EESC 9945
Geodesy with the Global Positioning
System
Class 7: Relative Positioning using Carrier-Beat Phase
GPS Carrier Phase
• The model for the carrier-beat phase observable for
receiver p and satellite r we developed, ignoring atmospheric and ionospheric propagation delays, was
r (t) + φr − φ◦ + N r + f [δ r (t) − δ (t)]
φrp(t) = − 1
ρ
◦
p
p
p
◦
λ p
• We’ve left off the “∆” since too many
• Subscript letters are different
2
Single Difference
• The single difference (or “single difference phase”)
is the difference between phase observations from
two sites for a single satellite:
φrpq (t) = φrp(t) − φrq (t)
i
1h r
r
= − ρp(t) − ρq (t) − φ◦p + φ◦q + Npr − Nqr
λ
+f◦[δq (t) − δp(t)]
• Satellite initial phase and clock error have differenced out
3
Baseline Vector
• Let ~bpq be the vector between two GPS site located
at ~
xp and ~
xq :
~bpq = ~
xq − ~
xp
• ~bpq is called the baseline vector (or “inter site vector”) between sites p and q
4
Single-difference geometry: Short baseline
limit
• The geometry term in the single-difference phase is
xr − ~
xp| − |~
xr − ~
xq |
∆ρrpq = ρrp − ρrq = |~
• Substituting ~
xq = ~
xp + ~bpq we have
r
r
r
r
r
~p − ~bpq x −~
xp − ~bpq = ρp − ρ
∆ρpq = ρp − ~
• The length of the vector ρ
~rp − ~bpq is
"
#1/2
2
r
ρ̂p ·~bpq
b
r ~ r
~p − bpq = ρp 1 − 2 ρr + ρpqr
ρ
p
p
5
Single-difference geometry: Short b limit
• Expanding for small |bpq | gives
"
2!#
r
~
ρ̂ ·bpq
bpq
r ~ ~p − bpq ' ρrp 1 − pρr + o
ρ
ρrp
p
• Dropping terms of order (bpq /ρrp)2 and smaller gives
r ~ ~p − bpq ' ρrp − ρ̂rp · ~bpq
ρ
• The geometry term in the single-difference phase
for bpq ρrp is thus
∆ρrpq = ρrp − ρ
~rp − ~bpq ' ρ̂rp · ~bpq
6
Single-difference geometry: Short b limit
Range Difference (km)
3000
Exact
Short BL Approx.
2500
2000
1500
1000
500
0
0
1000
2000
3000
4000
Baseline Length (km)
Difference is ∼10 cm @ 100 km
7
Implications of differencing
• Single-difference phase observations are much less
sensitive to position and much more sensitive to
difference of position
• In other words, sensitivity to coordinate origin has
been significantly decreased
8
Implications of differencing
• Sensitivity of phase to baseline error (as rule of
thumb) is
sat
δΦ ' δb + bδρ̂ ' δb + b δx
x
sat
• Thus, an error in the orbital position of a GPS satellite can compensated (i.e., ∆Φ ' 0) if
δb δxsat b'x
sat
• This is why orbit errors are often spoken of in terms
of fractional errors (e.g., parts-per-billion)
9
Errors in IGS Orbit Products
IGS Product
Name
Broadcast
Ultra-Rapid
Rapid
Final
Latency
0
0–9 h
17–41 h
12–18 d
Nominal
Accuracy (cm)
100
3–5
2.5
2.5
Nominal
Accuracy∗ (ppb)
40
0.1–0.2
0.09
0.09
1 ppb orbit error → ∼1 mm baseline error @ 1000 km
10
Error History for IGS Orbit Products
11
Double Difference
• Single difference for sites p and q to satellite r:
φrpq (t) = φrp (t) − φrq (t)
1
= − ρrp (t) − ρrq (t) − φ◦p + φ◦q + Npr − Nqr
λ
+f◦ [δq (t) − δp (t)]
• Double difference formed from two single differences to two
satellites:
r
s
φrs
pq (t) = φpq (t) − φpq (t)
1
= − ρrp (t) − ρrq (t) − ρsp (t) + ρsq (t) + Npr − Nqr − Nps + Nqs
λ
1
rs
= − ρrs
pq (t) + Npq
λ
• All site-only- and satellite-only-dependent terms have canceled
12
Working with Double Differences
• Some advantages:
1. No clocks, initial phases
2. Simple model
• Some disadvantages:
1. Bookkeeping
2. Estimating ambiguities
3. Further loss of sensitivity
4. Unable to reconstruct “one-way” phases
13
Triple Difference
• Double difference formed from two single differences to two
satellites:
r
s
φrs
pq (t) = φpq (t) − φpq (t)
1 r
r
s
s
= − ρp (t) − ρq (t) − ρp (t) + ρq (t) + Npr − Nqr − Nps + Nqs
λ
1
rs
= − ρrs
pq (t) + Npq
λ
• Triple difference formed from time differences of double difference
rs
rs
∆φrs
pq (tj , tk ) = φpq (tk ) − φpq (tj )
1
' − ρ̇rp (t) − ρ̇rq (t) − ρ̇sp (t) + ρ̇sq (t) (tk − tj )
λ
• Useful for cycle-slip detection
14
GPS Data Analysis: Outline
1. Perform “clock” solution
(a) Estimate receiver clock errors
(b) Estimate site position if no prior better (< 1 m)
is available
15
Clock solution showing “clock steering”
Clock Error (km)
300
250
200
150
100
50
0
0.0
0.2
0.4
0.6
0.8
1.0
0.08
0.10
Clock Error (km)
300
250
200
150
100
50
0
0.00
0.02
0.04
0.06
Epoch (fraction of day)
300 km ⇐⇒ 1 msec
16
LC Pseudorange Residual (m)
Pseudorange Residuals
40
30
20
10
0
-10
-20
-30
-40
0.0
0.2
0.4
0.6
0.8
1.0
Epoch (fraction of day)
RMS = 5.4 m
17
GPS Data Analysis: Outline
1. Perform “clock” solution
(a) Estimate receiver clock errors
(b) Estimate site position if no prior better (< 1 m)
is available
2. Using best model, repair cycle slips
18
Data Processing: Cycle Slips
• We have assumed that the integer bias N in the phase model
is not a function of time
• In fact, “cycles slips” can occur in systems—such as GPS
receivers—that employ phase-locked loops
• A phase-locked loop is a feedback system that uses the phase
difference of the received and internal signals to modify the
internal frequency in an attempt to match frequency variations
of the incoming signal and to keep the signal “locked”
• Such systems are highly sensitive to SNR to be able to separate the signal (which should be locked onto) from the noise
(which should be ignored)
• Also, if the phase varies rapidly in a noise-like manner (e.g.,
ionosphere, multipath), the receiver can lose lock
19
L1 DD Residuals (b = 10 km)
Units: cycles/hours
20
L1 DD Residuals (b = 10 km) with cycle slip
Units: cycles/hours
21
Rectifying cycle slips
• These days, cycle slip detection and removal is done
using automatic processing
• In old days, done by hand
• One useful data combination is the MelbourneWübbena wide lane:
∆mw = φ1 − φ2 −
1 (f1 − f2)
[f1R1 + f2R2]
c (f1 + f2)
(1)
22
Melbourne-Wübbena Wide-Lane
-23500000
MW_WL_07_cycles
-24000000
MW_WL_07_cycles
-24500000
-25000000
Cycle slip
-25500000
18.0
19.0
20.0
21.0
Time_Hrs
22.0
23.0
24.0
From Herring, Principles of the Global Positioning System
23
Next time
• Regional solutions using fixed orbits
• Orbit integration
• Orbit parameter estimation
• The IGS network
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