## Preconditioning for the Hessian-free Gauss-Newton full-waveform inversion

### Wenyong Pan, Kristopher A. H. Innanen, Wenyuan Liao

Full-waveform inversion (FWI) has emerged as a powerful strategy for estimating the subsurface model parameters by iteratively minimizing the difference between the synthetic data and observed data. The gradient-based methods promise to converge globally but suffer from slow convergence rate. The Newton-type methods provide a quadratic convergence, but the computation, storage and inversion of the Hessian are beyond the current computation ability for large-scale inverse problem. The Hessian-free (HF) optimization method represents an attractive alternative to these above-mentioned optimization methods. At each iteration, it obtains the search direction by approximately solving the Newton linear system using a conjugate-gradient (CG) algorithm with a matrix-free fashion. One problem of the HF optimization method is that the CG algorithm requires many iterations. The main goal of this paper is to accelerate the HF FWI by preconditioning the CG algorithm. In this research, different preconditioning schemes for the HF Gauss-Newton optimization method are developed. The preconditioners are designed as Hessian approximations (e.g., diagonal pseudo-Hessian and diagonal Gauss-Newton Hessian) or its inverse approximations. We also developed a new pseudo diagonal Gauss-Newton Hessian approximation for preconditioning based on the reciprocal property of the Greenâ€™s function. Furthermore, a quasi-Newton l-BFGS inverse Hessian approximation preconditioner with the diagonal Hessian approximation as initial guess is proposed and developed. Several numerical examples are solved to demonstrate the effectiveness of the preconditioning schemes. It is concluded that the quasi-Newton l-BFGS preconditioning scheme with the pseudo diagonal Gauss-Newton Hessian as initial guess shows the best performances in speeding up the HF Gauss-Newton FWI, improving the convergence rate and reducing the computation burden.