Wave propagation in anisotropic linear viscoelastic media: theory and simulated wavefields

SUMMARY The anisotropic linear viscoelastic rheological relation constitutes a suitable model for describing the variety of phenomena which occur in seismic wavefields. This rheology, known also as Boltzmann's superposition principle, expresses the stress as a time convolution of a fourth rank tensorial relaxation function with the strain tensor. The first problem is to establish the time dependence of the relaxation tensor in a general and consistent way. Two kernels based on the general standard linear solid are identified with the mean stress and with the deviatoric components of the stress tensor in a given coordinate system, respectively. Additional conditions are that in the elastic limit the relaxation matrix must give the elasticity matrix, and in the isotropic limit the relaxation matrix must approach the isotropic‐viscoelastic matrix. The resulting rheological relation provides the framework for incorporating anelasticity in time‐marching methods for computing synthetic seismograms. Through a plane wave analysis of the anisotropic‐viscoelastic medium, the phase, group and energy velocities are calculated in function of the complex velocity, showing that those velocities are in general different from each other. For instance, the energy velocity which represents the wave surface, is different from the group velocity unlike in the anisotropic‐elastic case. The group velocity loses its physical meaning at the cusps where singularities appear. Each frequency component of the wavefield has a different non‐spherical wavefront. Moreover, the quality factors for the different propagating modes are not isotropic. Examples of these physical quantities are shown for transversely isotropic‐viscoelastic clayshale and sandstone. As in the isotropic‐viscoelastic case, Boltzmann's superposition principle is implemented in the equation of motion by defining memory variables which circumvent the convolutional relation between stress and strain. The numerical problem is solved by using a new time integration technique specially designed to deal with wave propagation in linear viscoelastic media. As a first application snapshots and synthetic seismograms are computed for 2‐D transversely isotropic‐viscoelastic clayshale and sandstone which show substantial differences in amplitude, waveform and arrival time with the results given by the isotropic and elastic rheologies.