S3 41 1B 32
Pre-processing Noise Cross-correlations With Equalizing The Network Covariance Matrix Eigen-spectrum
[email protected] Leonard_Seydoux
leonard-seydoux.github.io
Léonard Seydoux , Julien de Rosny & Nikolai M. Shapiro 1
Theoretically, the extraction of Green’s functions from noise cross-correlation requires the ambient seismic wavefield to be generated by uncorrelated sources evenly distributed in the medium. Yet, this condition is often not verified. Strong events such as earthquakes often produce highly coherent transient signals. Also, the microseismic noise is generated at specific places on the Earth’s surface with source regions often very localized in space. Different localized and persistent seismic sources may contaminate the cross-correlations of continuous records resulting in spurious arrivals or asymmetry and, finally, in biased travel-time measurements. Pre-processing techniques therefore must be applied to the seismic data in order to reduce the effect of noise anisotropy and the influence of strong localized events. Here we describe a pre-processing approach that uses the covariance matrix computed from signals recorded by a network of seismographs. We extend the widely used time and spectral equalization pre-processing to the equalization of the covariance matrix spectrum (i.e., its ordered eigenvalues). This approach can be considered as a spatial equalization. This method allows us to correct for the wavefield anisotropy in two ways: (1) the influence of strong directive sources is substantially attenuated, and (2) the weakly excited modes are reinforced, allowing to partially recover the conditions required for the Green’s function retrieval. We also present an eigenvector-based spatial filter used to distinguish between surface and body waves. This last filter is used together with the equalization of the eigenvalue spectrum. We simulate two-dimensional wavefield in a heterogeneous medium with strongly dominating source. We show that our method greatly improves the travel-time measurements obtained from the inter-station cross-correlation functions. Also, we apply the developed method to the USArray data and pre-process the continuous records strongly influenced by earthquake-related signals and show that our approach significantly improves the recovered cross-correlations.
1 • Dataset
4 • Method: spatial equalization Beamforming 35 seismic stations vertical channel analysis
5 • Numerical experiment: spatial equalization in heterogenous medium 2-D acoustical simulation with finite-differences and heterogeneous velocity model
Planewave with testing slowness s
Eigenspectrum and beamforming with isotropic noise
Record from station i
Bensen, G. D., Ritzwoller, M. H., Barmin, M. P., Levshin, A. L., et al. (2007). Geophys. Journ. Int., 169(3), 1239-1260.
0
Cox, H. (1973). Journ. Acoust. Soc. Am., 54(5), 1289-1301.
0.25
0.5
0.75
1
Chew, W. C., & Liu, Q. H. (1996). Journal of Computational Acoustics, 4(04), 341-359.
Derode, A., Larose, E., Tanter, M., De Rosny, J., et al. (2003). Journ. Acoust. Soc. Am., 113(6), 2973-2976.
Moiola, A., Hiptmair, R., & Perugia, I. (2011). Zeitschrift für angewandte Mathematik und Physik, 62(5), 809-837.
Numerical crosscorrelations:
2 km/s
0.02 Hz
Normalized beam energy Noise emitted by source s
Green’s function between source s and sensor i
· 34 seismic stations
0
0.5
1
Seydoux, L., Shapiro, N. M., de Rosny, J., Brenguier, F., & Landès, M. (2016). Geophys. J. Int., 204(3), 1430-1442. Seydoux, L., Shapiro, N. M., Rosny, J., & Landès, M. (2016). Geophysical Research Letters, 43(18), 9644-9652. Seydoux, L., Rosny, J., & Shapiro, N. M., (2017). To be subm. to Geophysical Journal International Weaver, R. L., & Lobkis, O. I. (2001). Physical Review Letters, 87(13), 134301.
6 • Application to real data: spatial equalization of the M8.8 Maule earthquake Analysis of data around the M8.8 Maule, Chile earthquake (2010)
· 200 seismic sources · 20 perfectly-matched layers
1 month
Good-quality results from isotropic seismic noise
Array covariance matrix
Array covariance matrix spectrum
Eigenvectors
EVD
Fourier transform of uj(t )
= similarity between records
4 km/s
Normalized crosscorrelation
Gerstoft, P., Menon, R., Hodgkiss, W. S., & Mecklenbräuker, C. F. (2012). Journ. Acoust. Soc. Am., 132(4), 2388-2396.
6 km/s
(Cox 1973)
2 • Crosscorrelation and array covariance matrices FT
1,3
(1) Institut de Physique du Globe de Paris, UMR CNRS 7154, 75005 Paris, France (2) ESPCI Paris, CNRS, PSL Research University, Institut Langevin, 75005 Paris, France (3) Institute of Volcanology and Seismology FEB RAS, 9 Piip Boulevard, Petropavlovsk-Kamchatsky, Russia
Abstract
Crosscorrelation matrix
2
Eigenspectrum and beamforming with isotropic noise + source
Eigenvalues
= cross-spectra time average
Symmetrical crosscorrelations
M8.8 Maule (Chile)
Nearly-circular beamforming 0.02 Hz
0.02 Hz
Spectrum:
0.01 - 0.04 Hz
Green’s function retrieval from ambient seismic noise Isotropic and stationnary Ui(t)
Symmetrical crosscorrelation ≈ Earth Green’s function between i and j Rij(t)
Steadily decaying covariance matrix spectrum
Earthquake influence with spectral and temporal equalizations
Strong bias induced by the dominating source Spurious arrivals in the crosscorrelations
Spectral width: 4.59
Spurious arrivals induced by the earthquake
Earthquake-related waves dominate
0.02 Hz
Equalization of the covariance matrix eigen-spectrum
Uj(t)
Source-related waves dominate
0.02 Hz
Maximal rank
(e.g. Weaver & Lobkis 2001)
(Gerstoft 2012, Seydoux 2016a)
Equalized eigenvalues
0.01 - 0.04 Hz
0.01 - 0.04 Hz
Bias induced by anisotropic noise Strong source dominates Ui(t)
0.02 Hz
Asymmetric crosscorrelation ≠ Earth Green’s function between i and j
First eigenvalue dominates the covariance matrix spectrum
Attenuation of the dominating source with spatial equalization
Spectral width: 0.87
Rij(t)
Improved symmetry and attenuated spurious arrivals
Theoretical derivation of the covariance matrix rank
(Moiola 2011, Seydoux 2016c)
3 • Classical preprocessing: limitations and proposed improvements
Classical pre-processing Improvement
Spectral equalization
Improved symmetry and attenuated spurious arrivals
May be heterogeneous and anisotropic
Attenuate the impulsive events
Spatial equalization
We can also equalize the covariance matrix spectrum
0.01 - 0.04 Hz
0.01 - 0.04 Hz
Cylindrical harmomics decomposition of the wavefield (2D):
Performance of classical preprocessing Raw data
Travel-time measurements and inversion Only a finite number R = 2 + 1 of Bessel function contributes The covariance matrix rank cannot be higher than is 2 +1
Reduces the impact of strong sources
Temporal equalization
Earthquake equalized with background noise 0.02 Hz
0.02 Hz
Uj(t)
Raw data
Source amplitude equalized with the background noise
Earthquake influence with spectral, temporal and spatial equalizations
Isotropic noise
The strong bias induced by the source is highly attenuated with the spatial equalization.
Dominating source
Equalized source
Conclusions • We present a method that considers the equalization of the covariance matrix eigen-spectrum • Finite-difference simulations highlight that it significantly improves the travel-time measurements from isotropic noise polluted with highly-coherent source
Data with temporal and spectral equalizations
• Applied to real data, the method enables to equalize highly-coherent signals induced by earthquakes such as the M8.8 Maule (2010) megathrust earthquake
Coherent events may remain
Overal error: 37.64% M8.8 Maule, Chile (2010)
Overal error: 6.43%
The traveltimes are inverted using the open-source software Fatiando a Terra by Uieda et al. (2013).
• This method is of particular interest in the specific context of ambient noise-based monitoring of active zones, when the maximal amount of data is required
