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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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  • Terrestrial Geodesy


    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



    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:


    Network parameters (positions, baselines) areestimated using adjustment techniques








    Abccba cos2222





    22 )()( BABA yyxxa +=


    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


    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


    rZZYYXX ++=

    satellite 1



    satellite 3



    You are here



    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





    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 +


    = (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


    Download GPS measurements

    from receiver memory into

    computer, quality control, backups


    Large number/density of sites with

    few receivers

    Relatively low cost

    Problems: Transient deformation


    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)


    Better long-term precision Better detection of transient signals


    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


    Shelter with GPS

    receiver, solar


  • 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



  • GPS times series


    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


    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



    g T+a f


    2bg T+ rw



    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)