Full waveform inversion (FWI) is a recent powerful method in the area of seismic imaging where it used for reconstructing high-resolution images of the subsurface structure from local measurements of the seismic wavefield. This method consists in minimizing the distance between the predicted and the recorded data. The predicted data are computed as the solution of a wave-propagation problem. In this study, we investigate two algorithms Gauss-Newton and L-BFGS for solving FWI problems. We compare these algorithms in terms of its robustness and speed of convergence. Also, we implement the Tikhonov regularization for assisting convergence. Numerical results show that Gauss-Newton method performs better than L-BFGS method in terms of convergence of l2 -norm of misfit function gradient since it provides better convergence as well as the quality of high resolution constructed images. Yet, L-BFGS outperforms Gauss-Newton in terms of computationally efficiency and feasibility for FWI. © 2021, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.