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Lawrence et al., 2012: DRAFT: JGR: Ambient Noise Numerical Evaluation Most attempts that simplify this proof rely on significant assumptions and collapse to variations of equation 3 (e.g., equation 23 in Tsai [2011]). Under these assumptions, the NCF reveals a strong dependence on source and receiver distributions [Tsai, 2011]. Using the ambient seismic field displacement expressed in equation 8, the cross- spectrum for a receiver pair located in (x,y) becomes . 11 Equation 11 obviously differs from equation 5 or 6. The coherency presented in equation 9 then becomes Solving equation 12 analytically is laborious and we turn to numerical approaches to compute the coherencies. Numerical solution: For numerical calculations, we discretize equation 8 with a finite number of sources per time window Ns, . 13 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 (distributions and spectra), receiver distributions to the estimates the coherency. Another widely discussed factor entering analyses on Green’s function retrieval is the role of incoherent noise sources such as site effects, electrical noise, wind, thermal effects, cultural noise, micro-seismicity, and anything that can create vibrations well recorded at 12 8
Lawrence et al., 2012: DRAFT: JGR: Ambient Noise Numerical Evaluation one sensor compared to others. We include to each record incoherent sources, which could represent sources such as site effects, electrical noise, wind, thermal effects, cultural noise, micro-seismicity, and anything that can create vibrations well recorded at one sensor compared to others. To do so, we add to each simulated record of ambient seismic field a degree of local, white, and therefore incoherent noise that we keep with constant magnitude. We vary the level of source amplitudes to span wide range of coherent-signal-to-incoherent-noise ratios. To validate the technique of Prieto et al. [2009], we need to recover from dispersion measurements the phase velocities and attenuation coefficients used to generate the Re[γxy(ω)] values. We adopt similar approach and average all Re[γxy(ω)] for each station separation, sampled at a 2 km interval, regardless of the array aperture. We then conduct a 3-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 seconds). We explore through various noise source scenarios when equation 10 is valid. We find that it is not satisfied in the cases where: • There are not enough sources (azimuthal averaging is not possible). • The sources are at the center of the receiver array (insufficient azimuthal averaging and near field condition). • The incoherent noise is too high. • There are not enough receivers (poor azimuthal averaging and distance sampling). Consequently, equation 10 is validated when a) at least one source per time window (which is reasonable given the nature of the ambient seismic field), b) the sources are widespread, c) stacking over distances is sufficient (in our case, more than 2 station-pair per distance considered and d) the coherent signal from far-field sources is more energetic than the incoherent signal. Synthetic Sources: We seek to reflect the reality of the ambient seismic field to account for different 9
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