具有分数阶衰减的超声波伪微分模型
Pseudodifferential Models for Ultrasound Waves with Fractional Attenuation
AI总结：
本文推导含分数阶衰减波算子的伪微分分解，结合单向/双向扫描与广角 Padé 近似高效求解高频软组织超声 Helmholtz 方程，并分析误差界及数值验证。
摘要：
为在软组织超声波模拟中兼顾建模精度与计算效率，我们推导了具有分数阶衰减的波算子的伪微分分解。该分解使我们能够通过为高频波场定制的单向（透射）或双向（透射与反射）扫描格式近似求解 Helmholtz 方程。我们明确给出伪微分展开的三个最高阶项，以纳入众所周知的用于波传播的平方根一阶象征、用于由波速和阻尼变化引起的振幅调制的零阶象征，以及用于建模分数阶衰减的下一个象征。我们还针对对应于这三个最高阶象征的伪微分算子提出了广角 Padé 近似。我们的分析揭示了频率和 Padé 近似在误差界估计中的作用。我们还提供了所提方法的概念验证数值实现，并对误差估计进行了数值测试。
英文摘要：
To strike a balance between modeling accuracy and computational efficiency for simulations of ultrasound waves in soft tissues, we derive a pseudodifferential factorization of the wave operator with fractional attenuation. This factorization allows us to approximately solve the Helmholtz equation via one-way (transmission) or two-way (transmission and reflection) sweeping schemes tailored to high-frequency wave fields. We provide explicitly the three highest order terms of the pseudodifferential 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 Pade approximations for the pseudodifferential operators corresponding to these three highest order symbols. Our analysis provides insights regarding the role played by the frequency and the Pade 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.
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