Theory and modelling of constant-Q P- and S-waves using fractional spatial derivatives

We derive a time-domain differential equation for modelling seismic wave propagation in constant-Q viscoelastic media based on fractional spatial derivatives, specifically Laplacian differential operators of fractional order. The stress-strain relation is derived from the classical equation expressed in terms of fractional time derivatives. The new formulation has the advantage of not requiring additional field variables that increase the computer time and storage significantly. The spatial derivatives are calculated with a generalization of the Fourier pseudospectral method to the fractional-derivative case. The accuracy of the numerical solution is verified against an analytical solution in a homogeneous medium. An example shows that the proposed wave equation describes the constant-Q attenuation and velocity dispersion behaviour observed in Pierre Shale. Finally, we consider a plane-layer model and the Marmousi model to show how the new formulation applies to inhomogeneous media.