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 non-convexity 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 non-iterative 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.

翻译：波形反演旨在通过发射探测信号的震源和记录生成波的接收器，来估计由变系数波动方程建模的非均匀介质。这是一个历史悠久且被广泛研究但仍未达到理想效果的反问题，具有广泛的应用领域。其典型数学形式是非线性最小二乘数据拟合优化问题，困难源于目标函数的非凸性——该函数展现出大量局部极小值，导致局部优化方法在此停滞。这种病态行为至少存在三个不可避免的原因：（1）从未知系数到波场的映射复杂且非线性；（2）震源和接收器通常位于介质的同一侧，因此仅能测量到背向散射波；（3）探测信号带宽有限且包含高频成分。计算科学与工程领域正积极开展大量研究，致力于缓解通过数据拟合估计介质的困难。本文提出基于控制波传播的两个算子的降阶模型的不同视角。这些ROM被称为"数据驱动型"，因其直接由测量数据计算得出，无需了解不可达介质内部的波场信息。该计算采用标准数值线性代数方法且无需迭代。所得ROM以互补方式捕捉波传播的物理特征，并展现出令人惊讶的良好近似性质，从而促进了波形反演的实现。