Three-dimensional gravity inversion based on sparse recovery iteration using approximate zero norm
Meng Zhao-Hai1,4, Xu Xue-Chun2, and Huang Da-Nian3
1. Tianjin Navigation Instrument Research Institute, Tianjin 300131, China.
2. College of Earth Sciences, Jilin University, Changchun 130021, China.
3. College of Instrumentation and Electrical Engineering, Jilin University, Changchun 130021 China.
4. Key Laboratory of Petroleum Resources Research, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China.
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 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).
Cite this article:
. Three-dimensional gravity inversion based on sparse recovery iteration using approximate zero norm[J]. APPLIED GEOPHYSICS, 2018, 15(3-4): 524-535.
[1]
Abdelrahman, E. M., 1991, A least-squares minimization approach to invert gravity data: Geophysics, 56(1), 115-118.
[2]
Barbosa, V. C. F., and Silva, J. B., 1994, Generalized compact gravity inversion: Geophysics, 59(1), 57-68.
[3]
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.
[4]
Blakely, R. J., 1996, Potential theory in gravity and magnetic applications: Cambridge University Press.
[5]
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.
[6]
Candes, E. J., and Tao, T., 2005, Decoding by linear programming: IEEE Transactions on, Information Theory, 51(12), 4203-4215.
[7]
Cardarelli, E., and Fischanger, F., 2006, 2d data modelling by electrical resistivity tomography for complex subsurface geology: Geophysical Prospecting, 54(2), 121-133.
[8]
Chen, S. S., Donoho, D. L., and Saunders, M. A., 1998, Atomic decomposition by basis pursuit: SIAM J. Sci.Comput., 20(1), 33-61.
[9]
Chen, S., Saunders, M. A., and Donoho, D. L., 2001, Atomic decomposition by basis pursuit. Siam Review, 43(1), 129-159.
[10]
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.
[11]
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.
[12]
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.
[13]
Commer, M., 2011, Three-dimensional gravity modelling and focusing inversion using rectangular meshes: Geophysical Prospecting, 59(5), 966-979.
[14]
Commer, M., and Newman, G. A., 2008, New advances in three-dimensional controlled-source electromagnetic inversion: Geophysical Journal International, 172(2), 513-535.
[15]
Donoho, D. L., 2006, High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension: Discrete & Computational Geometry, 35(4), 617-652.
[16]
Donoho and David L., 2006, For most large underdetermined systems of linear equations the minimal $lsb 1$-norm solution is also the sparsest solution. Comm: Pure Appl. Math, 59, 797-829.
[17]
Elad, M., 2007, Optimized projections for compressed sensing. IEEE Transactions on Signal Processing, 55(12), 5695-5702.
[18]
Essa, K. S., 2007, Gravity data interpretation using the s-curves method: Journal of Geophysics and Engineering, 4(2), 204.
[19]
Farquharson, C. G., 2008, Constructing piecewise-constant models in multidimensional minimum-structure inversions: Geophysics, 73(1), K1-K9.
[20]
Fregoso, E., and Gallardo, L. A., 2009, Cross-gradients joint 3D inversion with applications to gravity and magnetic data: Geophysics, 74(4), L31-L42.
[21]
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.
[22]
Ghalehnoee, M. H., Ansari, A., and Ghorbani, A., 2017, Improving compact gravity inversion using new weighting functions: Geophysical Journal International, 208(2), 546-560.
[23]
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.
[24]
Guillen, A., and Menichetti, V., 1984, Gravity and magnetic inversion with minimization of a specific functional: Geophysics, 49(8), 1354-1360.
[25]
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.
[26]
Last, B. J., and Kubik, K., 1983, Compact gravity inversion: Geophysics, 48(48), 713-721.
[27]
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.
[28]
Li, Y., and Oldenburg, D. W., 1996, 3-D inversion of magnetic data: Geophysics, 61(2), 394-408.
[29]
Li, Y., and Oldenburg, D. W., 1998, 3-D inversion of gravity data: Geophysics, 63(1), 109-119.
[30]
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.
[31]
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.
[32]
Mallat, S. G., and Zhang, Z., 1993, Matching pursuits with time-frequency dictionaries: IEEE Transactions on Signal Processing, 41(12), 3397-3415.
[33]
Mohan, N. L., Anandababu, L., and Rao, S. V. S., 1986, Gravity interpretation using the mellin transform: Digital Library Home, 51(1), 114-122.
[34]
Mohimani, H., Babaie-Zadeh, M., & Jutten, C., 2009, A fast approach for overcomplete sparse decomposition based on smoothed l0, norm: IEEE Press.
[35]
Nettleton, L. L., 1976, Gravity and magnetics in oil prospecting. McGraw-Hill Companies.
[36]
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.
[37]
Portniaguine, O., and Zhdanov, M. S., 1999, Focusing geophysical inversion images: Geophysics, 64(3), 874.
[38]
Portniaguine, O., and Zhdanov, M. S., 2002, 3-D magnetic inversion with data compression and image focusing: Geophysics, 67(5), 1532-1541.
[39]
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), 4194-4206.
[40]
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.
[41]
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.
[42]
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.
[43]
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.
[44]
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.
[45]
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).
[46]
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).
[47]
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 gravity-gradiometry data: Applied Geophysics, 14(2), 301-313.
[48]
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 north-oriented frame: Earth Planets & Space, 70(1), 58.
[49]
Zhdanov, M. S., Ellis, R., and Mukherjee, S., 2004, Three-dimensional regularized focusing inversion of gravity gradient tensor component data: Geophysics, 69(4), 925-937.
2018, Vol. 15 
