1
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
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
Auto-correlation
Auto-correlation
Cross-correlation
Lag between two functions
Cross-correlation
5
Cross-correlation: Random functions
Auto-correlation: Random functions
7
Auto-correlation: Seismic signal
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) uy(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
9
Mw = 8.3 Tokachi-oki 25.09.2003
transverse acceleration – rotation rate
From Igel et al., GRL, 2005
Max. cross-corr. coefficient in sliding time window transverse acceleration – rotation rate
Small tele-seismic event
P-onset
S-wave Love waves Aftershock
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
Sumatra M8.3 12.9.2007
P
P Coda
13
… CC as a function of time …
observable for all events!
Rotational signals in the P-coda?
azimuth dependence
15
P-Coda energy direction
… comes from all directions …
correlations in P-coda window
Noise correlation - principle
17
Uneven noise distribution
Surface waves and noise
Cross-correlate noise observed over long time scales at different
locations
Vary frequency range, dispersion?
19
Surface wave dispersion
US Array stations
21
Recovery of Green‘s function
Disersion curves
All from Shapiro et al., 2004
23
Tomography without earthquakes!
Global scale!
25
Correlations and the coda
Velocity changes by CC
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
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
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)
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?
32
Correlations and random media:
Generation of random media:
Define spectrum
Random Phase
Back transform usig inverse FFT
Random media:
34
P-SH scattering
simulations with ADER-DG
translations
rotations
P-SH scattering
simulations with ADER-DG
36
Random mantle models
Random models
38
Convergence to the right spectrum
Mantle models
40
Waves through random models
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)