بهبود تخمین تابع گرین تجربی بین جفت ایستگاهی با استفاده از تکنیک جداسازی مقدار تکینه

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).