What Does a Waveform Obtained by Correlation of a Diffuse Anisotropic Wavefield Contain?

Paul Cupillard and Yann Capdeville (IPG Paris)


Recent developments have shown that the Green's function between two distant seismometers can emerge from the cross-correlation of several days of seismic noise recorded at the seismometers (Shapiro and Campillo, 2004). This provides new data that are greatly interesting for seismologists because they enable us to get information about the Earth structure in aseismic regions. Group-speeds on inter-station paths are now widely measured, and numerous high-resolution tomographic images appeared in the last three years.

Many theoretical developments tried to explain the phenomenon. All the theories only take into account the case of uniformly distributed noise sources. Now, an anisotropic flux as well as the absence of equipartition has to be considered to fully understand the limitations of the method. Indeed, noise consistently observed in seismic records mainly comes from the oceans (Longuet-Higgins, 1950), so that its distribution at the surface of the Earth clearly is nonhomogeneous. Here we study the effect of such a distribution by computing correlations of numerically generated seismic noise in an attenuating sphere.

Uniform Noise Sources Distribution

First, we compute synthetic noise to mimick the continuous oscillations that are consistently observed in seismic records. To create that noise, we randomly position three hundred sources on the Earth's surface. For each spatial component of each source, we generate a 24-hour time signal with random phase and flat spectrum filtered between 2 and 13 mHz. Using normal-modes summation in the Preliminary Reference Earth Model (Dziewonski and Anderson, 1981), the effect of all the sources is computed at three stations A, B and C located on the equator at longitude $\mbox{0}^{\circ}$, $\mbox{20}^{\circ}$ and $\mbox{70}^{\circ}$ respectively (Figure 2.41). Correlations between vertical components of displacement received at the stations are then calculated. That is the result of what we call a "realization". We perform 12,640 realizations (the total number of sources is then 3,792,000) and we stack all of them. Three different cases are studied, corresponding to different processes applied to the noise records : we distinguish raw noise (nothing is done), 1-bit noise (meaning that we completly disregard the amplitude) and whitened noise (meaning that the spectral amplitudes of each record are set to 1 in the chosen frequency band).

Figure: Location of receivers A, B and C (top). Station B (resp. C) is $\mbox{20}^{\circ}$ (resp. $ \mbox{70}^{\circ} $) away from A. Tiny dot pixels indicate location of 24,000 noise sources coming from 80 realizations of our numerical experiment.
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With the normal-modes summation method, the Green's tensors between A and B and between A and C can be easily calculated. Figure 2.42 shows the comparison between Green's functions and the derivatives of our synthetic correlations with and without power spectral density correction. We note that the waveform and relative amplitudes between the signal from A and B and the signal from A and C are conserved for each kind of noise processing. This means that information about geometrical as well as intrinsic attenuation is contained in correlations whatever the technique we use to process the noise recordings. Travel times have been preferentially considered so far on seismic noise correlations, but the use of amplitude is now more and more questioned, and the result we present here is essential in this perspective. Nevertheless, it is in disagreement with the experimental result from Larose et al (2007) who recover the geometrical spreading with raw data but lose it with 1-bit or whitened noise. We don't have an explanation for this difference yet.

Figure 2.42: a) Comparison, for each kind of process applied to the noise records (raw, 1-bit normalization and whitening), between the A-B Green's function (gray lines) and the derivatives of our synthetic cross-correlations with (dotted lines) and without (dashed lines) power spectral density (PSD) correction. b) Same as a) for stations A and C. The PSD correction improves the waveform fits.
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Non-Uniform Noise Sources Distribution

Here we perform the same experiment, except that noise sources are now confined in a $\mbox{50}^{\circ}$-radius disk at the surface of the sphere. We study cross-correlations from different azimuths by using records from 24 stations located in two circles around a central station A (Figure 2.43).

Figure: Location of the central station A (star) and the other receivers (numbered from 1 to 24). Stations 1 to 12 (resp. 13 to 24) are $\mbox{20}^{\circ}$ (resp. $\mbox{70}^{\circ}$) far from A and spaced by an angle of $\mbox{30}^{\circ}$, defining twelve azimuths. Great circles that link the stations to the centre A are plotted to highlight the different azimuths. In addition, tiny dots indicate location of 24,000 noise sources coming from 80 realizations of our numerical experiment.
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As in the previous section, we compare Green's functions and the derivatives of our correlations. Two azimuths are studied : $\mbox{90}^{\circ}$ (i.e. stations 12 and 24, see Figure 2.44) and $\mbox{30}^{\circ}$ (i.e. stations 2 and 14, see Figure 2.45). For the first azimuth, results are very similar to those observed in the uniformly distributed noise sources case: the power spectral density correction improves the waveforms' fit, overtones are not well excited, and attenuation is retrieved for the three different noise-processing procedures. Results from azimuth $\mbox{30}^{\circ}$ are very different. An important phase shift between the Green's functions and the correlations is observed, both for station 2 and station 14. This is because the emergence of the signal is only due to sources far from the vicinity of the great circle, and contribution of such sources provides incorrect travel times.

Figure 2.44: a) Comparison between the derivative of the correlation of station 12 by A and the corresponding Green's function. Gray lines correspond to the Green's function, whereas dotted and dashed lines are correlations respectively with and without power spectral density correction. b) Same as a) for station 24.
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Figure 2.45: Same as Figure 2.44 for stations 2 and 14.
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Dziewonski, A.M. and D.L. Anderson, Preliminary Reference Earth model, Phys. Earth Planet. Inter., 25, 297-356, 1981.

Longuet-Higgins, M.S., A theory on the origin of microseisms, Philo Trans. R. Soc. Lond. A., 243, 1-35, 1950.

Shapiro, N.M. and M. Campillo, Emergence of broaband Rayleigh waves from correlation of the ambient seismic noise, Geophys. Res. Lett., 31, L07614, 2004.

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