Summary
Seismic interferometry is routinely used to image and characterize underground geology. The vertical component cross-correlations (CZZ) are often analysed in this process; although one can also use radial component and multicomponent cross-correlations (CRR and CZR, respectively), which have been shown to provide a more accurate Rayleigh-wave Green’s function than CZZ when sources are unevenly distributed. In this letter, we identify the relationship between the multicomponent cross-correlations (CZR and CRR) and the Rayleigh-wave Green’s functions to show why CZR and CRR are less sensitive than CZZ to non-stationary phase source energy. We demonstrate the robustness of CRR with a synthetic seismic noise data example. These results provide a compelling reason as to why CRR should be used to estimate the dispersive characteristics of the direct Rayleigh wave with seismic interferometry when the signal-to-noise ratio is high.
1 INTRODUCTION
Characterizing underground geological structure is important for a variety of applications (e.g. geological hazard assessment, resource exploration, contaminant monitoring, etc.). Nowadays one commonly uses seismic interferometry (SI) to characterize elastic and anelastic properties of the subsurface. Vertical component (Z) data are often used to compute CZZ cross-correlations (e.g. Shapiro et al.2005), where CZZ indicates that the vertical channel at both stations is used. From CZZ, one can estimate an approximate fundamental-mode Rayleigh-wave Green’s function (GZZ) if the seismic sources are distributed evenly (Snieder 2004; Roux et al.2005) or if the wavefield is diffuse (Lobkis & Weaver 2001; Weaver & Lobkis 2006). However, seismic sources are usually not evenly distributed, nor is the wavefield diffuse (Mulargia 2012), and CZZ leads to a biased estimate of GZZ (e.g. Halliday & Curtis 2008; Yao & van Der Hilst 2009; Froment et al.2010). One can correct the biased GZZ using multidimensional deconvolution (Wapenaar et al.2011), the C3 method (Stehly et al.2008; Froment et al.2011), information about the source distribution (e.g. Yao & van Der Hilst 2009; Nakata et al.2015), or signal processing methods (e.g. Baig et al.2009; Stehly et al.2011; Melo et al.2013). One can also use radial component (R) data to retrieve GRR or a combination of vertical and radial components to retrieve GZR (e.g. Campillo & Paul 2003; Lin et al.2008; Stehly et al.2009), where the R direction is the in-line direction between the two receivers. van Wijk et al. (2011) (empirically) and Haney et al. (2012) (theoretically) determined that CZR and CRZ are less sensitive than CZZ to out-of-line sources, where out-of-line sources mean the non-stationary phase sources. Stationary-phase sources are defined as sources that constructively interfere to produce the Green’s function during correlation; these are sources that have an absolute phase difference less than π/4 when compared to the real Green’s function.
In this letter, we investigate the reliability of cross-correlations affected by an uneven source-energy distribution. Truncating the boundary of sources in seismic interferometry leads to coherent noise (i.e. artefacts or spurious arrivals; e.g. Snieder et al.2006; Mikesell et al.2009). We investigate why CZR and CRR are more robust than CZZ to estimate the fundamental-mode Rayleigh wave from a theoretical standpoint and determine why previous studies often find that CZZ has the largest signal-to-noise ratio (SNR). We first review the relationship between the fundamental-mode Rayleigh-wave Green’s function and the cross-correlation function. We then analyse how the source-energy distribution contributes to the cross-correlation and the estimate of the Green’s functions. We find that CZR and CRR attenuate the non-stationary-phase source energy and provide more reliable Rayleigh-wave Green’s functions than CZZ. We further the discussion with a synthetic data example where seismic noise sources are unevenly distributed. We consider how the uneven noise-source distribution affects the virtual shot records and coherent and incoherent noise, as well as the resulting Rayleigh-wave dispersion images. We demonstrate that coherent noise is present prior to the direct-wave arrival, and therefore, this type of noise is often not taken into account when the signal-to-noise ratio of correlations is computed using incoherent noise that arrives after the direct wave.
2 THE GREEN’S FUNCTIONS AND MULTICOMPONENT CROSS-CORRELATIONS
Diagram of the location of a point source (star) and the receivers (triangles). The R direction is parallel to the line linking the two sensors, rA and rB.
3 THE SIGNIFICANCE OF THE SOURCE ANGLE
The source angle contributes to the three different kinds of cross-correlations, CZZ, CZR and CRR, in different ways. One can assess the role of the source angle by considering the integrands of the cross-correlations (e.g. Fan & Snieder 2009). The source distribution area can be divided into two parts: a stationary-phase area (near θ = 0, π, 2π in Fig. 2a) and a non-stationary-phase area (the rapid oscillation area in Fig. 2a). The sources in the stationary-phase area are important for retrieving the Green’s functions; they contribute significantly to the integral in eq. (1) (Snieder 2004; Snieder et al.2008; Mikesell et al.2012). If the sources are evenly distributed, the integrands of the CZZ, CZR and CRR oscillate evenly in the non-stationary-phase area and completely cancel the non-stationary-phase energy in the integral from 0 to 2π. However, we are interested in the sources in the non-stationary-phase area; thus we consider an isolated number of sources in small angular range.
The amplitudes of the integrands of CZZ, CZR and CRR (eqs 4, 6 and 8) change with the source angle (θ). The black solid line represents the real part of the integrand, and the grey dashed line represents the imaginary part. These examples are computed with a frequency (ω) of 5 Hz, a phase velocity (c) of 200 m s−1 and an interstation distance (r) of 120 m.
At a constant receiver separation, the stationary-phase area increases as frequency decreases; therefore, more sources can contribute to retrieval of the low frequency Green’s function. However, the integrand of cross-correlations (eqs 4, 6 and 8) oscillates slower as frequency decreases (Fig. 3). Therefore, if the sources only exist in some small part of the non-stationary-phase area, frequency-dependent energy will remain after the integration and lead to spurious waves (i.e. artefacts) in the retrieved GZZ (e.g. Yang & Ritzwoller 2008). In contrast, at high frequencies the integrand oscillates rapidly (Fig. 3), and the non-stationary-phase source energy cancels over small angular ranges (Xu et al.2017). If we consider the integrands of CZR and CRR (Figs 2b and c, respectively), we observe an interesting relationship between source angle and the amplitude of the integrand.
The envelope of the integrand of CZZ (black line), CZR (blue line) and CRR (red line) at 5 Hz (a), 10 Hz (b) and 20 Hz (c). The envelope is the L2 norm of the real and imaginary part of the integrands in eqs (4), (6) and (8). The grey line is the real part of the integrand of CRR weighted by cos 2θ. The oscillation rate of the phase of CZZ and CZR is identical to CRR, and the phase varies much faster than the weighting term. Here we assume the phase velocity is 200 m s−1 and the interstation distance is 120 m.
The non-stationary-phase sources are spatially down weighted in the CZR and CRR cross-correlations due to the occurrence of the cos θ in eqs (6) and (8). For each source, the Rayleigh-wave energy is projected to the R direction and decreases from the maximum to 0 as the source angle increases from θ = 0 to π/2. Therefore the integrand amplitude of CZR and CRR is reduced in the non-stationary-phase area compared to the amplitude of CZZ (Fig. 2). Furthermore, the CRR amplitudes are down weighted more than CZR outside the stationary-phase area due to the cos 2θ term. Because of the projection in the R direction, CRR is theoretically the most robust Rayleigh-wave estimation for uneven source distributions. Haney et al. (2012) pointed out that the cos θ term acts as a spatial filter for the CZR and CRZ components in the spatial autocorrelation (SPAC) method. The idea of the spatial filter does not only apply to CZR, but also to CRR (Fig. 2).
The envelopes of the integrands also demonstrate that CZR and CRR attenuate the non-stationary-phase energy equally for all frequencies (Fig. 3). The stationary-phase energy in CRR and CZR is preferentially weighted more than the non-stationary-phase energy, and thus act as a spatial filter on the source distribution. This spatial filter is identical for different frequencies (Fig. 3), different interstation distances and different phase velocities because cos θ is independent of these parameters. Furthermore, the filter does not affect the stationary-phase sources because cos θ and cos 2θ vary slower than the integrand (Fig. 3). Finally, in the limit that the frequency goes to zero, or the interstation distance goes to zero, the correlation function becomes an autocorrelation, and all space becomes the stationary-phase area. In that case, the spatial filter no longer plays a significant role in the accuracy of the retrieved Green’s function.
4 A SYNTHETIC-NOISE SOURCE EXAMPLE
The integral on the right hand side of eq. (1) also represents the cross-correlation between noise records of two receivers, rA and rB, if the noise sources are independent of each other (i.e. mutually uncorrelated) (Wapenaar & Fokkema 2006). One can then use eq. (1) to estimate the Rayleigh-wave Green’s functions GZZ, GZR and GRR from seismic noise (e.g. Halliday & Curtis 2008). We demonstrate the reliability of CZZ, CZR and CRR with a synthetic example, where noise sources are unevenly distributed. We compute virtual shot records along a linear array from correlations of the noise. The noise sources are randomly distributed within two angle ranges (Fig. 4): from −π/12 to π/12 (the stationary-phase area) and from π/4 to 5π/12 (the non-stationary-phase area). The number of noise sources is used as a proxy for the noise energy strength, and the non-stationary-phase noise energy is twice as strong as the stationary-phase noise energy in this example.
The experiment geometry indicates the location of noise sources (dots) and geophones (triangles). The noise sources are located away from the origin between 100 and 500 m. See the text for more details.
The Earth model we use has two layers (Table 1) and is from Bonnefoy-Claudet et al. (2006). All noise sources emit the same wavelet, and we model only the fundamental-mode Rayleigh wave. Each noise source is randomly activated during a 1 hr recording time. We simulate the response for every source using the algorithm proposed by Michaels & Smith (1997) and project the response to the Z and R components of the sensors. Then we stack all of these source projections to create a 1 hr long synthetic noise recording at each of the 24 geophones, which are 5 m apart from each other (Fig. 4), with H13 near the origin.
The two-layer Earth model parameters used in the simulation.
| Layer | Vp | Vs | Density | Thickness |
|---|---|---|---|---|
| number | (m s−1) | (m s−1) | (kg m−3) | (m) |
| 1 | 1350 | 200 | 1900 | 25 |
| 2 | 2000 | 1000 | 2500 | ∞ |
| Layer | Vp | Vs | Density | Thickness |
|---|---|---|---|---|
| number | (m s−1) | (m s−1) | (kg m−3) | (m) |
| 1 | 1350 | 200 | 1900 | 25 |
| 2 | 2000 | 1000 | 2500 | ∞ |
The two-layer Earth model parameters used in the simulation.
| Layer | Vp | Vs | Density | Thickness |
|---|---|---|---|---|
| number | (m s−1) | (m s−1) | (kg m−3) | (m) |
| 1 | 1350 | 200 | 1900 | 25 |
| 2 | 2000 | 1000 | 2500 | ∞ |
| Layer | Vp | Vs | Density | Thickness |
|---|---|---|---|---|
| number | (m s−1) | (m s−1) | (kg m−3) | (m) |
| 1 | 1350 | 200 | 1900 | 25 |
| 2 | 2000 | 1000 | 2500 | ∞ |
We assess the accuracy of the three cross-correlations by comparing virtual shot records and comparing the Rayleigh-wave phase-velocity dispersion images to the true dispersion. We build virtual shot records (Figs 5a–c) from individual cross-correlations (e.g. Halliday et al.2008) and then map the data to the frequency-velocity domain using the phase-shift method (Song et al.1989) to generate phase-velocity dispersion images (Figs 5d–f). The virtual shot records and the dispersion images indicate that CRR is the most robust among the three cross-correlations. The dominate waveforms in the three cross-correlations are from the stationary-phase area noise sources, and the high-velocity spurious wave before the main waveform is due to the non-stationary-phase area noise energy. We find that CRR contains lower-amplitude spurious waves than CZR and CZZ (Fig. 6). The spurious waves in CZZ lead to the spurious energy trends at frequencies less than 7 Hz (Fig. 5d), which is fully discussed in Xu et al. (2017). We also find that CZR does not provide accurate information below 5 Hz (Fig. 5e). However, we observe accurate Rayleigh-wave phase velocities in the frequency-velocity domain of the CRR below 5 Hz (Fig. 5f), which matches the theoretical prediction in Section 3.
CZZ, CRZ and CRR virtual shot records (a–c) and the corresponding phase-velocity dispersion images (d–f). The dominant energy trends in (a–c) represent the Rayleigh wave. Black dots represent theoretical Rayleigh-wave phase velocities (Haskell 1953) in (d–f). The black dash lines in (d–f) indicate the resolvable image area, where the wavelength is less than the array length. All dispersion images are normalized per frequency.
The amplitude normalized CZZ, CRZ and CRR functions between receivers H00 and H20. The inset shows a zoom of the spurious-energy time window from −0.1 to −0.3 s. A π/2 phase shift has been applied to CRZ to facilitate the comparison with CZZ and CRR. The values in the legend indicate the maximum amplitude of each cross-correlation function.
5 DISCUSSION
Although CZR and CRR attenuate non-stationary sources, the amplitudes of these two cross-correlations are determined by the H/V ratio (eqs 6 and 8). The H/V ratio is normally less than 1; therefore, the CZZ amplitude is normally larger than CZR and CRR. In our synthetic data example, the 3–15 Hz frequency-averaged H/V ratio is 0.41, the standard deviation is 0.21, and the CRR peak amplitude is an order of magnitude smaller than the CZZ peak amplitude (Fig. 6). Relative to the maximum amplitude of each correlation, the coherent noise (Fig. 6, t > −0.4 s and inset) is much larger in CZZ than CRR, while the incoherent noise (Fig. 6, t < −0.6 s) is approximately the same. Therefore, when discussing notions of SNR, one needs to consider both coherent and incoherent noise. Artefacts due to an uneven source distribution should be considered coherent noise, while random fluctuations should be considered incoherent noise.
In most studies, authors compute SNR as the ratio between the maximum Rayleigh-wave amplitude and the incoherent noise (e.g. Bensen et al.2007; Lin et al.2008). The incoherent noise is measured based on a window of data after the direct arrival (e.g. Bensen et al.2007; Lin et al.2008). If we assume that the random fluctuation (i.e. incoherent noise) amplitude is the same on the Z component and the R component, then the SNR of CZR and CRR will be less than that of CZZ any time the Rayleigh wave H/V ratio is less than 1. Thus in practice, people observe (compute) that CZZ has a higher SNR than CZR and CRR (e.g. Lin et al.2008). However, this SNR metric does not take into account the coherent noise that precedes the direct Rayleigh wave. One approach to monitor the coherent noise is to use a continuous SNR computation method (e.g. Larose et al.2007; Clarke et al.2011).
Finally, CZR and CRR can also aid the identification of fundamental and higher-model surface waves when the two surface-wave dispersion curves are very close in the frequency-velocity domain (Boué et al.2016; Ma et al.2016). The fact that Rayleigh wave modes have different H/V ratios and particle motions enables one to identify (e.g. Boaga et al.2013) and separate these modes (e.g. Gribler et al.2016) to improve the reliability of dispersion estimation.
6 CONCLUSIONS
We present the relationships between the fundamental-mode Green’s functions (GZZ, GZR and GRR) and cross-correlation functions (CZZ, CZR and CRR) within the far-filed approximation. When estimating the fundamental-mode Rayleigh-wave Green’s functions, the CZZ cross-correlation weights source energy equally from all directions. In contrast, the CZR and CRR cross-correlations attenuate source energy in the non-stationary-phase area for all frequencies and thus act as spatial filters on the source distribution. Therefore, more accurate Green’s functions (i.e. fewer spurious arrivals or reduced coherent noise) are retrieved from CZR and CRR compared to CZZ when the source energy is unevenly distributed. We demonstrate the validity of this theoretical inference with a synthetic seismic noise example. Those interested in characterizing velocity structure from ambient noise Rayleigh waves should use CRR whenever possible to limit the effect of non-homogeneous noise source distributions on the frequency-dependent direct-wave phase velocity. Finally, we note that the analysis presented here pertains to the direct-wave Rayleigh wave; we have neglected how the multicomponent cross-correlations influence scattered waves.
