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We study surface wave anisotropy using three-component frequency-wave number analysis of 1 year (2012) of ambient seismic noise measured by the Southern California Seismic Network. Significant 2θ and 4θ Rayleigh wave anisotropy is observed over most of the frequency range 15 to 100 mHz (Millihertz). The wide Rayleigh wave illumination and large data volume allow for relatively high precision and sensitivity: the estimation variability above 35 mHz as well as the magnitude of the weakest significant detections is about 0.1%. The estimates are consistent with previous anisotropy studies of the region. We also show preliminary results for Love waves, but ambient Love wave illumination in Southern California may not be sufficient for the approach.
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Seismic anisotropy in the crust and mantle can elucidate geodynamic processes in the Earth [e.g., Long, and references therein]. One observational constraint on such anisotropy is the azimuthal variation of surface wave phase velocities. Surface waves offer the advantages of depth resolution [Wüstefeld et al., 2009] and illumination of areas with low seismicity. Smith and Dahlen  showed that for laterally homogeneous media a weak but otherwise arbitrary anisotropy at depth results in relatively simple azimuthal perturbations of phase velocity: a sum of two sinusoids with 180° and 90° periodicity, typically dubbed 2θ and 4θ anisotropy. These parameters may be inverted for a laterally homogeneous and anisotropic subsurface model [Montagner and Nataf, 1986].
Azimuthal anisotropy has been estimated at various scales using surface wave tomography with earthquakes [Trampert and Woodhouse, 2003; Zhang et al., 2007; Deschamps et al., 2008] or noise [Fry et al., 2010; Gallego et al., 2011; Pawlak et al., 2012; Moschetti et al., 2010]. Tomographic techniques effectively use only a limited amount of data to constrain anisotropy at any given point on the surface. Uncertainty in those estimates is accordingly high. Alvizuri and Tanimoto  analyzed teleseismic Rayleigh waves on the Southern California Seismic Network (SCSN) with array beamforming. Using 190 events no 2θ anisotropy could be detected below 30 mHz and 4θ anisotropy was not detected at all despite evidence for it in the region [Montagner and Tanimoto, 1990; Trampert and Woodhouse, 2003; Deschamps et al., 2008].
In this work we demonstrate how using three-component frequency-wave number (FK) analysis on ambient seismic noise [Riahi et al., 2013] allows for a much more sensitive detection and precise estimation of surface wave anisotropy. Statistical significance is assessed with F tests and the estimation precision due to the uncertainty of the array processor is captured by bootstrapping. The study highlights the potential offered by array processing of ambient noise to provide high-precision estimates of array-averaged seismic parameters.
2 Data and Method
We study ambient seismic data recorded by the Southern California Seismic Network (SCSN) shown in Figure 1. Data from vertical channels of this network has been used for array studies in the past, both to analyze teleseismic events [Tanimoto and Prindle, 2007; Alvizuri and Tanimoto, 2011] as well as ambient surface and body waves [Gerstoft et al., 2006; Gerstoft and Tanimoto, 2007; Gerstoft et al., 2008]. We refer to Alvizuri and Tanimoto  for a more detailed analysis of the SCSN network as a seismic array.
In this study, we use all three-components from the high-gain, long-period channels (LHZ, LHE, and LHN) in the frequency range 10 to 100 mHz. The raw data of each station was conservatively quality checked by analyzing the distribution of spectral amplitudes for each month of the year 2012 [McNamara and Buland, 2004]: If any component of a station contained gaps, instrument failures, or narrow band spectral peaks, the station was excluded for that month. The three-component beamforming method is a relatively straight-forward generalization of conventional single-component beamforming and is described in detail by Riahi et al. . We only provide a very brief summary of the processing steps here. When a polarized plane wave impinges on a three-component array, it will impose relative phase and amplitude variations between the 3N array channels. This variation is described in the Fourier domain by a 3N dimensional response vector k, the Kronecker product of two complex-valued vectors:
c(ξ) is a complex-valued three element vector of relative amplitudes and phase shifts between the East, North, and Vertical component, and following Samson  defines a polarization ellipse represented by ξ which in this approach can be tuned to represent transverse motion (Love waves) or elliptical motion within the propagation plane (Rayleigh waves). The complex-valued N element vector a(k) describes the relative phase shifts between the N station locations due to the wave vector , where f is frequency, v is phase velocity, and n is the wave propagation direction. The amplitudes and relative phases between all 3N channels of the array are captured by the spectral density matrix:
where s3C is a column vector containing the 3N Fourier amplitudes of all channels, ·† means conjugate transpose, and 〈·〉 indicates time averaging. The beamformer output is
Back azimuth, phase velocity and polarization of dominant wave trains can now be estimated by searching for maxima in R(k,ξ). We compute Fourier amplitudes on synchronous 512 s windows of all 3N channels between 10 and 100 mHz. The analyzed windows started on 1 January and were advanced by 256 s, scanning the entire year 2012. The spectral density matrix S3C is estimated by averaging over 11 consecutive windows, with one estimate made every 6 windows. The data are thus characterized every 25 min, using 51 min worth of data each time. A radial wave number grid is used with wave number intervals of 8.1310−5 km−1 and angular intervals of 5°. The search includes one SH polarization state and 11 Rayleigh polarization states with varying ellipticity and orientation of motion (prograde/retrograde). At each window the back azimuth, phase velocity, and polarization of sufficiently strong maxima in R are stored (the number of retained detections varies by signal quality).
3 Ambient Surface Wave Summary
For the analyzed year 2012 the array analysis yielded between about 11·103 and 33·103 detections per Fourier bin and polarization type. To investigate anisotropy, we study the distribution of phase velocity as a function of azimuth for each frequency bin. Figure 2 shows histograms of phase velocity against back azimuth of detected Rayleigh wave trains for 21.5, 43.0, 60.5, and 95.7 mHz. Each graph shows a 2-D histogram with the color indicating the logarithm of the counts made in any bin. The statistical variation in estimated velocities is similar for different back azimuths and a pattern is apparent, in particular for frequency bins 43.0, 60.5, and 95.7 mHz. Graphs as those in Figure 2 were computed for all 46 frequency bins from 11 to 100 mHz but the four bins shown here are representative in terms of illumination and variability. The dashed red line is discussed in the next section.
Figure 3 shows the same histograms for the Love wave detections, which numbered between 5·103 and 8·103 depending on frequency. Below 40 mHz Rayleigh waves dominated over the Love waves making their detection less reliable, which is why these frequencies are not shown.
4 Estimation of Anisotropy Parameters
For a laterally homogeneous half-space with anisotropic seismic properties, surface wave phase velocities would at first order be subject to azimuthal variation as follows [Smith and Dahlen, 1973]:
where v is surface wave phase velocity (km·s−1), θ is the azimuth of propagation clockwise from North, and ai are five parameters that depend on the subsurface. We fit the anisotropy model equation (4) using the estimates of phase velocity and back azimuth at every frequency bin and surface wave type (dashed lines in Figures 2 and 3). As shown later, the distribution of phase velocities at any given back azimuth is generally more heavy tailed than for a normal distribution. The heavy tails can bias conventional least squares fitting procedures, and we therefore use the robust ℓ1 norm minimization approach that minimizes the sum of absolute deviations [Bloomfield and Steiger, 1983]. A visual inspection of the residuals from the fits (not shown) reveals that they tend to follow a distribution which is more centered and has heavier tails compared to the normal distribution, in particular below 53 mHz and between 66 and 82 mHz.l
Using the model parameters a0,…,a4 we estimate the isotropic phase velocity (a0), the magnitude of 2θ and 4θ anisotropy (and , respectively) and the azimuth of fastest surface wave velocity. The statistical variability of the measured phase velocities will be inherited by these anisotropy measures. This variability should be low since the number of model parameters (five) is much smaller than the number of data points used to constrain them (thousands). We assess the variability using a bootstrap approach [Efron and Tibshirani, 1993]: The N data points are randomly sampled with replacement to produce a resample of the same size N. The anisotropy measures are estimated for this set and the entire procedure is repeated B=100 times, which approximates the sampling distribution of the estimated anisotropy parameters given the variability in the velocity estimates of the year 2012. The value for B is defined heuristically as the smallest value above which the distribution stabilizes.
The procedure is repeated for every frequency bin and surface wave type and the final result is summarized in the spectra in Figures 4a–4d. The vertical lines show the range of bootstrap estimates. No vertical lines are visible for the isotropic phase velocity spectrum (Figure 4a) because the variability is smaller than the line width. The Rayleigh wave mode there is consistent with the study from Alvizuri and Tanimoto  where the same array was used.
The range of bootstrap estimates in Figures 4b and 4d merely provides an idea on the precision of anisotropy estimates if the full model in equation (4) is used but does not test whether simpler models would explain the data better. To test if 2θ or 4θ terms are significantly constrained by the data we use a sequential F test. We compare fits between the full model (2θ+4θ), one with parameters a0,a1, and a2 (2θ model), one with parameters a0,a3, and a4 (4θ model), and an isotropic model with only a0 (0θ model). Each model's misfit to the data is captured by the sum of its squared residuals: , where is the phase velocity estimate of the best fit of that model. Increasing the number of parameters of a model typically reduces residuals of the best fit and a direct comparison of SSR measures is therefore misleading. Assuming that a simple model i (e.g., 2θ) is true and nested in a more complicated model j (e.g., 2θ+4θ) the F statistic offers a means to see this:
where ki and kj are the number of parameters of models i and j. The F statistic measures the improvement in the sum of squared residuals per additional parameter in the bigger model, normalized by an estimate of the variability of the data. It follows a Fisher distribution with degrees of freedom ν1=kj−ki and ν2=N−kj−1 [Miller, 1990]. In theory the F test requires the residuals from the two models to be normally distributed but for large values of N the distributional character of the data points is less important. If under the simpler model the probability of observing a value at or above the observed F statistic is less likely than a certain threshold probability, say p=0.01, then the simple model is rejected in favor of the richer model. If the 2θ model or 4θ model better fits the data than the 0θ model, we, respectively, conclude that 2θ or 4θ anisotropy is significantly constrained. To conclude that both terms are present in the data we additionally require that the 2+4θ model fits better than the 2θ or 4θ model. This procedure is applied to the data in every frequency bin using a threshold level of p = 0.01. The gray markers in Figures 4b and 4d indicate statistically insignificant terms.
As seen in Figure 4b), the variability of the Rayleigh wave anisotropy estimates above ∼35 mHz is almost negligible, as indicated by the small vertical bars at the measurement points (<0.1%). Second, both 2θ and 4θ anisotropy are statistically significant over much of the frequency band above 16 mHz. Only the frequency bands 60–70 mHz and 95–98 mHz show insignificant 4θ anisotropy. Note that anisotropy as small as 0.1% was significantly detected in the data. Furthermore, the fast direction for Rayleigh waves (Figure 4c) is mostly contained between 275° and 300°, with a more westward orientation (278–285°) around 40–70 mHz.
Figure 4d shows the results for Love wave anisotropy. Most frequencies have no significant 2θ anisotropy and also the 4θ term is often insignificant. The reliability of the Love wave results are discussed below.
As summarized in Figure 4b, 1 year of ambient seismic noise provides evidence of significant Rayleigh wave 2θ and 4θ anisotropy over most of the frequency band 15–100 mHz. Anisotropy magnitudes as small as 0.1% are significantly constrained. This extends a previous study of Alvizuri and Tanimoto  based on teleseismic arrivals over a 9 year period where Rayleigh wave 2θ anisotropy from 30 to 60 mHz was significantly constrained but 4θ anisotropy could not be detected between 9 and 60 mHz. Compared to hand-picked teleseismic surface wave arrivals, ambient noise is expected to have a lower signal-to-noise ratio and therefore provide less reliable estimates. We attribute the increased precision to the larger amount of data points (up to tens of thousands as opposed to hundreds) and broad illumination provided by the use of ambient seismic data. Consequently, for Love waves where the number of detections is only in the thousands and illumination is poorer, our anisotropy estimates are less conclusive.
The fact that 4θ anisotropy is found for Rayleigh waves is consistent with global tomography for Southern California in the analyzed frequency range [Montagner and Tanimoto, 1990; Trampert and Woodhouse, 2003; Deschamps et al., 2008]. The fast axis revealed in this study (278-284° in the range 40-60 mHz) is close to the direction given by Alvizuri and Tanimoto  for a similar frequency band, albeit trending slightly more westward. A shear-wave splitting study in the region by Polet and Kanamori  found similar fast directions but we note that the connection between shear-wave splitting and surface wave anisotropy is not well understood [Kosarian et al., 2011]. For Rayleigh waves the results confirm that 2θ anisotropy has higher magnitude than the 4θ type, as argued by Montagner and Nataf .
The bootstrap procedure only captures variability of parameters assuming a fixed 2θ+4θ model. The approach does not account for deviations of the true velocity-vs-azimuth model from that of equation (4) or systematic bias in the phase velocity and back azimuth estimations. Both errors may occur due to lateral heterogeneities or multipathing [Rost and Thomas, 2002] and are hard to quantify. In particular, Alvizuri and Tanimoto  considered multipathing possible at the SCSN above 60 mHz. For the same reasons, also the F test must be interpreted as being approximate and our choice of a 99% confidence level should not be confused as confidence in the test's strict admissibility.
We find significant 2θ and 4θ Rayleigh wave anisotropy in Southern California for most frequencies between 15 and 100 mHz. Our analysis uses three-component frequency-wave number analysis of ambient seismic noise from the year 2012. The broad illumination and large data volume make the approach both precise and sensitive: above 35 mHz the anisotropy estimation variability as well as the magnitude of the weakest significant detections is about 0.1%. Results for Love waves are less conclusive, probably because of the substantially lower illumination seen by the array processing.
Our observations confirm regional and global studies that found 4θ Rayleigh wave anisotropy in Southern California in this frequency range. We conclude that given sufficient ambient illumination, array processing of ambient noise on dense sensor networks can be a powerful tool to constrain surface wave anisotropy.
The authors are grateful to Peter Gerstoft and Ravi Menon for helpful comments. The seismic data used in this study were made available by the Southern California Earthquake Center (SCEC), which is funded by NSF Cooperative agreement EAR-0529922 and USGS Cooperative agreement 07HQAG0008.
The Editor thanks an anonymous reviewer for his assistance in evaluating this paper.