Improved Hessian-Free optimization for acoustic full waveform inversion

Abstract

Seismic full waveform inversion (FWI) is a method to reconstruct material properties of subsurface structures by minimizing the objective function based on residuals between modeled and observed seismic data. For seismic inverse problem, various kinds of optimization methods have been introduced. The truncated Newton method, also known as the Hessian-free (HF) optimization method, has been chosen to optimize large-scale inverse problems. The HF does not need to explicitly compute, store and invert the Hessian matrix. Instead of the Hessian matrix itself, the product of Hessian matrix and column vector is used for the linear conjugate-gradient loop during FWI process. To calculate the product of the Hessian matrix and column vector, the second-order adjoint (SOA) method or finite difference approximation (FDA) method has been widely used. The FDA is easy and intuitive to use in the linear conjugate-gradient method compared with SOA. The accuracy of FDA is dependent on not only the approximation interval but also the inversion settings, such as the model parameter, initial model, frequencies, etc. To overcome dependency of HF optimization on the approximation method and inversion setting, an improved method is proposed for a stable HF optimization method. The derivations of the improved method are based on not the FDA method but the limit of a function, which is independent of epsilon value. In other words, the improved HF method stably and accurately approximates the matrix-vector product of the Hessian matrix and column vector without any selection of epsilon value. In addition, computational cost of the improved HF optimization method is much lower than the conventional HF optimization method because additional construction and factorization of modeling operator are not needed during the linear conjugate-gradient method in the improved HF optimization method. To demonstrate the feasibility of the improve HF method, numerical examples for the Marmousi and acoustic Overthrust models are performed. Numerical examples indicate that the improved HF method shows better computational efficiency and stability than the conventional HF method without any degradation of inversion results.