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7 Frequency-Domain Numerical Modelling of Visco-Acoustic Waves with Finite-Difference and Finite-Element Discontinuous Galerkin Methods

Seismic exploration is one of the main geophysical methods to extract quantitative inferences about the Earth’s interior at different scales from the recording of seismic waves near the surface. Main applications are civil engineering for cavity detection and landslide characterization, site effect modelling for seismic hazard, CO2 sequestration and nuclearwaste storage, oil and gas exploration, and fundamental understanding of geodynamical processes. Acoustic or elastic waves are emitted either by controlled sources or natural sources (i.e., earthquakes). Interactions of seismic waves with the heterogeneities of the subsurface provide indirect measurements of the physical properties of the subsurface which govern the propagation of elastic waves (compressional and shear wave speeds, density, attenuation, anisotropy). Quantitative inference of the physical properties of the subsurface from the recordings of seismic waves at receiver positions is the so-called seismic inverse problem that can be recast in the framework of local numerical optimization. The most complete seismic inversion method, the so-called full waveform inversion (Virieux & Operto (2009) for a review), aims to exploit the full information content of seismic data by minimization of the misfit between the full seismic wavefield and the modelled one. The theoretical resolution of full waveform inversion is half the propagated wavelength. In full waveform inversion, the full seismic wavefield is generally modelled with volumetric methods that rely on the discretization of the wave equation (finite difference, finite element, finite volume methods). In the regime of small deformations associated with seismic wave propagation, the subsurface can be represented by a linear elastic solid parameterized by twenty-one elastic constants and the density in the framework of the constitutive Hooke’s law. If the subsurface is assumed isotropic, the elastic constants reduce to two independent parameters, the Lamé parameters, which depend on the compressional (P) and the shear (S) wave speeds. In marine environment, the P wave speed has most of the time a dominant footprint in the seismic wavefield, in particular, on the hydrophone component which records the pressure wavefield. The dominant footprint of the P wave speed on the seismic