Exact line search method for using the L1-norm misfit function in full waveform inversion

Full waveform inversion (FWI) is a non-linear inverse problem that can be sensitive to noise. The tolerance of the noise-interference characteristics depends on the types of misfit functions. To date, different misfit functions, such as the least-squares norm (L2), the least-absolute-value norm (L1), and combinations of the two (e.g., the Huber and hybrid criteria), have been applied to FWI. The L2 norm is highly sensitive to non-Gaussian errors in the data and gives rise to high-amplitude artifacts in reconstructed models. For non-Gaussian noise data, the L1 norm and the Huber and hybrid criteria always reliably reconstruct models. However, the Huber and hybrid criteria require tedious error investigations to estimate their threshold criterion. Thus, the L1 norm is adopted here to improve the anti-noise ability of the FWI. The step length is closely related to the misfit function, and an optimal step-length estimation method can rapidly make the FWI algorithm reach the global minimum, with a reduced number of iterations and fewer extra forward modeling simulations during each iteration. The step length can usually be obtained using the exact or inexact line search method. Generally, the exact line search method is faster than the inexact one. Therefore, we derived an exact line search method for the L1 norm in the FWI process. Its effectiveness was tested using noise-free data from Overthrust and the SEG/EAGE salt models. The results demonstrate that this method can recover high-resolution velocity models with low computational costs. Numerical tests using the synthetic Overthrust model contaminated by strong noise were used to further validate the robustness of this exact line search method.