Inexact line search method in full waveform inversion

Full waveform inversion (FWI) is a nonlinear data fitting process that can derive high-resolution model parameters through iteration. In this process, step length is related to inversion accuracy and computational efficiency. It can be calculated efficiently with the inexact line search method, which does not require a misfit function to achieve the exact minimum. This method is aimed toward obtaining the appropriate descent using evaluation conditions and initial step length. Moreover, it does not depend on the form of the misfit function. In the inexact line search method, the evaluation condition and initial step length are obviously important factors. In this work, the classical Armijo, Wolfe, and Goldstein evaluation conditions in solving optimization problems in mathematics are studied and compared in detail. Numerical examples from the synthetic data of the overthrust model show that the convergence characteristics of Armijo and Goldstein are similar and that the computational efficiency is high and conducive to seismic FWI. In addition, the adaptive Barzilai-Borwein (ABB) method is adopted in FWI. The ABB method maximizes the changes in model parameters and gradients to adaptively calculate the initial step length. The threshold value of the ABB method for the initial step length estimation is also studied to explore a suitable threshold value that can ensure that large and small step lengths are frequently adopted in FWI. Numerical examples from the synthetic data of the overthrust model demonstrate the validity of the ABB method. Moreover, the inversion is superior when the threshold value is less than 0.5.