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Earth introduces strong attenuation and dispersion to propagating waves. Thetime-fractional wave equation with very small fractional exponent, based onKjartansson's constant-Q theory, is widely recognized in the field ofgeophysics as a reliable model for frequency-independent Q anelastic behavior.Nonetheless, the numerical resolution of this equation poses considerablechallenges due to the requirement of storing a complete time history ofwavefields. To address this computational challenge, we present a novelapproach: a nearly optimal sum-of-exponentials (SOE) approximation to theCaputo fractional derivative with very small fractional exponent, utilizing themachinery of generalized Gaussian quadrature. This method minimizes the numberof memory variables needed to approximate the power attenuation law within aspecified error tolerance. We establish a mathematical equivalence between thisSOE approximation and the continuous fractional stress-strain relationship,relating it to the generalized Maxwell body model. Furthermore, we prove animproved SOE approximation error bound to thoroughly assess the ability ofrheological models to replicate the power attenuation law. Numericalsimulations on constant-Q viscoacoustic equation in 3D homogeneous media andvariable-order P- and S- viscoelastic wave equations in 3D inhomogeneous mediaare performed. These simulations demonstrate that our proposed techniqueaccurately captures changes in amplitude and phase resulting from materialanelasticity. This advancement provides a significant step towards thepractical usage of the time-fractional wave equation in seismic inversion.

Compressing the memory variables in constant-Q viscoelastic wave propagation via an improved sum-of-exponentials approximation