We derive a pseudo-differential factorization of the wave operator with fractional attenuation. This factorization allows us to approximately solve the Helmholtz equation via a two-way (transmission and reflection) sweeping scheme tailored to high-frequency wave fields. We provide explicitly the three highest order terms of the pseudo-differential expansion to incorporate the well-known square-root first order symbol for wave propagation, the zeroth order symbol for amplitude modulation due to changes in wave speed and damping, and the next symbol to model fractional attenuation. We also propose wide-angle Pad\'e approximations for the pseudo-differential operators corresponding to these three highest order symbols. Our analysis provides insights regarding the role played by the frequency and the Pad\'e approximations in the estimation of error bounds. We also provide a proof-of-concept numerical implementation of the proposed method and test the error estimates numerically.

翻译：我们推导了具有分数衰减的波算子的伪微分分解。该分解使得我们能够通过适用于高频波场的双向（透射和反射）扫描方案来近似求解亥姆霍兹方程。我们显式给出了伪微分展开的三个最高阶项，以纳入用于波传播的著名平方根一阶符号、因波速和阻尼变化引起的振幅调制的零阶符号，以及用于建模分数衰减的下一阶符号。我们还针对对应这三个最高阶符号的伪微分算子提出了广角帕德近似。我们的分析揭示了频率和帕德近似在误差界估计中所起的作用。我们还对所提方法进行了概念验证的数值实现，并通过数值测试验证了误差估计。