  Full-waveform inversion (FWI) is an effective method for imaging subsurface
properties using sparsely recorded data. It involves solving a wave propagation
problem to estimate model parameters that accurately reproduce the data. Recent
trends in FWI have led to the development of extended methodologies, among
which source extension methods leveraging reconstructed wavefields to solve
penalty or augmented Lagrangian (AL) formulations have emerged as robust
algorithms, even for inaccurate initial models. Despite their demonstrated
robustness, challenges remain, such as the lack of a clear physical
interpretation, difficulty in comparison, and reliance on difficult-to-compute
least squares (LS) wavefields. This paper is divided into two critical parts.
In the first, a novel formulation of these methods is explored within a unified
Lagrangian framework. This novel perspective permits the introduction of
alternative algorithms that employ LS multipliers instead of wavefields. These
multiplier-oriented variants appear as regularizations of the standard FWI, are
adaptable to the time domain, offer tangible physical interpretations, and
foster enhanced convergence efficiency. The second part of the paper delves
into understanding the underlying mechanisms of these techniques. This is
achieved by solving the FWI equations using iterative linearization and inverse
scattering methods. The paper provides insight into the role and significance
of Lagrange multipliers in enhancing the linearization of FWI equations. It
explains how different methods estimate multipliers or make approximations to
increase computing efficiency. Additionally, it presents a new physical
understanding of the Lagrange multiplier used in the AL method, highlighting
how important it is for improving algorithm performance when compared to
penalty methods.
