Seismic waves in earth media usually undergo attenuation, causing energy losses and phase distortions. In the regime of high-frequency asymptotics, a complex-valued eikonal is an essential ingredient for describing wave propagation in attenuating media, where the real and imaginary parts of the eikonal function capture dispersion effects and amplitude attenuation of seismic waves, respectively. Conventionally, such a complex-valued eikonal is mainly computed either by tracing rays exactly in complex space or by tracing rays approximately in real space so that the resulting eikonal is distributed irregularly in real space. However, seismic data processing methods, such as prestack depth migration and tomography, usually require uniformly distributed complex-valued eikonals. Therefore, we have developed a unified framework to Eulerianize several popular approximate real-space ray-tracing methods for complex-valued eikonals so that the real and imaginary parts of the eikonal function satisfy the classic real-space eikonal equation and a novel real-space advection equation, respectively, and we dub the resulting method the Eulerian partial-differential-equation method. We further develop highly efficient high-order methods to solve these two equations by using the factorization idea and the Lax-Friedrichs weighted essentially nonoscillatory schemes. Numerical examples demonstrate that our method yields highly accurate complex-valued eikonals, analogous to those from ray-tracing methods. Our methods can be useful for migration and tomography in attenuating media.

中文翻译：

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衰减介质中复值特征的欧拉偏微分方程方法

地球介质中的地震波通常会衰减，导致能量损失和相位失真。在高频渐近学领域，复值 eikonal 是描述衰减介质中波传播的重要组成部分，其中 eikonal 函数的实部和虚部分别捕获地震波的色散效应和振幅衰减。传统上，这种复值 eikonal 主要是通过在复空间中精确地跟踪光线或通过近似地在真实空间中跟踪光线来计算的，这样得到的 eikonal 在真实空间中的分布是不规则的。然而，地震数据处理方法，如叠前深度偏移和层析成像，通常需要均匀分布的复值征象。所以，我们开发了一个统一的框架来欧拉化几种流行的近似实空间光线追踪方法，以便复值特征函数的实部和虚部满足经典的实空间特征方程和新的实空间对流方程，我们将所得方法称为欧拉偏微分方程方法。我们通过使用分解思想和 Lax-Friedrichs 加权基本非振荡方案进一步开发了高效的高阶方法来求解这两个方程。数值例子表明，我们的方法产生了高度准确的复值 eikonals，类似于光线追踪方法中的那些。我们的方法可用于衰减介质中的迁移和断层扫描。