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Terrestrial Geodesy Triangulation Optical measurement of horizontal angles with a theodolite Accuracy ~ 10 -4 degrees ~10 mm at 10 km Distance measurement Optical measurements of distances with laser-ranging Electronic Distance 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

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Page 1: Terrestrial Geodesy

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

Page 2: Terrestrial Geodesy

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

Page 3: Terrestrial Geodesy

Terrestrial Geodesy put to work…

Lisowski et al., 1991

Page 4: Terrestrial Geodesy

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.,Earth’s rotation axis)

– Longitude: time difference withGreenwich --> angles between starsand Moon, or precise chronometer

• Astronomy…

Peter Apian - Geographia, 1533

Page 5: Terrestrial Geodesy

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

Page 6: Terrestrial Geodesy

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

Page 7: Terrestrial Geodesy

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

Page 8: Terrestrial Geodesy

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)

Page 9: Terrestrial Geodesy

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 + ioni

k (t) + tropik (t) Ni

k +

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

Page 10: Terrestrial Geodesy

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

Page 11: Terrestrial Geodesy

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

Page 12: Terrestrial Geodesy

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

Page 13: Terrestrial Geodesy

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

Page 14: Terrestrial Geodesy

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)

Page 15: Terrestrial Geodesy

From positions to velocities

Page 16: Terrestrial Geodesy

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?

Page 17: Terrestrial Geodesy

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

Page 18: Terrestrial Geodesy

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)