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)