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

6. Modeling block motions, fault locking, strain rates, transients
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. Data from PANGA
Block model can be used in non-steady state settings to separate kinemtics from transients.
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. Inter-event velocities are not independent
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. j=1,2 i=1,N [- HF  Qi ]j i Gjk (X, Xi)
GPS velocity vectors and uplift rates Vk(X) = [ RG  X ]k + [ RB  X ]k + ekk DXk + ekl DXl +
j=1,2 i=1,N [- HF  Qi ]j i Gjk (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. T(X) = [ Vz(X+X) - Vz(X - X) ] / (2 X )
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 options 1. Define all block outlines and faults separately
2. Program builds blocks from faults

36. Method 1. Define Blocks and Faults
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
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.

51. Total long-term (linear) velocities are the sum of the 4 terms:
Velocity field rotation
Block rotation
Anelastic strain rate within block
Elastic strain rates from fault locking

52. Sample control file:

53. Run 1 - get poles and strain rates

54. Run 2: use PBO field, rotate into PNW field reference frame

55. Run 3: Multiple fields; strain rates, rotation rates and reference frames.

56. Reference frame adjustments for PNW1 and PBO.

57. Models - a particular set of input parameters, designated by 4-char name. Multiple models can be in a single control file.
--- Model input
These lines pertain for mod1, mod2, and mod3
--- First mo signals start of MO: - EM: structure
mo: mod1 mod2
These line pertain to mod1 and mod2
mo: mod1
These lines pertain to mod1 only
mo: mod2
These lines pertain to mod2 only
mo: mod3
These lines pertain to mod3 only
em: end of models
These lines pertain to mod1, mod2, and mod3
en: end of input

58. All segments must end at another segment or an error occurs.
-- Fault input, blocks from faults
FA: for fault segments
-- For Fault1, it dips to east so start in south
Fa: Fault1 1
2 3 Blk2 Blk
0.0
Zd: 5 89
Zd: 10 89
-- If making blocks from faults, (+mkb flag) make pseudo-faults from remaining borders. They will be free-slip boundaries.
Fa: Blk1_bndry 2
4 1 Blk1 Blk
0.0
Fa: Blk2_bndry 3
4 1 Blk2 Blk
-- Give interior point of block to identify it, and specify pole and strain tensor for each
BC: Blk
BC: Blk
-100, 40
-90, 40
-80, 40
Blk1
Blk2
Fault1
All segments must end at another segment or an error occurs.
-80, 30
-100, 30
-90, 30

59. All block and fault points must coincide or an error occurs. Fault1
--- Block and Fault input
BL: for closed blocks
FA: for fault segments
For Fault1, it dips to east so start in south
Fa: Fault1 1
2 3 Blk2 Blk
0.0
Zd: 5 89
Zd: 10 89
If inputting blocks, make polygon of borders of blocks. They will be free-slip boundaries.
BL: Blk1 1 0 4
BL: Blk2 2 0
-- BP: is alternative way to specify poles and strain tensors
BP: Blk1 1 0
BP: Blk2 2 0
-100, 40
-90, 40
-80, 40
Blk1
Blk2
All block and fault points must coincide or an error occurs.
Fault1
-80, 30
-100, 30
-90, 30

60. Nodes - slip or locking on nodes can be represented in several ways
Locking parameter is (x,z) or (x,w)
Independent nodes with or without smoothing;
decreases down-dip; or
 is specified function of z
Boxcar
Gaussian
Exponential V
 is specified function of x,z x
- 2D Gaussian
- Uniform Polygon
z, w

61. The fault below has 6 surface nodes and 5 downdip for a total of 30.
For independent nodes (fault type FT: 0 or 1) we specify the inter-dependence of the nodes (NN:) and their starting values (NV:).
FT: 1 0
NNg: 1 6 5
NV: x z V x
z, w

62. The fault below has 6 surface nodes, so 6 downdip ‘profiles’
The fault below has 6 surface nodes, so 6 downdip ‘profiles’. For each one the function (z) can have different parameters. For example the function may be: V
# DD Prof
FT:
PN:
PV: 1 3 x  zu zl
z, w

63. Approximating ‘locking depth’; using downdip boxcar fixing upper depth (0 km) and locking amplitude (1); solve for lower depth x
Locked nodes
z, w
Unlocked nodes

64. Variable locking depths along San Andreas

65. Types of downdip (1D) functions:
Exponential (Type 2)
Boxcar (Type 3)
Gaussian (Type 4)
Boxcar with cosine taper (Type 5)
Types of 2D functions:
Gaussian (Type 6)
Boxcar (Type 7)
Irregular polygon (Type 8)
Types of off-fault functions (not on block boundary)
Planar shear slip (Type 9)
Mogi (Type 10)
Planar expansion crack (Type 11)

66. Interseismic; I recommend locking the updip edge and forcing monotonic decrease in locking downdip

67. Time dependence
Earth is linearly elastic (no viscous relaxation built in)
All sources are super-imposed
Every datum has a time stamp (except for now the linear velocities)
Time dependence of transients are represented by slip rate histories
Time dependence parameterized in several ways
Viscoelastic relaxation can be included by generating Green’s functions with VISCO-1D (Pollitz)

68. From block model, due to block motion, elastic and anelastic strain rates

70. Data GPS velocity vectors or displacements - 3 components
- use East, North, Up velocities and standard errors
- use NE covariance
- specify time frame
- psvelo and other formats
GPS time series - 3 components
- E, N, U positions (in mm relative to start)
- times in decimal years (e.g., )
- standard error for each point
- can be decimated by program
- offsets and seasonal signals estimated
InSAR line-of-sight changes (LOS)
- resampled to reduce numbers
- planar frame parameters estimated
- matched at times of differenced images

71. Earthquake slip vectors / fault slip azimuths
- specify fixed and moving blocks
Fault slip rates
- specify relative blocks
- can be min/max or Gaussian types
- can specify azimuth of measurement (e.g., for spreading rates)

72. Transients ES: option to enter transient parameters
EF: option to specify which parameters are adjusted
EX: assign bounds to parameters
EI: specify which transients are used
ET: specify time function elements
ER: specify polygon radii
EI: 1 2
ES: 1 fa 1 sp 4 ts 0 to ln lt 22.1 zh 10.0 xw 20.0 xx 50.0 am 500.0
EX: 1 ln lt
EF: 1 ln lt zh xw xx am
ES: 2 sp 10 ts 2 to ln lt 22.1 zh 5.0 am 500.0
EX: 2 ln lt zh
EF: 2 ln lt zh am
Codes: 'fa' fault number
'sp' spatial transient type (0 to 11)
'ts' temporal type (0 to 6)
'sa' slip azimuth control (0 or 1)
'ln' longitude (deg)
'lt' latitude (deg)
'zh' depth (km)
'xw' along-strike width (km)
'ww' down-dip width (km)
'az' azimuth of Gaussian X-width (deg)
'am' slip rate amplitude (mm/yr)
'to' origin time (dcecimal years)
'tc' time constant (days)
'mr' migration rate (km/day)
'ma' migration azimuth (deg)
'st' strike (deg)
'dp' dip (deg)
'rk' rake or slip azimuth (deg)
'ga' 1D Gaussian amplitude (mm/yr)
'gm' 1D Gaussian mean depth (km)
'gs' 1D Gaussian sigma (km)
'rd' polygon radii (as flag in EF: )
'ta' tau amplitude (as flag in EF: )
'mo' moment (Nm, in EX: option)
Transients

73. Arguments for sp, sa, ts
Spatial source type (sp)
= 1 Independent nodes
= 2 Wang exp() function for phi(z) (uses parm types 5,6,7)
= 3 1D boxcar phi(z) (uses parm types 4,6,7)
= 4 Gaussian phi(z) slip (uses parm types 4,8,9)
= 5 not used
= 6 Gaussian 2D slip source (uses parms 24, 10, 11, 12, 13,14,15,16,17)
= 7 2D Boxcar slip source
= 8 Polygon, uniform slip source (use ER: also)
= 9 earthquake slip source (double couple not on fault)
= 10 Mogi slip source (not on fault)
= 11 Planar expansion source (not on fault)
Slip azimuth type (sa)
= 0 if slip direction from block model (poles)
= 1 if azimuth of slip specified or estimated
Time dependence type (ts)
= 0 impulse
= 1 Gaussian A exp( [(t-To)/Ts]**2 )
= 2 triangles (set Ntau also; use ET:)
= 3 exponential
= 4 boxcar
= 5 negative boxcar (loading)
= 6 Omori (A/(t +Ts)

75. 2D Gauss
1D Gauss profiles

76. Time functions:
Also log and Omori functions (not shown)

77. COMMANDS (++ new to TDEFNODE;
COMMANDS (++ new to TDEFNODE; ** not for general use or in development; -- not used any more)
AV: ++ add block/fault surface points
BC: ++ specify point within block; also name of block and pole/strain indices
BL: outline of elastic rotating plate polygon
BP: specify pole and strain tensor indices for a block
CF: connect 2 faults (remove overlap or gap from subsurface intersection of two faults)
CL: clear specified data type
CO: continue reading from input file (used sith SK: option)
DD: set depth and dip to nodes (use only within FA: section; similar to ZD:)
DR: ++ set region for data
DS: ++ displacements input file
DT: ++ time interval for synthetic time series
DV: ++ delete block/fault surface points
EC: ++ elastic constants
EF: ++ flags for the individual transient parameters
EI: ++ flags to invert transient events
EM: end of model input section
EN: end of input data
EQ: equate two nodes on different faults (set their phi's equal)
ER: ++ polygon source information
ES: ++ transient source parameters
ET: ++ transient source time function information
EX: ++ constraints on transient source parameters
FA: fault geometry input
FB: ++ flag faults to use to make blocks
FD: -- fix depths
FF: fault flags (turn faults on and off)
FL: set miscellaneous flags
FO: -- fault orientation
FS: calculate and output relative block velocities at specified points
FT: fault parameterization type
FV: -- fix Xo or V for listed time series
FX: specify position of a particular fault node - overrides all other specifications
GD: specify Green's functions directory and other GF parameters
GF: -- combined with GD:
GI: GPS velocity fields (relative to reference frame) to be adjusted
GP: GPS velocity input data file
GR: grid of vectors to calculate
GS: parameter grid search controls

78. COMMANDS (++ new to TDEFNODE;
COMMANDS (++ new to TDEFNODE; ** not for general use or in development; -- not used any more)
GW: -- global weight
HC: hard constraints
IN: interpolation lengths for fault segments between nodes (for final forward run)
IS: ++ Insar data input file
LA: ** layered structure
LL: ** line length data
MF: merge faults at T-junction
MM: range of seismic moments allowed for a fault
MO: model experiment name, used for output filenames
MS: ++ merge two time series
MV: move block/fault surface points
NI: number of iterations
NN: node parameter index numbers (same as old NF:)
NV: node values (same as old NO:)
NX: indices of fixed nodes
OP: ** output poles relative to a block
PE: scaling factors for penalty functions
PF: parameter and model I/O file
PG: initialize pole of rotation for GPS vector file
PI: block poles to be adjusted
PM: parameter min and max values allowed
PN: node z-profile parameter index numbers
PO: block pole of rotation values
PR: surface profile line
PT: ** file of lon,lat points to compute displacements
PV: node z-profile parameter values
PX: fix node z-profile parameters
RC: remove sites within a specified circular area (e.g., volcanic region)
RE: reference block for vectors
RF: rotate reference frame for vector output
RM: remove named GPS sites or blocks from data
RO: rotation rates input data file

79. COMMANDS (++ new to TDEFNODE;
COMMANDS (++ new to TDEFNODE; ** not for general use or in development; -- not used any more)
RQ: remove equates with list of names to use
RS: reference site for GPS vectors
SA: simulated annealing inversion controls
SE: ++ select sites from GPS file
SI: strain rates tensors to be adjusted
SK: skip following lines of input data until a CO: line is encountered
SM: apply smoothing to fault locking
SN: snap block boundary points together
SR: fault slip rate / spreading rate data file
SS: strain rate tensor data file
ST: initialize strain rate tensor values and origin
SV: slip vector / transform azimuth data file
TI: tilt rate data file
TS: ++ time series input file
UP: -- uplift rate data file (see GP:)
ZD: set depth and dip to nodes (use only within FA: section); similar to DD:

80. Input
PI: Poles to invert
PI: 1 2
SI: Strain tensors to invert
SI: 1 14
RE: reference frame block
RE: Blk1
GD: Green’s functions
GD: gf
PF: Parameter file
PF: “mod1/pio” 3
GP: GPS vector files
GP: NORA "nora_2003.vec"
GI: rotate GPS files
GI: 2
SV: slip vector data
SV: cr.svs FORE COCO 5.0
SVd: Tual BHed C E
SR: fault slip rate data
SR: saf_rate.dat NOAM PACI 1 0 0

81. Input
PO: Pole of rotation
PO:
ST: Strain rate tensor
ST:
FL: set flags
FL: +mkb -cov
FF: fault flags (turns on/off elastic strain)
FF:
FB: fault flag (removes fault from blocks)
FB:
SA: simulated annealing controls
SA: 0 80
GS: grid search controls
GS:
IC: iteration control (1=SA, 2-GS)
IC:

82. Input
RM: remove site from velocity field
RM: PNW1 TILL DAYV
MV: move node to new position
MV:
TS: time series data file
TS: PBO1 "PNW.gts ”
DS: Displacement data file
SM: Smoothing fault slip
SE: Select specific sites from file, use with GP:, RM:, TS: component flags to select site-components

83. Inversion:
Minimize penalty function: sum of chi**2 (weighted data misfit) and penalties for parameter constraints
Simulated annealing or grid search - both methods require many solutions to forward model
Uses Green’s functions for elastic deformation; convolve locking or slip distribution with GFs
SA:
GS:

84. Iteration controls SA: 0.0 20 500 Simulated Annealing control:
Temperature
Number of iterations
Number of calls to ‘amoeba’ for each iteration
GS:
Grid Search control:
Number of steps away from current value
Nominal size of step (in parameter’s units)
Number of times to run through each parameter
Grid search type
Decrease in step size for each run through
IC:
Iteration Control:
Run simulated annealing
Run grid search

85. While iterating press:
‘q’ - quit iterating and finish program
‘s’ - go to next step in IC: sequence
‘n’ - if in GS: to go to next run through parameters

86. Examples
Costa Rica block model
Cascadia SSE
Taupo volcanic source

87. Example:
Costa Rica
block model
GPS velocities from 2 separate studies (rotate into common frame), uplift rates, solve for block motions and fault locking.
Cocos Plate subducts beneath the forearc. Forearc sliver moves to NW along possible strike-slip fault near arc. Use defnode to solve for locking on subduction thrust and motion of forearc.

89. Cascadia 2010 slow-slip event
Select continuous GPS sites in region
Solve for block motions, interseismic locking on subduction zone, slip distribution in SSE, Gaussian time history of SSE
2D Gaussian slip distribution
1D Gaussian downdip slip distribution

94. Taupo 2007 volcanic event
GPS time series, select time around event
Use a small block around event to remove steady velocity
Assume a Mogi source with Gaussian time history
Solve for source amplitude, depth, location, duration