Skip to main content
SpringerLink
Log in

Three-dimensional gravity inversion based on sparse recovery iteration using approximate zero norm

  • Published:
Applied Geophysics Aims and scope Submit manuscript

Abstract

This research proposes a novel three-dimensional gravity inversion based on sparse recovery in compress sensing. Zero norm is selected as the objective function, which is then iteratively solved by the approximate zero norm solution. The inversion approach mainly employs forward modeling; a depth weight function is introduced into the objective function of the zero norms. Sparse inversion results are obtained by the corresponding optimal mathematical method. To achieve the practical geophysical and geological significance of the results, penalty function is applied to constrain the density values. Results obtained by proposed provide clear boundary depth and density contrast distribution information. The method’s accuracy, validity, and reliability are verified by comparing its results with those of synthetic models. To further explain its reliability, a practical gravity data is obtained for a region in Texas, USA is applied. Inversion results for this region are compared with those of previous studies, including a research of logging data in the same area. The depth of salt dome obtained by the inversion method is 4.2 km, which is in good agreement with the 4.4 km value from the logging data. From this, the practicality of the inversion method is also validated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abdelrahman, E. M., 1991, A least–squares minimization approach to invert gravity data: Geophysics, 56(1), 115–118.

    Article  Google Scholar 

  • Barbosa, V. C. F., and Silva, J. B., 1994, Generalized compact gravity inversion: Geophysics, 59(1), 57–68.

    Article  Google Scholar 

  • Bertete–Aguirre, H., Cherkaev, E., and Oristaglio, M., 2002, Non–smooth gravity problem with total variation penalization functional: Geophysical Journal International, 149(2), 499–507.

    Article  Google Scholar 

  • Blakely, R. J., 1996, Potential theory in gravity and magnetic applications: Cambridge University Press.

    Google Scholar 

  • Bijani, R., Ponteneto, C. F., Carlos, D. U., and Silva Dias, F. J. S., 2015, Three–dimensional gravity inversion using graph theory to delineate the skeleton of homogeneous sources. Geophysics, 80(2), G53–G66.

    Article  Google Scholar 

  • Candes, E. J., and Tao, T., 2005, Decoding by linear programming: IEEE Transactions on, Information Theory, 51(12), 4203–4215.

    Article  Google Scholar 

  • Cardarelli, E., and Fischanger, F., 2006, 2d data modelling by electrical resistivity tomography for complex subsurface geology: Geophysical Prospecting, 54(2), 121–133.

    Article  Google Scholar 

  • Chen, S. S., Donoho, D. L., and Saunders, M. A., 1998, Atomic decomposition by basis pursuit: SIAM J. Sei. Comput., 20(1), 33–61.

    Article  Google Scholar 

  • Chen, S., Saunders, M. A., and Donoho, D. L., 2001, Atomic decomposition by basis pursuit. Siam Review, 43(1), 129–159.

    Article  Google Scholar 

  • Chen, G. X., Chen, S. C., and Wang, H. C., 2013, Geophysical data sparse reconstruction based on l0–norm minimization: Applied Geophysics, 10(2), 181–190.

    Article  Google Scholar 

  • Chen, G. X., and Chen, S. C., 2014, Based on Lp norm sparse constrained regularization method of reconstruction of potential field data: Journal of Zhejiang University (Engineering Science) (4), 748–785.

    Google Scholar 

  • Cheng, X. L., Zheng, X., and Han, W. M., 2013, the sparse algorithm to solve the underdetermined linear equations: Applied Mathematics A Journal of Chinese Universities, No. 2, 235–248.

    Google Scholar 

  • Commer, M., 2011, Three–dimensional gravity modelling and focusing inversion using rectangular meshes: Geophysical Prospecting, 59(5), 966–979.

    Google Scholar 

  • Commer, M., and Newman, G. A., 2008, New advances in three–dimensional controlled–source electromagnetic inversion: Geophysical Journal International, 172(2), 513–535.

    Article  Google Scholar 

  • Donoho, D. L., 2006, High–dimensional centrally symmetric polytopes with neighborliness proportional to dimension: Discrete & Computational Geometry, 35(4), 617–652.

    Article  Google Scholar 

  • Donoho and David L., 2006, For most large underdetermined systems of linear equations the minimal $lsb l$–norm solution is also the sparsest solution. Comm: Pure Appl. Math, 59, 797–829.

    Google Scholar 

  • Elad, M., 2007, Optimized projections for compressed sensing. IEEE Transactions on Signal Processing, 55(12), 5695–5702.

    Article  Google Scholar 

  • Essa, K. S., 2007, Gravity data interpretation using the s–curves method: Journal of Geophysics and Engineering, 4(2), 204.

    Article  Google Scholar 

  • Farquharson, C. G., 2008, Constructing piecewiseconstant models in multidimensional minimum–structure inversions: Geophysics, 73(1), K1–K9.

    Article  Google Scholar 

  • Fregoso, E., and Gallardo, L. A., 2009, Cross–gradients joint 3D inversion with applications to gravity and magnetic data: Geophysics, 74(4), L31–L42.

    Article  Google Scholar 

  • Figueiredo, M. A., Nowak, R. D., and Wright, S. J. 2007, Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems: IEEE Journal of Selected Topics in Signal processing, 1(4), 586–597.

    Article  Google Scholar 

  • Ghalehnoee, M. H., Ansari, A., and Ghorbani, A., 2017, Improving compact gravity inversion using new weighting functions: Geophysical Journal International, 208(2), 546–560.

    Article  Google Scholar 

  • Gholami, A., and Siahkoohi, H. R., 2010, Regularization of linear and non–linear geophysical ill–posed problems with joint sparsity constraints: Geophysical Journal International, 180(2), 871–882.

    Article  Google Scholar 

  • Guillen, A., and Menichetti, V., 1984, Gravity and magnetic inversion with minimization of a specific functional: Geophysics, 49(8), 1354–1360.

    Article  Google Scholar 

  • Kim, H. J., Song, Y., & Lee, K. H., 1999, Inequality constraint in least–squares inversion of geophysical data: Earth Planets & Space, 51(4), 255–259.

    Article  Google Scholar 

  • Last, B. J., and Kubik, K., 1983, Compact gravity inversion: Geophysics, 48(48), 713–721.

    Article  Google Scholar 

  • Lelièvre, P. G., and Oldenburg, D. W., 2009, A comprehensive study of including structural orientation information in geophysical inversions: Geophysical Journal International, 178(178), 623–637.

    Article  Google Scholar 

  • Li, Y., and Oldenburg, D. W., 1996, 3–D inversion of magnetic data: Geophysics, 61(2), 394–08.

    Article  Google Scholar 

  • Li, Y, and Oldenburg, D. W., 1998, 3–D inversion of gravity data: Geophysics, 63(1), 109–119.

    Article  Google Scholar 

  • Li, Y., Cichocki, A., and Amari, S. I., 2003, Sparse component analysis for blind source separation with less sensors than sources: Independent Component Analysis, 89–94.

    Google Scholar 

  • Ma, G. Q., Du, X. J., and Li, L. L., 2012, Comparison of the tensor local wavenumber method with the conventional local wavenumber method for interpretation of total tensor data of potential fields: Chinese Journal of Geophysics, 55(7), 2450–2461.

    Google Scholar 

  • Mallat, S. G., and Zhang, Z., 1993, Matching pursuits with time–frequency dictionaries: IEEE Transactions on Signal Processing, 41(12), 3397–3415.

    Article  Google Scholar 

  • Mohan, N. L., Anandababu, L., and Rao, S. V. S., 1986, Gravity interpretation using the mellin transform: Digital Library Home, 51(1), 114–122.

    Google Scholar 

  • Mohimani, H., Babaie–Zadeh, M., & Jutten, C., 2009, A fast approach for overcomplete sparse decomposition based on smoothed 10, norm: IEEE Press.

    Google Scholar 

  • Nettleton, L. L., 1976, Gravity and magnetics in oil prospecting. McGraw–Hill Companies.

    Google Scholar 

  • Oruç, B., 2010, Location and depth estimation of point–dipole and line of dipoles using analytic signals of the magnetic gradient tensor and magnitude of vector components: Journal of Applied Geophysics, 70(1), 27–37.

    Article  Google Scholar 

  • Portniaguine, O., and Zhdanov, M. S., 1999, Focusing geophysical inversion images: Geophysics, 64(3), 874.

    Article  Google Scholar 

  • Portniaguine, O., and Zhdanov, M. S., 2002, 3–D magnetic inversion with data compression and image focusing: Geophysics, 67(5), 1532–1541.

    Article  Google Scholar 

  • Qi, G., Lv, Q. T., Yan, J. Y., Wu, M. A., and Liu, Y., 2012, Geologic constrained 3D gravity and magnetic modeling of Nihe Deposit–A case study: Chinese Journal of Geophysics, 55(12), 4194?206.

    Google Scholar 

  • Salem, A., Ravat, D., Mushayandebvu, M. F., and Ushijima, K., 2004, Linearized least–squares method for interpretation of potential–field data from sources of simple geometry: Geophysics, 69(3), 783–788.

    Article  Google Scholar 

  • Shaw, R. K., and Agarwal, B. N. P., 1997, A generalized concept of resultant gradient to interpret potential field maps: Geophysical Prospecting, 45(6), 1003–1011.

    Article  Google Scholar 

  • Silva Dias, F. J. S., Barbosa, V. C., and Silva, J. B. C., 2009, 3D gravity inversion through an adaptive–learning procedure: Geophysics, 74(3), I9–I21.

    Article  Google Scholar 

  • Silva Dias, F. J. S., Barbosa, V. C. F., and Silva, J. B. C., 2011, Adaptive learning 3d gravity inversion for salt–body imaging: Geophysics, 76(3), 149–157.

    Article  Google Scholar 

  • Silva, J. B. C., and Barbosa, V C. F., 2006, Gravity inversion of basement relief and estimation of density contrast variation with depth: Geophysics, 71, J51–J58.

    Article  Google Scholar 

  • Tontini, F. C., Cocchi, L., and Carmisciano, C., 2009, Rapid 3–D forward model of potential fields with application to the Palinuro Seamount magnetic anomaly (southern Tyrrhenian Sea, Italy): Journal of Geophysical Research: Solid Earth, 114(B2).

    Google Scholar 

  • Tontini, F., Ronde, C. E. J., Yoerger, D., Kinsey, J., and Tivey, M., 2012, 3–D focused inversion of near–seafloor magnetic data with application to the Brothers volcano hydrothermal system, Southern Pacific Ocean, New Zealand: Journal of Geophysical Research: Solid Earth, 117(B10).

    Google Scholar 

  • Wang, T. H., Huang, D. N., Ma, G. Q., Meng, Z. H., and Li, Y., 2017, Improved preconditioned conjugate gradient algorithm and application in 3D inversion of gravitygradiometry data: Applied Geophysics, 14(2), 301–313.

    Article  Google Scholar 

  • Zhang, Y., Wu, Y., Yan, J., Wang, H., Rodriguez, J. A. P., and Qiu, Y., 2018, 3d inversion of full gravity gradient tensor data in spherical coordinate system using local northoriented frame: Earth Planets & Space, 70(1), 58.

    Article  Google Scholar 

  • Zhdanov, M. S., Ellis, R., and Mukherjee, S., 2004, Threedimensional regularized focusing inversion of gravity gradient tensor component data: Geophysics, 69(4), 925–937.

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank reviewers Profs. Yao Changli, Jiang Puyu, and Luo Zhicai for their valuable comments and suggestions for the final paper, and also would like to extend their sincere thanks to Li Fengting, Geng Meixia, Qin Pengbo, et al. for the support of this research.

Author information

Authors and Affiliations

  1. Tianjin Navigation Instrument Research Institute, Tianjin, 300131, China

    Zhao-Hai Meng

  2. College of Earth Sciences, Jilin University, Changchun, 130021, China

    Xue-Chun Xu

  3. College of Instrumentation and Electrical Engineering, Jilin University, Changchun, 130021, China

    Da-Nian Huang

  4. Key Laboratory of Petroleum Resources Research, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, 100029, China

    Zhao-Hai Meng

Authors
  1. Zhao-Hai Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xue-Chun Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Da-Nian Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhao-Hai Meng.

Additional information

This work was supported by the Development of airborne gravity gradiometer (No. 2017YFC0601601) and open subject of Key Laboratory of Petroleum Resources Research, Institute of Geology and Geophysics, Chinese Academy of Sciences (No. KLOR2018-8).

Meng Zhao-Hai, Engineer, received his B.S. (2011) in Geophysics at the College of Geo- Exploration Science and Technology, Jilin University, and received his Ph.D (2016). in petrology, mineralogy, and geology at the College of Earth Science, Jilin University. He is currently working in Tianjin Navigation Instrument Research Institute, and his major research interests are geophysical instrumentation development and geophysical data processing.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meng, ZH., Xu, XC. & Huang, DN. Three-dimensional gravity inversion based on sparse recovery iteration using approximate zero norm. Appl. Geophys. 15, 524–535 (2018). https://doi.org/10.1007/s11770-018-0688-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11770-018-0688-3

Keywords

Search

Navigation