Recovering the Attenuation of Surface Waves from One-Bit Noise Correlations: A Theoretical Approach

Paul Cupillard, Laurent Stehly and Barbara Romanowicz


Cross-correlation of ambient seismic noise recorded by a pair of stations is now commonly recognized to contain the Green's function between the stations. Travel times extracted from such data have been extensively used to get images of the Earth interior. Some studies have also attempted to explore the information contained in the amplitude (Larose et al., 2007; Gouedard et al., 2008; Matzel, 2008; Prieto et al., 2009). In a recent work, Cupillard and Capdeville (2009) carried out numerical experiments showing that the attenuation of surface waves can be recovered from one-bit noise correlations in the case of a uniform distribution of noise sources on the surface of the Earth (Figure 2.44). We here provide a theoretical explanation for such a surprising result.

Figure 2.44: Results from Cupillard and Capdeville (2009). These authors generate synthetic noise on the surface of a 1D Earth model and compute correlations using a stream of twelve stations. Then, they compare the amplitude decay of the correlations (gray line) with the amplitude decay of the Rayleigh wave Green's functions (dotted line). The two curves match well, even when applying one-bit normalization to the noise records.
\epsfig{file=cupillard09_1_1.eps, width=8cm}\end{center}\end{figure}

Amplitude of the raw noise correlation

Consider two receivers A and B that are separated by a distance $\Delta$ in a homogeneous medium in which the wave speed is $v$. Noise sources are distributed in this medium, and each receiver therefore records a time-signal. We denote by $A(t)$ (respectively $B(t)$) the recording from $A$ (respectively $B$).

We assume that these signals can be decomposed in the following manner:

A(t) = A^c(t) + A^i(t)
\end{displaymath} (25.1)

B(t) = B^c(t) + B^i(t)
\end{displaymath} (25.2)

such that the cross-correlation $C_{AB}(t)$ between the signals can be written
C_{AB}(t) = \frac{1}{T} \int_0^T A^c(\tau) B^c(t+\tau) \, d\tau \mbox{,}
\end{displaymath} (25.3)

where $T$ is the length of the signals. We designate $A^c(t)$ and $B^c(t)$ as coherent, whereas $A^i(t)$ and $B^i(t)$ are called incoherent.

We also assume that

B^c(t_0+\tau) \propto A^c(\tau) \mbox{,}
\end{displaymath} (25.4)

where $t_0=\Delta/v$, so we have
C_{AB}(t_0) = \sigma_{A^c} \sigma_{B^c} \mbox{,}
\end{displaymath} (25.5)

where $\sigma_{A^c}$ (respectively $\sigma_{B^c}$) is the standard deviation of $\vert A^c(t)\vert$ (respectively $\vert B^c(t)\vert$).

Amplitude of the one-bit noise correlation

Using notations and assumptions made in the previous section, we can now find out what the amplitude of the one-bit noise correlation contains.

One-bit normalization consists of retaining only the sign of the raw signal by replacing all positive amplitudes with a 1 and all negative amplitudes with a -1. Thus, we can write

$\displaystyle C_{AB}(t)$ $\textstyle =$ $\displaystyle \int sgn[A(\tau)] sgn[B(t+\tau)] \, d\tau$ (25.6)
  $\textstyle =$ $\displaystyle n_1(t) - n_{-1}(t) \mbox{,}$ (25.7)

where $n_1(t)$ (resp. $n_{-1}(t)$) is the number of samples for which $sgn[A(\tau)] = sgn[B(t+\tau)]$ (resp. $sgn[A(\tau)] \neq sgn[B(t+\tau)]$).

For some samples $\tau$, $\vert A^i(\tau)\vert > \vert A^c(\tau)\vert$ or $\vert B^i(t+\tau)\vert > \vert B^c(t+\tau)\vert$: at one of the two stations, the incoherent noise has a larger amplitude than the coherent noise and so controls the sign of the sample for this station. As the incoherent noise is random, the two events $sgn[A(\tau)] = sgn[B(t+\tau)]$ and $sgn[A(\tau)] \neq sgn[B(t+\tau)]$ have the same probability, so for this population of samples we have $n_1(t) = n_{-1}(t)$.

For the other samples, $\vert A^i(\tau)\vert < \vert A^c(\tau)\vert$ and $\vert B^i(t+\tau)\vert < \vert B^c(t+\tau)\vert$: the coherent noise controls the sign of both $A(\tau)$ and $B(t+\tau)$, so $sgn[A(\tau)] = sgn[A^c(\tau)]]$ and $sgn[B(t+\tau)] = sgn[B^c(t+\tau)]$. Equation 2.4 yields $sgn[B(t_0+\tau)] = sgn[A^c(\tau)]$, so for this population of samples we have $n_{-1}(t_0)=0$.

Now we can write

C_{AB}(t_0) = n P_1 P_2 \mbox{,}
\end{displaymath} (25.8)

where $n$ is the total number of samples in the correlation, $P_1$ is the probability that $\vert A^c(t)\vert > \vert A^i(t)\vert$ and $P_2$ is the probability that $\vert B^c(t)\vert > \vert B^i(t)\vert$. Assuming that coherent and incoherent noise are both gaussian, we are able to express $P1$ and $P2$. Denoting by $\sigma_{A^i}$ (respectively $\sigma_{B^i}$) the standard deviation of $\vert A^i(t)\vert$ (respectively $\vert B^i(t)\vert$), we find
C_{AB}(t_0) = n \left[1-\frac{2}{\pi}tan^{-1}\left(\frac{\si...
\end{displaymath} (25.9)

The case of a uniform distribution of noise sources

Equation 2.9 has been established with no hypothesis on the distribution of noise sources. We have found that the amplitude of the one-bit noise correlation is related to physical quantities. In this section, we evaluate these quantities in the case of a uniform distribution of noise sources.

We assume that $A^c(t)$ is due to the contribution of all the noise sources in the coherent zone (denoted by $\Omega^c$ in the following) as defined by Snieder (2004). This coherent zone is an hyperboloid whose parameters depend on inter-station distance and frequency. Using the central-limit theorem, we write

\sigma_{A^c}^2 = \int_{\Omega^c} \sigma_A^2(x) dx \mbox{.}
\end{displaymath} (25.10)

In this equation, $\sigma_A(x)$ is the standard deviation of the signal recorded in A due to a source in $x$. Considering surface waves at the angular frequency $\omega$, we have
\sigma_A(x) \propto \frac{1}{\sqrt{x}} \exp\left(-\frac{\omega x}{2vQ}\right) \mbox{,}
\end{displaymath} (25.11)

where $Q$ is the quality factor of the medium.

Using equation 2.11 in equation 2.10, we obtain

\sigma_{A^c}^2 \propto \Delta \, cos^{-1}\left(\frac{\Delta-...
...}\right) \left[{K_1(\beta)+K_2(\beta)}\right] e^\beta \mbox{,}
\end{displaymath} (25.12)

where $\lambda$ is the wavelength, $K_1$ and $K_2$ are Bessel functions of the second kind and $\beta = \frac{\omega \Delta}{2vQ}$.

The same procedure enables us to compute $\sigma_{B^c}$, $\sigma_{A^i}$ and $\sigma_{B^i}$. We finally get two analytical expressions for $C_{AB}(t_0)$: one for the raw noise correlation and one for the one-bit noise correlation. We do not provide these expressions because they are too long, but we plot them in Figure 2.45. It is clear that the two amplitude decays correspond to the decay of the Rayleigh wave Green's function.

Figure 2.45: The amplitude decay of the correlations predicted by our theory (gray line) is compared with the decay of the Rayleigh wave Green's function (dotted line). The curves have been obtained using $\lambda=50\,km$ and $Q=300$.
\epsfig{file=cupillard09_1_2.eps, width=8cm}\end{center}\end{figure}


Cupillard, P. and Y. Capdeville, On the amplitude of surface waves obtained by noise correlation and the capability to recover the attenuation: a numerical approach, Geophys. J. Int., submitted.

Gouédard, P., P. Roux, M. Campillo and A. Verdel, Convergence of the two-points correlation function toward the Green's function in the context of a prospecting dataset, Geophysics, 73(6), V47-V53, 2008.

Larose, E., P. Roux and M. Campillo, Reconstruction of Rayleigh-Lamb dispersion spectrum based on noise obtained from an air-jet forcing, J. Acoust. Soc. Am., 122(6), 3437, 2007.

Matzel, E., Attenuation tomography using ambient noise correlation, Seism. Res. Lett., 79(2), 358, 2008.

Prieto, G. A., J. F. Lawrence and G. C. Beroza, Anelastic Earth structure from the coherency of the ambient seismic field, J. Geophys. Res., 144, B07303, doi:10.1029/2008JB006067, 2009.

Snieder, R., Extracting the Green's function from the correlation of coda waves: a derivation based on stationary phase, Phys. Rev. E, 69, 046610, 2004.

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