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Terrestrial Geodesy
Triangulation
Optical measurement of horizontalangles with a theodolite
Accuracy ~ 10-4 degrees ~10
mm at 10 km
Distance measurement
Optical measurements of distanceswith laser-ranging ElectronicDistance Meter (EDM)
Accuracy ~ 1 ppm ~ 10 mm at
10 km
Errors and limitations:
Setup errors
Atmospheric refraction
Line-of-sight
Labor-intensive
Triangulation measurements on Mt St Helens
Two-color EDM in operation at Parkfield, California
Terrestrial Geodesy
Total station = theodolite + EDM
Post-processing = network adjustment Network decomposed in triangles
Basic trigonometry formulas used to compute triangleswhen at least 3 elements are known:
Law of cosines:
Law of sines:
Lenghts:
Network parameters (positions, baselines) areestimated using adjustment techniques
RC
c
B
b
A
a2
sinsinsin===
Abccba cos2222
+=
a
c
b
22 )()( BABA yyxxa +=
Result:
Best-fit network geometry (x,y) and changes in geometry (ux,uy, strain components)
Triangulation only Scale and orientation have to be fixed
Triangulation + distance measurement orientation has to be fixed
Terrestrial Geodesy put to work
Lisowski et al., 1991
Space Geodesy
Overcome limitations of terrestrialgeodesy:
Line-of-sight requirement
Accurate over long distances
3-D measurement (horizontal +vertical)
Continuous measurements possible
Automated measurements possible
Not a new concept!
Global positioning using spaceobjects has been made for 100eds ofyears
Latitude: elevation of Polaris (i.e.,Earths rotation axis)
Longitude: time difference withGreenwich --> angles between starsand Moon, or precise chronometer
Astronomy
Peter Apian - Geographia, 1533
Space Geodesy:VLBI
Very Long Baseline Interferometry
Radio-astronomy technique, used to locateand map stars, quasars, etc = sources
Wavelength = 1-20cm
Measures the time difference between thearrival at two Earth-based antennas of aradio wavefront emitted by a distant quasar
If the source positions are known =>ground baseline => geodetic VLBI
Time measurements precise to a fewpicoseconds, => relative positions of theantennas to a few millimeters
Advantages: ultimate accuracy
Problems: infrastructure, cost
VLBI antenna at Algonquin, Canada
Space Geodesy:GPS
Global Positioning System
GPS = VLBI with a man-madesignal
Three steps:1. Satellites broadcast a radio signal
towards the Earth
2. Receivers record the signal andconvert it into satellite-receiverdistances
3. Post-processing consist ofconverting these distances intopositions
Precision:$100 receiver 100 m
$10,000 receiver 1 mm
Principle of GPS positioning
Satellites broadcast signals on 1.2 GHzand 1.5 GHz frequencies:
Satellite 1 sends a signal at time te1 Ground receiver receives it signal at time tr The range measurement 1 to satellite 1 is:
1 = (tr-te1) x speed of light
We are therefore located on a spherecentered on satellite 1, with radius 1
3 satellites => intersection of 3 spheres
Or use the mathematical model:
A! The receiver clocks are mediocre andnot synchronized with the satellite clocks
Time difference between the satellite clocksand the receiver clock
Additional unknown => we need 4observations = 4 satellites visible at thesame time
222 )()()(rsrsrs
s
rZZYYXX ++=
satellite 1
Earth
1
satellite 3
3
2
You are here
x
2
satellite 2
Principle of GPS positioning
GPS data = satellite-receiverrange measurements ( )
Range can be measured by:
Measuring the propagation time ofthe GPS signal:
Easy, cheap
Limited post-processing required
As precise as the time measurements~1-10 m
Counting the number of cycles of thecarrier frequency
More difficult
Requires significant post-processing
As precise as the phase detection ~1mm
Earth
x
te
tr
data = (tr-te) x c data = x n
~ 20 cm
From codes: From carrier:
(unit = meters) (unit = cycles)
Principle of GPS positioning
GPS phase equation (units of cycles):
Range model:
Phase equation linearized
Form a system of n_data equations for n_unknowns (positions,phase ambiguities, tropospheric parameters)
Solve using weighted least squares (or other estimationtechniques)
End product: position estimates + associated covariance
ik (t) = i
k (t)f
c+ hk (t) hi(t)( ) f + ionik (t) + tropik (t) Nik +
ik
= (X k Xi)2
+ (Y k Yi)2
+ (Zk Zi)2
= phase measurement = DATA
ik = geometric range = CONTAINS UNKNOWNS Xi,Yi,Zi
Xk,Yk,Zk = satellite positions (GIVEN)
t = time of epoch
i = receiver, k = satellite
f = GPS frequency, c = speed of light
hk = satellite clock error, hi = receiver clock error
ionik = ionospheric delay, tropi
k = tropospheric delay
Nik = phase ambiguity, = phase noise
Principle of GPS positioning
Precise GPS positioning requires:
Dual-frequency equipment Rigorous field procedures Long (several days) observation sessions Complex data post-processing
MagnitudeTreatmentError source
~ 1 cmUse correction tablesAntenna phase center
~ 0.5 mChoose good sites!Multipath
???Choose good operators!Site setup
centimetersPrecession, Nutation, UT, Polar motionGeodetic models
centimetersTides (polar and solid Earth), Ocean loadingGeophysical models
2 cm to 100 mGet precise (2-3 cm) orbitsSatellite orbits
1-50 mDual frequency measurementsIonospheric refraction
0.5-2 mExternal measurement or estimation of troposphericparameters
Tropospheric refraction
metersDouble difference or direct estimationReceiver clock errors
~1 mDouble difference or direct estimationSatellite clocks errors
< 1 mmNonePhase measurement noise
Two measurement strategies: Repeated campaigns Continuous measurements at
permanent sites
Campaign GPS measurements
Field strategy: Network of geodetic benchmarks
perfectly attached to bedrock
Separation typically 10-100 km
Dual frequency GPS receivers
2 to 3 measurement sessions of
24 hours, sampling at 30 sec
Then move to next site.
Usually several crews operate
simultaneously.
Download GPS measurements
from receiver memory into
computer, quality control, backups
Advantages:
Large number/density of sites with
few receivers
Relatively low cost
Problems: Transient deformation
Monumentation
Antenna setup
Typical GPS campaign setup using a spike
mount, Dominican Republic
Typical GPS campaign setup
using a tripod, Mongolia
PermanentGPS sitesTypical setup:
Dual frequency GPS receivers Phase and pseudorange measurements at
30 sec rate, continuously, 24h/day, 365days/year
GPS antenna mounted permanently on astable geodetic monument
Site protected and unattended Receiver, power supply and modem in a
shelter by the antenna Data downloaded daily or more frequently if
needed (and if possible)
Advantages:
Better long-term precision Better detection of transient signals
Problems:
Cost and number of sites Power supply Lightning Vandalism Sites not as stable as originally thought
Permanent GPS
site, antenna on
concrete pillar
anchored in
bedrock
Shelter with GPS
receiver, solar
panels
GPS time series
1 position per day (per site)
Associated uncertainty
Successive daily positions
times series
Times series:
Slope = long-term sitedisplacement due to tectonicmotions
Noise
http://sideshow.jpl.nasa.gov/mbh/series.html
GPS times series
http://sideshow.jpl.nasa.gov/mbh/series.html
Time series can also show:
Steps => earthquakes (orequipment change?)
Transient features =>postseismic deformation (orlocal site effect?)
Periodic features, typicallyseasonal (loading)
From positions to velocities
Noise models
150 km baseline across the San
Jacinto active fault, 2.5 years of
continuous GPS observations
Linear trend fault slip rate: 16.9
0.6 mm/yr
Comparison with "simulated
campaigns":
Difference between continuous up to
10 mm/yr
10 mm/yr >> uncertainties estimates
Long period fluctuations in the
continuous time series
What noise model?
Noise models Power spectrum of GPS time series => non random noise (colored)
= time correlated noise
Consequences on velocity uncertainties (Mao et al., 1999):
T=time span
g=# meas. per year
=magnitude of noise
a,b=empirical constants
r12 w
2
3
g T+a f
2
2bg T+ rw
2
T
1/ 2
Noise models
Uncertainty as a function oftime: Continuous GPS
measurements in blue
Campaign measurements inred
Conclusion: Continuous GPS: 2 years at
least to reach 1mm/yrprecision
4 time longer for campaignmeasurements
(This simple experiment does not account for temporalcorrelation between errors)