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12.540 Principles of the Global
Positioning System
Lecture 20
Prof. Thomas Herring
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GPS Models and processing
–
? Rank deficiencies
– Processing methods:
? Differencing of data
? Cycle slip detection
? Bias fixing and cycle slip repair
? Summary:
Finish up modeling aspects
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Rank deficiencies
?
that can not be separately estimated.
? In GPS, there are several rank deficiencies:
satellite orbits, effectively can not be separated.
orbits could be used to align the orbits in a solar system frame,
setting the mean longitude change of stations to ITRF
coordinates. Longitude is standard problem because choice of
Greenwich as origin is arbitrary.
Ranks deficiencies are combinations of parameters
– UT1, Longitudes of all the stations and the nodes of the
– In theory, orbit perturbations by the moon/sun on the GPS
but effect is too small to be useful (I think: never really tested)
– Separation is solved by adopting UT1-AT from VLBI, and
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Rank deficiencies
? Other rank deficiencies:
– Pole position can not separated from over all rotation of
coordinates. Again resolved either by adopting polar motions
on one day or on average having zero rotation of the
coordinates relative to an initial frame.
but effect is too small.
estimated. Again there is sensitivity due moon/sun
perturbations but these are too small. (Later we will see how
differencing data, implicitly eliminates this problem). Solution,
if clocks are explicitly estimated, is to adopt one clock as
reference or set an average of the clock differences to be
zero.
– In principle could be separated by gravity field perturbations
– All station and satellite clocks can not be simultaneously
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Rank deficiencies
–
motions from secular drift of pole and secular UT1-
AT changes. (Remember pole has drifted 10
meters in 100 years--10 cm/yr comparable to plate
motions).
– IERS polar motion is referred to a no-net-rotation
geologic frame (Nuvel-1A).
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Subtle rank deficiencies
? Phase center patterns for satellites and ground
receivers can not separately determined using just
GPS antennas.
? Because the satellites point towards the center of the
Earth; a given elevation angle at a GPS receiver can
be mapped to an off-bore-sight angle on the satellite
and two effects can not separated.
?
(so no longer pointing at the center of the Earth), the
two effects could be separated.
? Even with low precision satellite phase center
? Velocity rank deficiency:
It is not possible to separate “absolute” station
? There are some other rank deficiencies with
nutations and orbits, but the apriori nutation
series is very well defined by VLBI
Interestingly, if the GPS satellites could be “rocked”
positions can be estimated assuming “point” antenna
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Estimated Satellite Z-offsets
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
5 1015202530
Z-offset No Choke ring phase center
Z offset Hannover Choke ring model
Satellite Z-phase center position (m)
PRN
Block IIR
Block IIR
Apriori Block II/IIA offset
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Time series estimates
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
1999 2000 2001 2001 2002 2002 2003
PRN 11 Abs 1.44±0.03 m
PRN 14 Abs 1.71±0.06 m
PRN 28 Abs 1.45±0.07 m
PRN 11 Rel -2.14±0.04 m
PRN 14 Rel -1.87±0.06 m
PRN 28 Rel -2.18±0.05 m
Z Phase center offset (m)
Year
Block IIR satellites
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Zoom of Absolute series only
0.5
1.0
1.5
2.0
2.5
3.0
2000.8 2001.0 2001.2 2001.4 2001.6 2001.8 2002.0 2002.2 2002.4
Z Phase center offset (m)
Year
PRN 11 Abs 1.44±0.03 m
PRN 14 Abs 1.71±0.06 m
PRN 28 Abs 1.45±0.07 m
Block IIR satellites. Absolute PC model only
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Effects on radial orbit position of satellite
-0.4
-0.2
0.0
0.2
0.4
0.6
2000.0 2000.4 2000.8 2001.2 2001.6 2002.0
s 0.20 m
s 0.21 m
No Satellite PC
Rel s 0.18 m
D
Semimajor Axis PRN 11 (m)
Year
Apriori orbit: No satellite or choke ring PC, sites constrained
Abs Mean -0.01±0.03 RMS 0.15 m
Rel Mean -0.01±0.03 RMS 0.15 m
Mean 0.12±0.02 RMS 0.13 m
Orbit Adjustment relative to constrained model
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Summary of phase center
? The effects of ground antenna phase center model
only satellite phase center estimates are large (~3.6
meters)
? Block II/IIA definitely different from Block IIR and some
same type (differences are a few centimeters)
? Radial orbit changes are small (<1 cm on average).
Interestingly better agreement of loose solution with
constrained when satellite PC estimated (10 cm
differences),
indication of differences between satellites within the
Scale effects
? From the different analyses and VLBI analysis we can estimate
scale and its rate of change:
Soln Scale +- Srate +-
Abs -6.04 0.25 -0.24 0.06
Rel 11.99 0.25 -0.22 0.06
VLBI -0.21 0.04 -0.02 0.01
? Scale in ppb and scale rate ppb/yr (1ppb=6mm)
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Processing methods
?
biggest deviations in the model of GPS phase and
range data.
? These terms can be explicitly handled by estimation of
is a very large estimation problem). Can be attacked
with sequential LSQ or a Kalman filter.
? When multiple sites see the same satellite, the
satellite clocks can also be estimated, but at every
or an ensemble average of cocks set to have zero
mean adjustment.
The clock and local oscillator phase variations are the
clock variations (but if done brut-force in least squares
epoch of measurement, one clock needs to be fixed,
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Differencing
? An alternative approach to explicit estimation is
differencing data.
? Single differences: two forms:
satellite. Eliminates error due to satellite clock.
Eliminates the ground receiver clock.
? Double differences:
satellites clocks are eliminated.
in the number of cycles of phase between the combination of
two satellites and two stations. This difference should be an
integer.
– Difference measurements from two sites that see the same
– Difference measurements from two satellites at the one site:
– By differencing a pair a single differences, but the ground and
– The local oscillator phases also cancel except the differences
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Differencing
? There are subtle problems with the exact times that
?
phase to all the satellites can be made at exactly the
same (within electronics noise)
? But signals measured at the same time receivers
separated by large distances must have been
transmitted from the satellite at different times due to
the light propagation time..
measurements are made with differencing.
In the receivers, the measurements of range and
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? This effect can lead to 20 ms differences in the
transmission times. When SA was on and satellite
clocks had frequency drifts of ~1Hz, this lead to errors
of 0.02 cycles (~4mm). Not such a problem anymore
and even with SA was not severe.
?
problems. Normally receivers stay with in 1 ms of
GPS time (by resetting their clock counters). Older
receivers could be off by up to 80 ms:
nobreakspace
Light propagation time and differencing
Non-synchronized receiver sampling can cause
Feigl, K. L, R. W. King, T. A. Herring, and M. Rotchacher, A scheme for reducing the
effect of selective availability on precise geodetic measurements from the Global
Positioning System, Geophys. Res. Lett., 1289–1292, 1991.
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Cycle slip detection
?
problem. You can look at this in HW2 data set. The
L1 and L2 phase values are in the L1 and L2 slots in
The have a large offset from the range
should be a integer value)
?
missing but not always), a cycle slip occurs and this
parameter
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Cycle slip detection
?
variations are dominated by clocks in receiver.
?
Removes geometry but affected
by ionospheric delay (opposite sign on phase and range) and
noise in range measurements
Some times called wide-lane. Affected
by ion-delay but is common detector
removes ionosphere and if good apriori positions are known,
should be a smooth function of time. Often used to estimate
When processing phase, cycles slips are a potential
the rinex file.
values (initial ambiguity, which in double differences
When the receiver looses lock (typically range will be
must be re-fixed to an integer or left as an unknown
When o-minus-c is computed in one-ways for phase,
Multiple techniques are used to detect cycle slips:
– Ln phase - Ln range (n=1,2).
– L1 phase - L2 phase.
– Double difference phase residuals: On short baselines,
number of cycles in sip and resolve to integer value.
– Melbourne-Webena wide lane (ML-WL) (see over)
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MW wide lane
? Very useful combination of data that is often
used in kinematic GPS where receiver
coordinates are changing.
? The MW WL should equal number of cycles of
phase between between L1 and L2 and is
calculated, effectively, be computing the
expected L1 and L2 phase difference from the
pseudorange data.
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MW Wide lane
? From the equations for range and phase with the
phase offsets for cycle offsets you can derive:
? The MW-WL should be constant if there are no cycle
slips. When the phase and range values are double
differences, N
2
-N
1
should be integer.
? The factor that scales range is ~0.1 and so range
noise is reduced.
?
geometry changes.
MW -WL = N
1
- N
2
=f
L 2
-f
L1
+ (P
1
+ P
2
)
f
L1
- f
L 2
f
L1
+ f
L 2
Average values of the MW-WL are used to esimate
L1/L2 phase difference independent of ion-delay and
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Ambiguity resolution
? The MW-WL is often used to get N1-N2 and then N1
squares fit to the phase data.
? If the sigma of the estimate is small, and the estimate
is close to an integer then it can be resolved to an
integer values.
?
position estimate by typically a factor of two and
makes it similar to the North sigma. Bias fixing has
little effect on North and Up sigma except for short
sessions.
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is estimated, as non-integer value, from the least-
Fixing ambiguities, improves the sigma of the east
Summary
? Today’s lecture examined:
? rank deficiencies
? Differencing of data to eliminate clock errors
? Cycle slip detection and bias fixing (also called
ambiguity resolution).
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