A numeric evaluation of attenuation from ambient noise correlation functions

1 Introduction

[2] In order to better understand the Earth structure and to predict ground shaking, seismologists have to face the foremost challenge of explaining seismic-wave amplitudes from scattering and anelasticity. Affecting in particular surface waves, attenuation is traditionally measured from earthquake records. Earthquakes occur at specific times and location around the Earth, which limits the path coverage of surface waves and, thus, the areas of potential studies. To overcome this shortage of data, recent developments in seismology have led to the use of the ambient seismic field to extract elastic and anelastic propagation effects.

[3] The cross correlation of the ambient seismic field recorded at two stations, which we refer as the noise correlation function (NCF), shows remarkably similar properties to the Green's function between the two receivers [e.g., Weaver and Lobkis, 2001, 2004; Derode et al., 2003a, 2003b; Shapiro and Campillo, 2004; Wapenaar, 2004; Wapenaar et al., 2006; Sanchez-Sesma and Campillo, 2006]. This powerful novel technique provides new opportunities to extensively explore the Earth [e.g., Shapiro et al., 2005; Sabra et al., 2005a, 2005b; Yao et al., 2006; Zheng et al., 2008; Prieto et al., 2009]. Among a wide range of applications, researchers have been able to produce higher-resolution images of the crustal and upper mantle structure than with traditional approaches [e.g., Shapiro et al., 2005; Sabra et al., 2005b; Yao et al., 2006; Lin et al., 2008; Gudmundsson et al., 2007; Yao and van der Hilst, 2009; Stehly et al., 2009].

[4] Ambient seismic noise is a ubiquitous signal that is recorded worldwide at seismic stations. At periods between 5 and 30 s, it is excited by the interaction between the ocean and the crust [e.g., Longuet-Higgins, 1950; Hasselmann, 1963; Tanimoto et al., 1998; Rhie and Romanowicz, 2004, 2006; Shapiro et al., 2006; Kedar et al., 2008; Tanimoto, 2007; Zheng et al., 2008; Nishida et al., 2009; Roux et al., 2010; Ardhuin et al., 2011]. At short periods, Longuet-Higgins [1950] suggested that the interaction of the oceanic waves with the shallow seafloor forms the first microseism peak (10–20 s) and that the interference of those waves with their coastal reflection forms the second microseism peak (5–10 s). More recently, Kedar et al. [2008] and Landés et al. [2010] have found strong spatial and temporal correlations between energetic microseism and powerful oceanic storms.

[5] The quality of the NCFs seems to depend on the distance separating the receivers [e.g., Young et al., 2011; Bensen et al., 2008], with faster convergence toward the Green's function for short distances [Sabra et al., 2005a, 2005b; Larose et al., 2005]. This is true when using long-duration 1 bit normalization or running-normalization techniques [e.g., Bensen et al., 2008], as compared to the largely increased convergence rate when using short time windows [Seats et al., 2012]. Scattering due to complex 3-D structure and intrinsic attenuation, however, strongly affects the bandwidth of the coherent content. The observed NCF amplitude variations are extremely similar to the relative variation in Green's function amplitudes [Weaver and Lobkis, 2001; Larose et al., 2005; Prieto and Beroza, 2008; Cupillard and Capdeville, 2010; Denolle et al., 2012; Y. Colin de Verdière, Mathematical models for passive imaging. I: General background, 2006, arXiv:math-ph/0610043v1, hereinafter referred to as Colin de Verdière, 2006].

[6] The purely elastic surface-wave amplitude decay with distance, or geometrical spreading, is expressed in cylindrical coordinates with a Bessel function of the first kind. The spatial autocorrelation (SPAC) method introduced by Aki [1957] predicts the variation of the amplitudes with frequency and distance found with the cross-correlation techniques [e.g., Sanchez-Sesma and Campillo, 2006; Ekström et al., 2009; Prieto et al., 2011]. The relation between the spectrum of the vertical-to-vertical NCF, or the spectrum of the vertical-to-vertical elastic Green's function, and the Bessel function is

(1)

where r is the receiver-receiver separation (later referred to as r_xy), and c(ω) is the phase velocity for a laterally homogeneous medium [e.g., Sanchez-Sesma and Campillo, 2006].

[7] Whether or not the attenuation can be extracted from the ambient noise SPAC technique is still controversial. Prieto et al. [2009] observed stronger amplitude decay with increased station separation than predicted by pure elasticity in homogenous media. From a 1-D approximated medium [Prieto et al., 2009] or a laterally varying medium [Lawrence and Prieto, 2011], the additional decay is modeled as a decreasing exponential [equivalent to attenuation (α) measurements with traditional earthquake studies] as follows:

(2)

[8] This method provides reasonable geological and geophysical results from the scales of reservoirs [Weemstra et al., 2011a, 2013] to continents [Lawrence and Prieto, 2011]. Lin et al. [2011] confirmed that the NCF amplitude decays with receiver separation, in agreement with numerical evaluations of Cupillard and Capdeville [2010]. Further support has been presented by Zhang and Yang [2013] using the C³ (C-cube) methodology [Stehly et al., 2008]. Nakahara [2012] showed that the expression above [equation 2] is approximately correct for small attenuation, providing a theoretical basis to estimate attenuation from the coherency of ambient noise records.

[9] With concerns about the effects of ambient noise source distribution on NCF amplitudes, the accuracy of the NCF anelastic measurements is actively discussed. Weaver [2011], Tsai [2011], and Buckingham [2012] suggested that NCF amplitudes depend on noise source distribution and that azimuthal averaging of coherency estimates may not be enough for retrieving reliable attenuation coefficients. Numerical simulations [Cupillard and Capdeville, 2010] and theoretical derivations [Tsai, 2011; Nakahara, 2012] corroborated that for uniformly distributed noise sources, NCF amplitude decays depend on the attenuation of the medium.

[10] Much of the debate regarding the recovery of attenuation from NCFs may largely depend on the different assumptions made regarding the distribution of the ambient source field [Weaver, 2011]. We stress that the Earth's ambient source field is not uniformly distributed; rather, its origin varies spatially and temporally, depending on seasonal effects and proximity to the coasts [e.g., Schulte-Pelkum et al., 2004; Stehly et al., 2006; Chevrot et al., 2007; Aster et al., 2008; Kedar et al., 2008; Ardhuin et al., 2011]. In practice, normalizing the ambient seismic field records and averaging of normalized correlations over time are thought to alleviate the strong directionality of the ambient seismic field. The NCFs can be considered as a summation of displacements measured after propagation (with geometrical spreading and attenuation) from many sources.

[11] In this work, we present a numerical approach to express our understanding of estimated attenuation coefficients from NCFs and the effect of appropriate signal processing to reduce biases introduced by nonuniform source distributions. We first build the synthetic ambient seismic field for a dispersive medium. We then compute synthetic coherencies (or spectra of the NCFs) at all station pairs for comparison with a theoretical model of the Green's function (or the Bessel function). We evaluate the impact of the noise source and receiver distributions on attenuation measurements by numerically demonstrating that synthetic NCF amplitudes are consistent with the expected results from Prieto et al. [2009]. We show that for a wide range of noise source distributions, we are able to recover accurate attenuation coefficients. We further show that the attenuation measured is sensitive to the local structure beneath the array, instead of the structure between the sources and the receiver array.

2 Analytical Framework

[12] Realistic ambient seismic field recorded at a given receiver can be considered as the sum of all wavefields propagating from all sources to the receiver. We assume a laterally homogenous medium with vertical-only variations and, hence, dispersive properties. In the far-field approximation, the response of the medium to a single point-force source located at s and of spectrum F becomes the displacement spectrum u, i.e.,

(3)

where ω is the frequency, k is the wave number, t₀ is the source time [after Sanchez-Sesma and Campillo, 2006], α(ω) is the frequency-dependent attenuation coefficient, and r_sx is the source-receiver distance. In cylindrical coordinates, this is a far-field approximation to the elastic surface-wave displacement eigenfunction with attenuation incorporated as an exponential decay. As in Prieto et al. [2009], we consider only the vertical component of the Rayleigh waves.

[13] We follow the conventions in Sanchez-Sesma and Campillo [2006] with two receivers, located at x and at the origin y, that record the response of the medium to a point source s of origin time t = 0. The cross spectrum (or cross correlation in the frequency domain) of those two records becomes

(4)

where θ and ϕ are the respective azimuths between receivers in y and in x to the source. Posing r_xy = |x|−|y| = |x| gives r_sx = r_sy + r_xycos(ψ − θ) and yields . In practice, we compute the cross correlation of time windows with finite duration, which limits the number of source per time window. We overcome this by summing correlation functions over large duration of time. We indicate this using the notation 〈 〉 and write the cross spectrum that results from the contribution of sources uniformly distributed on a circle centered in y, i.e.,

(5)

[14] Without attenuation (α = 0), assuming a uniform source field with a white source spectrum [i.e., F(s,ω) = =1 and r_sx = , constant for all sources], and within the far-field approximation (r_xy < < r_sx ~ r_sy), equation 5 becomes

(6)

[15] The assumption of r_sx ≈ r_sy likely introduces some degree of inaccuracy (see Figure 1) but is needed to simplify equation 6 into the analytical solution of Sanchez-Sesma and Campillo [2006] that converges to the Green's function as follows:

(7)

where J₀ is the Bessel function of the first kind, order zero; and the wave number k(ω) = ω/c(ω) with c(ω) as the phase velocity of the medium. Sanchez-Sesma and Campillo [2006] expressed the linear relation between the real part of the cross spectrum 〈C_xy(ω)〉 and the imaginary part of the Green's function between the two receivers (the Bessel function). This remarkable result led to direct imaging of the North American crustal structure with 2-D images of phase velocity [Ekström et al., 2009].

[16] In most analytical expressions of the noise correlation function, noise sources are assumed to be uncorrelated in time and space, and the contribution of each source to the correlation function can be determined independently. In this study, because of attenuation, we do not make this assumption and calculate the correlation, including the contribution of all sources in each time window. For realistic noise sources and seismic arrays, the geometrical spreading and attenuation terms affect the amplitudes of the recorded ambient seismic field, i.e.,

(8)

with the surface integral representing the summation of a continuum of noise sources that occur at origin times t_s and locations on ds.

[17] In order to measure the appropriate amplitudes for attenuation analysis, Prieto et al. [2009] computed the time-averaged coherence, i.e.,

(9)

where | | indicates the power spectral density in the L2 sense, and { } refers to a whitening process using running average (to ensure deconvolution stability). 〈 〉 indicates the time-averaged ensemble, which is accomplished by dividing a pair of long (>90 days) time series into many short (1800–7200 s) time series and averaging the coherence for all such time series [see Seats et al. [2012] for more discussion of equation 9]. Even though the ambient noise source distribution is not uniform on Earth, Prieto et al. [2009] empirically showed that the real part of equation 9 matches a Bessel function [equation 7] modified by an attenuation term [equation 2 repeated here], i.e.,

(10)

[18] Prieto et al. [2009] used a raw signal and did not preprocess the data with the typical approaches of 1 bit or running normalizations [e.g., Bensen et al., 2007]. The raw coherency provides reliable Green's function amplitude variations for short duration of stacked correlations.

[19] Unfortunately, deriving equation 10 from equations 8 and 9 is not trivial [e.g., Tsai, 2011, equations 12, (16), and (18); Colin de Verdière, 2009, equation 9]. Most attempts that simplify this proof rely on significant assumptions (e.g., independent sources) and collapse to variations of equation 3 [e.g., Tsai, 2011, equation (23)]. Under these assumptions, the NCF reveals strong dependence on source and receiver distributions [Tsai, 2011].

[20] Using the ambient seismic field displacement expressed in equation 8, the cross spectrum for a receiver pair located at (x,y) becomes

(11)

[21] Equation 11 differs from equation 5 or 6 in that we assume that multiple sources are contributing to the records at both stations (x and y). The coherency [equation 9] then becomes

(12)

[22] Solving equation 12 analytically is laborious, and we turn to numerical approaches to compute the coherencies.

3 Numerical Solution

[23] For numerical calculations, we discretize equation 8 with a finite number of sources per time window N_s as follows:

(13)

[24] The medium is laterally homogeneous, and we assume Rayleigh wave phase velocity C(ω) and attenuation α(ω) dispersions from Prieto et al. [2009] for Southern California. We explore the impact of different source scenarios (varying distributions and spectra) on the stacks of coherency NCFs of numerically generated data calculated with equation 13.

[25] Another widely discussed factor entering analyses on Green's function retrieval is the role of incoherent (or noncorrelated) noise sources such as site effects, electrical noise, wind, thermal effects, cultural noise, microseismicity, and anything that can create vibrations well recorded at one sensor compared to others. We add incoherent signal to each numerically generated record, which could represent site effects, electrical noise, wind, thermal effects, cultural noise, microseismicity, and anything that can create vibrations well recorded at one sensor compared to others. The incoherent noise is added as a white uniformly random signal with constant magnitude at all times. We vary the level of source amplitudes to span a wide range of coherent-signal-to-incoherent-noise ratios.

[26] To validate the technique of Prieto et al. [2009], we need to recover from Re[γ_xy(ω)] the phase velocities and attenuation coefficients used to generate the numerical records. We adopt a similar approach to Prieto et al. [2009] and average all Re[γ_xy(ω)] for each station separation, sampled at a 2 km interval, regardless of the array aperture. We then conduct a three-stage grid search for the appropriate value of phase velocity and attenuation coefficient by minimizing the L1 norm residuals derived from equation 10 at a given frequency. Our analysis focuses on the period band of the microseism ambient field (7–24 s) used in prior studies [Prieto et al., 2009; Lawrence and Prieto, 2011].

[27] We explore through various noise source scenarios when equation 10 is valid. We identified the cases where this condition failed and stress the need to incorporate sufficient sources (found to be above 16 sources per time window) to overcome local incoherent noise; the sources need to be in the far field, and the number of receivers has to be appropriate to average spatially the coherency and form the Bessel function.

3.3 Generalized Parameterization

[31] We explore the parameter domain within which equation 10 is valid to characterize the appropriate ambient seismic field. The three dimension scales, namely, the source-length scale L_s, the receiver-length scale L_x, and the source-receiver separation Δ_sx reflect the radiation aperture with the ratios L_s/Δ_sx and L_s/L_x (i.e., Figure 1). Another factor to consider is the number of sources, N_s, relative to the magnitude distribution, M_s. The larger the range of source amplitudes used to generate the synthetic spectra, the fewer the sources dominate any given seismogram. Finally, due to geometrical spreading, the range of source-to-receiver distances also plays an important role in the number of dominant sources: [max(r_sx) − min(r_sx)].

[32] For a given scenario, we generate synthetic ambient noise seismograms which we divide into 1800 s time windows, as found to be optimal for fastest convergence toward a stable NCF [Seats et al., 2012]. We then compute and stack the coherencies (in the frequency domain) for each station pair and stack Re[γ_xy(ω)] over all station pairs to estimate the coherency at various distances. For each frequency, we estimate phase velocity and attenuation coefficients, and we finally evaluate the match between the synthetic, or “observed,” coherency and the predicted values from equation 2. Table 1 summarizes the parameters simulated by the various scenarios (see also Figure 2).

Table 1. Description of Synthetic Scenarios

Source array geometrya	Ns/1800 s	Days	Overlap	Lx (km)	Ls (km)	Δsx (km)	Ms
A (151 runs)	25	30	50%	100	101–104	101–104	2
B (151 runs)	25	30	50%	100	101–104	101–104	2
C (151 runs)	25	30	50%	100	101–104	101–104	2
D (151 runs)	25	30	50%	100	101–104	101–104	2
B (24 runs)	20−8	30	50%	100	103	5 × 102	1,2,4,8
B (24 runs)	20−4	365	50%	100	103	5 × 102	1,2,4,8

• ^a The labels correspond to Figures 2(a)–2(d). A is a circle source. B is a linear source. C is a uniformly random filled circle. D is a uniformly random filled circular shell.

[33] The synthetic noise seismograms are a good visual control of similarity between the generated noise and the true ambient seismic field. Figure 3 shows examples of the generated ambient seismic field with different noise source distributions and the resulting variability in the signals. Figure 3 presents example synthetic noise seismograms for the four source distributions illustrated in Figure 2, namely, circular, linear, uniform, and uniform shell. We imposed 128 sources per time window (N_s = 128) because it provides a reasonable balance between synthetic seismograms with identifiable phases above the noise and those without. With 128 sources per time window, Δ_sx = 500 km, L_s = 1000 km, and L_x = 100 km, only % of the time windows contain events that have a short-term average (STA) over 4 s to long-term average (LTA) over 60 s ratio (STA/LTA) greater than 2. We find that the number of detectable phases increases () as N_s decreases (1 < N_s < 64) but only marginally decreases (%) as N_s increases (N_s > 256).

[34] Figure 4 shows encouraging results for the time domain NCFs for selected receiver pair for the uniformly random source distribution. The move out is consistent with the input phase velocity, and the amplitudes decrease with increasing station separation as expected due to geometrical spreading and attenuation. In Figure 5, we compare the real portion of the synthetic coherency [equation 12] with the theoretical elastic [equation 7] and the attenuated Bessel function [equation 10]. If the coherencies qualitatively fit the expected Bessel function, their amplitudes better match the attenuated Bessel function. We proceed with a three-stage grid search for phase velocity and attenuation following Prieto et al. [2009] and Lawrence and Prieto [2011].

[35] Figure 6 qualitatively illustrates the accuracy of the recovered dispersion and attenuation curves for all of the source types. Figure 6 confirms that retrieving correct and accurate phase velocities is possible for a wide range of source distributions. The estimated attenuation coefficients exhibit less accuracy than the phase velocities, but nonetheless reliable measurements. Most importantly, our attenuation estimates are consistent for all source distributions tested.

4 Discussion

[36] Our results suggest that it is possible to retrieve reliable attenuation coefficient estimates from the ambient seismic field using the coherency technique. We tested a wide range of source distribution length scales, varying the source-receiver distance Δ_sx and source lengths L_s, with respect to a fixed receiver distribution. We estimate the best fitting estimates of phase velocity and attenuation measurements by minimizing the misfit in the L2 norm sense of the objective function Φ(c,α) = ||Re(γ(ω)) − J₀(ωr/c(ω))e^−α(ω)r||₂.

[37] The L2 norm misfit between the initial and estimated phase velocities and attenuation coefficients provides a quantitative measure of the method's accuracy/precision, which we illustrate in Figure 7. Despite those encouraging results, we note particular scenarios for which the coherency method will fail to provide accurate attenuation measurement.

[38] The first and obvious scenario that fails to provide the correct answer occurs when the sources are confined in the near field (Δ_sx < L_x and L_s < L_x) and the sources are seen from a narrow angle. This result further supports the analytical studies. The second case for which the coherency is poorly reconstructed highlights the destructive role of incoherent phases. With a fixed level of incoherent or local noise, the signal-to-noise ratio of coherent signal decreases as the distance between the receivers and the noise sources increases (very far field). We confirmed this results by simply increasing the magnitude of the very far field sources (as tested with factors from 10 to 1000) or by further stacking over time (>90 days), thus retrieving correct attenuation coefficients when amplitudes are high.

[39] The source-receiver geometries under which we obtained accurate velocities from the NCF techniques span a large range of length scales. Receiver array scales vary from dozens of meters [e.g., de Ridder and Dellinger, 2011; Young et al., 2011] to thousands of kilometers [e.g., Lin et al., 2008; Bensen et al., 2008, 2009]. This suggests that our configurations, where L_s/L_x and Δ_sx/L_r ratios are large, may be more representative of the Earth's ambient noise source field. Although none of the idealized distributions are likely to be best representing the Earth's ambient field, we obtain the most accurate estimations when the sources are randomly distributed over a broad range of distances and azimuths.

[40] One very critical part of testing this technique is the effect of the number of noise sources compared with the level of incoherent noise surrounding the receiver array. We vary the number of noise sources per time window from 1 to 2048 and the maximum amplitude of the incoherent noise from 10⁻⁶ to 10¹ (Figure 8). We obtain more accurate estimates for simulations with more than ~16 noise sources per window (e.g., N_s > 16) and inaccurate estimates for simulations with fewer noise sources per window (N_s < 16). When the incoherent noise level is below ~1, the accuracy is sufficient to recover accurate attenuation. Within these two confines, there is little trade-off between these parameters for a wide range of dimensional values.

[41] Ultimately, a combination of the proposed source configuration is more plausible than any individual one. We therefore created a case that evenly draws from the four source scenarios shown in Figure 2. This general source distribution provides results similar to those of the uniformly random (shell) distribution (Figure 9), where the phase velocity and attenuation results are reliable for a broad range of parameters. This strongly suggests that sources with a sufficient distribution, in both aperture and range, will be sufficient to construct Green's functions with reliable amplitude information.

[42] One major result from these simulations reveals that our synthetic coherencies calculated with two or fewer sources per time window better match the unattenuated Bessel function. This is particularly true for nearly symmetric far-field distributions (i.e., circular and uniform sphere). Since we tested those cases for long durations of stacking (>4 years), we cannot explain this result by simply stating the lack of sources. It is worth noting that two sources per time window delimit a sharp boundary between the best fit to the Bessel function with or without attenuation. We recall the simple solution of equation 7 for a single source (N_s = 1). Theoretical analysis where N_s = 1 suggests that the stacked coherency is independent of attenuation. Our results are then in good agreement with both the analytic work of Weaver [2011] and Tsai [2011] and the empirical work of Prieto et al. [2009] and Lawrence and Prieto [2011].

[43] Including attenuation inherently alters the recorded ambient field. Due to attenuation, small local sources contribute to high frequency more than larger far-field sources. The synthetic seismogram is a sum of sources triggered at various locations, and each source is recorded with a different amplitude due to attenuation. Normalization by a smoothed amplitude spectrum “whitens” the records of all sources, not each source individually. Consequently, the coherency represents a normalization that preserves the relative damping of each source's propagated spectrum contained with the data record. While the above examples show the importance of including multiple sources in equation 12, there is no straightforward analytical solution proving that stacked coherency should equate to a damped Bessel function.

[44] It remains to be shown that the estimated attenuation coefficient is sensitive to the structure near the receivers, not near the sources or the path between sources and receivers. We compare the attenuation coefficients measured from the synthetic NCFs calculated with different attenuation coefficients locally around the sources and the receivers. In the first case, we set attenuation coefficients to zero for a 200 km radius region surrounding the receivers, while maintaining the same attenuation coefficients for the source region as above, and opposite in the second case. We perform this test only for the uniform shell scenario with a small source offset (Δ_sx = 100 km) and an intermediate-to-large source length (L_s = 1000 km) in order to separate the sources and receivers. This simple test confirms that the method measures attenuation near the receivers, and not the attenuation near the sources, as illustrated in Figure 10.

[45] Future analyses with the method presented in Prieto et al. [2009] should benefit from considering additional amplitude constraints. For example, 3-D focusing effects bias amplitudes significantly within a heterogeneous medium [e.g., Lin et al., 2011]. Prieto et al. [2009] and Lawrence and Prieto [2011] attempted to reduce the focusing and defocusing effects by azimuthally averaging over large arrays (and all directions). However, this averaging does not reduce all amplification effects for most 3-D heterogeneous structures. Accounting for amplification with adjoint or finite frequency kernels for amplitude may help reduce such biases. Similarly, repeating the analyses conducted here with full 3-D elastic [e.g., Stehly et al., 2011] wave propagation with intrinsic attenuation decay may illuminate potential biases in attenuation estimates due to structures between sources and receivers.

[46] While the results of this study support the link between the empirical interpretations of NCF energy decay as seismic attenuation, there remains a lack of any analytical proof linking the increased amplitude decay to the functional form . A recent paper by Nakahara [2012] has given theoretical proof that this assumption is nearly correct. Our simulations indicate that the functional form, which is the intuitive solution, holds for a large variety of noise source distributions.

5 Conclusion

[47] This study numerically demonstrates that the Prieto et al. [2009] method to measure seismic attenuation from ambient noise correlation functions is viable. In contrast, these results disagree with analytical studies, which may suggest the following: (1) assuming a “white homogeneous ambient seismic field” is not equivalent to whitening the recorded ambient field; (2) correlating multiple sources is different from independently correlating each source; and (3) incoherent noise between the sensors may contribute to the NCF distance dependant decay. Our numerical results illustrate that attenuation coefficients are recoverable for a wide range of appropriate source conditions: (1) a reasonable azimuthal distribution of far-field sources, (2) multiple sources recorded per time window, (3) sufficient stacking, and (4) a receiver distribution that provides ample station-to-station azimuths and station separations. For a variety of source conditions, including those with nonuniform source distributions, we accurately recover the phase velocity and attenuation from synthetic NCFs. However, when we include few sources per time window (N_s ≤ 2) in the simulations, our measured attenuation coefficients are negligible, which corresponds to the analytical assessment of Weaver [2011] that single-source NCFs are independent of attenuation. With more sources per time window (e.g., N_s = 128), the NCFs are only sensitive to the attenuation in the region of the receiver array. In elasticity, the correspondence between the well-constructed NCF and the elastic Green's function is well accepted [Weaver and Lobkis, 2004]. In anelasticity, we can now account for attenuation and include the exponential decay to the Green's function.

[48] Along with other studies [e.g., Yang and Ritzwoller, 2008; Cupillard and Capdeville, 2010], our synthetic ambient field seismograms yield NCFs that contain appropriate amplitude information (e.g., Figures 3 and 4). We investigated the accuracy to which those amplitudes are retrieved with respect to frequency and the scale lengths for the different source distribution configurations (Figure 7). Given the scenario listed above, our synthetic coherencies match really well the attenuated Bessel functions, and we estimate correctly the attenuation of the medium, which disagrees with the analytical results of Tsai [2011] and Weaver [2011]. The origin of the ambient seismic field sources still needs further investigation due to the complexity of the spatial and temporal distributions of noise sources and their respective contribution to the ambient seismic field [Longuet-Higgins, 1950; Cessaro, 1994; Chevrot et al., 2007; Kedar et al., 2008; Ardhuin et al., 2011]. Furthermore, Seats et al. [2012] demonstrated that coherency stacks from more overlapping short time windows (30 min to 1 h rather than daylong) yield better source distributions than those from longer time windows. This can result from a better averaging process where large-magnitude sources do not dominate over the other sources recorded in successive time windows.