一阶速度-压力常分数阶黏滞声波方程及其数值模拟

A novel constant fractional-order Laplacians viscoacoustic wave equation and its numerical simulation method

与传统的整数阶黏滞波动方程相比,分数阶拉普拉斯算子黏滞方程能更准确地匹配目前广泛使用的常Q模型,而且分数阶黏滞波动方程中控制振幅衰减和相位变化的算子是显式分离的,这对于发展稳定的衰减补偿逆时偏移算法至关重要。首先基于时间域二阶位移形式的常分数阶拉普拉斯算子黏滞声波方程,推导了一阶速度-压力形式常分数阶拉普拉斯算子黏滞声波方程;为了模拟更加真实的振幅变化信息,在新的黏滞声波方程中考虑了密度空变的影响;为了避免由傅里叶变换的周期性而引入的虚假反射,提出了一种适用于分数阶黏滞声波方程的卷积型完全匹配层(CPML)吸收边界加载方法;最后采用交错网格伪谱法进行数值模拟。均匀介质中数值解与解析解的对比证实了该一阶速度-压力常分数阶黏滞声波方程能准确描述常Q模型,BP盐丘模型的地震波场模拟结果证实了其对复杂介质的适用性。

We develop a viscoacoustic wave equation with fractional-order Laplacians,which is better than the traditional integral-order viscoacoustic equation because the new equation more accurately describes the widely used constant-Q model.The operators related to amplitude attenuation and phase change are explicitly independent of each other,which is important for the robust reverse time migration with attenuation compensation.We formulate the first-order velocity-pressure viscoacoustic wave equation with the constant fractional-order Laplacians based on the second-order displacement viscoacoustic equation with the constant fractional-order Laplacians in time domain.To better model amplitude variation,space-varying density is involved in the new equation.To avoid spurious reflections caused by the periodicity of the Fourier transform,a convolutional perfectly matched layer(CPML)is employed as the absorbing boundary for the fractional-order Laplacian viscoacoustic equation.Numerical simulations are fulfilled using staggered-mesh pseudo-spectral method.We compare the numerical solution with the analytic solution for the homogeneous medium,and we find the new equation accurately describes the constant-Q model.We also verify its feasibility for complicated media through seismic wave field simulation using the BP salt dome model.