A constant-Q model for general viscoelastic media

When seismic waves propagate through viscoelastic media, the viscoelastic response can be presented as a fractional-order derivative of the strain. This fractional order $\beta $ controlling the degree of viscoelasticity of subsurface media is referred to as the viscoelastic parameter. However, the viscoelasticity is conventionally quantified by the quality factor Q, and there is a gap between the viscoelastic parameter $\beta $ and the Q factor. Here this paper bridges the gap by establishing a relationship between these two parameters. An exact Q model is derived analytically based on the viscoelastic parameter $\beta $. Since the exact Q model is frequency dependent, a constant-Q model which is frequency independent is proposed under a small-dissipation assumption. This constant-Q model is applicable to seismic data with a narrow frequency band and is consistent with Kolsky's attenuation model. Furthermore, an inverse function of the constant-Q model is presented for evaluating the viscoelastic parameter $\beta $ from any given Q factor. Thus, the viscoelastic parameter $\beta $ has an intuitive physical meaning that is directly linked to the Q factor