Multiscale Modeling &Simulation, Volume 21, Issue 4, Page 1410-1456, December 2023.
Abstract. This work concerns the asymptotic analysis of high-frequency wave propagation in randomly layered media with fast variations and long-range correlations. The analysis takes place in the three-dimensional physical space and weak-coupling regime. The role played by the slow decay of the correlations on a propagating pulse is twofold. First we observe a random travel time characterized by a fractional Brownian motion that appears to have a standard deviation larger than the pulse width, which is in contrast with the standard O’Doherty–Anstey theory for random propagation media with mixing properties. Second, a deterministic pulse deformation is described as the solution of a paraxial wave equation involving a pseudodifferential operator. This operator is characterized by the autocorrelation function of the medium fluctuations. In case of fluctuations with long-range correlations this operator is close to a fractional Weyl derivative whose order, between 2 and 3, depends on the power decay of the autocorrelation function. In the frequency domain, the pseudodifferential operator exhibits a frequency-dependent power-law attenuation with exponent corresponding to the order of the fractional derivative, and a frequency-dependent phase modulation, both ensuring the causality of the limiting paraxial wave equation as well as the Kramers–Kronig relations. The mathematical analysis is based on an approximation-diffusion theorem for random ordinary differential equations with long-range correlations.

中文翻译：

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具有长程相关性的随机层状介质中波前的有效分数近轴波方程

多尺度建模与仿真，第 21 卷，第 4 期，第 1410-1456 页，2023 年 12 月。
摘要。这项工作涉及具有快速变化和长程相关性的随机分层介质中高频波传播的渐近分析。分析在三维物理空间和弱耦合状态下进行。传播脉冲相关性的缓慢衰减所起的作用是双重的。首先，我们观察到以分数布朗运动为特征的随机传播时间，其标准偏差似乎大于脉冲宽度，这与具有混合特性的随机传播介质的标准 O'Doherty-Anstey 理论相反。其次，确定性脉冲变形被描述为涉及伪微分算子的近轴波动方程的解。该算子的特点是介质波动的自相关函数。在存在长程相关性波动的情况下，该算子接近分数外尔导数，其阶数在 2 到 3 之间，取决于自相关函数的幂衰减。在频域中，伪微分算子表现出频率相关的幂律衰减，其指数对应于分数阶导数的阶数，以及频率相关的相位调制，既保证了极限近轴波方程的因果关系，又保证了克莱默斯-克罗尼格关系。数学分析基于具有长程相关性的随机常微分方程的近似扩散定理。