  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.
