1.3.2 Biased Ranges

CLOCK BEHAVIOUR

 

Frequency and Time Stability

A time scale is defined by the period of the basic oscillation of the frequency-determining element (be it the earth's rotation in the case of Sidereal Time, or the oscillation of atoms in the case of atomic time, or of a crystal in the case of quartz clocks) which is measured, and the origin of the time scale, which may be either arbitrarily defined or agreed upon by international convention. Individual clocks maintain their own time scale, however, for one-way (passive) ranging BOTH the ground and satellite clocks need to be synchronised.

How well must this synchronisation be made?

          >1 nanosecond (10-9 sec)  30 centimetres in distance!

To indicate the "absolute" magnitude of clock "error" it is necessary to introduce the notion of "perfect" or "true" time. Hence it is possible to "measure" clock error as an instantaneous "offset" from this perfect time scale. A time-varying clock error t(ti) can have the following form:

Figure 1. Time-varying clock error measured against a time scale standard.

 

Today's high precision clocks are all based on some form of frequency standard or oscillator. In the context of precise ranging systems these belong to one of two classes:

Time intervals are most precisely defined by the cycle counter of a frequency standard. (For example, the second is now defined as 9192631770 cycles of the fundamental resonance of the cesium atom.) Hence it suffices to establish the relationship between frequency and the phase output of an oscillator, and their errors, as the time scale can be directly obtained from such a relationship. The reading on a frequency cycle counter i can be represented by:

(ti) - (toi) = fi . (ti - toi) (1.3-1)

 

The wavelength of the phase cycle is ( c / fi ), where c is the velocity of electromagnetic radiation (299792458m/s in a vacuum). Substituting the appropriate scaling for the phase change to convert it into a time interval, and defining the origin of the time scale at the arbitrary reference epoch of the cycle counter, the following expression for a "clock reading" is obtained:

(1.3-2)

where:

to is the reference epoch,
toi is the clock reading at the reference epoch,
fi(t) is the frequency of the oscillator, and
fo is the nominal oscillator frequency.

 

A standard model for the frequency of an oscillator is:

fi(t) = f0 = f + (t - t0) + fr(t) (1.3-3)

 

where:

f is the frequency bias,
is the frequency drift, and
fr(t) are unmodelled random frequency errors.

 

Substituting eqn (1.3-3) into eqn (1.3-2) gives:

(1.3-4)

 

Rearranging terms into a representation of the error of the clock i as a time polynomial:

(1.3-5)

where:

ao is the clock bias term,
a1 is the clock drift term,
a2 is the clock drift-rate, and
is the integrated random fractional frequency error.

 

Expressed as "phase" error this is:

(1.3-6)

 

Clock error, whether expressed in terms of phase (eqn (1.3-6)), time (eqn (1.3-5)) or frequency (eqn (1.3-3)) instability, consists essentially of two distinct components:

Therefore, in addition to exhibiting deterministic deviations from a "true" time scale, they also undergo stochastic variations in both time (or phase) and frequency. An example of a realisation of the time difference between a commercial cesium clock and a time scale generated from an ensemble of atomic clocks is shown in the Figure 2 below.

Figure 2. Time difference between a commercial atomic clock and the U.S. Naval Observatory Time Scale. (JONES & TRYON, 1987)

(The middle graph has the linear trend removed, and the bottom graph has a quadratic component removed.)

 

The top graph shows how the clock has a frequency bias so that the time scale appears to drift linearly away from the "true" time (defined here by the group of atomic clocks). This linear drift (approximately 50 msec/year) does not affect the clock's ability to keep accurate time as long as the rate of the drift (coefficient a1 ) is known or can be estimated very well. The middle graph shows the residuals after fitting a straight line to the top graph. The variation is now of the order of 5 msec/year, but the quadratic appearance indicates a higher order effect still present, in this case a significant frequency drift in the clock (the coefficient a2 representing "ageing"). That is, the frequency of the clock appears to change linearly with time. The bottom graph shows the residuals after removing a quadratic function. These are now primarily stochastic variations (a higher order polynomial could be used to remove residual "systematic" trends, but the "stochastic" variations are assumed here to be those that remain after second order polynomial modelling).

The component of eqn (1.3-3) attributable to random error sources is known as the random fractional frequency deviation, or the integrated random fractional frequency error (eqn (1.3-5)). The total fractional frequency deviation (systematic + random), or just the random part, can be analysed. The standard approach is to deal with the sample variance of only the random fractional frequency fluctuations, and model the systematic part by an explicit polynomial-like function. As the frequency count or time difference over some elapsed time interval  can be measured it is possible to define the mean value of the fractional frequency deviation:

(1.3-7)

 

where tk+1 = tk + T, k=0,1,2,...T, is the repetition interval for measurements of duration , and t is arbitrary.

Now forming the sample variance of y(t):

(1.3-8)

where < g > denotes the infinite time average of "g".

 

A particular variance measure is chosen so that N=2, T=. This is the so-called Allan Variance (HELLWIG, 1979):

(1.3-9)

 

This two-sample variance of the fractional frequency error is the standard measure of clock stability. One of its special advantages is its relative simplicity: it is only a function of  and can be plotted in the form of stability graphs such as those in Figure 3 below (taken from HELLWIG, 1979). The units for y(t) are dimensionless. Note that the linear drifts of the quartz crystal and rubidium oscillator of 1 part in 1010 and 1 part in 1012 have been removed.

How can frequency stability plots such as these be interpreted? The clock stability is defined as a function of the time interval between the monitoring of a particular clock. If it is assumed that at the start of the interval the clock is synchronised with (or has been compared to) a "true" time scale, the amount by which the clock has "deviated" (on average) after a certain time interval  is given by the square-root of the Allan Variance y() times :

x() =  . y() ( 1.3-10 )

Figure 3. Square-root of the Allan Variance of typical oscillators (after removal of linear trend for crystal and rubidium oscillators). (HELLWIG, 1979)

 

For example, quartz crystal oscillators are as precise as hydrogen masers if the time intervals are less than approximately 5 seconds. In the short term, up to 104 seconds, cesium standards fare badly in comparison with other frequency standards. However, their medium to long term performance is superior to all but the hydrogen maser, which it rivals after approximately 106 seconds. The behaviour of the oscillators in Figure 3 above can therefore be characterised by three regimes:

(1) short-period, where the Allan Variance decreases as the time interval  increases, according to the relation:

y() = K1 . 1 (1.3-11)
where 1 is positive (=1 for hydrogen maser or crystal oscillator clocks, =0.5 for cesium and rubidium clocks).

 

(2) medium-period, where the Allan Variance is a constant:

y() = yF = constant (1.3-12)

 

(3) long-period, where the Allan Variance increases with increasing time interval  according to the relation:

y() = K2 . -0.5 (1.3-13)

Representative values of K1 ,  and K2 are given in table below.

Typical performance data for commercially available oscillators.
( Adapted from HELLWIG, 1979 )
  K1 K2 Drift 
(sec/sec)
H (active) 1 x 10-12 1 x 10-14 3 x 10-17 10-15
Cs 5 x 10-11 
7 x 10-12		 1 x 10-13 
5 x 10-14		 3 x 10-17 
3 x 10-17		 10-15 
10-15 - 10-14
Rb 5 x 10-12 5 x 10-13 3 x 10-15 10-12
X-tal 1 x 10-12s 5 x 10-13 3 x 10-15 10-10

 

The Allan Variance is a measure of the performance of a clock. The Allan Variance is, however, not based on any physical model of the oscillator, but rather the stability graphs are produced from the results of testing actual oscillators. (For the long-period portion of the graphs, long measurement time periods are required, and hence the results are not as reliable as for the short- and medium-period portions.) Nevertheless it can be used to predict the behaviour of clocks in a positioning system such as GPS over time spans ranging from fractions of a second to several hours, and hence the likely build up of clock error (phase, time or range equivalent). This is important for various aspects of GPS observable modelling (section 6.1.1).

 

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© Chris Rizos, SNAP-UNSW, 1999