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
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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.,Earth’s 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 + 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