Approximating Constant-Q Seismic Wave Propagations in Acoustic and Elastic Media Using a Cole–Cole Model


 Seismic wave propagation in the Earth’s interior inevitably encounters attenuation and dispersion effects, which usually can be represented by a constant-Q model. However, solving the constant-Q wave equations formulated by fractional Laplacians is computationally intensive. Alternatively, the Cole–Cole model provides an optimal description of seismic attenuation. Because of the fractional time derivatives of both stress and strain in the expression, this method exhibits good adaptability and flexibility. In this article, we investigate the performance of the Cole–Cole model to approximate constant-Q behaviors with different fractional orders in acoustic and elastic media. The phase velocity and quality factor are compared to determine an optimal fractional order. After that the Cole–Cole model can be easily represented by the other three parameters (i.e., relaxed modulus, minimum angular frequency, and minimum quality factor), which are similar to the constant-Q theory (reference modulus, reference angular frequency, and frequency-independent quality factor). The first-order viscoacoustic and viscoelastic wave equations are derived to implement seismic wavefield simulations by combining conservation equation and stress–displacement relation. Furthermore, a time-domain algorithm is developed to solve the wave equations based on the Grunwald–Letnikov approximation and finite-difference scheme. Numerical modeling results calculated by our proposed method have good consistencies with the reference solutions from the constant-Q theory, suggesting that a small fractional order can well approximate the constant-Q outputs in a broad frequency band.