I. Introduction

II. Methods in Morphotectonics

III. Methods in Geodesy an Remote sensing

IV. Relating strain, surface displacement and stress, based on elasticity

V. Fault slip vs time

VI. Learnings from Rock Mechanics

VII. Case studies

IV. Relating strain, surface displacement and stress

- Some elastostatic solutions (for elastodynamic solutions see a Seismology textbook, Aki& Richard for example)

- Inversion techniques

- The circular crack model

Some classical elasticity solutions of use in tectonics

• Elastic dislocations in an elastic half-space (Steketee, 1958, Cohen, Advances of geophysics, 1999; Segall)– Surface displacements due to a rectangular dislocation: Okada, 1985:– Displacements at depth due to a rectangular dislocation: Okada,1992:– The infinitely long Strike-slip fault (Segall, 2009)– 2-D model for a dip-slip fault: Manshina and Smylie, 1971, Rani and Singh,

1992; Singh and Rani, 1993, Cohen, 1996.

• The Boussinesq pb (normal point load at the surface of an elastic half-space); (Jaeger, Rock mechanics and Engineering)

• The Cerruti pb (shear point load at the surface of an elastic half-space); (Jaeger, Rock mechanics and Engineering)

• Point source of pressure: the ‘Mogi source’(Segall, 2010)

• The circular crack model (Scholz, 2002)

ReferencesSegall, P, Earthquake and Volcano Deformation, Princeton University Press, 2010.Scholz, C. (1990), The Mechanics of Earthquakes and Faulting, 439 pp.,

Cambridge University Press, New York.Cohen, S. C., Convenient formulas for determining dip-slip fault parameters from

geophysical observables., Bulletin of seismological society of America, 86, 1642-1644, 1996.

Cohen, S. C., Numerical models of crustal deformation in seismic zones, Adv. Geophys., 41, 134-231, 1999.

Okada, Y., Surface deformation to shear and tensile faults in a half space, Bull. Seism. Soc. Am., 75, 1135-1154, 1985.

Okada, Y., Internal Deformation Due To Shear And Tensile Faults In A Half-Space, Bulletin Of The Seismological Society Of America, 82, 1018-1040, 1992.

Kositsky, A. P., and J. P. Avouac (2010), Inverting geodetic time series with a principal component analysis-based inversion method, Journal of Geophysical Research-Solid Earth, 115.

Chanard, K., J. P. Avouac, G. Ramillien, and J. Genrich (2014), Modeling deformation induced by seasonal variations of continental water in the Himalaya region: Sensitivity to Earth elastic structure, Journal of Geophysical Research-Solid Earth, 119(6), 5097-5113.

In crack mechanics, 3 modes are distinguished

Mode I= Tensile or opening mode: displacement is normal to the crack walls

Mode II= Longitudinal shear mode: displacement is in the plane of the crack

and normal to the crack edge (edge dislocation)

Mode III= Transverse shear mode: displacement is in the plane of the crack

and parallel to the crack edge (screw dislocation)

I II III

Infinite Strike-Slip fault

Let’s consider a fault parallel to Oy, with infinite length, and surface deformation due to uniform slip, equal to Sy, extending from the surface to a depth h. (Slip vector is (0,Sy,0) // Oy)

h

Co-seismic displacementparallel to Oy

y

x

Infinite Strike-Slip fault

00 yCo-seismic slip ( ,for a left-lateral fault)0y

Co-seismic strain

NB: far-field displacements and strain decay with x- 1 and x-2 respectively

Infinite Thrust fault

2)(

)(

)(tan

cos.)(

221

xsignDxx

Dxx

D

xxSxx

D

PD

2

))(1(

)(tan

sin.)(

221 xsign

Dxx

Dx

xx

DSxz

DD

Surface displacements due to slip S on a fault dipping by θ

(Manshina and Smylie, 1971; Cohen, 1996)

where

sin .cosP

Dx

tanD

Dx

Infinite Thrust fault

2)(

)(

)(tan

cos.)(

221

xsignDxx

Dxx

D

xxSxx

P

PD

Surface displacements

Horizontal strain

22 2

2 ( )( )1 .cos( )

2 ( )

P Dxx

P

D x x x xx Sx

x x x D

2

))(1(

)(tan

sin.)(

221 xsign

Dxx

Dx

xx

DSxz

DD

Displacements are proportional to fault slip (lineraity)Note that the far-field displacements and strains decay with x- 1 and x-2

Infinite Thrust fault

(see Cohen, 1996)

Convention in Okada (1985, 1992)Function [ux,uy,uz] = calc_okada(U,x,y,nu,delta,d,len,W,fault_type,strike)

This function computes the displacement field [ux,uy,uz] on the grid [x,y] assuming uniform slip, on a rectangular fault withU: slip on the faultnu: Poisson Coefficient delta: dip angled: depth of bottom edgelen=2L: fault lengthW: fault widthfault_type: 1=strike,2=dip,3=tensile,4=inflation

NB:C is the middle point of bottom edge

• Surface displacements (Okada, 1985)

useful for inverting geodetic data

• Displacement and strain at depth (Okada, 1992)

useful for Coulomb stress change (ΔCFF) calculations

It is with images of ERS acquired before and after the Landers 1992 earthquake that the first interferogram of an earthquake was produced.

Observed phase

Massonnet et al., 1993

after USGS

Setting of the 1992 Mw 7.3 Landers and1999 Mw 7.1 Hector Mine Earthquakes

Co-seismic displacement field due to the 1992, Landers EQ

G. Peltzer

Here the measured SAR interferogram is compared with a theoretical interferogram computed based on the field measurements of co-seismic slip using the elastic dislocation theory

This is a validation that coseismic deformation can be modelled acurately based on the elastic dislocation theory

(based on Massonnet et al, Nature, 1993)

Co-seismic deformation during the Hector Mine earthquake

Grey areas are zones of low phase coherence

Courtesy of G. Peltzer, UCLA

Line Of Sight component of displacement

• The crack model works approximately in this example, In general the slip distribution is more complex than perdicted from this theory either due to the combined effects of non uniform prestress, non uniform stress drop and fault geometry.

• The theory of elastic dislocations can always be used to model surface deformation predicted for any slip distribution at depth,

IV. 2-Inverting Surface Displacement

• Principle:– Source is gridded– Linear (slip, imposed geometry) vs Nonlinear

inversions (slip+geometry)– Regularisation (generally inversion is ill-

posed)

• Inversion of times series– ENIF (Paul Segall, Jeff McGuire)– PCAIM

Linear inversion• Using Green functions G calculated

with Okada solve:

where X: geodetic displacement

S: slip on the gridded source

• Laplacian Regularisation:

• Weighting: - data uncertainties e.g., - λ?

Inversion of Time series

the PCAIM technique

(Kositsky and Avouac, JGR, 2010)

PCAIM available on-line at:

http://www.tectonics.caltech.edu/resources/pcaim

• Divide time series as principal components ordered amount of data variance explained

• PCA and Okada Formulation are linear and associative and thus you can switch their ordering

Elas

tic

Dis

loca

tion

Forw

ard

Mod

el

Elas

tic

Dis

loca

tion

Forw

ard

Mod

el

Inversion of Time series

the PCAIM technique

Principal Components

• Each component has several aspects:- Mode, time variation associated with the PC

(v)- Surface Displacement, left singular value associated with the PC (u)- Singular Value, a measure of the variance of

the data explained by this PC (s)- Slip Distribution, a slip map associated with

the PC (l)

Singular Value Decomposition

• First component explains maximal data variance• nth component maximal given n-1th component

Linear Inversion

• Using Green functions G calculatedwith Okada solve:

Slip Decomposition of X

Singular Value Decomposition

+

Slip Decomposition

Okada Formulation

=

Long Valley Caldara 1997-1998 Inflation Episode

• multiple inflation events since 1980's

•~10 cm uplift near the resurgent dome during 1997-98 episode

•8 EDM time series

•24 ERS scene and 65 interferograms

Original Data

Electronic Distance Measurements

SBAS Time series

PCA Decomposition

Spatial functionTime function

PCA Reconstruction

InSAR SBAS Time Series

PCA Reconstruction

Electronic Distance Measurements

Joint Inversion* 1st comp only

Summary

• PCAIM allows the joined analysis of multiple datasets with very different temporal and spatial resolutions

• The approach allows to filter out tropospheric effects in the InSAR data.

http://www.tectonics.caltech.edu/resources/pcaim

The Elastic crack model

See Pollard et Segall, 1987 or Scholz, 1990 for more details

A planar circular crack of radius a with uniform stress drop,Δσ, in a perfectly elastic body (Eshelbee, 1957)

NB: This model produces infinite stress at crack tips, which is not realistic

Slip on the crack

Stress on the crack

2 24 (1 )

(2 )u a x

See Pollard et Segall, 1987 or Scholz, 1990 for more details

A planar circular crack of radius a with uniform stress drop, Δσ, in a perfectly elastic body (Eshelbee, 1957)

NB: This model produces infinite stress at crack tips, which is not realistic

i. The predicted slip distribution is elliptical

ii. Dmean and Dmax increase linearly with fault length (if stress drop is constant).

Slip on the crack

Stress on the crack

2 24 (1 )

(2 )u a x

8 (1 )

3 (2 )meanD u a

max max

4 (1 )

(2 )D u a

The Elastic crack model

The 1999, Mw 7.1Hector Mine Earthquake

(Leprince et al, 2007)

Coseismic deformation due the 1999 Mw 7.1 Hector Mine earthquake measured from 10m GSD SPOT images.

N-S Component

Measurements of NS and EW displacement fields from the correlation of SPOT panchromatic images (pixel size 10m) taken before and after the EQ. Displacements as low of 1/10th of the pixel size (1m) can be measured from this technique

of the order of 5 MPa

(Treiman et al, 2002)

(Leprince et al, 2007)

Coseismic deformation due the 1999 Mw 7.1 Hector Mine earthquake measured from 10m GSD SPOT images.

N-S Component

Localized and off-fault distributed anelastic deformation add to form a smooth slip distribution

The 1999, Mw 7.1Hector Mine Earthquake