Correlations 1
Surface waves and correlations
Correlation of time series
Similarity
Time shifts
Applications
Correlation of rotations/strains and translations
Ambient noise correlations
Coda correlations
Scope: Appreciate that the use of noise (and coda) plus correlation techniques is one of the most innovative
directions 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
Correlations 3
Auto-correlation
Auto-correlation
Cross-correlation
Lag between two functions
Cross-correlation
Correlations 5
Cross-correlation: Random functions
Auto-correlation: Random functions
Correlations 7
Auto-correlation: Seismic signal
Auto-correlation length: example
a=20 b=20
a=20 b=10
Correlations 9
Random media
Correlation length
Correlations 11
Example: random mantle model
The cross-correlation coefficient
… normalizes the sequence such that the zero-lag cross-correlation is 1 … See also Matlab function xcorr(vector1,vector2,‘coeff‘)
Correlations 13
Waveform similarity: rotations and accelerations
Cross-correlation Cross-correlation
Translation Rotation
Corr. coeff.
Correlations 15
Strong rainfall on day 218 in the year of 2002 in Germany
Strong rainfall on day 218 in the year of 2002 in Germany
Correlation of processes
External driving of earthquakes?
External driving of earthquakes?
Correlations 17
Basic theory
Basic Theory
Correlations 19
Basic theory
Basic theory
Correlations 21
Noise correlation - principle
From Campillo et al.
Uneven noise distribution
Correlations 23
Theory
Green‘s function retrieval
Correlations 25
Noise on our planet
Stutzmann et al. 2009
Wavefield directions (winter-green, summer-red)
Geographical map showing at the station
location (black circles) the azimuths of the most abundant sources of secondary microseisms for months January and
February in green and July and August in red.
Correlations 27
Surface waves and noise
Cross-correlate noise observed over long time scales at different
locations
Vary frequency range,
dispersion?
Surface wave dispersion
Correlations 29
US Array stations
Recovery of Green‘s function
Correlations 31
Dispersion curves
All from Shapiro et al., 2004
Tomography without earthquakes!
Correlations 33
Global scale!
Nishida et al., Nature, 2009.
Time dependent changes in seismic velocity
Correlations 35
Time dependent changes in seismic velocity
Time-dependent changes
Correlations 37
Chinese network
Changes due to earthquake
Velocity changes in 1-3s period band
Chen, Froment, Liu and Campillo 2010
Correlations 39
Virtual sources
Industrial application
Correlations 41
Reflectivity from noise
Reflectivity
Wapenaar, Snieder, Physics Today, 2010
Correlations 43
Remote triggering of fault-strength changes on the San Andreas fault
Key message: Connection between significant
changes in scattering parameters and fault strength and dynamic stress
Taka’aki Taira, Paul G. Silver, Fenglin Niu & Robert M.
Nadeau Nature 461, 636-639 (1 October 2009) doi:10.1038/nature08395
How to
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)
Correlations 46
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)
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)
Correlations 48
Correlation/Convolution in a building
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)
