Full-waveform inversion with a vertical total variation constraint based on the Hinge loss function

Full waveform inversion can provide a high-precision velocity model for pre-stack depth migration imaging. However, this method has a non-linearity and dependence on the initial velocity model. Especially under the complex conditions of highly variable geology and discontinuous velocity changes, the degree of its nonlinearity increases, leading to an inversion trap in a local minima, lowering the accuracy of inversion. The total variation regularization method, a kind of unsmooth constraints which is widely used in image denoising, can preserve discontinuity interfaces and edges of the image during the denoising. In this paper we propose a full waveform inversion using the vertical total variation constraint based on the hinge loss function. By virtue of this constraint, we use the hinge loss function to control the model's updating direction, and get the iteration format of the optimization problem with the primal-dual hybrid gradient algorithm. This method improves the reconstruction accuracy of underground discontinuous interfaces effectively, and reduces the degree of dependence on the initial velocity model. Numerical examples demonstrate that this new full waveform inversion method is effective. Compared with the conventional full waveform inversion method, it can reconstruct the discontinuous interface effectively based on total variation constraint. Especially the reconstructing effect of the highspeed body edge is more obvious. Although the dependence of the initial velocity model of this method remains strong, It can reduce the degree of dependence on the initial velocity model and get a same accurate velocity model from a poor initial model eventually by means of cyclic iteration. In particular the high-speed body edge and discontinuous interfaces can be reconstructed accurately.