Elastic wavefield solutions computed by finite-difference (FD) methods in complex anisotropic media are essential elements of elastic reverse time migration and full-waveform inversion analyses. Cartesian formulations of such solution methods, though, face practical challenges when aiming to represent curved interfaces (including free-surface topography) with rectilinear elements. To forestall such issues, we have developed a general strategy for generating solutions of tensorial elastodynamics for anisotropic media (i.e., tilted transversely isotropic or even lower symmetry) in non-Cartesian computational domains. For the specific problem of handling free-surface topography, we define an unstretched coordinate mapping that transforms an irregular physical domain to a regular computational grid on which FD solutions of the modified equations of elastodynamics are straightforward to calculate. Our fully staggered grid (FSG) with a mimetic FD (MFD) (FSG + MFD) approach solves the velocity-stress formulation of the tensorial elastic wave equation in which we compute the stress-strain constitutive relationship in Cartesian coordinates and then transform the resulting stress tensor to generalized coordinates to solve the equations of motion. The resulting FSG + MFD numerical method has a computational complexity comparable with Cartesian scenarios using a similar FSG + MFD numerical approach. Numerical examples demonstrate that our solution can simulate anisotropic elastodynamic field solutions on irregular geometries; thus, it is a reliable tool for anisotropic elastic modeling, imaging, and inversion applications in generalized computational domains including handling free-surface topography.

中文翻译：

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各向异性介质的张量弹性动力学

在复杂的各向异性介质中通过有限差分 (FD) 方法计算的弹性波场解是弹性逆时偏移和全波形反演分析的基本要素。然而，当旨在用直线元素表示弯曲界面（包括自由表面地形）时，此类求解方法的笛卡尔公式面临实际挑战。为了防止此类问题，我们开发了一种通用策略，用于在非笛卡尔计算域中为各向异性介质（即，倾斜的横向各向同性或什至更低的对称性）生成张量弹性动力学的解决方案。对于处理自由表面地形的具体问题，我们定义了一个未拉伸的坐标映射，它将不规则的物理域转换为规则的计算网格，在该网格上可以直接计算修改后的弹性动力学方程的 FD 解。我们使用模拟 FD (MFD) (FSG + MFD) 方法的完全交错网格 (FSG) 解决了张量弹性波动方程的速度-应力公式，其中我们计算了笛卡尔坐标中的应力-应变本构关系，然后对所得结果进行变换应力张量到广义坐标以求解运动方程。由此产生的 FSG + MFD 数值方法具有与使用类似 FSG + MFD 数值方法的笛卡尔场景相当的计算复杂度。数值例子表明，我们的解决方案可以模拟不规则几何体上的各向异性弹性动力场解决方案；