变分数阶粘弹波动方程最小二乘快速解法

由于分数阶粘弹波动方程存在变分数阶拉普拉斯算子, 其数值求解需要对不同品质因子的空间任意点均进行全域的正反傅里叶变换, 因而计算量巨大, 难以满足实际生产需求。通过引入最小二乘理论, 构建变分数阶空间-波数混合域算子与波数域算子间的逼近关系, 将空间-波数混合域变分数阶算子分解为波数域常分数阶算子与空间域算子的形式, 有效避免直接求取空间-波数混合域算子时计算量大的问题, 从而构建变分数阶粘弹波动方程的常分数阶求解形式, 实现变分数阶粘弹波动方程快速求解。数值模拟计算结果表明, 在品质因子非均值的情况下, 该方法的计算精度优于平均品质因子模拟方法, 计算量小于分块模拟方法, 且提速比随着地下品质因子复杂度的提高而更加明显, 在保证精度的前提下可大幅提高粘弹波场模拟效率, 有利于后续相应高效粘弹成像算法的开发。

The fractional-order viscoelastic wave equation accurately simulates the constant-Q viscoelastic attenuation phenomenon in the exploration seismic frequency band.Because the fractional-order varies with space, it is necessary to calculate the space-wavenumber mixed-domain operator for any point in space.The calculation amount is enormous, and it is challenging to meet the actual application requirements.An approximation relationship between the variable fractional-order space-wavenumber mixed-domain operator and the constant fractional-order wavenumber operator was constructed using the least squares theory.We decomposed the variable fractional-order space-wavenumber mixed-domain operator into a constant fractional-order wavenumber operator and a spatial operator.This avoids the drawback of requiring the Fourier transform in different quality factor value regions to solve the variable fractional-order viscoelastic wave equation.The solution form of the constant fractional-order viscoelastic wave equation was then constructed, enabling the variable fractional-order viscoelastic wave equation to be solved quickly.The numerical results show that the proposed method outperforms the simulation method with an average quality factor in terms of computational accuracy.The computational volume is less than that of the fractional-order simulation method, which requires the Fourier transform in different quality factor regions.Furthermore, the efficiency ratio becomes more evident as the subsurface quality factors become more complex.Finally, using the proposed method, the efficiency of viscoelastic wave field simulation can be significantly improved, which is conducive to developing the corresponding efficient viscoelastic imaging algorithm.