Transcript Slide 1

A quick GPS Primer (assumed knowledge on the course!)
Observables
Error sources
Analysis approaches
Ambiguities
If only it were this easy…
Review of GPS positioning
Dealing with errors
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•
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•
Orbit Error
Clock Error
Epsilon (SA)
Dither (SA)
Clock errors (review)
Ionosphere (review)
Troposphere (part review)
Earth body deformations (new)
Orbit errors (new)
Ionospheric
refraction
Receiver Noise
Clock Error
Tropospheric
refraction
Multipath
GPS Undifferenced observable
A somewhat simplified view, but all these need to be dealt with (at least)
for precise GPS geodesy
True range
Carrier
phase
ambiguity
Tropospheric
Delay
Satellite j
    c  dt  dt    N  I  T  Orbit  other errors
S
R
Observed
range
Receiver
and Satellite
clock errors
(multiplied
by speed of
light)
Ionospheric
Delay
Includes
Multipath
Station A
Dealing with clock errors
Undifferenced observable
• Estimate both receiver and satellite clocks
• Precise Point Positioning – Fix prior satellite clocks and estimate only
receiver clocks
Satellite j
• Parameter hungry
Satellite k
Double-differenced observable
• Undifferenced observations to
two satellites at two stations
• Form two between-station
differences and then
double-difference:
  
   N  I  T 
Orbit  other errors
• Common clock terms difference
Station A
Station B
Dealing with orbit errors
These days somewhat easy
• Use the IGS final orbits (precise to 2-5cm)
• Use Rapid or Ultra-rapid if quick turnaround needed (precise to ~5cm)
• Probably no reason to use the broadcast orbits (precise to ~0.5-2m)
In practice
• Need orbits from adjacent days when processing against the day
boundary
• Orbit errors are rarely an error source when using IGS products (main
exception is pre-IGS data – earlier than 1994)
Dealing with the Tropospheric Delay (I)
Total delay
• ~2.3m at zenith, greater at horizon
• Elevation angle dependency may be relatively well modelled with a
mapping function (M) for each of two tropospheric components
Two components
• Hydrostatic – could be well modelled with accurate pressure
• Wet – not well modelled and must be parameterised
• Over very short (<<10km) and small elevation difference (<100-200m)
baselines, effect cancels in double-difference
General approach
• Model hydrostatic with standard pressure or (more accurate) use
ECMWF or station met data
• Parameterise zenith wet delay (Twet), which also absorbs any residual
Thydro , once per 1-2 h (static) or every epoch (kinematic)
TSlant  Thydro .M hydro (El )  Twet .M wet (El )
Dealing with the Tropospheric Delay (II)
Troposphere is not azimuthally uniform
• Horizontal gradients are common, particularly N-S
Highest precision static processing will further estimate horizontal
gradient terms
• 1-2 for each E-W and N-S per day common
In kinematic analysis, steps in estimated tropospheric zenith delay
suggest likely wrong ambiguity fixed and hence quality control
Dealing with Ionospheric Delay (I)
Different frequency signals (in L-band) delayed by different amounts
through Ionosphere
• Dual frequency GPS receivers allow 99.9% for effect to be removed
• Higher order terms may be important for most precise geodetic work
Use a linear combination of L1 and L2 measurements to form new
measurement ionosphere free combination for carrier (LC or
alternatively L3)

1 f2
 L1 (cycles)  f L1  N L1 
I  other errors
c
f L1 c

1 f2
 L 2 (cycles)  f L 2  N L 2 
I  other errors
c
fL2 c
f L1 , f L 2
Where
are frequency of the L1 and L2 carrier phase signals
Dealing with Ionospheric Delay (II)
Differencing and re-arranging cancels I term
Ionosphere-free phase Linear Combination LC is defined:

 f L21

 f L21
fL2
fL2


 L 2 (cycles)  2
   other errors  f L1   N L1 
NL2  2
(cycles)
 L1 (cycles) 
2
2
f
f

f
c
f
f

f



L1
 L1 L 2

L1
 L1 L 2
Note:
•
•
Ambiguity terms are no longer integers – ambiguity fixing is not an option with
LC
Noise (“other errors”) is scaled up
General approach
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•
Adopt LC for baselines >~10km
Fix ambiguities, where possible, using a different linear combination (e.g.,
wide-lane) then final solution using LC, holding ambiguities fixed
Matrix Form
Static case – solving for parameters x
A
Obs1
Obsn
 F ( x)
 X
AB
















F ( x)
YAB
F ( x)
Z AB
x
F ( x)
T1
0
1-4hrs
0
F ( x)
T2
F ( x)
N1
F ( x)
N 2
= b + V




  d X AB  
  d Y  
AB 

  d Z AB  


dT
1


 

 

ij
  d N AB  
  d N ik  
AB 

 
 





 
 
 
 
 
 
 

 
 
 
 
 
 
 















Multipath
Generally dealt with through
• Stochastic model by assumption of elevation-dependence and downweighting lower elevation observations (GAMIT examines the
residuals and allows iterative reweighting on a station-by-station
basis)
• Assuming to “average toward zero” over 24h sessions
• Possibly a blind spot in GPS geodesy today
Ambiguity Fixing
Ambiguity for each satellite pass and all cycle slips thereafter
• Dozens of ambiguity terms for a 24 h period
Ambiguity fixing process is essentially a series of statistical tests
• Can each ambiguity be confidently (given it’s uncertainty) be fixed to
an integer?
• Iteration required, since uncertainties will change (normally reduce) as
ambiguities are fixed and removed from the least squares parameters
set
Essential for kinematic (or stabilisation of real-valued estimates in,
e.g., Kalman Filter such as in Track)
• Not always possible to fix all ambiguities
Less impact for static
• Largest effect (normally <10mm) in E, then N & U (see Blewitt, 1989)
• Can change the way systematic errors propagate
Double Difference vs PPP
Similar precision possible in 24 h solutions
Software
• Few software do geodetic PPP (GIPSY mainly)
• GAMIT/Track are Double Difference
PPP is requires extra care
• modelling geophysical phenomena (e.g., ocean tide loading
displacements) which may be (partially) differenced in relative analysis
• orbit/clock errors (some periodic) map 1:1 into positioning
Kinematic PPP requires longer periods of data – ambiguity fixing is
not possible without a double difference second step
DD is more precise when short-baseline relative motion is all that is
required (e.g., glacier monitoring), but depends on base station
Further Reading
Reference Texts
• Hofmann-Wellenhof, B., H. Lichtenegger, and J. Collins. 2001. Global
Positioning System: theory and practice, Springer, Wien, 382 pp.
• Leick, A. 2004. GPS Satellite Surveying, John Wiley & Sons, New
York, 435 pp.
Review Paper
• Segall, P., and J.L. Davis. 1997. GPS applications for geodynamics
and earthquake studies, Annual Review of Earth Planet Science, 25,
301-336