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

Oceanic spreading center

creation of new oceanic crust

Ocean-continent subduction volcanism

Continental rift break-up of a continent

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

Transform fault

strike-slip motion

lithosphere

viscous mantle

lithosphere

viscous mantle

The Earth’s rigid shell (= lithosphere) is made of ~15 major platesNotice 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 satellite-receiver 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

satellite 1

Earth

satellite 3

You are here

x

satellite 2

• 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 1 to satellite 1 is: 1 = (tr-te1) x speed of light

– We are therefore located on a sphere centered on satellite 1, with radius 1

– 3 satellites => intersection of 3 spheres

• Or use the mathematical model:

• A! The receiver clocks are mediocre and not synchronized with the satellite clocks

– Time difference between the satellite clocks and the receiver clock

– Additional unknown => we need 4 observations = 4 satellites visible at the same time

222 )()()( rsrsrssr ZZYYXX −+−+−=

Principle of GPS positioning

• GPS data = satellite-receiver range measurements ()

• Range can be measured in two ways:1. Measuring the propagation time

of the GPS signal:• Easy, cheap• Limited post-processing required• As precise as the time

measurements ~1-10 m

2. Counting the number of cycles of the carrier frequency

• More difficult• Requires significant post-

processing• As precise as the phase

detection ~1 mm

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 estimation

techniques) End product: position estimates + associated covariance

€

Φik (t) = ρ i

k (t) ×f

c+ h k (t) − hi(t)( ) × f + ioni

k (t) + tropik (t) − N i

k + ε

€

ik = (X k − X i)

2 + (Y k −Yi)2 + (Z k − Zi)

2

Φ = phase measurement = DATA

ik = geometric range = CONTAINS UNKNOWNS Xi,Yi,Zi

Xk,Yk,Zk = satellite positions (GIVEN)t = time of epochi = receiver, k = satellitef = GPS frequency, c = speed of light

hk = satellite clock error, hi receiver clock error

ionikionospheric delay, tropi

ktropospheric 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

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! ???

Campaign measurements Continuous 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

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:

€

X si = Xcomb

i + (ts − tcomb ) ˙ X combi + T + DXcomb

i + RXcombi

6 7 4 4 4 8 4 4 4 + (ts − tcomb ) ˙ T + ˙ D Xcomb

i + ˙ R X combi

[ ]

6 7 4 4 4 4 4 8 4 4 4 4 4

knownposition

at epoch s(in reference

frame s)

unknown(final)

position(in final

referenceframe)

unknown(final)

velocity(in final

referenceframe)

transformation between final reference frame and reference frame at epoch s(T = translation, D = scale factor, R = rotation)

position velocity

• 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 :– 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 :

€

rV = R

r Ω ×

r P u[ ]

(R = mean Earth’s radius)

• 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:

Plate motions, inverse problem

• For a given site, linear equation:

• Or in matrix form:

€

vx

vy

vz

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟= R

0 Z −Y

−Z 0 X

Y −X 0

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

ωx

ωy

ωz

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

or

V = AΩ

€

=(ATCV−1A)−1 ATCV

−1L

€

CΩ = (ATCV−1A)−1

€

rV = R

ωX

ωY

ωZ

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟×

X

Y

Z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟= R

Zωy −Yωz

Xωz − Zωx

Yωx − Xωy

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

P

Plate kinematics and deep mantle structures

Behn et al., 2004: Arrows = mantle flow field, colors = seismic velocity anomalies. Top = map view at the base of the lithosphere (300 km), bottom: cross-section of S20RTS (Ritsema et al.)

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)

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 Nubia-Somalia extension rate increases from S to N

• Discrepancy at MBAR

(Work by Saria Elifuraha)

(Work by Sarah Stamps)

EARkinematics

• 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

Summary

• 2 major plates, divergence rate increases from 3 to 6 mm/yr from S to N.

• GPS + slip vector data consistent with:

– Strain focused along narrow rift valleys

– 2 undeformed domains: Victoria and Rovuma microplates

• WARNINGS:– 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)