GPS and its Application to
Geodynamics in East Africa
Eric Calais
Purdue University, West Lafayette, IN, USA
ecalais@purdue.edu
The distribution of earthquakes is not random
Ocean-ocean
subduction
 island arc
Transform
fault
 strike-slip
motion
lithosphere
viscous mantle
Oceanic spreading
center
 creation of new
oceanic crust
Ocean-continent
subduction
 volcanism
Continental rift
 break-up of a
continent
lithosphere
viscous mantle
Tectonic plates are rigid and float on a viscous mantle.
Earthquakes occur at their boundaries: divergent (rifts and oceanic
spreading centers), convergent (subductions), or strike-slip
The Earth’s rigid shell (= lithosphere) is made of ~15 major plates
Notice the lack of plate boundary through East Africa…!
In Summary…
• We know:
– Plate tectonics as a kinematic theory that
describes the motion of (rigid) plates at the surface
of the Earth
• We do not know:
– The present-day motion of all plates
– Why plates move the way they do (dynamics)
• We need:
– Accurate techniques to measure present-day
motions of the Earth’s lithosphere
– Physical models that explain the dynamics of the
system (= kinematics + rheology)
The Global
Positioning System
• Three steps:
1. Satellites broadcast a radio
signal towards the Earth
2. Receivers record the signal
and convert it into satellitereceiver distances
3. Post-processing consist of
converting these distances
into positions
• Precision:
 $100 receiver  100 m
 $10,000 receiver  1 mm
Principle of GPS positioning
• Satellites broadcast signals on 1.2 GHz
and 1.5 GHz frequencies:
– Satellite 1 sends a signal at time te1
– Ground receiver receives it signal at time tr
– The range measurement r1 to satellite 1 is:
r1 = (tr-te1) x speed of light
– We are therefore located on a sphere
centered on satellite 1, with radius r1
– 3 satellites => intersection of 3 spheres
satellite 2
rr2
2
• Or use the mathematical model:
r  ( X s X r )  (Ys  Yr )  (Z s  Z r )
s
r
2
2
– Time difference between the satellite clocks
and the receiver clock
– Additional unknown => we need 4
observations = 4 satellites visible at the
same time
r1
r3
2
• A! The receiver clocks are mediocre and
not synchronized with the satellite clocks
satellite 1
satellite 3
You are here
x
Earth
Principle of GPS positioning
• GPS data = satellite-receiver
range measurements (r)
• Range can be measured in two
ways:
te
~ 20 cm
1. Measuring the propagation time
of the GPS signal:
•
•
•
Easy, cheap
Limited post-processing required
As precise as the time
measurements ~1-10 m
tr
2. Counting the number of cycles
of the carrier frequency
•
•
•
More difficult
Requires significant postprocessing
As precise as the phase
detection ~1 mm
x
Earth
From codes:
data = (tr-te) x c
(unit = meters)
From carrier:
data = x n
(unit = cycles)
Principle of GPS positioning
 GPS phase equation (units of cycles):
ki (t)  rik (t) 
f
 h k (t)  hi (t) f  ion ik (t)  tropik (t)  N ik  
c
 Range model:


rik  (X k  X i )2  (Y k Yi )2  (Z k  Zi )2
 = phase measurement = DATA
rik = 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, tropik tropospheric delay
Nik = phase ambiguity,  = phase noise
 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 estimation
techniques)
 End product: position estimates + associated covariance
Principle of GPS positioning
Error source
Treatment
Magnitude
Phase measurement noise
None
< 1 mm
Satellite clocks errors
Double difference or direct estimation
~1 m
Receiver clock errors
Double difference or direct estimation
meters
Tropospheric refraction
External measurement or estimation of “tropospheric
parameters”
0.5-2 m
Ionospheric refraction
Dual frequency measurements
1-50 m
Satellite orbits
Get precise (2-3 cm) orbits
2 cm to 100 m
Geophysical models
Tides (polar and solid Earth), Ocean loading
centimeters
Geodetic models
Precession, Nutation, UT, Polar motion
centimeters
Antenna phase center
Use correction tables
~ 1 cm
Multipath
Choose good sites!
~ 0.5 m
Site setup
Choose good operators!
???
 Precise GPS positioning requires:
• Dual-frequency equipment
• Rigorous field procedures
• Long (several days) observation sessions
• Complex data post-processing
Campaign measurements
 Field strategy:
– Network of geodetic benchmarks perfectly attached
to bedrock -- Separation typically 10-100 km
– 2 to 3 measurement sessions of 24 hours
 Advantages:
– Large number/density of sites with few receivers
– Relatively low cost
 Problems:
– Transient deformation
– Monumentation and antenna setup
Continuous measurements
 Typical setup:
– Antenna mounted permanently on a stable geodetic
monument, measurements 24h/day, 365 days/year
– Site protected and unattended
– 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 and communication
GPS time series
• Processing strategy:
– GPS data (phase and
pseudorange) processed in
daily sessions
– Use of precise orbits and
Earth Orientation Parameters
from the IGS
– Use of additional continuous
sites with well-defined position
and velocity in ITRF
• Output:
– 1 position per day (per site)
– Associated uncertainty
– Successive daily positions 
times series
– Slope = long-term site velocity
due to tectonic motions
From positions to velocities
•
Velocity can be estimated by combining several measurement epochs
with the following model:
known
position
at epoch s
(in reference
frame s)
=
unknown
(final)
position
(in final
reference
frame)
+
unknown
(final)
velocity
(in final
reference
frame)
transformation between final reference frame and reference frame at epoch s
(T = translation, D = scale factor, R = rotation)
+
position
+
velocity
 i
i
i
i
i
i
X si  X comb
 (t s  t comb ) XÝcomb
 T  DX comb
 RX comb
 (t s  t comb )TÝ DÝX comb
 RÝX comb

•
•
The model is linear  X, X, T, D, R, T, D, R can be estimated using
standard least squares and error propagation.
As such, problem is rank deficient (datum defect)  define a frame by
fixing or constraining the position and velocity of a subset of sites to known
values, for instance from International Terrestrial Reference Frame (ITRF)
REVEL GPS plate model
Sella et al., JGR 2002
• Velocities are shown with arrows
• Expressed with respect to ITRF = absolute reference frame
• Can be used to quantify plate motions
From velocities to plate motions
• The motion of “plates” (= spherical
caps) can be described by:
– A pole of rotation (lat, lon), also
called Euler pole
– An angular velocity (deg/My)
• Or by a rotation vector W:
– Origin at the Earth’s center, passes
through Euler pole
– Length = scalar angular velocity
• Relation between horizontal
velocity at a given site (position
described by unit vector Pu) and
rotation vector W :

V  R W Pu

(R = mean Earth’s radius)
Plate motions, inverse problem
•
•
For a given site, linear equation:
 X  X  Z y  Y z 

    
V  RY  Y  RX z  Z x 
    

 Z  Z  Y x  X y 
W
•
W  (AT CV1 A)1 AT CV1L
P
Or in matrix form:
vx  0
Z Y  x 
  
 
vy

R
Z
0
X
  
 y 
  
 
vz  Y X 0  z 
or
V  AW
If at least 2 sites with velocities, the
problem is over-determined and can
be solved using least squares (L =
data vector, W = data weight matrix):

•
The model covariance matrix is:
CW  (AT CV1 A)1

Plate kinematics and deep mantle structures
Color arrows: motion of adjacent plates with respect to Nubia (Sella
et al., 2002). Black arrows: Nubian plate motion in a hot spot frame
(Gripp and Gordon, 2002)
Behn et al., 2004: Arrows = mantle flow field, colors =
seismic velocity anomalies. Top = map view at the
base of the lithosphere (300 km), bottom: crosssection of S20RTS (Ritsema et al.)
Regional tectonics and upper mantle structures
Nyblade et al. (2000): Top = cross-section of
tomographic model (Ritsema et al., 1998) with
stacked receiver function superimposed. Bottom:
schematic interpretation.
Nubia/Somalia
kinematics
• Very few continuous GPS sites
on Nubian and Somalian plates
 Nubia-Somalia relative motion
still poorly constrained
• Two plates:
– Nubia = MAS1, NKLG, SUTH,
SUTM, GOUG (ZAMB, HRAO,
HARB)
– Somalia = MALI, HIMO, SEY1,
REUN
• Euler pole between South Africa
and SW Indian Ridge  NubiaSomalia extension rate increases
from S to N
• Discrepancy at MBAR
(Work by Saria Elifuraha)
EAR
kinematics
• Seismicity + active faults  2
possible microplates within
the EAR
• Data:
– GPS, MBAR on Victoria +
SNG1 on Rovuma
– Earthquake slip vectors
• Invert GPS + slip vectors for
block motions
• Results:
– Somalia: consistent with
previous estimates
– Victoria: CCW rotation
– Rovuma: CW rotation
(Work by Sarah Stamps)
Summary
•
•
2 major plates, divergence rate
increases from 3 to 6 mm/yr
from S to N.
GPS + slip vector data
consistent with:
–
–
•
WARNINGS:
–
–
–
•
Strain focused along narrow rift
valleys
2 undeformed domains: Victoria
and Rovuma microplates
Model
Constrained by very few GPS
data
Needs to be tested/improved
Dynamics of Victoria pl.?
Conclusions
• Kinematics:
– Combination of (limited) GPS data set + earthquake slip vectors 
preliminary kinematic model for Nubia/Somalia + 2 microplates (Victoria
and Rovuma)
– Model will be refined using new GPS data in Tanzania.
– Next GPS campaigns = August 2008 and 2010.
• Dynamics:
– Combine kinematic model with other tectonic indicators, seismic
anisotropy data, mantle and lithospheric structures (tomography, xenoliths,
etc.)
– Geodynamic modeling: driving forces, mantle-lithosphere interactions.
• Broader impacts:
–
–
–
–
Establishment of new national geodetic network
Establishment of new IGS site
Training and collaborative research
Other research projects: geoid, datum transformations, vertical motions,
etc…
Acknowledgments
• Partners:
– University College for Lands and
Architectural Studies
– Survey and Mapping Division, Ministry of
Lands and Human Settlement
– Department of Geology, University of Dar
Es Salaam
– Department of Earth and Atmospheric
Sciences, Purdue University
– Department of Geology, Rochester
University
– Hartebeesthoek Radio Astronomy
Observatory, South Africa
– Royal Museum for Central Africa,
Belgium
– Universite de Bretagne Occidentale,
IUEM, France
• Technical Support from UNAVCO
(www.unavco.org)
• A project funded by the National Science
Foundation (www.nsf.org)