EVALUATION OF THREE FIRST-ORDER ISOTROPIC VISCOELASTIC FORMULATIONS BASED ON THE GENERALIZED STANDARD LINEAR SOLID

Stress and strain relaxation times in the generalised standard linear solid (GSLS) define the Q behavior as a function of frequency. Given Q(P) and Q(S) models, strain relaxation times are different for P- and S-waves; P- and S-wave stress relaxation times within each mechanism can be the same, when the tau method is used to estimate relaxation times from Q with multiple relaxation mechanisms, but can also be different (for example, when a single relaxation mechanism is used for both P- and S-waves). Both Robertsson's and Carcione's viscoelastic formulations are based on the GSLS. In Robertsson's formulation, the same stress relaxation times are used for both P- and S-waves; in Carcione's formulation, P- and S-wave stress relaxation times can be different. Moreover, the physical meanings of the P-wave stress and strain relaxation times in these two formulations are not the same. They are calculated from the P-wave quality factor, and the bulk quality factor, respectively; i.e., Robertsson's and Carcione's formulations have different parameterisations. To demonstrate that they are equivalent when the P- and S-stress relaxation times are the same, we derive the second-order viscoelastic equations in the frequency domain. We then generalise Robertsson's formulation to allow different stress relaxation times for P- and S-waves, by reformulating the memory variable equations. The generalised Robertsson's formulation is equivalent to Carcione's formulation. The seismograms modelled by Carcione's, Robertsson's and the generalised Robertsson's formulations are identical, when the input stress relaxation times are the same for P- and S-waves. With different P- and S-wave stress relaxation times in the input model, seismograms produced by Carcione's and the generalised Robertsson's formulations are indistinguishable, while there are obvious differences with seismograms produced by Robertsson's formulation. The limitation of using the same stress relaxation times for P- and S-waves in Robertsson's formulation produces the differences, and leads to errors in the effective Q modelled in the seismograms. Considering the accurate inclusion of intrinsic attenuation, Carcione's and the generalised Robertsson's formulation are equivalent choices as they both allow different P- and S-wave stress relaxation times. Robertsson's original formulation is equivalent only when the P- and S-wave stress relaxation times within each mechanism are the same. The three formulations have comparable computational cost.