Seismic modeling in viscoelastic media

Anelasticity is usually caused by a large number of physical mechanisms which can be modeled by different microstructural theories. A general way to take all these mechanisms into account is to use a phenomenologic model. Such a model which is consistent with the properties of anelastic media can be represented mechanically by a combination of springs and dash‐pots. A suitable system can be constructed by the parallel connection of several standard linear elements and is referred to as the general standard linear solid rheology. Two relaxation functions that describe the dilatational and shear dissipation mechanisms of the medium are needed. This model properly describes the short and long term behaviors of materials with memory and is the basis for describing viscoelastic wave propagation. This work presents two‐dimensional (2-D) and three‐dimensional (3-D) forward modeling in linear viscoelastic media. The theory implements Boltzmann’s superposition principle based on a spectrum of relaxation mechanisms in the time‐domain equation of motion by the introduction of the memory variables. The algorithm uses a polynomial interpolation of the evolution operator for time integration and the Fourier pseudospectral method for computation of the spatial derivatives. This scheme has spectral accuracy for band‐limited functions with no temporal or spatial dispersion, a very important fact in anelastic wave propagation. Examples are given of how to pose typical problems of viscoelastic forward modeling for geophysical problems in two and three dimensions. A model separating an elastic medium of a viscoelastic medium with similar elastic moduli but different attenuations shows that the interface generates appreciable reflected energy. A second example computes the response of a single interface in the presence of highly dissipative sandstone lenses, properly simulating the anelasticity of direct and converted P‐ and S‐waves. A common‐shot reflection survey over a gas cap reservoir indicates that attenuation significantly affects the bright spot response. 3-D viscoelastic modeling requires twice the memory storage of 3-D viscoacoustic modeling when one dissipation mechanism is used for each wave mode. Simulation in a 3-D homogeneous viscoelastic medium shows how the anelastic characteristics of the different modes can be controlled independently. The result of this simulation is compared to the analytical solution indicating sufficient accuracy for many applications.