地震成像与最优传输
Seismic Imaging and Optimal Transport
AI总结：
本文研究了地震成像中使用最优传输理论替代传统最小二乘法以提高FWI性能，提出用二次Wasserstein度量作为新的误差点估计函数。
摘要：
地震学已是一门活跃的科学。大约50年前，地球振动的测量在时间和空间上更加精确频繁。完整的波动场可以确定，偏微分方程（PDE）开始用于寻找地球内部性质的反演过程。本文将简要回顾早期技术，但主要聚焦于声学公式中的全波形反演（FWI）。FWI是一种PDE约束优化问题，在正向波动方程中调整变量速度，使解与表面测量数据匹配。最小化不匹配通常与共轭状态方法结合，后者包括求解共轭波动方程。最小二乘范数是传统的目标函数，用于测量模拟与测量数据的差异，但常常导致陷入局部极小值。一种缓解方法是选择具有更好凸性性质的其他误差点函数。本文提出使用二次Wasserstein度量作为FWI中的新误差点函数。二次Wasserstein度量的最优映射可通过求解Monge-Ampere方程来计算。将讨论使用最优传输相对于最小二乘范数的优势定理，并展示多个大规模计算示例。
英文摘要：
Seismology has been an active science for a long time. It changed character about 50 years ago when the earth's vibrations could be measured on the surface more accurately and more frequently in space and time. The full wave field could be determined, and partial differential equations (PDE) started to be used in the inverse process of finding properties of the interior of the earth. We will briefly review earlier techniques but mainly focus on Full Waveform Inversion (FWI) for the acoustic formulation. FWI is a PDE constrained optimization in which the variable velocity in a forward wave equation is adjusted such that the solution matches measured data on the surface. The minimization of the mismatch is usually coupled with the adjoint state method, which also includes the solution to an adjoint wave equation. The least-squares norm is the conventional objective function measuring the difference between simulated and measured data, but it often results in the minimization trapped in local minima. One way to mitigate this is by selecting another misfit function with better convexity properties. Here we propose using the quadratic Wasserstein metric as a new misfit function in FWI. The optimal map defining the quadratic Wasserstein metric can be computed by solving a Monge-Ampere equation. Theorems pointing to the advantages of using optimal transport over the least-squares norm will be discussed, and a number of large-scale computational examples will be presented.
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