Full waveform inversion (FWI) is a powerful tool to reconstruct subsurface parameters. This highly nonlinear inverse problem is normally solved iteratively to minimize a misfit function, which is usually defined as the distance between the observed and predicted data, by gradient-based method or Newton type method. Incorporating more nonlinearity within each update in FWI, especially for multiparameter reconstruction, may have very important consequences for convergence rates and discrimination of different parameter classes. In this study, we focus on acoustic media with variable density, and the goal is to simul-taneously update velocity and density, other parameterization is also discussed. We start from the physical interpretation of both the gradient and the Hessian of the misfit function, and derive one approach from the Newton method, to include the additional term of the Hessian, which contains the second-order partial derivative of the wavefield and related to the second-order scattering, into the gradient, to construct a new descent direction. A matrix-free scheme is used to efficiently calculate the product of the Hessian and a vector.
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