Introduction to the modelling of GPS results
• GPS provides
• Surface crustal velocities in a global reference frame, or with respect to
a block, realized through a set of stations (globk ‘plate’ command)
• Time dependent deformation (mainly CGPS & time series from glred)
What can be asked of geodetic results ?
Secular velocities
• Kinematic boundary conditions around deformation zones
• Deformation regime across deforming zones and its relation
to global plate motion – input for dynamic models
• slip rates along major faults*
• locking depth, spatial distribution of coupling*
* constraints on seismic hazard
What can be asked of geodetic results ?
Time dependent deformation
• co-seismic displacement: location, slip distribution, moment
• post-seismic deformation: afterslip, visco-elastic processes
• detection and quantification of slow slip events
Modelling velocities for tectonics
In actively deforming zones, the velocities are the sum of
several contributions:
global plate motion – estimated & removed during Globk
analysis to obtained velocities with respect to stable plate
interiors
•
• long-term tectonic motion – modeled using either blocks or
distributed deformation
• because most faults are locked during the period
separating two earthquakes (inter-seismic phase), they
induce surface elastic deformation
Rigid blocks approach
Assumptions
• deformation is localized along major faults
• small internal deformation (<< deformation across major
boundaries)
• blocks defined by closed boundaries
Methods

solve for rigid rotation rates (Euler poles)
Exemple: Nazca/South America convergence
Villegas, 2009
Refinement: elastic blocks approach
Same assumptions as for the blocks
But, interseismic elastic deformation accounted for
Results

slip rates along major faults
 locking depth, coupling coefficient
 large faults like subduction interface: spatial distribution of coupling
Figure from Z.-K. Shen, in Stein and Wysession, 2006
Strain rate
• We now consider the local change of velocity
•
the velocity gradient tensor is defined by the derivative of the
velocity components wrt to the coordinates
• It can be divided into:
• A local rotation
• A strain rate tensor (symetric)
Example of strain rate analysis using polygons
Aktug et al., 2009
Strain rate analysis using a regular grid
Kreemer et al., 2003
Thin viscous sheet approach
Assumptions
• deformation at scale (>>100 km) is driven by the balance of
stress distribution induced by boundary conditions and stress
arising from crustal thickness lateral variations
• Lithosphere modeled as fluid
• thin sheet approximation: all quantities are averaged over
lithosphere thickness
Example: large scale velocity field in Asia
Vergnolle et al., 2006
Using SHELLS,
http://peterbird.name
Example: large scale velocity field in Asia
Vergnolle et al., 2006
Using SHELLS,
http://peterbird.name
Elastic Block Models as a Tool for GPS Analysis
• Account for surface deformation from fully or
partially locked faults
• Provide secular constraints in estimating timedependent motion
• Create a kinematically consistent model for largescale motions
There are (at least) two well-developed, documented
software packages freely available:
DEFNODE / TDEFNODE
Rob McCaffrey, Portland State University (formerly at RPI)
http://web.pdx.edu/~mccaf/www/defnode/
BLOCKS
Brendon Meade, Harvard University (formerly at MIT)
http://summit.fas.harvard.edu/~meade/meade/Software.html
Region is divided into
‘blocks’, contiguous areas
that are thought to rotate
rigidly.
The relative long-term
slip vectors on the faults
are determined from
rotation poles.
Each block rotates
about a pole.
Back-slip is applied at
each fault to get surface
velocities due to locking.
Velocities due to fault locking are
added to rotations to get full
velocity field.
The rotating blocks are
separated by dipping faults.
Courtesy Rob McCaffrey
Okada model
applied at
boundaries
Meade et al., [2002]
Model velocities same for
any path integral
Program Flow for DEFNODE
Data
GPS velocities
InSAR line-of-site rates
Uplift rates
Tilt rates
Slip vectors
Transform azimuths
Spreading rates
Fault slip rates
Strain rates
Parameters
Block rotations
Reference frame
Fault locking
Uniform strain rates
Output
Text files
GMT mappable files
Uncertainties (linearized)
Solution
Grid search
Downhill simplex
McCaffrey [1995; 2007]
Example from the eastern Mediterranean
Seismicity and earthquake focal mechanisms provide a first-cut for
block boundaries
GPS velocities refine the boundaries
GMT representation of
DEFNODE output
GPS velocities
 observed
 modeled
 block motion
Residuals
Large-scale rotation with subduction lockingExample
superimposed
from the
western
Mediterranean :
GPS velocities from 10
years of CGPS and
SGPS measurements
Note rigid rotation of
Africa with respect to
Iberia and independent
motion of the Rif
(Morocco) and Betic
(Spain) mountains
Koulali et al. (2011)
Large-scale rotation with subduction locking superimposed
Velocity
residuals from
a 3-block
model
Error elliipses
are 70%
confidence
Example from Cascadia:
Large-scale rotation with subduction locking superimposed
Example from Cascadia:
Large-scale rotation with subduction locking superimposed
Surface velocities from subduction alone
Deep red is fully locked; deep blue freely
slipping
GPS velocities from continuous (red) and surveymode (blue) sites. Insert shows depth of the
subducting slap and fault nodes used in the
inversion. Triangles are volcanoes.
McCaffrey et al.
[2007]
A block model can be used in non-steady state settings to
separate kinemtics from transients
Example: Spatially propagating slow slip events (SSEs) in Cascadia
Time series data from PANGA
Model showing rotating blocks, subduction
locking, and rates of uniform strain