Plate Tectonics. convergent (subduction or collision zones) transcurrent (transform faults) divergent (oceanic spreading centers)

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1 Plate Tectonics Earth s lithosphere (~100 km thick) = 15 major plates Plate are rigid (no or very little intraplate seismicity) Deformation is essentially localized at plate boundaries Plate boundaries: convergent (subduction or collision zones) transcurrent (transform faults) divergent (oceanic spreading centers) Present-day plates and their boundaries

2 Oceanic vs. continental crust Oceanic crust Continental crust

3 Plate Kinematics Magnetic anomalies in the oceans (Vine and Matthew, 1963): Linear magnetic anomalies Parallel to ridges Symmetrical pattern w.r.t. the ridge Datable Provide spreading rate

4 Plate kinematics Transform faults: Link spreading centers Active part between spreading axis Fracture zone Provide spreading direction

5 Some history By late 60 s, most concepts are in place: oceanic spreading, magnetic anomalies, transform faults, subduction. 1967: McKenzie and Parker (Nature) propose the hypothesis of rigid plates, substantiated by the fact that a rotation pole fit to the San Andreas fault predicts observed earthquake slip vectors in Alaska, Kouriles, and Japan. 1968: Jason Morgan proposes proposes an angular rotation for the Atlantic based on transform faults and magnetic anomalies and a 5-plate model, but not fully quantified. 1968: Xavier Le Pichon proposes the first global model and fully quantified, with 6 plates (based on anomaly 5, ~10 My). Le Pichon, 1968: data used Le Pichon, 1968: the 6-plate model

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7 Describing motions on a sphere The motion of plates (= spherical caps) on a sphere can be described by: A pole of rotation (lat, lon), also called Euler pole An angular velocity (usually in units of deg/my) Pole of rotation and angular velocity are equivalent to a 3- component angular rotation vector (Ω) in a cartesian geocentric frame

8 From Euler parameters to rotation vector (and back) Rotation vector Ω(ω x,ω y,ω z ) is equivalent to Euler parameters (latitude λ, longitude φ, angular velocity a) Euler to Rotation: " x = acos(#)cos($) " y = acos(#)sin($) " z = asin(#) (same formulas to convert coordinates from spherical to cartesian geocentric and back, of course) Rotation to Euler: % $ " = tan #1 ' z ' $ 2 2 & x + $ y % + = tan $ ( #1 y ' * & ) $ x a = $ x 2 + $ y 2 + $ z 2 ( * * )

9 Plate motions on a sphere Relative plate motion direction given by transform fault directions: Transform faults = small circles about the rotation pole Conversely, rotation pole located on great circle perpendicular to transform direction Relative plate motion rate given by sea-floor magnetic isochrons. Relative plate motion direction also given by earthquake slip vectors (but sometimes more complicated). GPS velocities: Direct measure of both plate motion direction and magnitude Normal to great circle passing through pole of rotation

10 Plate motions on a sphere During a time interval t, P travels an angle ωt, which corresponds to a distance: d = " rad tr where r is the radius of the (small) circle perpendicular to the axis of rotation. Since: sin" = r R E one gets: d = " rad tr E sin# and the velocity at P is: v = d t = " rad / yrr E sin# The velocity is obviously maximum when θ=π/2 (i.e. at the equator to the rotation pole R), therefore: and the velocity at P becomes: v max = " rad / yr R E v = v max sin" The velocity at P is a vector tangent to a small circle perpendicular to (O,z) and its direction is obviously perpendicular to the great circle passing through R and P. (O,x,y,z) orthogonal geocentric coordinate system. Plate rotation axis coincides with (O,z) coordinate axis. Angular velocity of the plate is ω. Point R is the rotation pole of the plate. Point P is located at angle θ to the rotation pole. R E is mean Earth radius = 6,378,137 m.

11 Forward problem for geodetic velocities Since by definition the cross-product of 2 unit vectors is the sine of the angle between them, one can write: sin" = # r U $ P r With ω u and P u = unit vectors in the (O,z) and (O,P) directions, respectively. The expression for the velocity vector becomes: r v = "R E." r U # P r U = R E "" r U # r " rad / yr # r Since angular velocities are usually given in degree per My, one can write: r " r v = R E ( #6 $ deg/ Myr % P r U ) Velocity is a vector, with 3 components v x,v z,v z in the geocentric coordinate system defined above. Note that this equation is linear in ω and can therefore easily be inverted to solve for ω(ω x ω y ω z ) given an ensemble of known v s and P s (from geodetic measurements, for instance). U r ( P U ) = R E ( P U )

12 Computing velocities from a plate angular velocities Simply apply forward model derived above: computation much easier in geodetic frame using angular rotation vector than Euler parameters. Note that Ω (rotation vector) typically provided in degrees per million years (sometimes in radians per Myr). At location defined by the unit vector P u (X,Y,Z) in a geocentric, Cartesian, frame, the velocity vector V(vx,vy,vz) in a geocentric frame for a plate rotation defined by the rotation vector Ω(ω x, ω y, ω z ) is given by the cross-product: r V = R " r #6 $ deg/ Myr % r R = mean Earth radius = 6,378,137 m V is obtained in geocentric coordinates (v x,v y, v z ), units of m/yr Only possible issues: ( P u ) r V = R " #6 Conversion from geodetic to geocentric frame (lat/lon to X,Y,Z) Conversion from geocentric to local frame (X,Y,Z to NEU) or % Z$ y #Y$ z ( ' * X$ z # Z$ x ' & Y$ x # X$ * y )

13 From geodetic to geocentric coordinates X = (N + h)cos"cos# Y = (N + h)cos"sin# Z = [N(1$ e 2 ) + h]sin" with: N = a 1" e 2 sin 2 # e 2 = 2 f " f f = a " b a λ, φ, h = geodetic latitude, longitude and height (above ellipsoid) X,Y,Z = ECEF Cartesian coordinates a = ellipsoid semi-major axis b = ellipsoid semi-minor axis

14 Local topocentric datum Origin of this datum is any point you choose on the surface of the Earth. 3 right-handed orthogonal axis: u (for up ) is vertical (= perpendicular to the local equipotential surface) and points upwards n (for north ) is in the local horizontal plane and points to the geographic north e (for east ) is in the local horizontal plane and points to the geographic east. Units are meters.

15 From geocentric to local frame The conversion is a combination of the 3 rotations needed to align the geocentric frame with the NEU frame: r L = RV r %"sin#cos$ "sin#sin$ cos# ( ' * with R = ' "sin$ cos$ 0 * &' cos#cos$ cos#sin$ sin# )* Valid at location (λ,φ,h) L(North, East, Up) = velocities in local frame V(Vx,Vy,Vz) = ECEF velocities The inverse matrix can be used to convert from local to ECEF -- since R is a rotation matrix, its inverse is also its transpose: R "1 = R t r V = R r t L Variance propagation: Cx is the covariance matrix associated with vector X A is a transformation matrix such that Y = A.X Then the covariance matrix of the transformed parameters Cy is: C Y = A.C X.A t

16 Forward problem from marine data In most cases, the plate rotation axis does not coincide with (O,Z). Marine geophysicists measure: The strike of mid-ocean ridges. At a point P on a ridge, the velocity vector v is, by definition, perpendicular to the (P,R) direction. A priori, it can be at any angle to the ridge strike. The perpendicular distance between various magnetic anomalies and the ridge => we ll be interested in an expression for v n, the ridgenormal velocity component. From the figure, one can see that: cos" = v n v # v n = v cos" The angle δ is related to 2 the ridge strike and to the angle α via: " = STRIKE # $ Therefore, using the equation for v derived in step 1, we get: v n = v max sin" cos(strike # $)

17 Forward problem from marine data Now we need an expression for α and θ. Let s start with θ, which is the angle between two vectors: the rotation vector and P s position vector. By definition the dot product of 2 vectors is related to θ: r ". P r = "P cos# where ω and P are the norm of ω and P. Expanding the dot product in terms of the Cartesian coordinates of ω and P gives: "P cos# = x P x " + y P y " + z P z " Now let s write this in a polar coordinate system, which is simply the usual (latitude, longitude) system. See figure: the conversion between Cartesian and lat-lon coordinates for P s gives: x P = R cos" P cos# P y P = R cos" P sin # P z P = R sin" P Similarly, for ω : x " = " cos# R cos$ R y " = " cos# R sin $ R z " = " sin# R Once can therefore write: "Rcos# = "R(cos$ P cos% P cos$ R cos% R... +cos$ P sin% P cos$ R sin% R + sin$ P sin$ R ) Rearranging gives: cos" = cos# P cos# R ( cos$ P cos$ R + sin $ P sin $ R ) + sin# P sin# R [ ( )] %" = arccos sin# P sin# R + cos# P cos# R cos $ P & $ R

18 Forward problem from marine data For α, the easiest way around is to use the law of sines for a spherical triangle (see figure), which writes: sin a " = sin b # = sin c $ In our case, this becomes: sin" ( ) = sin ( 90$ % P) sin& sin # P $ # R ' sin& = sin (# P $ # R ) sin( 90$ % P ) ( ) cos (% P ) ( ' & = arcsin sin # $ # P R * ) sin" 1 sin" + -, Therefore: v n = v max sin" cos(strike # $) ( ) cos (% P ) ' " = arcsin sin # $ # P R ) ( sin& " = arccos[ sin# P sin# R + cos# P cos# R cos ( $ P % $ R )] *, +

19 Inverse problem from marine TBD data

20 Inverse problem from geodetic velocities We saw that, in geocentric frame, velocity is the cross-product of angular rotation vector Ω(ω x, ω y, ω z ) and position unit vector P u (X,Y,Z) : This set of linear equations can be written in matrix form as: " vx% " 0 Z (Y %" $ ' vy $ ' = $ ' $ (Z 0 X $ ' $ # vz& # Y (X 0 &# or r V = R " V = A* #6 % Z$ y #Y$ z ( ' * X$ z # Z$ x ' & Y$ x # X$ * y ) If at least 2 sites with velocities, the problem is over-determined and can be solved using least squares (V = data vector, C V = data covariance matrix matrix): ) x ) y ) z % ' ' & " = (A T C V #1 A) #1 A T C V #1 V The model covariance matrix is: C " = (A T C V #1 A) #1

21 Inverse problem, additional conditions Additional conditions can be used besides geodetic results: To impose reference frame To strengthen the estimation For example: No-Net-Rotation: see previous chapter. Plate circuit closure: AΩ C = AΩ B + BΩ C or ΣΩ = 0

22 Back to history First global inversions based on spreading rates and transform directions: Le Pichon (1968): 6 plates Chase (1972, 1978): 11 plates Minster and Jordan (1978): RM1 and RM2 models: 12 plates, 330 data More recently: DeMets et al. (1990): NUVEL1, 12 plates, 1122 data NUVEL1 = standard model for current plate motions

23 NUVEL1A Geologic Plate Motion Model Plate X Y Z afrc anta arab aust carb coco eura indi nazc noam soam phil NUVEL1A plate rotation vectors (w.r.t. Africa) De Mets et al., 1990, updated NUVEL1A in 1994 to account for change in magnetic time scale. Geological model: ocean floor magnetic anomalies, transform faults, earthquake slip vectors. Hypothesis: Limited number of rigid plates Constant velocity over the past 3 million years

24 Comparison Geodesy / Nuvel1A Argus, D.F., and M.B. Heflin, years of CGPS, 43 sites, 6 plate-model Standard deviation = 2 mm/yr Very good agreement GPS / NUVEL1 Except Pacific: 10% faster and Eu/Na pole has moved north

25 Interpretation: Plate motions are steady (over the past 3 My) Validation of the geological model Validation of the geodetic measurements NNR condition in ITRF applied using the Nuvel1A- NNR model Nuvel1A-NNR used to rotate a velocity field expressed in ITRF into a plate-fixed reference frame Nuvel1A-NNR used as a geodetic reference frame

26 Geodetic Plate Motion Models Sella et al., 2002 REVEL Global plate motion model can be computed directly from geodetic velocities Many attempts since ~1995, among them: Argus, D.F., and M.B. Heflin, 1995: 43 sites, 6 plates Larson et al Kreemer et al., 2003: includes deformation at plate boundaries, own NNR frame Sella et al., 2002: REVEL, 200 sites, 19 plates, ITRF97 Prawirodirjo and Bock, 2004, ITRF2000 Kogan et al., 2008: 71 sites, 10 plates, accounts for offset between center of mass and center of plate rotation, ITRF2005 Kogan et al., residuals

27 REVEL GPS plate model Sella et al., JGR 2002 Plate rigidity level ~200 CGPS velocities 19 plate-model, includes some small plates (Amuria, Anatolia, etc.) Some plates poorly sampled: Somalia, Arabia Or poorly defined: Amuria, South China Velocities are in ITRF97

28 Comparison geologic/geodetic plate motion model Geologic, 3 My average model = NUVEL1A Geodetic, instantaneous model = e.g. REVEL Very good agreement between the two for 2/3 of the plates. But: Missing plates in NUVEL1: Somalia Ca/Na, Ca/Sa, and Na/Pa disagree: errors in Nuvel1 Ar/Eu, Ar/Nu, and In/Eu slower than Nuvel1 Nz/Pa, Nz/Sa slower than Nuvel1 Eu/Na and Eu/Nu: Euler pole moved

29 Caribbean plate: twice faster than Nuvel1 De Mets et al., GRL, 2000 Nuvel1A = 12 mm/an GPS = 20 mm/an

30 Arabia: slower than NUVEL1 (Sella et al., 2000) Slowing down of Arabia as it collides with Eurasia?

31 Nazca plate: slowing down geologically Norabuena et al. (1999): deceleration back to at least 20 My, initiation of Andes growth Consequence of construction of the Andes? Increased friction and viscous drag as leading edge of Sa thickens?

32 < 3My change in Pa/Na motion DeMets and Dixon (1999) : Pacific-North America velocity increased significantly over the last 3 Million years.

33 Comparison geodesy/geology Calais et al., EPSL, 2003 Plate motions may have changed in the past 3 My `

34 Anatolia: not included in NUVEL1 GPS campaigns in Turkey (Reilinger et al., JGR, 1997; McClusky et al., JGR, 2000) NAF = great circle, ~30 mm/yr An/Eu Euler pole in NE Egypt

35 Conclusions Motion of rigid plates can easily be described by 3 parameters per plate. Plate motions can be given with respect to another plate, or with respect to an absolute frame (NNR or hotspots, see previous class on reference frames). Space geodesy has contributed to demonstrating that plates are essentially rigid, while deformation is concentrated at their boundaries. 3 My average geologic plate motion model agrees very well overall with geodetic model, but there are exceptions. Current challenges: Recent changes in plate motions? Can deforming continents be described by motions of rigid (micro-)plates? How rigid are plates?