Full Waveform Inversion (FWI) is a standard algorithm in seismic imaging. Its implementation requires the a priori choice of a number of "design parameters", such as the positions of sensors for the actual measurements and one (or more) regularisation weights. In this paper we describe a novel algorithm for determining these design parameters automatically from a set of training images, using a (supervised) bilevel learning approach. In our algorithm, the upper level objective function measures the quality of the reconstructions of the training images, where the reconstructions are obtained by solving the lower level optimisation problem -- in this case FWI. Our algorithm employs (variants of) the BFGS quasi-Newton method to perform the optimisation at each level, and thus requires the repeated solution of the forward problem -- here taken to be the Helmholtz equation. This paper focuses on the implementation of the algorithm. The novel contributions are: (i) an adjoint-state method for the efficient computation of the upper-level gradient; (ii) a complexity analysis for the bilevel algorithm, which counts the number of Helmholtz solves needed and shows this number is independent of the number of design parameters optimised; (iii) an effective preconditioning strategy for iteratively solving the linear systems required at each step of the bilevel algorithm; (iv) a smoothed extraction process for point values of the discretised wavefield, necessary for ensuring a smooth upper level objective function. The algorithm also uses an extension to the bilevel setting of classical frequency-continuation strategies, helping avoid convergence to spurious stationary points. The advantage of our algorithm is demonstrated on a problem derived from the standard Marmousi test problem.

翻译：全波形反演（FWI）是震源成像中的标准算法。其实现需要预先选择若干“设计参数”，例如实际测量中传感器的位置以及一个（或多个）正则化权重。本文提出一种新颖算法，通过（监督式）双层学习方法，从一组训练图像中自动确定这些设计参数。在我们的算法中，上层目标函数衡量训练图像重建的质量，其中重建结果通过求解下层优化问题——即FWI——获得。该算法采用（变体）BFGS拟牛顿法在每一层执行优化，因此需要反复求解正问题——此处为亥姆霍兹方程。本文聚焦于算法的实现。其新颖贡献包括：（i）一种用于高效计算上层梯度的伴随状态法；（ii）对双层算法进行复杂度分析，统计所需亥姆霍兹方程求解次数，并证明该次数与优化的设计参数数量无关；（iii）一种有效的预处理策略，用于迭代求解双层算法每一步所需的线性系统；（iv）一种离散化波场点值的平滑提取方法，这对于确保上层目标函数光滑性至关重要。该算法还将经典频率延拓策略扩展至双层框架，有助于避免收敛到伪稳态点。我们通过基于标准Marmousi测试问题推导的算例，展示了该算法的优势。