It was shown in Chapter 3 that we can estimate the travel times between passive sensors from the cross correlations of the signals generated by ambient noise sources and recorded by the sensors. This is possible provided that the illumination of the pair of sensors is suitable, in the sense that the ray joining the two sensors intersects the source region. If this is not the case then we show in this chapter that travel time estimation is still possible provided the medium is scattering. This is because the scatterers can act as secondary sources and this secondary illumination can generate peaks in the cross correlation at the inter-sensor travel time. However, scattering can also increase the fluctuation level of the cross correlations. In this chapter we analyze these two competing phenomena that are both involved in inter-sensor travel time estimation.
We first introduce, in Section 7.2, a simple model for a weakly scattering medium. We can then analyze the peaks of the cross correlation of the signals recorded by a pair of sensors, which shows that the scatterers can indeed play the role of secondary sources and can therefore provide an appropriate secondary illumination (Proposition 7.1). However the scatterers are also responsible for additional fluctuations in the cross correlation that can be quantified by a variance calculation (Proposition 7.2). When the trade-off between illumination diversity enhancement and signal-to-noise ration reduction is not good enough for travel time estimation with the cross correlations, it may be possible to estimate the travel time between two sensors by looking at the main peaks of a special fourth-order correlation, as shown in Section 7.4.
As in Chapter 3, given estimates of travel times between sensors in a network that covers well an extended region, it is possible to estimate the propagation speed of the waves in that region. As noted in Chapter 3, this can be done with travel time tomography (Berryman, 1990) using Fermat's principle, as is done in Shapiro et al. (2005) with seismic data. It can also be done using the eikonal equation (Lin et al., 2009; Gouédard et al., 2012; de Ridder, 2014). The use of fourth-order cross correlations may improve the estimated travel times and, therefore, may also improve background velocity estimation.