Determination of the viscoelastic parameters for the generalized viscoelastic wave equation


 The viscoelasticity of subsurface media is succinctly represented in the generalized wave equation by a fractional time derivative. This generalized viscoelastic wave equation is characterized by the viscoelastic parameter and the viscoelastic velocity, but these parameters are not well formulated and therefore unfavorable for seismic implementation. Here, we prove that the generalized wave equation is causal and stable by deriving the rate-of-relaxation function. Causality and stability are two necessary conditions for the applicability of the wave equation in seismic simulations. On this basis, we determine the physical parameters for the application of the generalized wave equation. First, we formulate the relationship between the viscoelastic parameter and the constant Q model. The proposed frequency-independent relation agrees with the theoretical solution and fits the field data. Second, we formulate the viscoelastic velocity in terms of the reference velocity and the viscoelastic parameter. These two formulations adequately represent the viscoelastic effect in seismic wave propagation and lead to an improvement in the accuracy of the numerical simulation of the generalized viscoelastic wave equation.