Analysis of the deformation of the Earth’s surface

1. Analysis of the deformation of the Earth’s surface Overarching goal is to find out what is going on at depth - motions, forces, rheology Observations largely confined to near Earth’s surface - we make numerous assumptions about what goes on below Approaches: Curve-fitting - estimate strain rates directly from data, no geologic model Continuum - surface deformation mirrors deformation in a continuum substrate (lower crust, mantle) Blocks - deformation can be discontinuous at surface, i.e, faults Viscoelastic - deformation continues longer than initial forces

2. Deformation analysis Velocity field V(x,y) = [ Vx(x,y), Vy(x,y) ] Solve for deformation gradient tensor: dVx/dx dVx/dy dVy/dx dVy/dy Where: Vx = x dVx/dx + y dVx/dy + Cx Vy = x dVy/dx + y dVy/dy + Cy

3. The strain rate tensor is: dVx/dx ½ (dVx/dy + dVy/dx) ½ (dVx/dy + dVy/dx) dVy/dy The vertical axis rotation rate is: ½ ( dVx/dy – dVy/dx ) This is done here in Cartesian coordinates (x,y) but can be done in spherical coordinates as well. TDEFNODE uses shperical coordinates (Savage).

4. Large-scale rotation with subduction locking superimposed Field through mid 2009

5. Rotation rates Strain rates Not computed with TDEFNODE; but a program is available.

6. TDEFNODE Modeling block motions, fault locking, strain rates, transients Use GPS velocities, displacements, time series, earthquake slip vectors, fault slip rates, InSAR

7. Acknowledgments Funding: NSF, NASA, USGS, GNS Science Routines: Chuck DeMets, Charles Williams, Steve Roecker, Bob King, W. Randolph Franklin, Dave Hollinger, Numerical Recipes Debuggers: Dave Hollinger, Larry Baker Guinea pigs (beta testers): Suzette Payne, Linette Prawirodirdjo, Laura Wallace, Zhang Zhuqi, and others

8. Defnode - modeling steady motions only, use linear velocities • Tdefnode - includes time-dependent motions and uses time series; data are time-sensitive

9. Motivation Velocity fields are superposition of multiple signals; rotations, strain rates, noise Time series showing strong non-linear (transient) effects

10. Not as scary as GAMIT

11. Calculate uniform strain and rotation rates in regions Figure has block rotations removed from vectors -0.15 deg/Myr -0.39 deg/Myr -0.25 deg/Myr -0.06 deg/Myr Courtesy of S. Payne

13. Non-linearity of time series is a major challenge. 2004 quake Steady velocity Afterslip 2005 quake millimeters Afterslip from both events East component of continuous GPS site SAMP The Sumatra quakes of 2004 and 2005, with afterslip

14. Block model can be used in non-steady state settings to separate kinemtics from transients. Data from PANGA In some cases the inter-event velocities are clear - short transients separated by long inter-event times. From McCaffrey 2009 GRL Data from GNS Science

15. In other cases, it is difficult to see the steady site velocity through the transients. Block models help by taking advantage of the spatial correlation among nearby sites Long-term velocity? Data from GNS Science

16. Parkfield quake TBLP P566 Inter-event velocities are not independent Data from PBO

17. Modeling estimated co-seismic offsets 2009 2002

18. Papua time series Occurrence of earthquakes results in non-linear GPS time series. We model the time series as a combination of the linear trend (kinematics) plus the steps from quakes.

19. Block model from inversion of GPS time series Thrusting ~17 mm/yr Mountain building comprises only ~10% of the action Oblique ~11 mm/yr Strike-slip ~10 mm/yr

20. Yellowstone

21. InSAR (M. Aly)

22. Time series with sinusoidal term InSAR data may overlap or have gaps in time

23. Multiple sill-like sources each with own time history

24. Complex GPS time series Invert simultaneously with InSAR

26. Blocks • Closed polygons on surface of Earth • Each characterized by angular velocity, uniform strain rate • Bounded by faults, or pseudo-faults

27. Faults • Surfaces dipping into the Earth described by nodes • Separate blocks in three dimensions • Coincide with block boundaries at surface • Slip according to relative velocities of blocks • Have locking or not • Can have transients

28. Transients • Spatial and time dependence types are specified • Many types can be modeled - quakes, after-slip, slow-slip, volcanic • Superimposed on long-term linear velocities

29. Data • GPS velocities (East, North, Up) • GPS displacements (E, N, U) • GPS time series (E, N, U) • InSAR interferograms (LOS changes) • Fault slip rates or directions • Earthquake slip vectors • Uplift rates or displacements (tidegauge, coral, etc.)

30. GPS velocity vectors and uplift ratesVk(X) = [RGX]k+ [ RBX ]k + ekk DXk+ ekl DXl+ • j=1,2i=1,N [- HFQi ]jiGjk (X, Xi) X is the position of the surface observation point,k represents the velocity component (x, y, or z),RB is the angular velocity of the block containing the observation point relative to the reference frame,RG is the angular velocity of the GPS velocity solution containing the observation point relative to the reference frame,e is the horizontal strain rate tensor (DX is the offset from strain rate origin) HF is the Euler pole of the footwall block of fault relative to the hangingwall block,N is the number of nodes along the fault,Qi is the position of node i, i is the coupling fraction at node i, Gjk (X, Qi) is the kth component of the response function giving the velocity at X due to a unit velocity along fault at Qi in the jth direction on fault plane (downdip or along strike)

31. Other data types Tilt rates: T(X) = [ Vz(X+X) - Vz(X - X) ] / (2 X ) (X is at the mid-point of the leveling line and X is the offset from the mid-point to the ends) Slip vector and transform fault azimuths: A(X) = arctan{[( HR - FR ) X]x / [( HR - FR ) X]y } Geologically estimated fault slip rates or spreading rates:   R(X) = | ( HR - FR ) X |

32. Compiling • TDEFNODE is written in fortran and has one C program to link • Edit tdefcom1.h - set dimensions of arrays • Edit tdefiles.h - set filenames for earthquakes and volcanoes to be included in profile lines • Edit Makefile provided, put in your compiler names and flags • gcc and gfortran work fine • Put the executable file ‘tdefnode’ in your path.

33. Control file • All input information (except data files) are put in a file that the program reads at startup • Each line has a 2-character key that signifies its purpose • Key characters are in first two columns, followed by a colon : • Order of lines does not matter except for repeated lines it uses the last instance

34. Models • Model names are specified by MO: option and are 4-characters long • The Control file can have multiple models using the MO: - EM: structure • The model to run is selected in the command line:% defnode control_file model

35. Building the Blocks Two options1. Define all block outlines and faults separately 2. Program builds blocks from faults

36. Method 1. Define Blocks and Faults Fault Block Use BL: to outline block; FA: to describe fault

37. Block boundaries are determined by seismicity, faulting, strain rates, … (reviewers and co-authors always ask for justification of block boundaries).

38. Block outline Fault segment • The block outline has the surface nodes and must coincide exactly with the fault surface nodes. • Not every edge of block has to be a defined fault. • But every fault must fall on a block edge.

39. Faults - defined by nodes Nodes are in an irregular grid. Confined to depth contours. Designated by (longitude, latitude, depth). Subsurface nodes can be generated by program.

40. Representation of fault slip • Nodes are specified along depth contours of fault • Slip at each node is jV, where j ranges from 0 to 1 and V is taken from poles • Area between nodes is broken into small patches • Surface deformation for each patch is determined and summed Response (Green’s) functions are determined by putting unit velocity at one node and zero at all other nodes, then calculating the surface velocities by integration. Pyramidical Bilinear

41. Half-space dislocation model (HSDM) to calculate surface deformation due to fault locking and slip events

42. Velocities from elastic strain rates arising from fault locking Use back-slip method to compute elastic deformation around locked fault. surface Locked fault Free slipping Integrate over fault using small patches, can represent non-planar fault and non-uniform locking

43. Angular velocities - AV (Euler poles) • Each block has an AV assigned • Multiple blocks can have same AV, in which case there is no slip between them • Long-term linear velocity V of each point in block is V =  x r • AVs can be fixed or adjusted in inversion • Entered as Cartesian or Spherical coords, always units of ‘degrees per Million years’ and right-hand rule • PO: option to input AV • BP:, BC: options assign AV to blocks • PI: option to adjust AV in inversion

44. Strain Rate Tensors (SRT) • Each block may have uniform SRT assigned (optional) • May arise due to small faults within block (anelastic, permanent deformation) • Multiple blocks can have same SRT (use common origin) • Long-term linear velocity V of each point in block is relative to specified origin • SRTs can be fixed or adjusted in inversion • Entered as nanostrain per year (10-9 / year) • Described by 3 components Exx, Eyy, Exy • ST: option to input SRT and origin • BP: or BC: option to assign SRT to blocks • SI: option to adjust SRT in inversion

45. Strain Rate Velocities Point (, ) Origin (o, o) Block

46. Assign AV and SRT to Blocks Block Blk1 Blk3 Blk2

47. Method 2. Define Faults, Build blocks Fault (extends to depth, can be locked) Pseudo-fault (surface boundary, free-slip) Block Set flag +mkb FA: to describe faults BC: to identify blocks

48. 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.

49. The strain rate tensor near a locked fault represents a spatial transition from the velocity of one block to the velocity of the other. In other words, a locked fault allows one block to communicate information about its motion into an adjacent block.

50. Rotate velocity fields (or time series) into common reference frame. Specify reference frame block with RE: option Velocity fields are rotated to minimize velocities of sites on that block. GI: option - list fields to be rotated Does not require all velocity fields to have sites on reference block, since all velocity fields must agree on all blocks.