Full-waveform inversion (FWI) is rapidly becoming a standard tool for high-resolution velocity estimation. However, the application of this method is usually limited to low frequencies due to the high computational cost of the wavefield propagation and the inversion scheme. To mitigate this problem, we have developed a rapid expansion method (REM) for numerical wavefield extrapolation inside the FWI workflow. This method approximated the partial time derivative of the wave equation using Chebyshev polynomials instead of the conventional finite-difference (FD) approximation. This allowed the REM to accurately propagate wavefields with bigger time steps, thus improving the computational efficiency of FWI. We have compared high-frequency FWI results using REM and the traditional FD approximation of the time derivative to illustrate the ability of REM to remain stable and free of numerical dispersion noise even with large grid and time samplings. In addition, we studied a step-length estimation scheme with the objective of avoiding an extra forward wavefield propagation during the line search at each FWI iteration. In this scheme, we estimated the step-length value based on prior iterations and validated this value using the Wolfe conditions. If the value was accepted, then the forward modeling was availed in the next iteration and no extra propagation was necessary. We tried seven different step-length estimation methods, from which we highlighted the adaptive Barzilai-Borwein method when combined with the steepest-descent inversion scheme, and the unitary step length for the L-BFGS algorithm. Through synthetic numerical results, we showed how this scheme could achieve convergence, while keeping the number of extra forward modelings way below the number of FWI iterations.

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