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In previous studies, the auxiliary differential equation (ADE) method has been applied to 2-D seismic-wave propagation modelling in viscoelastic media. This method is based on the separation of the wave propagation equations derived from the constitutive law defining the stress–strain relation. We make here a 3-D extension of a finite-difference (FD) scheme to solve a system of separated equations consisting in the stress–strain rheological relation, the strain–velocity and the velocity–stress equations. The current 3-D FD scheme consists in the discretization of the second order formulation of a non-linear viscoelastic wave equation with a time actualization of the velocity and displacement fields. Compared to the usual memory variable formalism, the ADE method allows flexible implementation of complex expressions of the desired rheological model such as attenuation/viscoelastic models or even non-linear behaviours, with physical parameters that can be provided from dispersion analysis. The method can also be associated with optimized perfectly matched layers-based boundary conditions that can be seen as additional attenuation (viscoelastic) terms. We present the results obtained for a non-linear viscoelastic model made of a Zener viscoelastic body associated with a non-linear quadratic strain term. Such non-linearity is relevant to define unconsolidated granular model behaviour. Thanks to a simple model, but without loss of generality, we demonstrate the accuracy of the proposed numerical approach.

Seismic wave propagation in nonlinear viscoelastic media using the auxiliary differential equation method