On the numerical implementation of time-reversal mirrors for tomographic imaging

International audienceA general approach for constructing numerical equivalents of time-reversal mirrors is introduced. These numerical mirrors can be used to regenerate an original wavefield locally within a confined volume of arbitrary shape. Though time-reversal mirrors were originally designed to reproduce a time-reversed version of an original wavefield, the proposed method is independent of the time direction and can be used to regenerate a wavefield going either forward in time or backward in time. Applications to computational seismology and tomographic imaging of such local wavefield reconstructions are discussed. The key idea of the method is to directly express the source terms constituting the time-reversal mirror by introducing a spatial window function into the wave equation. The method is usable with any numerical method based on the discrete form of the wave equation, for example, with finite difference (FD) methods and with finite/spectral elements methods. The obtained mirrors are perfect in the sense that no additional error is introduced into the reconstructed wavefields apart from rounding errors that are inherent in floating-point computations. They are fully transparent as they do not interact with waves that are not part of the original wavefield and are permeable to these. We establish a link between some hybrid methods introduced in seismology, such as wave-injection, and the proposed time-reversal mirrors. Numerical examples based on FD and spectral elements methods in the acoustic, the elastic and the visco-elastic cases are presented. They demonstrate the accuracy of the method and illustrate some possible applications. An alternative implementation of the time-reversal mirrors based on the discretization of the surface integrals in the representation theorem is also introduced. Though it is out of the scope of the paper, the proposed method also apply to numerical schemes for modelling of other types of waves such as electromagnetic waves