Nonlinear least squares data-fitting driven by physical process simulation is a classic and widely successful technique for the solution of inverse problems in science and engineering. Known as "Full Waveform Inversion" in application to seismology, it can extract detailed maps of earth structure from near-surface seismic observations, but also suffers from a defect not always encountered in other applications: the least squares error function at the heart of this method tends to develop a high degree of nonconvexity, so that local optimization methods (the only numerical methods feasible for field-scale problems) may fail to produce geophysically useful final estimates of earth structure, unless provided with initial estimates of a quality not always available. A number of alternative optimization principles have been advanced that promise some degree of release from the multimodality of Full Waveform Inversion, amongst them Wavefield Reconstruction Inversion, the focus of this paper. Applied to a simple 1D acoustic transmission problem, both Full Waveform and Wavefield Reconstruction Inversion methods reduce to minimization of explicitly computable functions, in an asymptotic sense. The analysis presented here shows explicitly how multiple local minima arise in Full Waveform Inversion, and that Wavefield Reconstruction Inversion can be vulnerable to the same "cycle-skipping" failure mode.

中文翻译：

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波场重建反演：一个例子

由物理过程模拟驱动的非线性最小二乘数据拟合是科学和工程中求解逆问题的经典且广泛成功的技术。在地震学应用中被称为“全波形反演”，它可以从近地表地震观测中提取地球结构的详细地图，但也存在其他应用中不常见的缺陷：最小二乘误差函数是其核心方法往往会产生高度的非凸性，因此局部优化方法（唯一适用于现场规模问题的数值方法）可能无法产生对地球结构有用的地球物理最终估计，除非提供了并非总是可用的质量的初始估计. 已经提出了许多替代优化原则，这些原则有望在一定程度上摆脱全波形反演的多模态，其中波场重建反演是本文的重点。应用于简单的一维声学传输问题时，全波形和波场重建反演方法在渐近意义上都简化为显式可计算函数的最小化。这里提出的分析明确显示了全波形反演中如何出现多个局部最小值，并且波场重建反演可能容易受到相同的“跳周期”故障模式的影响。在渐近的意义上，全波形和波场重建反演方法都减少到显式可计算函数的最小化。这里提出的分析明确显示了全波形反演中如何出现多个局部最小值，并且波场重建反演可能容易受到相同的“跳周期”故障模式的影响。在渐近的意义上，全波形和波场重建反演方法都减少到显式可计算函数的最小化。这里提出的分析明确显示了全波形反演中如何出现多个局部最小值，并且波场重建反演可能容易受到相同的“跳周期”故障模式的影响。