Multiple scattering in evolving media

techniques that are used, for example, in radar and in seismic exploration. If one knows the arrival time of a singly scattered wave, along with its velocity and direction of propagation, one can determine the location of the scatterer. For multiply scattered waves it is more difficult to determine the locations of the scatterers because the waves propagate over many possible scattering trajectories involving a number of scatterers. Thus, multiply scattered waves are not so useful for imaging. Multiply scattered waves are very useful, however, for detecting temporal changes in the medium through which they propagate. The changes may arise from the motion of the scatterers, changes in their scattering properties, or changes in the properties of the matrix in which the scatterers are embedded. The singly scattered wave in figure 1a traverses the medium at most twice. By contrast, the multiply scattered waves in figure 1b travel along much longer trajectories through the medium. Because of their much greater spatial sampling, multiply scattered waves are much more sensitive to changes in the medium than are singly scattered waves. This applies both to waves that are truly multiply scattered within the medium and to reverberant waves that bounce back and forth between the medium’s boundaries (figure 1c). We refer to both types of waves as multiply scattered waves, despite the difference in the mechanism that causes them to repeatedly sample the changes in the medium. An example of multiply scattered waves is given in figure 2, which shows the ground motion recorded on Mount Merapi, an active volcano in Indonesia. The waves, excited by an air gun placed on the side of the volcano and recorded on a nearby seismograph, do not consist of isolated discrete arrivals but rather of a wavetrain of interfering scattered waves. Seismologists refer to these scattered waves as coda waves, after the Latin word for “tail.” Volcanoes strongly scatter elastic waves, and the coda waves recorded on Mount Merapi are dominated by multiple scattering: The distance between scattering events, as measured by the scattering mean free path (about 400 m), is much smaller than the path length covered by the waves in figure 2 (up to about 50 km, or 125 scattering events).1