Energy is absorbed and attenuated when seismic waves propagate in real earth media. Hence, the viscoelastic medium needs to be considered. There are many ways to construct the viscoelastic body, in which the generalized standard linear viscoelastic body is the most representative one. For viscoelastic wave propagation and imaging, it is very important to obtain a compact and efficient viscoelastic equation. Because of this, we derived a set of simplified viscoelastic equations in isotropic media on the basis of the standard linear solid body and the constitutive relation for a linear viscoelastic isotropic solid. The simplified equations were composed of the linear equations of momentum conservation, the stress-strain relations, and the memory variable equations. During the derivation of the equations, the Lamé differentiation matrix, which has a similar form to the stiffness matrix and indicates the relations between viscoelastic and elastic stiffness matrices, was introduced to simplify the memory variable equations. Analogous to the elastic equations, the simplified equations have symmetrically compact forms and are very useful for efficient viscoelastic modeling, migration, and inversion. Applied to a 2D simple model and the 2D SEG/EAGE salt model, the results show that our simplified equations are more efficient in computation than Carcione’s equations.
Earthquake response of adjacent structures with viscoelastic and friction dampers