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Pure and Applied Geophysics

, Volume 176, Issue 1, pp 215–234 | Cite as

A Folding Calculation Method Based on the Preconditioned Conjugate Gradient Inversion Algorithm of Gravity Gradient Tensor

  • Yu Tian
  • Xiaoping KeEmail author
  • Yong Wang
Article

Abstract

To solve the non-uniqueness problem of gravity gradient inversion, we proposed a folding calculation method based on preconditioned conjugate gradient inversion. Compared with the original algorithm in which the entire study area is taken as the research subject and all grids are used simultaneously in the inversion, the proposed folding method divides the research area into several sub-areas. A prism unit from any of the four corner grids is selected for the first iteration, whose density anomaly result is taken as the initial density anomaly for the next iteration of the same sub-area. The folding in the left–right and up–down directions takes turns during the calculation until the inversion calculation has covered the entire research area. This folding algorithm demonstrates strong regularity. The inversion results of multiple synthetic models show that the folding calculation method performs multiple parameter corrections in the initial model. Meanwhile, on the basis that the relative fitting of standard errors of the observed values satisfies the convergence condition, the model errors are constrained, and thus, the model errors of the inversion results are consequently reduced. The 3-D density anomaly pattern over the Craton area in North China was obtained via a joint inversion of the four components (\(\varvec{T}_{\text{xx}} ,\)\(\varvec{T}_{\text{xz}} ,\)\(\varvec{T}_{\text{yy}} ,\) and \(\varvec{T}_{\text{zz}}\)) of the Gravity field and steady-state Ocean Circulation Explorer L2 gravity gradient after preprocessing. We compared the inversion results from the folding calculation method and the original method, and performed detailed analysis and discussion on the inversion results with existing geological and geophysical data. Our analysis shows that the improved calculation method is effectively applicable to the inversion of measured gravity gradient data, and the inversion results provide more detailed and reliable pattern information for the density anomaly.

Keywords

Gravity gradient tensor preconditioned conjugate gradient inversion algorithm folding calculation method North China Craton 

Notes

Acknowledgements

The authors are grateful to Prof. Josef Sebera in European Space Agency for providing the initial gravity gradiometry data. The authors are also grateful to Prof. Michael S. Zhdanov in University of Utah for providing valuable and constructive comments and suggestions that improved this work. This work was supported by the National Natural Science Foundation of China under Grant No. 41574073 and No. 41621091; the Major State Research Development Program of China under Grant No. 2016YFC0601101; the R&D of Key Instruments and Technologies for Deep Resources Prospecting (the National R&D Projects for Key Scientific Instruments) under Grant No. ZDYZ2012-1-04.

References

  1. Amante, C., & Eakins, B. W. (2009). ETOPO1 1 Arc-Minute Global Relief Model: Procedures, data sources and analysis. NOAA Technical Memorandum NESDIS NGDC, National Geophysical Data Center, NOAA.Google Scholar
  2. Bowin, C., Scheer, E., & Smith, W. (1986). Depth estimates from ratios of gravity, geoid, and gravity gradient anomalies. Geophysics, 51(1), 123–136.CrossRefGoogle Scholar
  3. Constable, S. C., Parker, R. L., & Constable, C. G. (1987). Occam’s inversion: A practical algorithm for generating smooth models from electromagnetic sounding data. Geophysics, 52(3), 289–300.CrossRefGoogle Scholar
  4. DeGroot-Hedlin, C. S., & Constable, S. (1990). Occam’s inversion to generate smooth, two-dimensional models from magnetotelluric data. Geophysics, 55(12), 1613–1624.CrossRefGoogle Scholar
  5. Hansen, P. C. (1992). Analysis of discrete ill-posed problems by means of the L-curve. Siam Review, 34(4), 561–580.CrossRefGoogle Scholar
  6. Huang, J. L., & Zhao, D. P. (2004). Crustal heterogeneity and seismotectonics of the region around Beijing, China. Tectonophysics, 385(1–4), 159–180.CrossRefGoogle Scholar
  7. Kelbert, A., Egbert, G. D., & Schultz, A. (2008). Non-linear conjugate gradient inversion for global EM induction: Resolution studies. Geophysical Journal International, 173(2), 365–381.CrossRefGoogle Scholar
  8. Kusky, T. M., & Li, J. H. (2003). Paleoproterozoic tectonic evolution of the North China Craton. Journal of Asian Earth Sciences, 22(4), 383–397.CrossRefGoogle Scholar
  9. Laske, G., Masters, G., Ma, Z. T., & Pasyanos, M. (2013). Update on CRUST1.0—a 1-degree global model of Earth’s crust. Geophysical Research Abstracts, 15, Abstract EGU2013-2658.Google Scholar
  10. Li, Y. G., & Oldenburg, D. W. (1996). 3-D inversion of magnetic data. Geophysics, 61(2), 394–408.CrossRefGoogle Scholar
  11. Li, Y. G., & Oldenburg, D. W. (1998). 3-D inversion of gravity data. Geophysics, 63(1), 109–119.CrossRefGoogle Scholar
  12. Li, Y. Y., & Yang, Y. S. (2011). Gravity data inversion for the lithospheric density structure beneath North China Craton from EGM 2008 model. Physics of the Earth and Planetary Interiors, 189(1–2), 9–26.CrossRefGoogle Scholar
  13. Li, S. L., Zhang, X. K., Zhang, C. K., Zhao, J. R., & Cheng, S. X. (2002). A preliminary study on crustal velocity structures of Maqin-Lanzhou-Jingbian deep seismic sounding profile. Chinese Journal of Geophysics, 45(2), 209–216. (in Chinese with English abstract).CrossRefGoogle Scholar
  14. Nagy, D., Papp, G., & Benedek, J. (2000). The gravitational potential and its derivative for the prism. Journal of Geodesy, 74(7–8), 552–560.CrossRefGoogle Scholar
  15. Pilkington, M. (1997). 3-D magnetic imaging using conjugate gradients. Geophysics, 62(4), 1132–1142.CrossRefGoogle Scholar
  16. Pilkington, M. (2009). 3D magnetic data-space inversion with sparseness constraints. Geophysics, 74(1), L7–L15.CrossRefGoogle Scholar
  17. Sebera, J., Šprlák, M., Novák, P., Bezděk, A., & Vaľko, M. (2014). Iterative spherical downward continuation applied to magnetic and gravitational data from satellite. Surveys In Geophysics, 35(4), 941–958.CrossRefGoogle Scholar
  18. Sun, W. J., & Kennett, B. L. N. (2017). Mid-lithosphere discontinuities beneath the western and central North China Craton. Geophysical Research Letters, 44(3), 1302–1310.CrossRefGoogle Scholar
  19. Tian, Y., Ke, X. P., & Wang, Y. (2018). DenInv3D: A geophysical software for three-dimensional density inversion of gravity field data. Journal of Geophysics and Engineering, 15, 354–365.  https://doi.org/10.1088/1742-2140/aa8caf.CrossRefGoogle Scholar
  20. Tian, X. B., Teng, J. W., Zhang, H. S., Zhang, Z. J., Zhang, Y. Q., Yang, H., et al. (2011). Structure of crust and upper mantle beneath the Ordos Block and the Yinshan Mountains revealed by receiver function analysis. Physics of the Earth and Planetary Interiors, 184(3–4), 186–193.CrossRefGoogle Scholar
  21. Tian, Y., & Zhao, D. P. (2011). Destruction mechanism of the North China Craton: Insight from P and S wave mantle tomography. Journal of Asian Earth Sciences, 42(6), 1132–1145.CrossRefGoogle Scholar
  22. Tian, Y., Zhao, D. P., Sun, R. M., & Teng, J. W. (2009). Seismic imaging of the crust and upper mantle beneath the North China Craton. Physics of the Earth and Planetary Interiors, 172(3–4), 169–182.CrossRefGoogle Scholar
  23. Uieda, L., & Barbosa, V. C. F. (2012). Robust 3D gravity gradient inversion by planting anomalous densities. Geophysics, 77(4), 55–66.CrossRefGoogle Scholar
  24. Uieda, L., Barbosa, V. C. F., & Braitenberg, C. (2016). Tesseroids: Forward-modeling gravitational fields in spherical coordinates. Geophysics, 81(5), 41–48.CrossRefGoogle Scholar
  25. Wang, X. S., Fang, J., & Hsu, H. (2012). Density structure of the lithosphere beneath North China Craton. Chinese Journal of Geophysics, 55(4), 1154–1160.  https://doi.org/10.6038/j.issn.0001-5733.2012.04.011. (in Chinese with English abstract).CrossRefGoogle Scholar
  26. Wang, X. S., Fang, J., & Hsu, H. (2013). 3D density structure of lithosphere beneath northeastern margin of the Tibetan Plateau. Chinese Journal of Geophysics, 56(11), 3770–3778.  https://doi.org/10.6038/cjg20131118. (in Chinese with English abstract).Google Scholar
  27. Wang, X. S., Fang, J., & Hsu, H. (2014). Three-dimensional density structure of the lithosphere beneath the North China Craton and the mechanisms of its destruction. Tectonophysics, 610(6), 150–158.CrossRefGoogle Scholar
  28. Wilson, G., Čuma, M., & Zhdanov, M. S. (2011). Massively parallel 3D inversion of gravity and gravity gradiometry data. Preview, 152(6), 29–34.CrossRefGoogle Scholar
  29. Zhai, M. G., & Santosh, M. (2011). The early Precambrian odyssey of the North China Craton: A synoptic overview. Gondwana Research, 20(1), 6–25.CrossRefGoogle Scholar
  30. Zhang, C., Mushayandebvu, M. F., Reid, A. B., Fairhead, J. D., & Odegard, M. E. (2000). Euler deconvolution of gravity tensor gradient data. Geophysics, 65(2), 512–520.CrossRefGoogle Scholar
  31. Zhang, Y. Q., Teng, J. W., Wang, F. Y., Zhao, W. Z., Li, M., & Wang, Q. S. (2011). Structure of seismic wave property and lithology deduction of the upper crust beneath the Yinshan orogenic belt and the northern Ordos block. Chinese Journal of Geophysics, 54(1), 87–97. (in Chinese with English abstract).CrossRefGoogle Scholar
  32. Zhao, G. C., Wilde, S. A., Cawood, P. A., & Sun, M. (2001). Archean blocks and their boundaries in the North China Craton: Lithological, geochemical, structural and P-T path constraints and tectonic evolution. Precambrian Research, 107(1–2), 45–73.CrossRefGoogle Scholar
  33. Zhdanov, M. S., Ellis, R., & Mukherjee, S. (2004). Three-dimensional regularized focusing inversion of gravity gradient tensor component data. Geophysics, 69(4), 925–937.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.State Key Laboratory of Geodesy and Earth’s Dynamics, Institute of Geodesy and GeophysicsChinese Academy of SciencesWuhanChina
  2. 2.University of Chinese Academy of SciencesBeijingChina
  3. 3.Consortium for Electromagnetic Modeling and Inversion (CEMI)University of UtahSalt Lake CityUSA

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