Full waveform inversion (FWI) updates the subsurface model from an initial model by comparing observed and synthetic seismograms. Due to high nonlinearity, FWI is easy to be trapped into local minima. Extended domain FWI, including wavefield reconstruction inversion (WRI) and extended source waveform inversion (ESI) are attractive options to mitigate this issue. This paper makes an in-depth analysis for FWI in the extended domain, identifying key challenges and searching for potential remedies towards practical applications. WRI and ESI are formulated within the same mathematical framework using Lagrangian-based adjoint-state method with a special focus on time-domain formulation using extended sources, while putting connections between classical FWI, WRI and ESI: both WRI and ESI can be viewed as weighted versions of classic FWI. Due to symmetric positive definite Hessian, the conjugate gradient is explored to efficiently solve the normal equation in a matrix free manner, while both time and frequency domain wave equation solvers are feasible. This study finds that the most significant challenge comes from the huge storage demand to store time-domain wavefields through iterations. To resolve this challenge, two possible workaround strategies can be considered, i.e., by extracting sparse frequencial wavefields or by considering time-domain data instead of wavefields for reducing such challenge. We suggest that these options should be explored more intensively for tractable workflows.

翻译：全波形反演(FWI)通过对比观测地震图与合成地震图，从初始模型更新地下模型。由于高度非线性，FWI易陷入局部极小值。扩展域FWI（包括波场重构反演WRI和扩展源波形反演ESI）是缓解该问题的有效方案。本文对扩展域FWI进行深入分析，识别关键挑战并探索面向实际应用的潜在解决方案。采用基于拉格朗日伴随状态法的统一数学框架建立WRI与ESI模型，特别关注使用扩展源的时域公式表述，同时阐明经典FWI、WRI与ESI之间的关联：WRI与ESI均可视为经典FWI的加权形式。鉴于海森矩阵的对称正定性，探索采用共轭梯度法以无矩阵方式高效求解正规方程，且时域与频域波动方程求解器均适用。研究发现主要挑战来自迭代过程中存储时域波场所需的庞大存储需求。针对该问题，可考虑两种替代策略：提取稀疏频域波场，或采用时域数据替代波场以降低存储需求。建议在可操作的工作流中更深入探索这些方案。