Full waveform inversion (FWI) is a large-scale nonlinear ill-posed problem for which computationally expensive Newton-type methods can become trapped in undesirable local minima, particularly when the initial model lacks a low-wavenumber component and the recorded data lacks low-frequency content. A modification to the Gauss-Newton (GN) method is proposed to address these issues. The standard GN system for multisource multireceiver FWI is reformulated into an equivalent matrix equation form, with the solution becoming a diagonal matrix rather than a vector as in the standard system. The search direction is transformed from a vector to a matrix by relaxing the diagonality constraint, effectively adding a degree of freedom to the subsurface offset axis. The relaxed system can be explicitly solved with only the inversion of two small matrices that deblur the data residual matrix along the source and receiver dimensions, which simplifies the inversion of the Hessian matrix. When used to solve the extended source FWI objective function, the Extended GN (EGN) method integrates the benefits of both model and source extension. The EGN method effectively combines the computational effectiveness of the reduced FWI method with the robustness characteristics of extended formulations and offers a promising solution for addressing the challenges of FWI. It bridges the gap between these extended formulations and the reduced FWI method, enhancing inversion robustness while maintaining computational efficiency. The robustness and stability of the EGN algorithm for waveform inversion are demonstrated numerically.

翻译：全波形反演（FWI）是一个大规模非线性不适定问题，传统牛顿类方法不仅计算成本高，还容易陷入不良局部极小值，尤其当初始模型缺乏低波数分量且记录数据缺少低频成分时更为显著。本文针对这些问题提出了一种改进的高斯-牛顿（GN）方法。首先将标准多震源多接收器FWI的GN系统重新表述为等效矩阵方程形式，其解为对角矩阵（区别于标准系统中的向量解）。通过放宽对角性约束，将搜索方向从向量转换为矩阵，从而在偏移距轴线上有效增加一个自由度。该松弛系统可通过仅反演两个小矩阵实现显式求解——这两个矩阵沿震源和接收器维度对数据残差矩阵进行去模糊处理，从而简化了Hessian矩阵的反演过程。当用于求解扩展震源FWI目标函数时，扩展高斯-牛顿（EGN）方法融合了模型扩展与震源扩展的双重优势。该方法有效结合了简化FWI方法的计算效率与扩展公式的鲁棒性特征，为解决FWI难题提供了可行方案。它弥合了扩展公式与简化FWI方法之间的鸿沟，在保持计算效率的同时增强了反演鲁棒性。数值实验验证了EGN算法在波形反演中的鲁棒性和稳定性。