Earth introduces strong attenuation and dispersion to propagating waves. The time-fractional wave equation with very small fractional exponent, based on Kjartansson's constant-Q theory, is widely recognized in the field of geophysics as a reliable model for frequency-independent Q anelastic behavior. Nonetheless, the numerical resolution of this equation poses considerable challenges due to the requirement of storing a complete time history of wavefields. To address this computational challenge, we present a novel approach: a nearly optimal sum-of-exponentials (SOE) approximation to the Caputo fractional derivative with very small fractional exponent, utilizing the machinery of generalized Gaussian quadrature. This method minimizes the number of memory variables needed to approximate the power attenuation law within a specified error tolerance. We establish a mathematical equivalence between this SOE approximation and the continuous fractional stress-strain relationship, relating it to the generalized Maxwell body model. Furthermore, we prove an improved SOE approximation error bound to thoroughly assess the ability of rheological models to replicate the power attenuation law. Numerical simulations on constant-Q viscoacoustic equation in 3D homogeneous media and variable-order P- and S- viscoelastic wave equations in 3D inhomogeneous media are performed. These simulations demonstrate that our proposed technique accurately captures changes in amplitude and phase resulting from material anelasticity. This advancement provides a significant step towards the practical usage of the time-fractional wave equation in seismic inversion.

翻译：地球对传播中的波产生强烈的衰减和色散效应。基于Kjartansson常Q理论的极小分数阶时间分数阶波动方程，在地球物理学领域被广泛认可为频率无关Q非弹性行为的可靠模型。然而，该方程的数值求解因需存储波场的完整时间历史而面临重大挑战。为应对这一计算难题，我们提出一种新方法：利用广义高斯求积技术，对极小分数阶Caputo分数阶导数进行近乎最优的指数和（SOE）近似。该方法在指定误差容限内，最小化逼近幂律衰减所需记忆变量的数量。我们建立了该SOE近似与连续分数阶应力-应变关系之间的数学等价性，并将其关联至广义Maxwell体模型。进一步，我们证明了改进的SOE近似误差界，以全面评估流变模型复现幂律衰减的能力。针对三维均匀介质中的常Q粘声波方程及三维非均匀介质中的变阶P波与S波粘弹性波动方程，开展了数值模拟。这些模拟表明，所提技术精确捕捉了材料非弹性引起的振幅与相位变化。该进展为时域分数阶波动方程在地震反演中的实际应用迈出了关键一步。