BIASED RANGES |
The satellite clock scale and the receiver clock scale are not synchronised
at the instant of measurement (Figure 1). It is assumed that the satellite-receiver
range is affected only by a clock error (d
) caused by the receiver
oscillator (for this discussion satellite time is taken to be "true"
time). The relationship between the measured range 
* and the true
range is:
| *(t)
= r(t) + d(tr).c |
(1.3-14) |
where c is the speed of electromagnetic radiation. Note the time dependence
of the range and clock error, and that they are tagged with the time-of-reception.
This implies that if the receiver clock is slow (positive value
of din eqn (1.3-14)), then the measured range is too long;
and if the receiver clock is fast, then the measured range is too short.
If measurements are made simultaneously to several satellites, although the time of transmission of the signals is different for each satellite (and hence the flight time), they will be biased by the same amount.
Figure 1. Receiver clock error in one-way ranging.
A series of observation equations can be constructed:
| *s1
= s1 + d.c |
(1.3-15) |
| *s2
= s2 + d.c |
|
| *s3
= s3 + d.c |
|
| *s4
= s4 + d.c |
|
| ............................ | |
| *si
= si + d.c |
where si refers to the i'th satellite. Figure 2 illustrates the situation.
Figure 2. Receiver-biased ranges affecting all measurements at a receiver.
Satellite-Biased Ranges
The satellite clock scale and the receiver clock scale are not synchronised
at the instant of signal transmission:
Figure 3. Satellite clock error in one-way ranging.
Assuming that the satellite-receiver range is affected by only a clock
error (
) associated with the satellite oscillator (for this discussion
receiver time is assumed to be "true" time). The relationship
between the measured range * and the true range is:
| *(t)
= (t) – (Ts).c |
(1.3-16) |
Note the time dependence of the range and clock error, and that the measurements
are tagged with the time-of-reception, but the clock error is tagged with
the time-of-transmission. This implies that if the satellite clock is slow
(positive value of ), then the measured range is too short; and
if the satellite clock is fast, then the measured range is too long.
If measurements are made simultaneously at several ground receivers the time-of-transmission of the signals must have been different, and the measurements will be biased by slightly different amounts:
| *r1
= r1 – (Tr1).c |
(1.3-17) |
| *r2
= r2 – (Tr2).c |
|
| *r3
= r3 – (Tr3).c |
|
| *r4
= r4 – (Tr4).c |
|
| ......................... | |
| *rj
= rj – (Trj).c |
where rj refers to the j'th receiver. Figure 4 illustrates this situation.:
Figure 4. Satellite-biased ranges affecting measurements made at two receivers.
How different are the values of (Trj)? For receiver
separations of the order of 103km the maximum difference in arrival
time from a satellite at the horizon collinear with the interstation vector
is of the order of 0.003 seconds. If the satellite oscillator is a cesium
standard, then from Table in section1.3.2 it
can be seen that (Trj) may vary by up to 5x10-11
3x10-12, or approximately one millimetre in range equivalent!
Clearly this is negligible, and hence it can be assumed that the satellite
clock error is the same, even for widely separated ground receivers.
Ambiguous Ranges
Assume that each measurement made by a receiver to a satellite is ambiguous
because only a fraction of a wavelength (in other words, a fraction of the
time scale resolution) can be measured (Figure 5).
Figure 5. An ambiguous time delay measurement.
The measurement would be modelled as:
| *(t)
= (t) – n(Trjsi) |
(1.3-18) |
where n(Trjsi) is dependent on the receiver, the
satellite and the time. This is not a very useful measurement!
If it were assumed that the ambiguity n is a constant
over time, the measurement *(t) in effect contains the change
in distance since some initial epoch t0:
| *(t)
= (t) – nrjsi |
(1.3-19) |
Figure 6 illustrates this situation:
Figure 6. Ambiguous ranges where the ambiguity is constant with time.
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© Chris Rizos, SNAP-UNSW, 1999