Improving estimation of inter-station empirical Green's function using SVD technique


Assistant Professor, Department of Physics, Islamic Azad university Damavand branch, Damavand, Iran


Various studies have shown that the cross-correlation (Wapenaar, 2004), cross-convolution (Slob and Wapenaar, 2007) and de-convolution (Wapenaar et al., 2008) can provide empirical Green’s functions (EGFs) between receiver pairs. These approaches, which are attributed to seismic interferometry, assume that one of the receiver acts as a source, whereas the other one is instated as a virtual receiver. The resulted EGFs allowed many studies to be applied in different regions even though (including) areas with low seismicity.
The main assumption of interferometry approach is based on completely diffuse signals which are generated by a closed surface of sources (Schuster et al. 2004). In other word, the distribution of sources and theirs energy in a medium are usually uniform. This condition ensured that inter-station EGF is extracted accurately. In general, the sources (left and right of the receivers) located on or near lines which is passed through both receivers are in the stationary region, and the sources above and below are in non-stationary regions. Also, Snieder (2004) indicated that the Fresnel zone of receivers surrounded all the sources which are located in stationary region. In this study, we referred to these sources as stationary sources. Consequently, all sources outside the Fresnel zone were referred to as non-stationary sources. It is generally accepted that the stationary sources play a major role/contribution to retrieve the inter-station EGF. Stationary sources and their energies are characterized by coherency and small wavenumber. In contrast, non-stationary sources and their energies are characterized by incoherency, larger wavenumber. We used this difference in order to separate stationary and nonstationary sources.
In the Earth, distribution of noise sources and theirs energy are strongly non-uniform, which contravenes the theoretical interferometry requirements (Stehly et al. 2006). In other words, cross correlations from non-stationary sources in stacking procedure do not cancel completely if the source coverage is incomplete. Consequently, incomplete source coverage leads to retrieve unreliable inter-station EGF.
In this study, we used 144 sources on circle environment (r=40 km) surrounded by two receivers which are located/installed in A(-4 , 0) and B(4 , 0) as shown in Figure 1. Moreover, synthetic time series were generated using Mexican-hat source time function (see left panel of Figure 2). All recorded waveform signals of these sources in station A and B are shown in middle and right panel of Figure 2, respectively. After the preprocessing procedure and cross-correlation performances, we constructed a cross-correlogram matrix, which is called CC, using a collection of cross-correlation function signals (see left panel of Figure 3). The dimensions of this matrix include time (number of point in signal data set, npts) and source counter/numerator. In brief, inter-station EGF is retrieved using stacking the cross-correlogram matrix signals along the source counter dimension. We followed the analysis and preprocessing of the cross-correlogram before stacking outlined in Poliannikov and Willis (2011). Thus, we decompose cross-correlogram matrix using singular value decomposition (SVD) technique to separate the stationary and non-stationary energies. This idea illustrates/explains that the cross-correlogram matrix could be calculated by its eigenvalues and eigenvectors. Poliannikov and Willis (2011) indicated that the large eigenvalues (singular values) are associated with events which is located in Fresnel zone. Afterward, we constructed lower-rank approximations of the cross-correlogram matrix using two larger eigenvalues, which is called CC2, and then stack CC2 along the source counter dimension to retrieve inter-station EGF (see Figures 4, 5 and 6).


Main Subjects

Bensen, G. D., Ritzwoller, M. H., Barmin, M. P., Levshin, A. L., Lin, F., Moschetti, M. P., Shapiro, N. M. and Yang, Y., 2007, Processing seismic ambient noise data to obtain reliable broad-band surface wave dispersion measurements. Geophys. J. Int., 169, 1239–1260.
Cho, K. H., Herrmann, R. B., Ammon, C. J. and Lee, K., 2007, Imaging the upper crust of the Korean Peninsula by surface-wave tomography, Bull. seism. Soc. Am., 97, 198–207.
Golub, G. and van Loan, C., 1996, Matrix Computations, The Johns Hopkins Univ. Press, Baltimore, MD.
Halliday, D. and A. Curtis, 2008, Seismic interferometry, surface waves and source distribution, Geophysical Journal International, 175, 1067–1087, doi: 10.1111/gji.2008.175.issue-3.
Hansen, P. C., Kilmer, M. E. and Kjeldsen, R., 2006, Exploiting residual information in the parameter choice for discrete ill-posed problems, BIT, 46, 41–59.
Menke M., 1989, Geophysical data analysis discrete inverse theory. Academic Press.
Poliannikov, O. V. and Willis, M. E., 2011, Interferometric correlogramspace analysis, Geophysics, 76(1),SA9–SA17.
Schuster, T. G., Yu, J., Sheng, J. and Rickett, J., 2004, Interferometric/daylight seismic imaging, Geophys. J. Int., 157, 838–852.
Shapiro, N. M. and Campillo, M., 2004, Emergence of broadband Rayleigh waves from correlations of the ambient seismic noise, Geophys. Res. Lett., 31, L07614, doi:10.1029/2004GL019491.
Slob, E. and Wapenaar, K., 2007, Electromagnetic Green’s functions retrieval by cross-correlation and cross-convolution in media with losses. Geophysical Research Letters, 34, L05307, doi: 10.1029/2006GL029097.
Slob, E., Draganov, D. and Wapenaar, K., 2007, Interferometric electromagnetic Green’s functions representations using propagation invariants, Geophysical Journal International, 169, 60–80, doi: 10.1111/gji.2007.169.issue-1.
Snieder, R., 2004, Extracting the Green’s function from the correlation of coda waves: A derivation based on stationary phase. Physical Review E, 69, 046610, doi: 10.1103/PhysRevE.69.046610.
Stehly, L., Campillo, M. and Shapiro, N., 2006, A study of seismic noise from its long-range correlation properties, J. geophys. Res., 111, 10306, doi:10.1029/2005JB004237.
Ulrych, T. J., Sacchi, M. D. and Graul, J. M., 1999, Signal and noise separation: art and science, Geophysics, 64(5), 1648–1656.
Wapenaar, K., 2004, Retrieving the elastodynamic Green’s function of an arbitrary inhomogeneous medium by cross correlation. Physical Review Letters, 93, 254301, doi: 10.1103/PhysRevLett.93.254301.
Wapenaar, K. and Fokkema, J., 2006, Green’s function representations for seismic interferometry, Geophysics, 71, no. 4, SI33–SI46, doi:10.1190/1.2213955.
Wapenaar, K., van der Neut, J. and Ruigrok, E., 2008, Passive seismic interferometry by multidimensional deconvolution. Geophysics, 73 (6), A51–A56.
Wessel, P. and Smith, W. H. F., 1998, New, improved version of the Generic Mapping Tools released, EOS. Trans. Am. Geophys. Union, 79, 579.