Full Waveform Inversion (FWI) is a large-scale nonlinear ill-posed problem for which implementation of the Newton-type methods is computationally expensive. Moreover, these methods can trap in undesirable local minima when the starting model lacks low-wavenumber part and the recorded data lack low-frequency content. In this paper, the Gauss-Newton (GN) method is modified to address these issues. We rewrite the GN system for multisoure multireceiver FWI in an equivalent matrix equation form whose solution is a diagonal matrix, instead of a vector in the standard system. Then we relax the diagonality constraint, lifting the search direction from a vector to a matrix. This relaxation is equivalent to introducing an extra degree of freedom in the subsurface offset axis for the search direction. Furthermore, it makes the Hessian matrix separable and easy to invert. The relaxed system is solved explicitly for computing the desired search direction, requiring only inversion of two small matrices that deblur the data residual matrix along the source and receiver dimensions. Application of the Extended GN (EGN) method to solve the extended-source FWI leads to an algorithm that has the advantages of both model extension and source extension. Numerical examples are presented showing robustness and stability of EGN algorithm for waveform inversion.

翻译：全波形反演（FWI）是一个大规模非线性不适定问题，采用牛顿型方法实现时计算成本高昂。此外，当初始模型缺乏低频波数成分且记录数据缺少低频信息时，这些方法可能陷入不理想的局部极小值。本文对高斯-牛顿（GN）方法进行了改进以解决上述问题。我们将多震源多接收器FWI的GN系统重写为等效的矩阵方程形式，其解为对角矩阵，而非标准系统中的向量形式。随后我们松弛了对角性约束，将搜索方向从向量提升至矩阵。这一松弛相当于在搜索方向的子表面偏移轴中引入额外的自由度。此外，它使Hessian矩阵变得可分离且易于求逆。通过显式求解松弛系统计算所需搜索方向，仅需对两个小矩阵求逆，以沿震源和接收器维度对数据残差矩阵进行去模糊处理。将扩展GN（EGN）方法应用于扩展源FWI，形成了一种兼具模型扩展和震源扩展优势的算法。数值算例展示了EGN算法在波形反演中的鲁棒性和稳定性。