GEODESY%20and%20GEODYNAMICS

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Geodesy and Geodynamics By Christophe
Vigny National Center for scientific Research
(CNRS) Ecole Normale Supérieure (ENS) Paris,
France http//www.geologie.ens.fr/vigny
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DEFORMATION PATTERN IN ELASTIC CRUST

• Stress and force in 2D
• Strain normal and shear
• Elastic medium equations
• Vertical fault in elastic medium gt arctangent
• General elastic dislocation (Okadas formulas)
• Fault examples

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Stress (s) in 2D
- Normal stress sii
- Shear stress sij
- Force s x surface

• no rotation gt sxy
  syx

• only 3 independent .components
  ..sxx , syy , sxy

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Applied forces
Normal forces on x axis
sxx(x). dy - sxx(xdx). dy
dy sxx(x). - sxx(xdx)
- dy dsxx/dx . dx (1)
Shear forces on x axis
syx(y). dx - syx(ydy). dx
- dx dsyx/dy . dy (2)
Total on x axis (1)(2)
dsxx/dx dsyx/dy dx dy

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Forces Equilibrium
Total on x axis dsxx/dx dsyx/dy dx dy
Total on y axis dsyy/dy dsyx/dx dy dx
dsyy/dy dsyx/dx 0 dsxx/dx dsyx/dy
0
Equilibrium gt
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Normal strain change length (not angles)
- Change of length proportional to length
- exx, eyy, ezz are normal component of strain
nb If deformation is small, change of volume
is exx eyy ezz (neglecting quadratic terms)
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Shear strain change angles
exy -1/2 (F1 F2) 1/2 (dwy/dx dwx/dy )
exy eyx (obvious)
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Solid elastic deformation (1)

• Stresses are proportional to strains

• No preferred orientations

sxx (l2G) exx l eyy l ezz
syy l exx (l2G) eyy l ezz
szz l exx l eyy (l2G) ezz

• l and G are Lamé parameters

The material properties are such that a principal
strain component e produces a stress (l2G)e in
the same direction and
stresses le in mutually perpendicular directions
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Solid elastic deformation (2)
Inversing stresses and strains give
exx 1/E sxx - n/E syy - n/E szz
eyy -n/E sxx 1/E syy -n/E szz
ezz -n/E sxx -n/E syy 1/E szz

• E and n are Youngs modulus and Poissons ratio

a principal stress component s produces
a strain 1/E s in the same direction
and strains n/E s in mutually
perpendicular directions
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III/ Elastic deformation at plate boundaries
Two plates (red and green) are separated by a
vertical strike-slip fault. If the fault was
slick then the two plates would slide freely
along each other with no deformation. But because
its surface is rough and there is some friction,
the fault is locked. So because the plates keep
moving far away from the fault, and dont move on
the fault, they have to deform.
What is the shape of the accumulated deformation ?
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Mathematical formulation
The plates are made of elastic crust above a
viscous mantle.
The viscous flow localizes the deformation in a
narrow band just beneath the elastic layer
We can compute the deformation of the elastic
layer using the elastic equations detailed before
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Mathematical formulation

• Symetry gt

all derivative with y 0
eyy 0
szz 0

• No gravity gt

• What is the displacement field U in the elastic
  layer ?

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Mathematical formulation

• Elastic equations

(1) sxx (l2G) exx l ezz (2) syy l
exx l ezz (3) szz l exx
(l2G) ezz
sxy 2G exy sxz 2G exz syz 2G eyz
_________________________
(3) szz 0 gt l exx l ezz -2G ezz
and (2) gt syy l exx l ezz -2 G ezz
gt exx - (2G l)/ l ezz
and (1) gt sxx - (l2G)2/l l ezz
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Mathematical formulation

• Force equilibrium along the 3 axis

x
(x) dsxx / dx dsyx / dy dsxz / dz
0
x
(y) dsxy / dx dsyy / dy dsyz / dz
0
x
x
(z) dsxz / dx dsyz / dy dszz / dz
0
_________________________
d2sxx / dx2 0

• Derivation of eq. 1 with x and eq. 3 give

dsxy / dx dsyz / dz 0

• equation 2 becomes

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Mathematical formulation
relations between
stress (s) and displacement vector (U)
sxy 2G exy 2G dUx / dy dUy / dx .1/2
syz 2G eyz 2G dUz / dy dUy / dz .1/2
_________________________
Using dsxy / dx dsyz / dz 0 we
obtain
x
x
d/dxdUx/dy dUy/dx d/dzdUz/dy dUy/dz
0
d2Uy / dx2 d2Uy / dz2 0
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Mathematical formulation
d2Uy / dx2 d2Uy / dz2 0
What is Uy, function of x and z, solution of this
equation ?
Guess Uy K arctang (x/z) works fine !
Nb. datan(a)/da 1/(1a2 )
d2Uy/dx2 -2Kxz/(z2x2)
dUy/dxK/z(1x2/z2)
gt
d2Uy/dz2 2Kxz/(x2z2)
dUy/dz-Kx/z2 (1x2/z2)
gt
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Mathematical formulation
Uy K arctang (x/z)
Boundary condition at the base of the crust (z0)
Uy K . P/2 if x gt 0 K . P/2 if
x lt 0
And also
Uy V0 if x gt 0 V0 if
x lt 0
gt K 2.V0 / P
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Mathematical formulation
Uy K arctang (x/z)
at the surface (zh)
Uy 2.V0 / P arctang (x/h)
The expected profile of deformation across a
strike slip fault we should see at the surface of
the earth (if the crust is elastic) is shape like
an arctangant function. The exact shape depends
on the thickness of the elastic crust, also
called the locking depth.
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Arctang profiles
Uy 2.V0 / P arctang (x/h)
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Sagaing Fault, Myanmar
GPS measurement on the Sagaing fault fit well the
arctang profile
but with an offset of 10-15 km
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Palu Fault, Sulawesi
Part of the GPS data on Palu fault fits well an
arctang profile. But wee need a second fault to
explain all the data
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Altyn Tagh Fault, China
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Altyn Tagh Fault, China (INSAR)
Interferogram Nov. 1995/ Nov. 1999
Fault-parallel velocity
Slip rate V0 1.4 cm/yr Locking depth D 15 km
v v0/p atan(x/D)
1 color cycle 28 mm LOS displacement
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San Andreas Fault, USA (INSAR)
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Elastic dislocation (Okada, 1985)
Surface deformation due to shear and tensile
faults in a half space, BSSA vol75, n4,
1135-1154, 1985.
The displacement field ui(x1,x2,x3) due to a
dislocation D uj (x1,x2,x3) across a surface S in
an isotropic medium is given by
Where djk is the Kronecker delta, l and m are
Lamés parameters, nk is the direction cosine of
the normal to the surface element dS. uij is the
ith component of the displacement at (x1,x2,x3)
due to the jth direction point force of magnitude
F at (x1,x2,x3)
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Elastic dislocation (Okada, 1985)
(1) displacements
For strike-slip
For dip-slip
For tensile fault
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Elastic dislocation (Okada, 1985)
Where
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Subduction modeling
Velocity component // to convergence direction
In the case of a subduction (dippping fault with
downward slip) we use Okadas formulas.
We find a very large deformation area (gt 500 km)
because the dipping angle is only 22
With oblique slip we predict the surface vector
will start to rotate above the end-tip of the
subduction plane
The profile of the velocity component // to the
convergence shows a plateau at this location
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Subduction elastic curves
Normalized horizontal and vertical curves

.
The applied dislocation
on the subduction plane is equivalent to 1 cm/yr
accumulation
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GPS measurements in Chile and S-America
fiducial stations on the stable South American
craton (Brazil, Guyana, Argentina, Urugay,) allow
to define the South-American reference frame.
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GPS measurements in Chile
Deformation (elastic def. induced by coupling on
the subduction) is visible in Chile
(Constitucion) And reaches far inland TUCU
(Tucuman) and CFAG (Coronel Fontana) in Argentina
show deformation more than 400 km away from the
trench
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Velocity profiles
As expected from elastic coupling, velocities
decrease Eastward (from 35-45 mm/yr along the
coast to 10-15 mm/yr at the cordillera) and
vector directions rotate from a direction // to
plate convergence to East-West trending.
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Subduction parameter adjustments
Model and data
Residuals
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Elastic curves fit
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Along strike comparison
Conception 37S
Coquimbo 30S
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Along strike comparison
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Patial coupling model
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Oblique subduction strike slip faulting at
Arakan trench and Sagain fault in Myanmar, SE-Asia
plane dip angle 15 . locked at
50km slip 2 2.5 cm/an . N
30-35

• potential for an Eq . Mw 8.5 every 100y

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END OF CHAPTER