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Localizing change with Coda Wave Interferometry : Derivation and validation of the sensitivity kernel

Previous formulations of coda wave interferometry make it possible to assess the average change of the medium, but they do not allow for the spatial localization of this change. We present a new theory for localizing changes in the medium using strongly scattered waves, and test it with numerical models for different velocity perturbations. The technique is tested for 2D scalar waves. If the diffusion approximation for the energy transport is valid, we can accurately predict the perturbation in the traveltimes given a perturbation in the velocity field. Using an integral representation for the diffuse wavefield, we derive an expression relating the mean traveltime perturbation due to a small perturbation in the slowness field. These results suggest that we can detect and localize changes in the propagation velocities of the multiple scattering medium. We study the sensitivity of our derived expression for the traveltime change in a multiple scattering medium to different perturbations in the slowness. We validate our theory using synthetic seismograms calculated with a finite-difference algorithm. For the special case of coincident source and receiver, we obtain an analytical solution of the mean traveltime change of the diffuse wavefield. In this case, we are able to relate traveltime changes to changes in the propagation velocity that are located close (distance r < 4l, where l is the scattering mean free path) to the source and receiver position. For larger distances, the perturbation is difficult to detect because the changes in the traveltime become too small to be detectable for realistic perturbations in the slowness. For the more general case of a non-coincident source and receiver, we are able to detect the change in the traveltimes for longer distances when the perturbation is located somewhere between the source and the receiver. In general, for localized slowness perturbations, our theory predicts the mean traveltime change of the diffuse wavefield in a multiple-scattering medium. Thus, we need to perform averaging over many realizations of the experiment or over different source-receiver pairs to accurately estimate the mean traveltime change associated with a localized slowness perturbation. The technique presented here can be used in many applications such as medical imaging, non-destructive testing, and reservoir monitoring, to infer temporal changes of the multiple scattering medium.