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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 Computational Geophysics and Data Analysis
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: Computational Geophysics and Data Analysis
Auto-correlation Auto-correlation Computational Geophysics and Data Analysis
Cross-correlation Cross-correlation Lag between two functions Computational Geophysics and Data Analysis
Cross-correlation: Random functions Computational Geophysics and Data Analysis
Auto-correlation: Random functions Computational Geophysics and Data Analysis
Auto-correlation: Seismic signal Computational Geophysics and Data Analysis
Theoretical relation rotation rate and transverse accelerationplane-wave propagation Acceleration Plane transversely polarized wave propagating in x-direction with phase velocity c Rotation rate Rotation rate and acceleration should be in phase and the amplitudes scaled by two times the horizontal phase velocity Computational Geophysics and Data Analysis
Mw = 8.3 Tokachi-oki 25.09.2003transverse acceleration – rotation rate From Igel et al., GRL, 2005 Computational Geophysics and Data Analysis
Max. cross-corr. coefficient in sliding time windowtransverseacceleration – rotation rate P-onset Aftershock Love waves S-wave Small tele-seismic event Computational Geophysics and Data Analysis
M8.3 Tokachi-oki, 25 September 2003phase velocities ( + observations, o theory) Horizontal phase velocity in sliding time window From Igel et al. (GRL, 2005) Computational Geophysics and Data Analysis
Sumatra M8.3 12.9.2007 P P Coda Computational Geophysics and Data Analysis
… CC as a function of time …observable for all events! Computational Geophysics and Data Analysis
Rotational signals in the P-coda?azimuth dependence Computational Geophysics and Data Analysis
P-Coda energy direction… comes from all directions … correlations in P-coda window Computational Geophysics and Data Analysis
Noise correlation - principle From Campillo et al. Computational Geophysics and Data Analysis
Uneven noise distribution Computational Geophysics and Data Analysis
Surface waves and noise Cross-correlate noise observed over long time scales at different locations Vary frequency range, dispersion? Computational Geophysics and Data Analysis
Surface wave dispersion Computational Geophysics and Data Analysis
US Array stations Computational Geophysics and Data Analysis
Recovery of Green‘s function Computational Geophysics and Data Analysis
Disersion curves All from Shapiro et al., 2004 Computational Geophysics and Data Analysis
Tomography without earthquakes! Computational Geophysics and Data Analysis
Global scale! Nishida et al., Nature, 2009. Computational Geophysics and Data Analysis
Correlations and the coda Computational Geophysics and Data Analysis
Velocity changes by CC Computational Geophysics and Data Analysis
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 Computational Geophysics and Data Analysis
Seismic network 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 Computational Geophysics and Data Analysis
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) Computational Geophysics and Data Analysis
True? • 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) Computational Geophysics and Data Analysis
Correlations and random media: Generation of random media: • Define spectrum • Random Phase • Back transform usig inverse FFT Computational Geophysics and Data Analysis
Random media: Computational Geophysics and Data Analysis
P-SH scattering simulations with ADER-DG translations rotations Computational Geophysics and Data Analysis
P-SH scatteringsimulations with ADER-DG Computational Geophysics and Data Analysis
Random mantle models Computational Geophysics and Data Analysis
Random models Computational Geophysics and Data Analysis
Convergence to the right spectrum Computational Geophysics and Data Analysis
Mantle models Computational Geophysics and Data Analysis
Waves through random models Computational Geophysics and Data Analysis
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) Computational Geophysics and Data Analysis
