Humility and nonlinearity

In seismic experiments, waves that bounce back and forth between layers are called multiple reflections. Since these waves interact repeatedly with the heterogeneities in the earth, the relation between the multiple reflections and the earth structure is nonlinear. Present-day seismic processing is geared toward suppressing these multiples rather than extracting the information that is contained in this part of the wavefield. The main motivation for this is that the retrieval of earth structure from multiply-scattered waves is assumed to be an inherently unstable process so that random noise or unwarranted assumptions magnify under the nonlinear inversion steps needed to map the multiple reflected waves onto a model for the earth’s interior. It is for these reasons that Claerbout (1985, 363) notes that: “Multiple reflections is a good subject for nuclear physicists, astrophysicists, and mathematicians who enter our field. Those who are willing to take up the challenge of trying to carry theory through to industrial practice are rewarded by some humility.” At the Ecole Normal Supérieure de Physique et de Chimie Industrielles de la Ville de Paris a group working under the leadership of Mathias Fink has taken up this challenge. This group has pioneered the development of acoustic Time Reversal Mirrors (TRM) [for an overview see Fink (1997)]. The idea behind TRM is that since the wave equation is second-order in time, solutions are invariant under time reversal: if ψ(r, t) is a solution then so is ψ(r,−t). This property is quite familiar to exploration seismologists, where reverse-time propagation of recorded seismic data forms the basis of migration algorithms. However, migration involves simulated propagation on a computer using a model of the subsurface elastic properties. The work of Fink’s group, in contrast, involves the propagation of real physical waves in reverse time and doesn’t require knowledge of the medium. How is this achieved? Waves digitally recorded via an array of piezoelectric transducers (the TRM), are reversed in time and played back using the receivers as a phased-array source. The conceptual simplicity of the TRM belies an impressive experimental achievement, one which has found many important applications. For instance, back propagating the waves scattered by an object focuses the waves on the scatterer, even though the location of the scatterer and the properties of the medium are unknown [see for example the measurements of Prada et al. (1996)]. This ability has led to