TDEFNODE
Modeling block motions, fault locking, strain rates, transients
Use GPS velocities, displacements, time series, earthquake
slip vectors, fault slip rates, InSAR
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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
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Download
ppt, web page, source, two examples
% ftp chandler.mit.edu
Login mitg, no pwd
cd incoming/workshop_miami
mget td*1117*
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• Defnode - modeling steady motions
only, use linear velocities
• Tdefnode - includes time-dependent
motions and uses time series; data are
time-sensitive
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Motivation
• Velocity fields are superposition
of multiple signals; rotations,
strain rates, noise
• Time series showing strong nonlinear (transient) effects
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Not as scary as GAMIT
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Large-scale rotation with subduction locking superimposed
Field through mid 2009
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-0.15 deg/Myr
Calculate
uniform strain
and rotation
rates in regions
Figure has block
rotations
removed from
vectors
-0.39 deg/Myr
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-0.25 deg/Myr
-0.06 deg/Myr
Courtesy of S. Payne
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Non-linearity of time series is a major challenge.
2004 quake
millimeters
Steady velocity
Afterslip
2005 quake
Afterslip from both events
East component of continuous GPS site SAMP
The Sumatra quakes of 2004 and 2005, with afterslip
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Data from PANGA
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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.
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From McCaffrey 2009 GRL
Data from GNS Science
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
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Long-term velocity?
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Data from GNS Science
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Parkfield quake
TBLP
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P566
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Inter-event velocities
are not independent
Data from PBO
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Modeling estimated co-seismic offsets
2002
2009
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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.
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Block model from inversion of GPS time series
Mountain building
comprises only ~10%
of the action
Strike-slip
~10 mm/yr
Oblique
~11 mm/yr
Thrusting
~17 mm/yr
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Yellowstone
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InSAR (M. Aly)
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Time series with sinusoidal term
InSAR data may overlap or have gaps in time
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Multiple sill-like sources each with own time history
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Complex GPS time series
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Invert simultaneously with InSAR
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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
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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).
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Rotation rates
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Strain rates
Not computed with TDEFNODE; but
Program is available.
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Blocks
• Closed polygons on surface of Earth
• Each characterized by angular velocity,
uniform strain rate
• Bounded by faults, or pseudo-faults
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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
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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
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Data
•
•
•
•
•
•
•
GPS velocities (East, North, Up)
GPS displacements (E, N, U)
GPS time series (E, N, U)
InSAR interferograms
Fault slip rates or directions
Earthquake slip vectors
Uplift rates or displacements (tidegauge, coral, etc.)
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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)
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Other data types
Tilt rates:
T(X) = [ Vz(X+DX) - Vz(X - DX) ] / (2 DX )
(X is at the mid-point of the leveling line and DX 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 |
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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.
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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
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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
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Building the Blocks
Two options
1. Define all block outlines and faults
separately
2. Program builds blocks from faults
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Method 1. Define Blocks and Faults
Fault
Block
Use BL: to outline block;
FA: to describe fault
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Block boundaries are determined by seismicity, faulting, strain
rates, … (reviewers always ask for justification of block
boundaries).
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Fault segment
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Block outline
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• 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.
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Faults - defined by nodes
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Nodes are in an irregular grid.
Confined to depth contours.
Designated by (longitude, latitude, depth).
Subsurface nodes can be generated by program.
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Representation of fault slip
•
•
•
•
Pyramidical
Bilinear
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. 40
Half-space dislocation model (HSDM) to calculate surface
deformation due to fault locking and slip events
41
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 nonuniform locking
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Angular velocities - AV (Euler poles)
• Each block has an AV assigned
• Multiple blocks can have same AV, in which case there is no
fault 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
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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
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Strain Rate Velocities
Point (j, )
Origin (jo, o)
Block
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Assign AV and SRT to Blocks
Block
Blk1
Blk3
Blk2
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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
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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.
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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.
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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.
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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
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Examples
• Update tdefnode - td_1116.zip
• Oregon - oregon_example.zip
• Costa Rica - cr_example.zip
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Sample control file:
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Run 1 - get poles and strain rates
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Run 2: use PBO field,
rotate into PNW field
reference frame
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Run 3: Multiple fields;
strain rates, rotation
rates and reference
frames.
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Reference
frame
adjustments for
PNW1 and
PBO.
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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
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-- Fault input, blocks from faults
-- If making blocks from faults,
(+mkb flag) make pseudo-faults
from remaining borders. They
will be free-slip boundaries.
FA: for fault segments
-- For Fault1, it dips to east so
start in south
Fa: Blk1_bndry 2
4 1 Blk1 Blk1 1 0 0
0.0
-90.0 30.0
-80.0 30.0
-80.0 40.0
-90.0 40.0
Fa: Fault1 1
2 3 Blk2 Blk1 1 0 0
0.0
-90.0 30.0
-90.0 40.0
Zd: 5 89
Zd: 10 89
-100, 40
-90, 40
-80, 40
-- Give interior point of block
to identify it, and specify pole
and strain tensor for each
Blk2
Blk1
BC: Blk1 -95.0 35.0 1 0
BC: Blk2 -85.0 35.0 2 0
Fault1
-100, 30
-90, 30
Fa: Blk2_bndry 3
4 1 Blk2 Blk2 1 0 0
0.0
-90.0 30.0
-100.0 30.0
-100.0 40.0
-90.0 40.0
-80, 30
All segments must end
at another segment or
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an error occurs.
--- Block and Fault input
If inputting blocks, make
polygon of borders of blocks.
They will be free-slip
boundaries.
BL: for closed blocks
FA: for fault segments
For Fault1, it dips to east so start
in south
BL: Blk1 1 0
4
-90.0 30.0
-80.0 30.0
-80.0 40.0
-90.0 40.0
Fa: Fault1 1
2 3 Blk2 Blk1 1 0 0
0.0
-90.0 30.0
-90.0 40.0
Zd: 5 89
Zd: 10 89
-100, 40
-90, 40
-80, 40
BL: Blk2 2 0
4
-90.0 30.0
-100.0 30.0
-100.0 40.0
-90.0 40.0
-- BP: is alternative way to
specify poles and strain tensors
BP: Blk1 1 0
BP: Blk2 2 0
Blk2
Blk1
Fault1
-100, 30
-90, 30
-80, 30
All block and fault
points must coincide or
an error occurs.
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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
- 2D Gaussian
- Uniform Polygon
x
z, w
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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 interdependence of the nodes (NN:) and their starting values (NV:).
z
V
FT: 1 0
NNg: 1 6
1 1 2 2
1 1 2 2
4 4 5 5
4 4 5 5
0 0 0 0
NV: 1
5
3
3
6
6
0
3
3
6
6
0
x
1.0 1.0 1.0 0.8 0.8 0.8
x
z, w
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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:
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V
# DD Prof 1 2 3 4 5 6
FT: 1 2 1 1 1
PN: 1
1 1 2 2 3 3
PV: 1 3

5.0 5.0 5.0
5.0 5.0 5.0
zu
15.0 15.0 15.0
x
zl
z, w
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Downdip function 2 for variable , Zu = 6 Zl = 59 km
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Types of downdip (1D) functions:
Exponential (Type 2)
Boxcar (Type 3)
Gaussian (Type 4)
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)
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Interseismic; I
recommend locking the
updip edge and forcing
monotonic decrease in
locking downdip
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Approximating ‘locking depth’; using downdip
boxcar (FT: 3) fixing upper depth (0 km) and locking
amplitude (1); solve for lower depth
x
Locked nodes
z, w
Unlocked nodes
# locking depth approx.
FT: 1 3 0 0 1
PN: 1 1 1 2 2 3
PV: 1 3
1.0 1.0 1.0
0.0 0.0 0.0
15.0 15.0 15.0
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