| Low-frequency seismic data has traditionally been of great importance in full waveform inversion in order to mitigate problems of local minima. However, here we perform full waveform inversion with only high-frequency seismic data. Motivated by the Holder stability estimates for geometric data, we introduce an objective function, defined by the residual of the structure tensor, which penalizes the difference in the directionality between the observed and synthetic data. For 2D seismic data, the structure tensor is a 2 x 2 symmetric matrix at each point of the data, and contains the convolution of a regularization function with data's first order partial derivatives. Since the structure tensor breaks down in the presence of caustics, we apply it to directional boundary sources, which are defined by Gaussian wave packets which can be used to decompose the data. Each directional source contains no intersecting wavefronts. A variable regularization function contained in the structure tensor then naturally leads us to an iterative scale-evolving inversion scheme. We apply the method to piecewise smooth models, and the results show that the structure tensor approach using just high-frequency data is able to recover discontinuities and smooth velocities in these types of models. |
