SIAM Review, Volume 66, Issue 3, Page 501-532, May 2024.
Waveform inversion is concerned with estimating a heterogeneous medium, modeled by variable coefficients of wave equations, using sources that emit probing signals and receivers that record the generated waves. It is an old and intensively studied inverse problem with a wide range of applications, but the existing inversion methodologies are still far from satisfactory. The typical mathematical formulation is a nonlinear least squares data fit optimization and the difficulty stems from the nonconvexity of the objective function that displays numerous local minima at which local optimization approaches stagnate. This pathological behavior has at least three unavoidable causes: (1) The mapping from the unknown coefficients to the wave field is nonlinear and complicated. (2) The sources and receivers typically lie on a single side of the medium, so only backscattered waves are measured. (3) The probing signals are band limited and with high frequency content. There is a lot of activity in the computational science and engineering communities that seeks to mitigate the difficulty of estimating the medium by data fitting. In this paper we present a different point of view, based on reduced order models (ROMs) of two operators that control the wave propagation. The ROMs are called data driven because they are computed directly from the measurements, without any knowledge of the wave field inside the inaccessible medium. This computation is noniterative and uses standard numerical linear algebra methods. The resulting ROMs capture features of the physics of wave propagation in a complementary way and have surprisingly good approximation properties that facilitate waveform inversion.

中文翻译：

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当数据驱动的降阶建模遇到全波形反演时


《SIAM 评论》，第 66 卷，第 3 期，第 501-532 页，2024 年 5 月。

波形反演涉及估计异质介质，通过波动方程的可变系数建模，使用发射探测信号的源和记录生成的波的接收器。这是一个古老且深入研究的反问题，具有广泛的应用前景，但现有的反演方法还远远不能令人满意。典型的数学公式是非线性最小二乘数据拟合优化，其困难源于目标函数的非凸性，该函数显示出许多局部最小值，在该处局部优化接近停滞。这种病态行为至少有三个不可避免的原因：（1）未知系数到波场的映射是非线性的、复杂的。 (2) 源和接收器通常位于介质的单侧，因此仅测量反向散射波。 (3)探测信号是频带受限的并且具有高频成分。计算科学和工程界开展了大量活动，试图减轻通过数据拟合估计介质的难度。在本文中，我们基于控制波传播的两个算子的降阶模型（ROM）提出了不同的观点。 ROM 被称为数据驱动，因为它们是直接根据测量结果计算的，无需了解不可访问介质内的波场。该计算是非迭代的并且使用标准数值线性代数方法。由此产生的 ROM 以互补的方式捕获波传播的物理特征，并具有令人惊讶的良好近似特性，有利于波形反演。