  Implementation of the standard full waveform inversion (FWI) poses
difficulties as the initial model offsets from the true model. The wavefield
reconstruction inversion (WRI) was proposed to mitigate these difficulties by
relaxing the wave-equation constraint. In this abstract, working on the
nonlinear term in the Hessian matrix of FWI, we develop a new approximate
Hessian as an Augmented Gauss-Newton (AGN) Hessian including second-order
derivative information. Moreover, we establish an intimate connection between
an updating formula which results from approximate solve of the Newton's method
with the AGN Hessian on the FWI problem and the WRI method. Our analysis opens
new perspectives for developing efficient algorithms for FWI based on the
Newton's method and highlights the importance of the nonlinear term in the
Hessian matrix, which is ignored in most cases.
