In the previous chapter we have shown how the Green's function is related to the cross correlation of noise signals recorded by passive sensors, provided the illumination by the noise sources is uniform in all directions. In many applications, however, the noise source distribution is spatially limited and the illumination is directional. The signals recorded by the sensors are dominated by the flux coming from the direction of the noise sources. The cross correlations of the recorded signals depend on the orientation of these sensors relative to the direction of the energy flux. This affects significantly the quality of the estimate for the Green's function, as seen with seismic data in Stehly et al. (2006).

In this chapter we analyze travel time estimation by cross correlation of signals when the noise source distribution is spatially limited. We use the stationary phase method to estimate the travel time between two sensors from the cross correlation of recorded noise signals. This asymptotic approach is valid when the decoherence time of the noise sources is small compared to the travel time between the two sensors. With the stationary phase method we can analyze systematically the dependence of the estimate of the travel time on the orientation of the ray between the sensors and the direction of the energy flux coming from the noise sources.

Given estimates of travel times between sensors in a network that covers an extended region, it is possible to estimate, in turn, the propagation speed of the waves as a function of the spatial coordinates. This is usually done with travel time tomography (Berryman, 1990) using Fermat's principle, as is done in Shapiro et al. (2005) with seismic data. It is also possible to recover the background propagation speed by inversion using the estimated travel times with the eikonal equation (Lin et al., 2009; Gouédard et al., 2012; de Ridder, 2014).

We can also estimate travel times between sensors over consecutive time periods. This lapse-time travel time estimation can identify temporal changes in the background propagation speed (Sens-Schönfelder and Wegler, 2006; Stehly et al., 2007). In particular, this technique can be used in forecasting volcanic eruptions by monitoring changes over time of their geological structure (Brenguier et al., 2008b; Anggono et al., 2012; Brenguier et al., 2014).