1. 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)

2. 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

3. 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

4. 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

5. 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)

6. Exemple: Nazca/South America convergence Villegas, 2009

7. 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

8. 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)

9. Example of strain rate analysis using polygons Aktug et al., 2009

10. Strain rate analysis using a regular grid Kreemer et al., 2003

11. 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

12. Example: large scale velocity field in Asia Vergnolle et al., 2006 Using SHELLS, http://peterbird.name

13. Example: large scale velocity field in Asia Vergnolle et al., 2006 Using SHELLS, http://peterbird.name

14. Elastic Block Models as a Tool for GPS Analysis Account for surface deformation from fully or partially locked faults Provide secular constraints in estimating time- dependent motion Create a kinematically consistent model for large- scale motions

15. 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

16. Region is divided into ‘blocks’, contiguous areas that are thought to rotate rigidly. Each block rotates about a pole. The rotating blocks are separated by dipping faults. Velocities due to fault locking are added to rotations to get full velocity field. The relative long-term slip vectors on the faults are determined from rotation poles. Back-slip is applied at each fault to get surface velocities due to locking. Courtesy Rob McCaffrey

17. Okada model applied at boundaries Model velocities same for any path integral Meade et al., [2002]

18. 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] Program Flow for DEFNODE

19. Example from the eastern Mediterranean Seismicity and earthquake focal mechanisms provide a first-cut for block boundaries

20. GPS velocities refine the boundaries

21. GPS velocities  observed  modeled  block motion Residuals GMT representation of DEFNODE output

22. Large-scale rotation with subduction locking superimposed Example 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)

23. Velocity residuals from a 3-block model Error elliipses are 70% confidence Large-scale rotation with subduction locking superimposed

24. McCaffrey et al. [2007] Example from Cascadia: Large-scale rotation with subduction locking superimposed

25. McCaffrey et al. [2007] Example from Cascadia: Large-scale rotation with subduction locking superimposed GPS velocities from continuous (red) and survey- mode (blue) sites. Insert shows depth of the subducting slap and fault nodes used in the inversion. Triangles are volcanoes. Surface velocities from subduction alone Deep red is fully locked; deep blue freely slipping

26. 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