Memory-efficient displacement-based internal friction for wave propagation simulation

The characterization of anelastic losses due to material internal friction has become increasingly important in geophysical exploration and other seismological applications, as these losses greatly affect the amplitude and dispersion of seismic waves. Anelasticity is usually specified in terms of the material's quality factor, Q. Different viscoelastic models have been used to represent Q as a function of frequency. Most of these models are defined in terms of stresses and strains as the primary variables. Thus, in three dimensions, a separate model needs to be associated with each of the six strain components. We introduce an internal friction model that uses, instead, displacements as primary variables. For a fiber, the proposed model consists of a set of three distinct elements in parallel with different relaxation mechanisms: namely, two elements that consist of a spring and a dashpot in series (Maxwell) and a third element that consists of a spring and a dashpot in parallel (Voigt). In addition to saving memory, this formulation is particularly suitable for finite-element schemes. The model exhibits an almost constant quality factor within the frequency range of interest, with a tolerance of 5% with respect to the target Q value, and provides a close approximation to the variation of the phase-velocity with frequency - as has been observed in empirical data. The extension of this model to 3D anelastic problems and its use in idealized cases, such as an infinite-space, a half-space, and a layered half-space, and the comparison of results with semi-analytical reference solutions obtained from theory and previous, similar studies, corroborates the validity of the proposed model for incorporating anelastic losses in wave propagation problems.