correlationscomputational geophysics and data analysis 1 correlations correlation of time series...

Computational Geophysics and Data Analysis1

Correlations

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

Computational Geophysics and Data Analysis2

Correlations

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

Nk

fgN

rkN

iikik

Computational Geophysics and Data Analysis3

Correlations

Auto-correlation

Auto-correlation

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Correlations

Cross-correlation

Lag between two functions

Cross-correlation

Computational Geophysics and Data Analysis5

Correlations

Cross-correlation: Random functions

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Auto-correlation: Random functions

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Auto-correlation: Seismic signal

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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

kctkxftxu y /)(),( kctkxftxu y /)(),(

)(),(),( 2 tkxftxutxa yy )(),(),( 2 tkxftxutxa yy Acceleration

ctxtxa 2),(/),( ctxtxa 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

1,0,00,,0

2

1),( tkxfkutx y

)(

2

1,0,00,,0

2

1),( tkxfkutx y

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Mw = 8.3 Tokachi-oki 25.09.2003transverse acceleration – rotation rate

From Igel et al., GRL, 2005

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Correlations

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

Small tele-seismic event

P-onset

S-waveLove waves Aftershock

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Correlations

M8.3 Tokachi-oki, 25 September 2003phase velocities ( + observations, o theory)

From Igel et al. (GRL, 2005)

Horizontal phase velocity in sliding time window

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Correlations

Sumatra M8.3 12.9.2007

P

P Coda

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… CC as a function of time …observable for all events!

Computational Geophysics and Data Analysis14

Correlations

Rotational signals in the P-coda?azimuth dependence

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P-Coda energy direction… comes from all directions …

correlations in P-coda window

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Correlations

Noise correlation - principle

From Campillo et al.

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Correlations

Uneven noise distribution

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Correlations

Surface waves and noise

Cross-correlate noise observed over long

time scales at different locations

Vary frequency range, dispersion?

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Correlations

Surface wave dispersion

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US Array stations

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Recovery of Green‘s function

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Disersion curves

All from Shapiro et al., 2004

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Tomography without earthquakes!

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Global scale!

Nishida et al., Nature, 2009.

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Correlations and the coda

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Velocity changes by CC

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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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Correlations

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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Correlations

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 all two times above 2004 Sumatra)

True?

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Correlations and random media:

Generation of random media:

Define spectrum Random Phase Back transform usig

inverse FFT

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Random media:

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P-SH scattering simulations with ADER-DG

translations

rotations

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P-SH scatteringsimulations with ADER-DG

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Random mantle models

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Random models

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Convergence to the right spectrum

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Mantle models

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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)