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 torwards 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）通过比较观测与合成地震图，从初始模型更新地下模型。由于高度非线性，FWI容易陷入局部极小值。扩展域FWI（包括波场重建反演WRI和扩展源波形反演ESI）是缓解该问题的有吸引力方案。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补救措施。基于拉格朗日的伴随状态法，在统一数学框架下公式化WRI和ESI，重点聚焦使用扩展源的时域公式化，同时建立经典FWI、WRI与ESI之间的关联：WRI和ESI均可视为经典FWI的加权版本。由于海森矩阵对称正定，本文探索共轭梯度法以无矩阵方式高效求解正规方程，同时时域与频域波动方程求解器均具可行性。研究发现，最显著的挑战来自迭代过程中存储时域波场所需的庞大存储需求。为应对该挑战，可考虑两种可能的规避策略：提取稀疏频率域波场，或使用时域数据替代波场以降低存储压力。我们建议应在可行工作流中对这些方案进行更深入探索。