This paper is devoted to theoretical and numerical investigation of the local minimum issue in seismic full waveform inversion (FWI). This paper provides a mathematical analysis of optimal transportation (OT) type objective function's differentiability and proves that the gradient obtained in the adjoint-state method does not depend on the particular choice of the Kantorovich potentials. A novel approach using the softplus encoding method is presented to generalize and impose the OT metric on FWI. This approach improves the convexity of the objective function and mitigates the cycle-skipping problem. The effectiveness of the proposed method is demonstrated numerically on an inversion task with the benchmark Marmousi model.

中文翻译：

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最优传输和Softplus编码的地震反演分析

本文致力于地震全波形反演（FWI）中局部极小值问题的理论和数值研究。本文对最优运输 (OT) 型目标函数的可微性进行了数学分析，并证明了伴随状态方法获得的梯度不依赖于 Kantorovich 势的特定选择。提出了一种使用 softplus 编码方法的新方法来对 FWI 泛化和强加 OT 度量。这种方法改善了目标函数的凸性并减轻了循环跳跃问题。所提出方法的有效性在具有基准 Marmousi 模型的反演任务上得到了数值证明。