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Abstract
The interpretation of wave scattering data has led, and continues to lead, to a wide range of scientific discoveries and practical engineering applications in domains as varied as materials science, biomedical imaging, geophysics, atmospheric science, oceanography, plasma physics, archaeology, astrophysics or quantum information. A special case is tomography: the process of reconstructing the volume properties of a sample from scattered wave measurements collected on its boundary. Under various assumptions linked to the physical nature of the waves (acoustic, elastic, electromagnetic, etc.) and the properties of the sample, tomography mathematically reduces to the solution of an inverse wave scattering problem. When the scattering is weak and receivers can be placed around the sample, explicit analytical solutions can be obtained using filtered back projection, a solution procedure based on the inverse Radon transform [125]. If the scattering is stronger or if sources and receivers can only be placed on a limited part of the boundary, iterative reconstruction must be used instead. Basically any iterative reconstruction method consists in minimizing the mismatch between the measured wave scattering data and some simulated data, obtained through a forward wave propagation model of the scattering phenomenon [33, 29, 40, 141]. Full waveform inversion (also called waveform tomography or wave field inversion) falls into this second category. The particular propagation model used in that case is a full wave propagation model, e.g. the Helmholtz equation for acoustics in the frequency domain, Maxwell’s equations for electromagnetics and the Navier equation for elastodynamics, at the opposite for example of ray tomography which are based on eikonal equations, i.e. on asymptotic approximations of the full model [10, 126, 206]. Considering a full model instead of a simplified one is computationally much more expensive but also promises a much higher resolution, as sub-wavelength accuracy is expected [7, 39, 54, 198, 122]. Another specificity of full waveform inversion is its brute force approach to the inverse problem: the mismatch is simply minimized w.r.t. the coefficients appearing in the full model. While simple in principle, this optimization nevertheless remains challenging: firstly because it is hard to design a meaningful mismatch functional for oscillatory signals such as wave scattering data [19, 124, 166, 203]; secondly because observed data are contaminated by noise, a perfect match with the simulated data is therefore often impossible and 1 2 does not mean that the true coefficients are recovered anyway; finally because the coefficients are unknown at every location of the sample volume under investigation, hence large-scale optimization methods must be used. However, thanks to ever increasing available computational power, full waveform inversion is now becoming feasible and is therefore being investigated for a growing number of engineering applications. Three of them are described in Figure 1.
(a) Marine seismic survey. These surveys aim to image the structure and nature of the rock layers and the reservoirs beneath the seabed. Sound waves are sent into the earth from an excitation source (¨), which is pulled in the water behind the survey vessel. Reflections from the different structures beneath the crust are traveling back to the surface and are recorded on a receiver streamer (•) towed behind the excitation source. Receivers can also be placed on the seafloor, using ocean-bottom cables or nodes.
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