Using the SEM to Simulate Random Wavefields and Improve Noise Tomography

Paul Cupillard, Laurent Stehly and Barbara Romanowicz


The correlation of a random wavefield recorded by distant receivers contains the Green's function (GF) of the medium. This makes the measurement of seismic wave travel times between any pair of receivers a possibility. These measurements can then be inverted to image the Earth's interior (e.g. Shapiro et al, 2005).

This result is valid in any medium but relies on a strong assumption: The wavefield has to be equipartitioned, that is, all the modes of the medium have to be excited with the same level of energy and a random phase (Sánchez-Sesma and Campillo, 2006). Equipartition is achieved, for instance, if there are white noise sources everywhere in the medium or if few sources are present in a highly scattering medium.

The seismic ambient noise comes from interactions between the atmosphere, the ocean and the sea floor. Even averaged over one year, the distribution of noise sources at the surface of the Earth is not homogeneous and does not completely match the requirements of the theories that relate noise correlations to the GF of the medium.

The uneven distribution of noise sources, the fact that one uses a finite amount of data to compute noise correlations, and the way noise correlations are inverted imply several limitations on ambient noise tomography. (i) In most of the noise tomographic studies, less than one interstation path out of three is used. Other paths are rejected because either surface waves cannot be identified unambiguously, or the surface wave travel times measured on the positive and negative side of the correlation are not consistent. This is mostly due to the uneven distribution of noise sources. (ii) The velocity of surface waves can be systematically over- or under- estimated for certain azimuths, since noise sources are not evenly distributed. This could be erroneously interpreted as anisotropy of the medium. (iii) Most of the time, surface wave dispersion curves are inverted using the Path Average approximation. This procedure does not account for the complexity of the wave propagation within the Earth. When using noise correlations, this is an important problem because noise correlations are very sensitive to the crust, which is a very heterogeneous structure.

Figure 2.28: SNR of noise correlation surface waves as a function of the number of days of correlated noise (plain lines). Four interstation distances are considered. Each SNR curve is compared to the semi-empirical expression from Larose et al (2008) (dashed lines).
\begin{figure}\centering\epsfig{file=cupillard10_1_1.eps, width=8.5cm}\setlength{\abovecaptionskip}{0pt}\setlength{\belowcaptionskip}{0pt}\end{figure}

Simulating seismic ambient noise using the Spectral Element Method

In this work, we explore the possibility of taking into account the distribution of noise sources when inverting noise correlations. A first step towards such a goal is to perform a forward modeling which includes noise sources. To do so, we use the Spectral Element method (SEM). Computing synthetic correlations by simulating a random wavefield using the SEM would eliminate most of the biases arising from the uneven distribution of noise sources during the forward problem. We use the SEM instead of other methods such as the normal mode summation technique (Cupillard and Capdeville, 2010) because it enables us to solve the wave equation with no restriction on the velocity contrast of the model.

The code we use enables us to impose a three-component random traction on the top surface of the region. Because it makes the implementation very easy, the traction is discretized over the grid points of the spectral element mesh. For each grid point, three random signals are generated: one to define the normal traction and two to define the tangential traction. In the present work, we only use the normal component (the two tangential components are set to zero). All the random signals are filtered in the 25 - 80s period band and are then used in the SEM simulation as an external surface force.

We start by investigating the easiest case: an homogeneous distribution of noise sources at the free surface of an attenuating and spherically symmetric Earth. Our simulation consists of 37 numerical runs computed in PREM and lasting 4000s each. We consider a 75x35 degree wide region surrounded by absorbing boundaries and a bottom lying at 600km. An array of 40x5 receivers separated by 60km records the vertical displacement. The GF is retrieved between each station pair by correlating the background seismic noise records. No processing is performed on the noise records, such as frequency whitening or one-bit normalization.

Since we consider a 1D model, the GF only depends on the source-receiver distance. This implies that all correlations computed between station pairs separated by the same distance converge towards the same GF. Therefore, we consider that averaging the correlation over time $T$ or over station pairs is equivalent. Since we perform 37 runs of 4000s, each receiver records $37*4\,000s=1.7$ days of noise. As we have, for instance, 140 pairs of stations separated by 720km, summing the corresponding correlations provides a result similar to correlating $37*140*4\,000\,s$ = 237 days of noise.

The convergence towards the GF

For scalar waves in a 2D homogeneous medium with a uniform source distribution, the signal-to-noise ratio (SNR) of the correlations is (Larose et al, 2008):

\mbox{SNR}(T,r) = C \sqrt{\frac{n\,T\,\Delta\!f\,c}{r\,f}},
\end{displaymath} (16.1)

where $r$ is the interstation distance, $n$ the number of noise sources, $c$ the velocity of the medium, $f$ the frequency, $\Delta f$ the bandwidth and $C$ a constant.

We compare this theoretical prediction with our observations. Figure 2.28 shows how the SNR of the correlations evaluated in the 25-60s period band evolves with the amount of data used for stations separated by 240, 480, 840, and 1440km. For a short offset, less than two days are required to get a SNR$\,>10$, and 20 days are required for stations separated by 1440km. We fit this SNR vs time curve with a function of the form $ a \left(\frac{n\,T\,\Delta\!f\,c}{r\,f}\right)^{b}$, $a$ and $b$ being two free parameters we want to determine. We find that our SNR measurements are best fitted with $a$=27.8 and $b$=0.45, i.e the SNR increases with $T^{0.45}$ whereas theoretically one expects $T^{0.5}$. However, this difference is only an artifact, coming from our assumption that averaging correlation over time is equivalent to averaging them over the interstation pairs.

Figure 2.29: Correlations of 11 days of noise between stations separated by 320, 720 and 1440km (dashed lines) vs corresponding GF (plain lines). For each comparison, we show the misfit of the correlation and GF waveforms measured in the shaded area and the surface wave travel time difference $dt$. Top: raw correlations. Bottom: correlations de-noised by curvelet filters.
\begin{figure}\epsfig{file=cupillard10_1_2.eps, width=8.5cm}\epsfig{file=cupillard10_1_3.eps, width=8.5cm}\setlength{\abovecaptionskip}{0pt}\end{figure}

Improving synthetic noise correlations using curvelet filters

Our numerical results show that (i) it takes only a few days to reconstruct accurately the surface wave part of the Green's function and (ii) several months of data are required for the random fluctuations of the correlation to disappear. If most of the information on the medium is already present in a correlation averaged over a few days, then it should be possible to isolate this information from the random fluctuations of the correlation. This would allow us to not only measure surface wave travel times more accurately, but also reconstruct the full waveform of the GF using a smaller amount of data. Figure 2.29 shows that curvelet filters achieve this goal very well.


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., 181(3), 1687-1700, 2010.

Larose, E., P. Roux, M. Campillo and A. Derode, Fluctuations of correlations and Green's function reconstruction: role of scattering, J. Appl. Phys., 103, 114907, 2008.

Sánchez-Sesma, F. J. and M. Campillo, Retrieval of the Green's function from cross-correlation: The canonical elastic problem, Bull. Seism. Soc. Am., 96, 1182-1191, 2006.

Shapiro, N. M., M. Campillo, L. Stehly and M. H. Ritzwoller, High-resolution surface wave tomography from ambient seismic noise, Science, 307, 1615-1618, 2005

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