Pure and Applied Geophysics

, Volume 176, Issue 12, pp 5315–5335 | Cite as

Reflection-Waveform Inversion Regularized with Structure-Oriented Smoothing Shaping

  • Gang Yao
  • Nuno V. da Silva
  • Di WuEmail author


Restricted by the limited length of the receiver arrays, estimating the seismic properties of deep targets requires inverting reflection data. Reflection-waveform inversion (RWI) utilizes reflection data to recover the background velocity. RWI splits the Earth model into a long-wavelength background velocity and short-wavelength reflectors. Because of the limited apertures of reflection data illuminating targets, and the trade-off between the depth of reflectors and the average velocity above the reflectors, RWI is inherently ill-posed and contains a large null space. One manifestation of the ill-posedness is the existence of characteristic artificial blob-shaped or column-like anomalies in the velocity models estimated with RWI. That erroneous background velocity, which still fits the travel time of reflection arrivals in the recorded data, can lead to inaccurate velocity models and distorted migration images in the subsequent processes. In order to mitigate those artifacts and improve the conditioning of RWI, we incorporate a piece of prior information into RWI, i.e. slow property variation along geological structures. This prior information is expressed mathematically as structure-oriented smoothing. We achieved this smoothing by solving an anisotropic diffusion equation with a finite-difference method. Thus, we formulated a method for RWI regularized with structure-oriented smoothing shaping by alternating the following three steps: building temporary reflectors, updating background velocity, and solving the diffusion equation. We successfully tested this scheme on the Marmousi model and a modified overthrust model. The results show that the regularized RWI yields a stable and accurate background velocity, and it is more effective than conventional Tikhonov regularization approaches. The estimated background model leads to a high-quality final velocity model when used as an initial model in conventional full-waveform inversion.


Full-waveform inversion reflection-waveform inversion structure-oriented smoothing shaping regularization diffusion equation 



The authors are grateful to the National Key Research and Development Program of China (No. 2017YFC1500303), NSFC (Grant Nos. 41504106, 41630209), NSF (Grant No. EAR-1547228), Science Foundation of China University of Petroleum, Beijing (No. 2462018BJC001), and the sponsors of the FULLWAVE consortium for supporting this research. The authors would like to thank the editor, Professor C. Farquharson, and two anonymous reviewers for their comments and suggestions, which helped to improve and clarify the manuscript significantly. Finally, the authors would also like to express their gratitude to Dr. J. Cao for the discussion on Tikhonov regularization.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.The Unconventional Natural Gas Institute, State Key Laboratory of Petroleum Resources and ProspectingChina University of Petroleum (Beijing)BeijingChina
  2. 2.Department of Earth ScienceRice UniversityHoustonUSA
  3. 3.Department of Earth Science and EngineeringImperial College LondonLondonUK

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