TDEFNODE

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TDEFNODEModeling 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

web page, source

http://web.pdx.edu/~mccaf/www/defnode/

ppt, two examples

% ftp chandler.mit.eduLogin 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 non-linear (transient) effects

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Not as scary as GAMIT

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Field through mid 2009

Large-scale rotation with subduction locking superimposed

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-0.06 deg/Myr

-0.25 deg/Myr

-0.39 deg/Myr

-0.15 deg/Myr

Calculate uniform strain and rotation rates in regions

Figure has block rotations removed from vectors

Courtesy of S. Payne

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East component of continuous GPS site SAMP

mill

imet

ers

Non-linearity of time series is a major challenge.

2004 quake

2005 quakeAfterslip

Steady velocity

Afterslip from both events

The Sumatra quakes of 2004 and 2005, with afterslip

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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.

Data from GNS Science

Data from PANGA

From McCaffrey 2009 GRL

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Long-term velocity?

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

Data from GNS Science

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TBLP

P566

Inter-event velocities are not independent

Parkfield quake

Data from PBO

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Modeling estimated co-seismic offsets

2002 2009

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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.

Papua time series

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Block model from inversion of GPS time series

Thrusting~17 mm/yrOblique

~11 mm/yrStrike-slip~10 mm/yr

Mountain building comprises only ~10% of the action

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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

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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Strain rates

Rotation rates

Not computed with TDEFNODE; butProgram 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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X is the position of the surface observation point,k represents the velocity component (x, y, or z),

RB is the angular velocity of the block containing the observation point relative to

the reference frame,

RG is the angular velocity of the GPS velocity solution containing the observation

point relative to the reference frame,is the horizontal strain rate tensor (X is the offset from strain rate origin)

HF 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)

GPS velocity vectors and uplift rates

Vk(X) = [ RG X ]k + [ RB X ]k + kkXk+klXl +

j=1,2 i=1,N [- HF Qi ]j i Gjk (X, Xi)

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Other data typesTilt 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{[( HR - FR ) X]x / [( HR - FR ) X]y }

Geologically estimated fault slip rates or spreading rates:

  R(X) = | ( HR - FR ) 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 segmentBlock outline

• 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

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

• Nodes are specified along depth contours of fault

• Slip at each node is V, where ranges fromto 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

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

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Velocities from elastic strain rates arising from fault locking

surface

Locked fault

Free slipping

Use back-slip method to compute elastic deformation around locked fault.

Integrate over fault using small patches, can represent non-planar fault and non-uniform 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

Block

Origin (o, o)

Point (, )

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Assign AV and SRT to Blocks

Block

Blk1

Blk2

Blk3

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Method 2. Define Faults, Build blocks

Fault (extends to depth, can be locked)

Block

Set flag +mkbFA: to describe faultsBC: to identify blocks

Pseudo-fault (surface boundary, free-slip)

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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.

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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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--- 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

Models - a particular set of input parameters, designated by 4-char name. Multiple models can be in a single control file.

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-- Fault input, blocks from faults

FA: for fault segments

-- For Fault1, it dips to east so start in south

Fa: Fault1 12 3 Blk2 Blk1 1 0 00.0 -90.0 30.0 -90.0 40.0Zd: 5 89Zd: 10 89

Blk1

-100, 30

-100, 40 -90, 40

-90, 30

-80, 40

-80, 30

Fault1

Blk2

-- If making blocks from faults, (+mkb flag) make pseudo-faults from remaining borders. They will be free-slip boundaries.

Fa: Blk1_bndry 24 1 Blk1 Blk1 1 0 00.0 -90.0 30.0 -80.0 30.0 -80.0 40.0 -90.0 40.0

Fa: Blk2_bndry 34 1 Blk2 Blk2 1 0 00.0 -90.0 30.0-100.0 30.0-100.0 40.0 -90.0 40.0

-- Give interior point of block to identify it, and specify pole and strain tensor for each

BC: Blk1 -95.0 35.0 1 0BC: Blk2 -85.0 35.0 2 0

All segments must end at another segment or an error occurs.

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--- Block and Fault input

BL: for closed blocksFA: for fault segments

For Fault1, it dips to east so start in south

Fa: Fault1 12 3 Blk2 Blk1 1 0 00.0 -90.0 30.0 -90.0 40.0Zd: 5 89Zd: 10 89

Blk1

-100, 30

-100, 40 -90, 40

-90, 30

-80, 40

-80, 30

Fault1

Blk2

If inputting blocks, make polygon of borders of blocks. They will be free-slip boundaries.

BL: Blk1 1 04 -90.0 30.0 -80.0 30.0 -80.0 40.0 -90.0 40.0

BL: Blk2 2 04 -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 0BP: Blk2 2 0

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

x

z, w

Locking parameter is (x,z) or (x,w)

Independent nodes with or without smoothing; decreases down-dip; or is specified function of z

V-Boxcar-Gaussian-Exponential

is specified function of x,z- 2D Gaussian- Uniform Polygon

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x

z, w

V

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 0NNg: 1 6 5 1 1 2 2 3 3 1 1 2 2 3 3 4 4 5 5 6 6 4 4 5 5 6 6 0 0 0 0 0 0

NV: 1 1.0 1.0 1.0 0.8 0.8 0.8

x

z

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x

z, w

V

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:

# DD Prof 1 2 3 4 5 6FT: 1 2 1 1 1PN: 1 1 1 2 2 3 3PV: 1 3 5.0 5.0 5.0 5.0 5.0 5.015.0 15.0 15.0

zu

zl

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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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x

z, w

Locked nodes

Approximating ‘locking depth’; using downdip boxcar (FT: 3) fixing upper depth (0 km) and locking amplitude (1); solve for lower depth

Unlocked nodes# locking depth approx.FT: 1 3 0 0 1PN: 1 1 1 2 2 3PV: 1 3 1.0 1.0 1.0 0.0 0.0 0.0 15.0 15.0 15.0