ABSTRACT This paper is the second in a sequel of two papers and dedicated to the computation of paraxial rays and dynamic characteristics along the stationary rays obtained in the first paper. We start by formulating the linear, second‐order, Jacobi dynamic ray tracing equation. We then apply a similar finite‐element solver, as used for the kinematic ray tracing, to compute the dynamic characteristics between the source and any point along the ray. The dynamic characteristics in our study include the relative geometric spreading and the phase correction due to caustics (i.e. the amplitude and the phase of the asymptotic form of the Green's function for waves propagating in 3D heterogeneous general anisotropic elastic media). The basic solution of the Jacobi equation is a shift vector of a paraxial ray in the plane normal to the ray direction at each point along the central ray. A general paraxial ray is defined by a linear combination of up to four basic vector solutions, each corresponds to specific initial conditions related to the ray coordinates at the source. We define the four basic solutions with two pairs of initial condition sets: point–source and plane‐wave. For the proposed point–source ray coordinates and initial conditions, we derive the ray Jacobian and relate it to the relative geometric spreading for general anisotropy. Finally, we introduce a new dynamic parameter, similar to the endpoint complexity factor, presented in the first paper, used to define the measure of complexity of the propagated wave/ray phenomena. The new weighted propagation complexity accounts for the normalized relative geometric spreading not only at the receiver point, but along the whole stationary ray path. We propose a criterion based on this parameter as a qualifying factor associated with the given ray solution. To demonstrate the implementation of the proposed method, we use several isotropic and anisotropic benchmark models. For all the examples, we first compute the stationary ray paths, and then compute the geometric spreading and analyse these trajectories for possible caustics. Our primary aim is to emphasize the advantages, transparency and simplicity of the proposed approach.

中文翻译：

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3d 异质各向异性介质中的特征射线：第二部分——动力学

摘要 本文是两篇论文的续集中的第二篇，致力于计算沿第一篇论文中获得的静止光线的近轴光线和动态特性。我们首先制定线性、二阶、雅可比动态光线追踪方程。然后，我们应用类似的有限元求解器，如用于运动学射线追踪，来计算源和沿射线的任何点之间的动态特性。我们研究中的动态特性包括相对几何扩展和由于焦散引起的相位校正（即波在 3D 异质一般各向异性弹性介质中传播的格林函数的渐近形式的振幅和相位）。雅可比方程的基本解是在垂直于射线方向的平面内沿中心射线的每一点处的近轴射线的位移矢量。一般的近轴射线由最多四个基本矢量解的线性组合定义，每个解对应于与光源处的射线坐标相关的特定初始条件。我们用两对初始条件集定义了四个基本解：点源和平面波。对于建议的点源射线坐标和初始条件，我们推导出射线雅可比矩阵并将其与一般各向异性的相对几何扩展相关联。最后，我们引入了一个新的动态参数，类似于第一篇论文中提出的端点复杂性因子，用于定义传播波/射线现象的复杂性度量。新的加权传播复杂度不仅在接收点处，而且在整个静止射线路径上解释了归一化的相对几何扩展。我们提出了一个基于此参数的标准，作为与给定光线解决方案相关的限定因素。为了演示所提出方法的实现，我们使用了几个各向同性和各向异性基准模型。对于所有示例，我们首先计算静止光线路径，然后计算几何扩散并分析这些轨迹以寻找可能的焦散。我们的主要目标是强调所提议方法的优势、透明度和简单性。我们提出了一个基于此参数的标准，作为与给定光线解决方案相关的限定因素。为了演示所提出方法的实现，我们使用了几个各向同性和各向异性基准模型。对于所有示例，我们首先计算静止光线路径，然后计算几何扩散并分析这些轨迹以寻找可能的焦散。我们的主要目标是强调所提议方法的优势、透明度和简单性。我们提出了一个基于此参数的标准，作为与给定光线解决方案相关的限定因素。为了演示所提出方法的实现，我们使用了几个各向同性和各向异性基准模型。对于所有示例，我们首先计算静止光线路径，然后计算几何扩散并分析这些轨迹以寻找可能的焦散。我们的主要目标是强调所提议方法的优势、透明度和简单性。