Studies on modified limited-memory BFGS method in full waveform inversion

Full waveform inversion (FWI) is a non-linear optimization problem based on full-wavefield modeling to obtain quantitative information of subsurface structure by minimizing the difference between the observed seismic data and the predicted wavefield. The limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method is an effective quasi-Newton method in FWI due to its high inversion efficiency with low calculation and storage requirements. Like other conventional quasi-Newton methods, the approximation of the Hessian matrix in the L-BFGS method satisfies the quasi-Newton equation, which only exploits the gradient and model information while the available objective function value is neglected. The modified quasi-Newton equation considers the gradient, model, and objective function information together. Theoretical analysis reveals that the modified quasi-Newton equation is superior to the conventional quasi-Newton equation as it achieves higher-order accuracy in approximating the Hessian matrix. The modified L-BFGS method can be obtained by using the modified quasi-Newton equation to modify the L-BFGS method. This modification improves the accuracy of the Hessian matrix approximation with little increase of calculation for each iteration. We incorporate the modified L-BFGS method into FWI, numerical results show that the modified L-BFGS method has a higher convergence rate, achieves better inversion results, and has stronger anti-noise ability than the conventional L-BFGS method.