Research of step-length estimation methods for full waveform inversion in time domain

Full waveform inversion (FWI) is a powerful tool for reconstructing high-resolution subsurface parameters estimated by iteratively minimising the difference between calculated and observed data. Step-length estimation is a key step in the successful implementation of the FWI algorithm. An optimal step-length value rapidly causes the FWI algorithm to reach the global minimum with reduced iterations and fewer extra forward modelling simulations during each iteration. Step-length can typically be calculated using an inexact or an exact line-search method. The backtracking line-search method (BLSM) is a typical inexact method. Initial methods of guessing the step-length and evaluation conditions determine the efficiency of a BLSM scheme. Here, we propose a quadratic extrapolation value as the initial guess in a BLSM scheme, and then compare it with other initial-guess approaches by using the first Wolfe condition to evaluate step-length values. Exact line-search methods include a parabolic fitting search method through three points (PFSM-3), a parabolic fitting search method through two points (PFSM-2) and the analytical step-length method (ASLM). To find optimal and stable step-length estimation methods for FWI, we compare four step-length estimation methods: BLSM, PFSM-3, PFSM-2 and ASLM. Numerical examples using synthetic data demonstrate that quadratic extrapolation values perform better than first-order change and adaptive values in the BLSM scheme, in terms of resolution of the reconstructed model and computational costs. Of the four step-length estimation methods, ASLM and BLSM are both efficient for noise-free data, and robust ASLM is more efficient for noisy data. However, PFSM-2 and PFSM-3 are less efficient because of low accuracy and high computational cost.