Full waveform seismic inversion (FWI) in the viscoelastic regime entails the task of identifying parameters in the viscoelastic wave equation from partial waveform measurements. Traditionally, one frames this nonlinear problem as an operator equation for the parameter‐to‐state map. Alternatively, in an all‐at‐once approach, one augments the nonlinear operator by the viscoelastic wave equation as an additional component and considers the states as additional variables. Hence, parameters and states are sought for simultaneously. In this article, we give a mathematically rigorous all‐at‐once version of FWI in a functional analytical formulation. Further, the corresponding nonlinear map is shown to be Fréchet differentiable, and the adjoint operator of the Fréchet derivative is given in an explicit way suitable for implementation in a Newton‐type/gradient‐based regularization scheme.

中文翻译：

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粘弹性状态下一次全部波形反转的方法

粘弹性体系中的全波形地震反演（FWI）要求根据部分波形测量结果来确定粘弹性波动方程中的参数。传统上，将这种非线性问题构架为参数到状态图的算子方程。另外，在一次全部方法中，通过将粘弹性波动方程作为附加分量来扩充非线性算子，并将状态视为附加变量。因此，同时寻找参数和状态。在本文中，我们在功能分析公式中给出了数学上严格的一次性FWI版本。此外，相应的非线性映射显示为Fréchet可微的，