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Correlations

Correlation of time series

Similarity

Time shitfs

Applications

Correlation of rotations/strains and translations

Ambient noise correlations

Coda correlations

Random media: correlation length

Scope: Appreciate that the use of noise (and coda) plus correlation techniques is one of the most innovative

direction in data analysis at the moment: passive imaging

(2)

Discrete Correlation

Correlation plays a central role in the study of time series.

In general, correlation gives a quantitative estimate of the degree of similarity between two functions.

The correlation of functions g and f both with N samples is defined as:

Correlation plays a central role in the study of time series.

In general, correlation gives a quantitative estimate of the degree of similarity between two functions.

The correlation of functions g and f both with N samples is defined as:

1 ,

, 2 , 1 , 0

1 1

0

N k

f N g

r N k

i

i k i k

(3)

3

Auto-correlation

Auto-correlation

(4)

Cross-correlation

Lag between two functions

Cross-correlation

(5)

5

Cross-correlation: Random functions

(6)

Auto-correlation: Random functions

(7)

7

Auto-correlation: Seismic signal

(8)

Theoretical relation

rotation rate and transverse acceleration plane-wave propagation

Plane transversely polarized wave propagating in x-direction with phase velocity c Plane transversely polarized wave propagating in x-direction with phase velocity c

k c

t kx

f t

x

uy(x ,t ) f (kx t ) c /k uy( , ) ( ) /

) (

) , ( )

,

(x t u x t 2 f kx t

ay(x,t) uy(x,t) 2 f (kx t) ay y 

Acceleration

c t

x t

x

a(x ,t )/ (x ,t ) 2c a( , )/ ( , ) 2

Rotation rate and acceleration should be in phase and the amplitudes scaled by two times the horizontal phase velocity

Rotation rate and acceleration should be in phase and the amplitudes scaled by two times the horizontal phase velocity

Rotation rate  ( )

2 , 1 0 , 0 0

, , 2 0

) 1 ,

(x t uy kf kx t

  

( )

2 , 1 0 , 0 0

, , 2 0

) 1 ,

(x t uy kf kx t

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9

Mw = 8.3 Tokachi-oki 25.09.2003

transverse acceleration – rotation rate

From Igel et al., GRL, 2005

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Max. cross-corr. coefficient in sliding time window transverse acceleration – rotation rate

Small tele-seismic event

P-onset

S-wave Love waves Aftershock

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11

M8.3 Tokachi-oki, 25 September 2003

phase velocities ( + observations, o theory)

From Igel et al. (GRL, 2005)

Horizontal phase velocity in sliding time window

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Sumatra M8.3 12.9.2007

P

P Coda

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13

… CC as a function of time …

observable for all events!

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Rotational signals in the P-coda?

azimuth dependence

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15

P-Coda energy direction

… comes from all directions …

correlations in P-coda window

(16)

Noise correlation - principle

(17)

17

Uneven noise distribution

(18)

Surface waves and noise

Cross-correlate noise observed over long time scales at different

locations

Vary frequency range, dispersion?

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19

Surface wave dispersion

(20)

US Array stations

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21

Recovery of Green‘s function

(22)

Disersion curves

All from Shapiro et al., 2004

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23

Tomography without earthquakes!

(24)

Global scale!

(25)

25

Correlations and the coda

(26)

Velocity changes by CC

(27)

27

Remote triggering (from CCs)

Taka’aki Taira, Paul G. Silver, Fenglin Niu &

Robert M. Nadeau:

Remote triggering of fault-strength changes on the San Andreas fault at Parkfield

Nature 461, 636-639 (1 October 2009) | doi:10.1038/nature08395;

Received 25 April 2009; Accepted 6 August 2009

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Remote triggering of fault-strength changes on the San Andreas fault at Parkfield

Taka’aki Taira, Paul G. Silver, Fenglin Niu & Robert M.

Nadeau Key message:

• Connection between significant changes in

scattering parameters and fault strength and dynamic stress

Seismic network

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29

Principle

Method:

• Compare waveforms of repeating earthquake sequences

• Quantity: Decorrelation index D(t) = 1-Cmax(t)

• Insensitive to variations in near-station environment

(Snieder, Gret, Douma & Scales 2002)

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Changes in scatterer properties:

•Increase in Decorrelation index after 1992 Landers earthquake (Mw=7.3, 65 kPa dyn.

stress)

•Strong increase in Decorrelation index after 2004 Parkfield earthquake (Mw=6.0,

distance ~20 km)

•Increase in Decorrelation index after 2004 Sumatra Earthquake (Mw=9.1, 10kPa dyn.

stress)

•But: No traces of 1999 Hector Mine, 2002 Denali and 2003 San Simeon (dyn. stresses

True?

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Correlations and random media:

Generation of random media:

Define spectrum

Random Phase

Back transform usig inverse FFT

(32)

Random media:

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34

P-SH scattering

simulations with ADER-DG

translations

rotations

(34)

P-SH scattering

simulations with ADER-DG

(35)

36

Random mantle models

(36)

Random models

(37)

38

Convergence to the right spectrum

(38)

Mantle models

(39)

40

Waves through random models

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Summary

The simple correlation technique has turned into one of the most important processing tools for seismograms

Passive imaging is the process with which noise recordings can be used to infer information on structure

Correlation of noisy seismograms from two stations allows in

principle the reconstruction of the Green‘s function between the two stations

A whole new family of tomographic tools emerged

CC techniques are ideal to identify time-dependent changes in the structure (scattering)

The ideal tool to quantify similarity (e.g., frequency dependent) between various signals (e.g., rotations, strains with translations)

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