Lawrence et al., 2012: DRAFT: JGR: Ambient Noise Numerical ...

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