

rt(i)= co(i), 0 < t < t1(i)

=-c(i,01 tl(i) < t < t2(i)

-Ch(0) th(i) < t < th II )

Since the Ck(l)'S and tk(i)'s are not known, they must beestimated from the data. Obviously, all estimation schemescan only identify frequency steps after they occur. Thefrequency step model leads to an iterative procedure forgenerating a time scale, as may be seen by the followingidealized example. Suppose that, at time tk(j), the set ofestimated rates is correct and that, in particular, the rate ofthe jth clock is correctly set to rtk(j)(i) = Ck- l(j). At timetk(j) + 1, clock j steps in frequency to its new value Ck(j). Wecontinue to generate the time scale, using the old rate forclock j, until time tk(j) + d. At this time, the frequency step inclock j, after time tk(j), is detected by some (as yetunspecified) step detection procedure. A new rate for clockjis estimated. Using this new rate, the time scale is thenreprocessed over the d units of time that have elapsed fromthe moment clock j changed frequency until the frequencychange was actually detected.When there are enough clocks in the time scale, so that

moderate frequency steps by a small number of clocks willnot greatly affect the mean time scale, frequency steps maybe detected by examining the time or frequency differencebetween a clock and the most recent estimate of the meantime scale. In the generation of A.1(USNO, MEAN),frequency steps are actually detected in the following way.Every five days, the five day average difference between thecurrent estimated rate (converted to fractional frequency)and the observed fractional frequency for each clock withnonzero weight is calculated:

k + 4

Z (Yt(i) -rt(i))5t = k

where the sampling time r is one day. Sequences of theseaverage differences are examined for each clock. If theestimated rate for a particular clock is good, then theaverage differences should vary at random about 0. If asequence of average differences becomes consistently posi-tive or negative by a small amount, a frequency step has beenidentified. The value for the new estimated rate is calculatedusing: 1) the old rate; 2) the 5 day average differences,themselves averaged over the entire period the newfrequency has been in effect; and 3) a correction designed tocompensate for the fact that the averaged 5 day averagedifferences are biased toward 0, due to the effect on the meantime scale of the frequency step. Finally, if a five day averagedifference is ever greater than 3 x 10 13 in magnitude for aweighted clock, it is deleted immediately from the time scale.

1114/1C -. L (USNO,MEAN)

43097 43147 43197 43247 43297 43347 43397 43447 43497MODIFIED JULIRN DRTE

Fig. 2. Decomposition of one day fractional frequency values intofrequency steps and white noise. (Absolute position of all three lineswith respect to the y axis is arbitrary.)

Figure 2 shows the decomposition, into frequency stepsand white noise, of the one day fractional frequency values ofCs 1114/1C minus A.1(USNO, MEAN), for 400 days. Cs1114/1C is a commercial cesium beam frequency standardwith conventional beam tube. It was a contributor to themean time scale for the entire period shown in Fig. 2. Interms of [5], the upper line represents the fractionalfrequency deviates y,(i); the middle line, the frequency stepcomponent yu(i); and the lower line, the white noise com-ponent a,(i). A Kolmogorov-Smirnov, cumulative period-ogram, white noise test on the at(i) series indicates that thehypothesis that this series is white noise is rejected at the 0.01critical level (5). This indicates that, in this case, frequencystep plus white noise model is only an approximation.

Other procedures may be devised to detect frequencysteps. Since changes in the mean fractional frequency trans-late into changes in the slope of time, one can identifyfrequency steps by looking for sudden changes in slope on atime or phase plot. Some simulation work indicates that caremust be exercised when using this method. Overfitting or thedetection of false frequency steps degrades the resulting timescale. An attempt to devise a completely objective detectionprocedure is given in [1]. Finally, when there are only a fewclocks in a time scale, one has to use rate correlationtechniques to identify frequency changes. A good discussionof the rate correlation method may be found in [2].For various reasons discussed in [3], all clocks in the

A.1(USNO, MEAN) time scale are currently assigned aweight of either 0 or 1. Before a new clock is entered into thetime scale, it is evaluated for about six weeks in order todetermine both its rate and its frequency stability. If itsstability is good with respect to the existing time scale, theclock is added with a weight of 1.

C. ARIMA Time-Scale Algorithm

The new time-scale algorithm uses autoregressive, in-tegrated, moving average (ARIMA) models for each clock.Box and Jenkins [5] have given a detailed exposition on thepractical application ofARIMA models. The application ofARIMA models to the time and frequency field is discussedby Barnes [8] and Percival [6].

379

zLLJ

C3LLJ

LL.

L1

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-27, NO. 4, DECEMBER 19'/,

TABLE IARIMA MODELS FOR 10 COMMERCIAL CESIUM BEAM CLOCKS

High PerformanceClock Beam Tube? Model i1 90 O1 62 a

Cs 346 no (0,2,1) - +095 .86 - 12.4

Cs 532 no (0,2,1) - 0 .84 - 10.4

Cs 549 no (0,2,2) - 0 .75 .05 10.7

Cs 571 yes (1,2,1) .30 +.026 .90 - 4.2

Cs 591 no (0,2,1) - +.112 .92 - 13.9

Cs 654 yes (0,2,2) - -.104 .60 .15 6.0

Cs 660 yes (1,2,1) .45 0 .80 - 7.0

Cs 783 yes (0,2,1) - -.086 .73 - 4.8

Cs 834 yes (1,2,1) .45 +.060 .90 - 4.7

Cs 837 yes (0,2,1) - -.199 .78 - 8.0

00 and ca are expressed in nanoseconds.

The general form of an ARIMA model for the ith clockminus the mean time scale is

p q

vt(i) - L 4j(i)v, j(i) = Oo(i) + a(i) - E Of(i)afj(i)j=l j=1

where v,(i) is the dth finite difference at time t of z,(i); +1(i),+2(i)., , Op(i) are p autoregressive parameters; 00(i) is aparameter which compensates for a systematic dth orderpolynomial drift in time; 01(i), 02(i), * *, O0(i) are q movingaverage parameters; and at(i), a,1-(i), , at ,-), are asequence of uncorrelated random variables with zero meanand constant variance, i.e., E(a,(i)) = 0, E(a7(i)a8(i)) = 0 forr + s, and E(a2(i)) = U2(i). A model of the above form iscalled an ARIMA model of order (p, d, q). Once p, d, and qhave been specified, there are p:+ q + 2 parameters to beestimated (including a2(i)).From historical data, a large number of commercial

cesium beam clocks compared against A.l(USNO, MEAN)have been modeled by the procedure discussed in Box andJenkins. It was found that, for a sampling time of one day,one of the following models is almost always adequate: (0, 2,1), (0, 2, 2), or (1, 2, 1). These are all submodels ofthe (1, 2, 2)model, which may be written in the following form:

zt(i) = (2 + 01 (i))zt - 1(i)- (1 + 241(i))zt -2(i)+ 01(i)Zt-3(i) + 00(i) + at(i)- 61(i)at (i) -02 at 2(i)

For d = 2, 00(i) compensates for the slow linear drift infrequency observed in some atomic frequency standards.Table I gives the adopted ARIMA models for 10 commercialcesium standards and the corresponding estimated par-ameters. (See [6] for more details. It should be noted that theARIMA models discussed in [6] are for series of fractionalfrequency values instead oftime difference values. Thus, a (p,1, q) model in [6] corresponds to a (p, 2, q) model in thispaper.)

Special attention should be given to the Oo(i) parameter. Ithas been well established that commercial cesium beam

CS 1114/IC - R.I(USNO,MERN)

43097 43147 43197 43247 43297 43347 43397 43447 43497MODIFIED JULIRN ORTE

Fig. 3. Decomposition of one day fractional frequency values intopredictable and unpredictable portions. (Absolute position of all threelines with respect to the y axis is arbitrary.)

frequency standards have a frequency drift [6], [7]. In mostcases, this drift is approximately linear. The 00(i) term doescompensate for a linear drift, but, for the drifts typicallyobserved, one must have at least a year's worth of data inorder to estimate it reasonably well. In short segments ofdata, the frequency drift is usually of the same order ofmagnitude as the frequency dispersion one would expectfrom typical random clock noise (see the upper line in Fig. 2as an example). The penalty for ignoring the Oo(i) term islarge, since any errors in the estimation of this term willresult in a time dispersion proportional to t2. It is not prac-tical to wait a year before adding a new clock to a time scalein order to estimate this term, since a year is a substantialproportion of the expected lifetime of a commercial cesiumstandard [9]. Frequency drift is currently the major limita-tion to time scales formed from commercial cesium stan-dards.One of the main practical uses of ARIMA models is

forecasting. Let z0(i, k) represent the forecast of zB±k(i) madeat time t. One can show that one-unit-in-advance forecastsusing the (1, 2, 2) model are given by

zt(i, 1) = (2 + q$1(i))zt(i)- (1 + 20 1 (i))z_ 1(i)+ 10(i)Zt-2(i) + 0o(i)- Oi(i)at(i) - 02(i)a l1(i) (6)

where the a,(i)'s now take on the following interpretation:

a,t(j) = Zt(j) t_ I (i, 1 )

i.e., the a,(i)'s are the one-unit-in-advance prediction errors.If the adopted ARIMA model is a good approximation, theforecasts made using (6) are almost minimum mean-squareerror forecasts. To gain some insight, we rewrite the aboveequation as

(7)The original time series zt(i) is thus decomposed into 2 parts:a predictable portion Zt- 1(i, 1) and an unpredictable portionat(i).

Fig. 3 shows the decomposition of the one day fractionalfrequency values of Cs 1114/1C minus A. 1(USNO, MEAN)

380

zLLJ

C3LLJa:

LLJ

cr

LXJcc

AZt(i) = Zt -I (i, 1) + a,(i).


TABLE IIRMS PREDICTION ERRORS (IN NANOSECONDS)

Clock 1 Day in Advance 8 Days in Advance

Cs 532 10.4 44.2

Cs 660 7.0 51.1

into predictable and unpredictable portions. SubtractingZt- 1(i) from both sides of (7) and dividing by the samplingtime, T, yields

yt(i) = (At_- Ji, 1 )- t-1()/ til

= Yt(i) + Yt'(i)

where yt(i) = (zt 1(i, 1) - Zt 1(i))/T represents the predict-able portion of the fractional frequency yt(i); and yO'(i)=a&(i)/T, the unpredictable portion [10]. The upper line in Fig.3 represents yt(i); the middle line y'(i); and the lower liney'(i). A Kolmogorov-Smirnov, cumulative periodogram,white noise test on the y'(i) series indicates that the hypoth-esis which considers this series to be white noise cannot berejected at a critical level less than 0.24.

It is desirable to remove the effect of the predictableportion of a clock's performance on the mean time scale. Thenatural way to do this, under the formulation of the basictime-scale equation, is by letting the relation,

r + 1(i) = (At(i, 1 -

be the estimate of the current rate of clock i minus the meantime scale. This substitution is the basis of the ARIMA timescale algorithm. Since rt+ 1(i) is calculated at time t, theARIMA time scale, formed using (4) and the above equa-

tion, is a real-time time scale, i.e., in contrast to the oldtime-scale algorithm, the final value for zt(0) is available attime t.

In a manner analogous to the weighting scheme for a

weighted least squares solution, one can weight the contrib-utions from each clock by assigning weights proportional tothe reciprocal of the variance of the one-unit-in-advanceprediction errors:

n-

w(i ) = ( l/U2(i ) )/ ( l/U2(i ))_i=m

Past experience with atomic clocks indicates that cautionshould be exercised in using the above weighting scheme.First, the poor stability of some clocks is due to unmodeledfrequency instabilities. Including these clocks in a time scale,even with a low weight, can degrade the stability of the timescale. It is better to assign a weight of 0 to such clocks.Second, the above weighting scheme is based on the one-

unit-in-advance prediction errors. With one unit equal toone day, the one day predictability of a clock governs itscontribution to the mean time scale. If long term stability is aprimary goal for a time scale, this weighting scheme may notbe the best. As an example, Table II reproduces the observedone and eight day prediction errors for two commercialcesium beam clocks. Cs 532 has a conventional beam tube,

while Cs 660 has a high performance beam tube. From theone-day-in-advance prediction errors, Cs 660 would have2.2 times the weight of Cs 532. From the eight day-in-advance prediction errors, Cs 660 would have 0.7 times theweight of Cs 532. The second weighting scheme wouldproduce a time scale with better long term stability.Appendix A gives the ARIMA algorithm in detail. The

weighting scheme is based on weights proportional to thereciprocal of the variance of the one-unit-in-advance predic-tion errors, with an option to give certain clocks a weight of0. Modifications to this algorithm that enable it to utilize adifferent weighting scheme should not be difficult to make.In addition, at each time t, the absolute prediction errorat(i) is compared to the 3 sigma limit 3a&(i). Ifone or more

clocks exceed this limit, the one with the largest relativeerror,

at(i) a(i)'is given a weight of 0. The time scale is recomputed for time t,until all weighted clocks are within the 3 sigma limit.

Extensive simulation work and reprocessing of historicaldata have been done to test the ARIMA algorithm. Thesimulation studies indicate that, as a rough rule of thumb,one can expect a decrease in expected time dispersion, in amean time scale, of about 2500/, by using the ARIMAalgorithm instead of a simple average (as defined above inSection II-A). In addition, the ARIMA algorithm has beenused to reprocess the data used to form A. 1 (USNO, MEAN)from the old time scale algorithm. In a 400-day segmentfrom MJD 43096 to 43496, the 2 algorithms produced timescales which differed by 2 ps after 400 days. Simulation workindicates that this time difference is somewhat large. Thus, inpractice, the difference between the two algorithms is nottrivial. A detailed investigation into the cause of thisdiscrepancy is in progress.Computer simulation also indicates that the ARIMA

algorithm is useful in forming a good time scale with as fewas three clocks. For those who desire to use this algorithm ona small set of clocks, the following discussion describes howto "bootstrap" up to the ARIMA algorithm and, if desired,to simplify it somewhat. Both the "bootstrap" procedureand the simplification have been verified as practical bysimulation studies and examination of historical data.Assume that the sampling time is set to one day and that

one has 100 or more days of intercomparison data for threeor more atomic clocks. In order to estimate the parametersnecessary to use the ARIMA algorithm, one first forms apreliminary time scale using a simple average (3). From themethods described by Box and Jenkins, a preliminary modelis formed for each clock minus the preliminary time scale.Due to the limited amount of data, only the (0, 2, 1) modelneed be considered at this point, and the 00(i) terms canusually be set to 0. Model fitting thus reduces to determiningtwo parameters for each clock, 01(i) and c2(i). Using thepreliminary model elements, one next uses the ARIMAalgorithm given in appendix A to generate a second pre-liminary time scale. A final model for each clock is nowdetermined using values for each clock minus the second

38X1


preliminary time scale. Using the final model elements, afinal time scale is formed by applying again the ARIMAalgorithm.

Several comments are in order. First, the above procedureconsists of two iterations of preliminary time scale forma-tion and model determination. One may be tempted to usemore iterations, but simulation studies indicate that morethan two iterations will degrade the parameter estimates,particularly for a2(i). Second, the resulting estimate of 52(i)for a clock contributing to the time scale will be biasedtoward a value that is smaller than its actual value. This biasis due to the fact that a clock is being compared to a timescale to which it is a contributor [7]. Simulation studiesindicate, however, that there will be little bias normally inthe resulting system of weights, which are only a function ofthe 6`(i)'s. One should also note that, due to this bias, theaddition or deletion of one or more clocks in a time scalewith a small number of clocks may necessitate a reevalua-tion of C`(i) for all clocks. Third, one should use care whenusing this procedure. For example, a pronounced frequencydrift in one frequency standard may have to be removedbefore the first preliminary time scale is calculated. Oneshould examine the prediction errors for each clock, at eachstage, in order to check for large errors which may affect theestimates of 01(i) and cr(i). If 200 or more days of data areavailable, cross validation is a valuable technique: in addi-tion to using the above procedure on all available data, onecan also take nonoverlapping blocks of 100 days each anduse the "bootstrap" procedure on each block. This techniquewill indicate whether estimates for the model parameters areconsistent with time. Additional problems may arise ifsubstantially less than 100 days of data are available.For various reasons, one may not wish to carry out the

parameter estimation procedure described by Box andJenkins. If an increase of roughly 10 percent in the expectedtime dispersion of the mean time scale can be tolerated, thefollowing simplified procedure may be used. Modeling ofnumerous commercial cesium beam frequency standardsindicates that the 0 1(i) parameter in the (0, 2, 1) model, for asampling time of one day, is almost always in the range of0.40 to 0.90. By setting 01 (i) = 0.75, there is usually, at most,a 10 percent increase in the sigma. With the 01(i)'s thusspecified, one needs only to determine estimates for the

2(i)'s. First, setting all the r2(i)'s equal to the same nominalvalue, one executes the ARIMA algorithm and produces aninitial estimate for 52(i) by letting

N

6a(i) = E a (i)/(N + 1)t=O

where N + 1 is equal to the number of days ofdata. Second,with these estimates for the a2(i)'s, the ARIMA algorithm isexecuted a second time in order to produce the finalestimates for o2(i)'s, using the same above equation. A finaltime scale is then formed by applying again the ARIMAalgorithm.

D. Brief Comparison of Old and New Time Scale AlgorithmApart from the important discussion of the merits of the

two noise models presented above [1], [3], [7], [8], there are

Fig. 4. Steering of U.S. Naval Observatory master clock by data acquisi-tion and control system via phase microstepper.

several operational considerations. Both algorithms areimplemented using equation (4), the basic time scale equa-tion. In both cases, estimation of rates attempt to filter outthe noise inherent in atomic clocks. The old time scalealgorithm requires no explicit modeling ofindividual atomicclocks. There are two major problems in using this algor-ithm. First, the resulting time scale is iterative in nature andthus is not final until one or two months after the fact.Second, experience and care are required in theidentification of frequency steps. The algorithm requiressubjective judgment and is not currently in a form suitablefor complete computer implementation.The ARIMA time scale algorithm does produce a real-

time time scale with little or no reprocessing required. Itdoes require an explicit model for each clock, although themodeling procedure can be shortened somewhat at theexpense of an increase in the expected time dispersion ofthe mean time scale. The ARIMA algorithm is ideally suitedfor a digital computer, and less subjective judgment isrequired on a day to day basis.

III. STEERING OF MASTER CLOCK TO UTC(USNO)Prior to late 1977 the UTC(USNO) time scale was

computed twice a week in a batch processing mode. Theresulting estimated values of the time difference betweenUTC(USNO, MC) and UTC(USNO) were used to setmanually a phase microstepper in order to steer the masterclock to UTC(USNO). With this procedure, time errors ofup to 25 ns were experienced, due to the sparsity of changesto the microstepper and due to errors in the estimation ofthefrequency of the reference system frequency standard.The improved steering procedure is shown in Fig. 4. Each

hour, a data acquisition system intercompares the 5-MHzsignals produced by all of the frequency standards at theObservatory. (For a detailed description ofthis data acquisi-tion system, see [4].) From this data, a provisionalUTC(USNO) time scale (described below) is generated.

382


Once each day the phase microstepper is updated automa-tically by the data acquisition and control system. A predic-tion of the frequency of the reference system frequencystandard versus the provisional UTC(USNO) for one day inadvance is made using an (0, 2, 1) ARIMA model, with01 = 0.75. To this predicted value is added a frequencycorrection in order to compensate for the time differencebetween where the master clock currently is with respect tothe provisional UTC(USNO) and where it should be oneday after the microstepper update. The microstepper is setindirectly to this new value by the data acquisition andcontrol system, through a serial to parallel converter(designed by P. Wheeler of the U.S. Naval Observatory).This serial to parallel converter not only reduces the numberof control lines required, but it also serves as a protection incase of a power failure in the data acquisition and controlsystem. (In fact, even if the serial to parallel converter itselfloses power, the operation ofthe phase microstepper will notbe affected.) One can determine how well this new steeringprocedure works by looking daily at the time differencebetween where the master clock is with respect to theprovisional UTC(USNO) and the destination toward whichthe data acquisition and control system was steering it onthe previous day. Based on five months of data the rootmean square of this time difference has been 4 ns.

Since 1968, A. l(USNO, MEAN) has been formed by theiterative time scale procedure described in Section II-B.Since UTC(USNO) is mathematically related toA.1(USNO, MEAN), final values for UTC(USNO, MC)-UTC(USNO) are two or three months in arrears. This delayis caused by the method used to identify rate changes in thefrequency standards contributing to A.1(USNO, MEAN).Once a rate change has been detected in a standard,A.l(USNO, MEAN) is recomputed from the time of theidentified change up to the present time. This recomputationintroduces artificial time steps in the real-time estimatedtime difference between UTC(USNO, MC) andUTC(USNO). These time steps have been as large as 100 ns.Steering the master clock in the past to eliminate theseartificial time steps within a week or so has degraded thestability of the UTC(USNO, MC) time scale for samplingtimes up to a few weeks.

There are three possible solutions to this problem. First,one could retain the old iterative time scale algorithm andattempt to compensate for the artificial time steps slowly,over a much longer period of time. Second, one couldimplement a real-time algorithm on the data acquisition andcontrol system which would better approximate the finalUTC(USNO) generated by the old iterative time scalealgorithm and which would result in less corrective steeringof the master clock. Finally, one could attempt to create analgorithm that would generate UTC(USNO) entirely on areal time basis, with little or no reprocessing.The first of these solutions was implemented in late 1977.

As noted above, a provisional UTC(USNO) time scale iscomputed each hour by the data acquisition and controlsystem. This provisional UTC(USNO) is the first approxi-mation to UTC(USNO) and uses the most recent set of ratecorrections for each contributing frequency standard. The

UTC(USNO,MC) - UTC(USNO)

10

-10 nsec-

-20

-30 nsec-

artificial timestep due torecomputation

Fig. 5. Illustration of artificial time step introduced by iterative time-scale algorithm.

old time-scale algorithm is still run twice a week andsupplies the estimated rates for the provisional time-scalealgorithm. As soon as a rate change is detected in one of thefrequency standards contributing to UTC(USNO), recom-putation of UTC(USNO) introduces an artificial time stepin the estimated current time difference betweenUTC(USNO, MC) and UTC(USNO).An example of such a time step is shown in Fig. 5. On

MJD 43619 a rate change was identified as having occurredat MJD 43607 in one of the frequency standards contribut-ing to UTC(USNO). Recomputation ofUTC(USNO) from43607 to 43619 introduced a 25 ns difference in the estimateofUTC(USNO, MC)-UTC(USNO) for MJD 43619. Beforethe recomputation, the data acquisition and control systemwas steering the master clock to keep the time differenceUTC(USNO, MC)-UTC(USNO) close to 0 ns. In order toremove the artificial time step slowly, the master clock wassteered on the following day to a target value forUTC(USNO, MC)-UTC(USNO) of -25 ns. This targetvalue was incremented slowly on subsequent days, at a rateof 1 ns per day, until it was zero. (If the target value is toolarge, it may be necessary to reduce it at a faster rate than 1ns per day.) By means of the target value, artificial time stepsin the UTC(USNO) time scale are now being removed in amuch smoother fashion.The next step in the improvement of UTC(USNO, MC)

will consist of the replacement of the provisional time-scalealgorithm by the ARIMA real-time time-scale algorithm.This algorithm will compensate in real time for rate changesin contributing frequency standards. Extensive simulationwork and reprocessing of historical data indicate that theARIMA algorithm does produce an improved, real-time,time scale. This improvement should translate into animproved UTC(USNO, MC), since there should now be lessdifference between the provisional and final UTC(USNO)time scales. Plans have been made to implement the ARIMAalgorithm after the data acquisition and control system hasbeen upgraded in late 1978.

383

IFEE iRANSACTIONS ON INSTRlJMENTATION AND MEASUREMENT, VOL. IM-27, NO. 4, DECEMBER 1978

Because of the large amount of historical data needed toestimate the frequency drift terms for commercial cesiumbeam clocks, the ARIMA algorithm cannot entirely com-pensate for these drifts on a real-time basis. Reprocessing isnecessary to remove the effect of these drifts. Rather thansimply letting the real-time ARIMA time scale be the finaltime scale, a different procedure is favored in which finalvalues for A.1(USNO, MEAN) would become availableonly after a much longer time interval. This procedurewould allow one to take as long as required to remove theeffects of frequency drifts on the final time scale. At thepresent time, no date has been set to implement thisprocedure.

APPENDIX A.ARIMA TIME-SCALE ALGORITHM

Input for each clock at time t:

measured values of master clockminus clock i at times t and t- 1;previous estimated values of clock iminus mean time scale at times t - 1

and t - 2;one-unit-in-advance prediction ofzL(i), made at time t- 1;previous prediction error at timet - 1;estimated rate;indicator function: if 0, clock i willhave no weight in the time scalecomputation; if 1, clock i will contrib-ute to the time scale if its currentprediction error is not too large;

ARIMA model parameters.

Output for each clock at time t:

zt(i)Zt(i, 1)

at(i)rt+ I(i)It(i)

estimated value of clock i minus mean time scale;one-unit-in-advance prediction of z,+ 1(i), madeat time t;current prediction error;updated estimated rate;indicator function: if 0, clock i did not contributeto the time scale; if 1, clock i did contribute.

All summations in this algorithm are from m to n-1,where m = 0 if the master clock is a possible contributor tothe time scale, and m = 1 if it is to be excluded.

1) Determine the weight of each clock:

t(i) 2(i)E (It (ij)/(A2 (i))

2) Using the basic time scale algorithm, calculate masterclock minus mean-time scale:

z,(0) = Zt -(0) + E3 wt(i)(xt(i) - x 1(i) + Tr,(i)).3) Determine clock i minus mean values:

Zt(i) = z(0) -xt(i)

and prediction errors:

at(i) =Zt(i-At_ I (is I )

4) For each clock i for which It(i)= 1, compare themagnitude of its prediction error to its 3 sigma limit. If

at(i) < 3ca(i)for all clocks with nonzero weight, then proceed to step 5;otherwise, determine the clock j with the largest relativeprediction error

| t(j) /&a(})

and delete it from the time scale by setting It(j) = 0 andreturning to step 1.

5) Generate one-unit-in-advance predictions:

zt(i, 1) = (2 + 1(i))zt(i) - (1 + 2k1(i))zt 1(i)+ (i)Zt- 2(i) + 00 0)- 0,(')at(i) - 02(i)at -lI(i)

and update estimated rate corrections:

rt + 1 ZiZ(,1) t(i))/TNotes

1) As a diagnostic aid, the relative rms prediction erroraveraged over all contributing clocks is a useful statistic tomonitor:

2) Occasionally a clock's rms prediction error will beginto increase. To guard against this occurrence, one could usean F-test to test for a significant difference in the meansquare prediction error of the n most recent values of a,(i)ascompared to W2(i).

3) Once every 100 or 200 units, one should check that theARIMA model elements have not changed significantly, byeither remodeling the clocks or by performing the variousdiagnostic checks on the prediction errors that Box andJenkins recommend. Typically, when a new clock with onlylimited historical data is added to the time scale, the (0, 2, 1)model will be completely adequate. As more data points areaccumulated, higher order models should be investigated todetermine if these models would significantly reduce the rmsprediction error. In addition, one should carefully check forindications of linear frequency drifts in new clocks andcompensate for them as soon as possible by using the 00 termin the ARIMA model.

ACKNOWLEDGMENT

The autlhor wishes to express his gratitude to Dr. J. A.Barnies of the National Bureau of Standards and to Dr. G.M. R. Winkler, R. E. Keating, Dr. D. D. McCarthy, Dr. R. G.Hall, and other colleagues at the U.S. Naval Observatoryfor their many helpful comments and corrections to thispaper. Special credit is due to P. Wheeler of the U.S. NavalObservatory, whose design of the serial to parallel convertermade steering of the master clock practical.

Xt(i), xt l (i)

Zt- (I), Zt -2(i)

zt-I(i, 1)

at - I (i)

rt(i)It(i)

0a(i), 1 (i) 00 (i)0 1(i), 02(i)

384


REFERENCES

[1] D. B. Percival, "A heuristic model of long-term atomic clock behav-ior," in Proc. 30th Annu. Symp. Frequency Control, pp. 414-419, June1976.

[2] J. C. Hafele, and R. E. Keating, "Around-the-world atomic clocks:Observed relativistic time gains," Science, vol. 177, pp. 168-170, July14, 1972.

[3] G. M. R. Winkler, R. G. Hall, and D. B. Percival, "The U.S. NavalObservatory clock time reference and the performance of a sample ofatomic clocks," Metrologia, vol. 6, pp. 126-134, Oct. 1970.

[4] K. Putkovich, "Automated timekeeping," IEEE Trans. Instrum.Meas., vol. IM-21, pp. 401-405, Nov. 1972.

[5] G. E. P. Box and G. M. Jenkins, Time Series Analysis: Forecastingand Control, revised ed. San Francisco, CA: Holden-Day, 1976.

[6] D. B. Percival, "Prediction error analysis of atomic frequency stan-dards," in Proc. 31st Annu. Symp. Frequency Control, pp. 319-326,June 1977.

[7] D. W. Allan, J. E. Gray, and H. E. Machlan, "The National Bureau ofStandards atomic time scale: Generation, stability, accuracy, andaccessibility," in National Bureau of Standards Monograph 140,Time and Frequency: Theory and Fundamentals, B. E. Blair, Ed., pp.205-231, May 1974.

[8] J. A. Barnes, "Models for the interpretation of frequency stabilitymeasurements," NBS Tech. Note 683, Aug. 1976.

[9] D. B. Percival and G. M. R. Winkler, "Timekeeping and the reliabil-ity problem," in Proc. 29th Annu. Symp. Frequency Control, pp. 412--416, May 1975.

[10] E. Parzen, "Some recent advances in time series modeling," IEEETrans. Automatic Control, vol. AC-19, pp. 723--730, Dec. 1974.

Precision RE Voltage Standard Usinga Thermistor Bridge Covering

the HF-UHF RangeFUTOSHI UCHIYAMA, KYOHEI YAMAMURA, AND ICHIRO YOKOSHIMA

Abstract-This paper describes a thermistor mount designed sothat residual inductances of leads in the thermistor are eliminated foruse in the UHF range. The output voltage of the thermistor bridge isevaluated by the dc substitution method with an accuracy of 0.3percent for 0.3-1 V over frequencies up to 1.5 GHz. The outputvoltage agrees with the voltage derived from power and impedancestandards within the accuracy.

I. INTRODUCTION

FOR THE primary RF voltage standard, two techniqueshave widely been used such as a bolometer-type volt-

meter [1] in which RF voltage is measured by a bolometerlocated in a coaxial line parallel with the electric field and apower meter whose impedance is known exactly [3].Reported accuracies of the primary voltage standard in theUHF range were 1 percent by the bolometer-type voltmeter[1], [2] and 0.25 percent by the power meter. Thus a higheraccuracy has been obtained by the power meter. Forcalibration of the voltmeter in terms of the power meter,however, a voltage source with a monitor to indicate voltageat the measurement port is required [3], which is calibratedto the power meter. On the other hand, the bolometer-typevoltmeter with a RF source can directly calibrate othervoltmeters, serving as a constant voltage source. Thus the

Manuscript received May 18, 1978; revised July 31, 1978.The authors are with the Electrochemical Laboratory, 5-4-1, Mukodai-

cho, Tanashi-shi, Tokyo, Japan.

bolometer-type voltmeter can provide a more straightfor-ward and convenient technique for RF voltage standardiza-tion. This paper describes a precision bolometer-typevoltmeter whose accuracy is comparable with that for thepower meter in the UHF range.A bolometer-type voltmeter should satisfy a condition

that the admittance of the bolometer in the voltmeterconsists of a pure conductance shunted by any value ofsusceptance. For realizing the pure conductance condition,the bolometer should be designed so that not only thecondition is satisfied in the bolometer, including electrodesof the mount, but also the conductance of the bolometer canbe measured exactly. The most exact impedance measure-ment has been realized in a 14-mm 50-Q coaxial system. Onthe other hand, the bolometer should be sensitive anddurable for as much overloads as possible for practicaluse. As the bolometer satisfies these conditions, the ther-mistor is most suitable among bolometers for room temper-ature use.By considering these facts, a precision bolometer mount

in which thermistors are used as the bolometer, i.e., athermistor mount, is designed for the 14-mm 50-Q coaxialline system.

II. STRUCTURE OF THE THERMISTOR MOUNT

Fig. 1 shows the structure of the thermistor mount. Thetwo-thermistors method [1] is adopted in the mount. For

0018-9456/78/1200-0385$00.75 ©) 1978 IEEE

the u. s. naval observatory clock time scales

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