基于分数阶时间导数的黏弹性衰减VTI介质中平面波传播

目前在地震勘探频带范围内通常假设品质因子Q与频率无关,且呈衰减各向同性.事实上,相比较速度各向异性,介质的衰减各向异性同样不可忽视.本文将衰减各向异性和速度各向异性二者与常Q模型相结合,建立了黏弹性衰减VTI介质模型,并基于分数阶时间导数理论,给出了对应的本构关系和波动方程.利用均匀平面波分析和Poynting定理,推导出准压缩波qP、准剪切波qSV和纯剪切波SH的复速度、相速度、能量速度以及品质因子的解析表达式.对模型的正确性进行了数值验证,并分析了qP,qSV和SH波在介质中的传播特性.数值试验结果表明:本模型能够实现理想的恒定Q行为,表现了品质因子和速度的各向异性特征,显示出黏弹性增强将导致能量速度和相速度的频散曲线变化剧烈;速度和衰减各向异性参数与传播角度之间的耦合效应对qP,qSV和SH波的速度和能量影响明显;qP,qSV和SH波的频散曲线和波前面随着衰减各向异性强度的改变发生显著变化,其中耦合在一起的qP和qSV波变化趋势相同,而SH波与它们呈现相反的变化规律.本研究为从常Q模型角度分析地震波在衰减各向异性黏弹性介质中的传播特征奠定了理论基础.

Plane wave propagation in viscoelastic attenuative VTI media based on fractional time derivatives

At present,it is generally assumed that the quality factor Q is frequency independent in the seismic exploration band and the attenuation is isotropic.In reality,like the velocity anisotropy,the attenuation anisotropy cannot be ignored.In this paper,we combine the attenuation anisotropy and velocity anisotropy with the constant-Q model to establish a viscoelastic vertical transversely isotropic(VTI)attenuative model.Based on the fractional time derivatives theory,the corresponding constitutive relation and wave equations are proposed.The analytical expressions of the quality factor,complex,phase,and energy velocities for the quasi-compressional wave qP,the quasi-shear wave qSV,and the pure shear wave SH,respectively,are derived from the homogeneous plane wave analysis and Poynting theorem.The correctness of the presented model is numerically verified,and the propagation characteristics of the qP,qSV,and SH waves are analyzed.The results of numerical experiments indicate that the given model can achieve the desirable constant-Q behavior,the attenuation and velocity show the anisotropic characteristics,and the variation of the dispersion effects depends on the degree of the viscoelasticity.The coupling effect between the attenuation and velocity anisotropic parameters and propagation angle has a great influence on the velocities and energy of the seismic waves.Moreover,the dispersion curves and wavefront of the qP,qSV,and SH waves vary significantly with the change of the anisotropic attenuation strength,where the coupled qP and qSV waves have the same trends,while the SH wave shows the opposite variation tendency.This study lays a theoretical foundation for analysing characteristics of wave propagation in the viscoelastic attenuative anisotropic media from the perspective of the constant-Q model.