Full-waveform inversion (FWI) includes both migration and tomography modes. The tomographic component of the gradient from reflection data is usually much weaker than the migration component. In order to use the tomography mode to fix background velocity errors, it is necessary to extract the tomographic component from the gradient. Otherwise, the inversion will be dominated by the migration mode. We propose a method based on non-stationary smoothing to extract the tomographic component from the raw gradient. By analyzing the characteristics of the scattering angle filtering, the wavenumber of the tomographic component at a given frequency is seen to be smaller than that of the migration component. Therefore, low-wavenumber-pass filtering can be applied to extract the tomographic component. The low-wavenumber-pass smoothing filters are designed with Gaussian filters that are determined by the frequency of inversion, the model velocity, and the minimum scattering angle. Thus, this filtering is non-stationary smoothing in the space domain. Since this filtering is carried out frequency by frequency, it works naturally and efficiently for FWI based on frequency-domain modeling. Furthermore, as the maximum opening angle of the reflections in a typical acquisition geometry is much smaller than the minimum scattering angle for the tomographic component, which is generally set to 160°, there is a relatively large gap between the wavenumbers of the tomographic and migration components. In other words, the non-stationary smoothing can be applied once to a group of frequencies for time-domain FWI without leaking the migration component into the tomographic component. Analyses and numerical tests show that two frequency groups are generally sufficient to extract the tomographic component for the typical frequency range of time-domain FWI. The numerical tests also demonstrate that the non-stationary smoothing method is effective and efficient at extracting the tomographic component for reflection waveform inversion.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledgements: This research was supported by the National Key Research and Development Program of China (No. 2017YFC1500303), NSFC (Grant No. 41630209), NSF (Grant No. 1547228), Science Foundation of China University of Petroleum, Beijing (No. 2462018BJC001, No. 2462015YJRC012), and the FULLWAVE consortium. The authors would like to thank Editors A. Baumstein and J. Shragge, and four anonymous reviewers, for their comments and suggestions, which helped to improve and clarify the manuscript significantly. Finally, the authors would also like to show gratitude to Veronica E. Tremblay for the proofreading。