论文标题
3D异质各向异性媒体中的本征：第四部分 - 从旅行期Hessian传播的几何形状
Eigenrays in 3D heterogeneous anisotropic media: Part IV -- Geometric spreading from traveltime Hessian
论文作者
论文摘要
考虑到3D异质性各向异性介质，在本研究的第三部分中，我们使用有限元方法获得了两个固定端点之间的固定射线路径的有限元溶液。它涉及计算全局（全节点）旅行时梯度向量和Hessian矩阵，相对于节点位置和方向组件。已解决的固定射线的全球旅行时间Hessian在沿射线沿线获得动态特征也起着重要作用。它是一个带矩阵，包括所有节点和节点对的旅行时间的空间，定向和混合的第二个衍生物。在本研究的第五部分，VI和VII中，我们明确使用全球旅行时间Hessian来求解动态射线追踪方程，以在沿射线的每个点上获得几何扩展。此外，动态射线追踪使沿射线识别/分类苛性含量成为可能。在此部分（第四部分）中，我们提出了一种原始的两阶段方法，用于计算整个射线路径的相对几何扩散而不明确执行动态射线跟踪。第一阶段由一种有效的算法组成，该算法将已经计算的全球旅行时间Hessian减少到了端点的空间旅行时间Hessian。第二阶段涉及应用已知的工作流以使用端点旅行时间Hessian计算几何扩张。请注意，此部分中提出的方法（第四部分）未提供有关苛性剂的信息，也不提供固定射线路径中间点的几何扩散。在第七部分中介绍的整个示例中，我们通过比较了基于旅行时间的Hessian还原与从动态射线示踪计算的方法，通过比较了该方法计算出的几何扩散，证明了本部分中提出的方法的准确性。
Considering 3D heterogeneous anisotropic media, in Part III of this study we obtain the finite-element solution for the stationary ray path between two fixed endpoints using the finite element approach. It involves computation of the global (all-node) traveltime gradient vector and Hessian matrix, with respect to the nodal location and direction components. The global traveltime Hessian of the resolved stationary ray also plays an important role for obtaining the dynamic characteristics along the ray. It is a band matrix that includes the spatial, directional and mixed second derivatives of the traveltime at all nodes and nodal pairs. In Parts V, VI and VII of this study we explicitly use the global traveltime Hessian for solving the dynamic ray tracing equation, to obtain the geometric spreading at each point along the ray. Moreover, the dynamic ray tracing makes it possible to identify/classify caustics along the ray. In this part (Part IV) we propose an original two-stage approach for computing the relative geometric spreading of the entire ray path without explicitly performing the dynamic ray tracing. The first stage consists of an efficient algorithm for reducing the already computed global traveltime Hessian into a endpoint spatial traveltime Hessian. The second stage involves application of a known workflow for computing the geometrical spreading using the endpoint traveltime Hessian. Note that the method proposed in this part (Part IV) does not deliver information about caustics, nor the geometric spreading at intermediate points of the stationary ray path. Throughout the examples presented in Part VII we demonstrate the accuracy of the method proposed in this part over a set of isotropic and anisotropic examples by comparing the geometric spreading computed from this method based on the traveltime Hessian reduction with the one computed from the dynamic ray tracing.
