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Lawrence et al., 2012: DRAFT: JGR: Ambient Noise Numerical Evaluation generated at t = 0, and located on s. The cross-spectrum (or cross-correlation in frequency) of those two records becomes: , 4 where θ is the azimuth from receiver y to the source, ψ is the angle from receiver x to the source. Posing rxy = |x - y| = |x| gives rsx = rxy cos(ψ−θ) and yields . We recall that to mimic the azimuthal average over the sources, the NCF technique calls for averaging the correlations (or cross spectra) over many time windows. This equivalence leads to: where 〈〉 indicates the ensemble average over time. , 5 Without attenuation (α = 0), assuming equipartition of the source field (i.e., F(s,ω) = and rsx = , constant for all sources), and the far field approximation (rxy
Lawrence et al., 2012: DRAFT: JGR: Ambient Noise Numerical Evaluation Bessel function). This remarkable result led to directly image crustal structure from phase velocity measurements [Ekström et al., 2009]. The key message of this work is that most analytical studies incorrectly simplify the correlation expression by assuming that the sum over the sources of the correlations is equal to the correlation of the sum. With attenuation, the operations of integration and multiplication are not permutable. For realistic noise sources and seismic arrays, the geometrical spreading and attenuation terms affect the amplitudes of the recorded ambient seismic field, , 8 with the surface integral representing the summation of a continuum of noise sources that occurred at origin time ts and located on ds. In order to measure appropriate amplitudes for attenuation analysis, Prieto et al. [2009] computed the time-averaged coherence, , 9 where | . | indicates the power spectral density in the L2 sense, { . } refers to a whitening process using running average (to ensure deconvolution stability), and 〈 . 〉 is the time averaged ensemble. Even though 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, Prieto et al. [2009] use raw signal and do not pre-process the data with the typical approaches of one-bit or running normalizations [e.g., Bensen et al., 2007], which provides reliable Green’s function amplitude variations for short amount of data. Unfortunately, deriving equation 10 from equations 8 and 9 is not trivial (e.g., equations 12, 16, and 18 from Tsai [2011] or equation 9 from Colin de Verdière [2006]). 10 7
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