Simulation of seismic waves at the earth's crust (brittle–ductile transition) based on the Burgers model

The earth's crust presents two dissimilar rheological behaviors depending on the in situ stress-temperature conditions. The upper, cooler part is brittle, while deeper zones are ductile. Seismic waves may reveal the presence of the transition but a proper characterization is required. We first obtain a stress–strain relation, including the effects of shear seismic attenuation and ductility due to shear deformations and plastic flow. The anelastic behavior is based on the Burgers mechanical model to describe the effects of seismic attenuation and steady-state creep flow. The shear Lamé constant of the brittle and ductile media depends on the in situ stress and temperature through the shear viscosity, which is obtained by the Arrhenius equation and the octahedral stress criterion. The P and S wave velocities decrease as depth and temperature increase due to the geothermal gradient, an effect which is more pronounced for shear waves. We then obtain the P−S and SH equations of motion recast in the velocity-stress formulation, including memory variables to avoid the computation of time convolutions. The equations correspond to isotropic anelastic and inhomogeneous media and are solved by a direct grid method based on the Runge–Kutta time stepping technique and the Fourier pseudospectral method. The algorithm is tested with success against known analytical solutions for different shear viscosities. A realistic example illustrates the computation of surface and reverse-VSP synthetic seismograms in the presence of an abrupt brittle–ductile transition.