gps and its application to geodynamics in east africa eric calais purdue university, west lafayette,...

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)