Exploration Geophysics Exploration Geophysics Society
Journal of the Australian Society of Exploration Geophysicists
RESEARCH ARTICLE

Fast inversion of gravity data using the symmetric successive over-relaxation (SSOR) preconditioned conjugate gradient algorithm

Zhaohai Meng 1 4 Fengting Li 2 Xuechun Xu 1 Danian Huang 3 Dailei Zhang 3
+ Author Affiliations
- Author Affiliations

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

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

3 College of GeoExploration Science and Technology, Jilin University, Changchun, Jilin 130021, China.

4 Corresponding author. Email: 526468457@qq.com

Exploration Geophysics 48(3) 294-304 https://doi.org/10.1071/EG15041
Submitted: 7 May 2015  Accepted: 13 January 2016   Published: 16 February 2016

Abstract

The subsurface three-dimensional (3D) model of density distribution is obtained by solving an under-determined linear equation that is established by gravity data. Here, we describe a new fast gravity inversion method to recover a 3D density model from gravity data. The subsurface will be divided into a large number of rectangular blocks, each with an unknown constant density. The gravity inversion method introduces a stabiliser model norm with a depth weighting function to produce smooth models. The depth weighting function is combined with the model norm to counteract the skin effect of the gravity potential field. As the numbers of density model parameters is NZ (the number of layers in the vertical subsurface domain) times greater than the observed gravity data parameters, the inverse density parameter is larger than the observed gravity data parameters. Solving the full set of gravity inversion equations is very time-consuming, and applying a new algorithm to estimate gravity inversion can significantly reduce the number of iterations and the computational time. In this paper, a new symmetric successive over-relaxation (SSOR) iterative conjugate gradient (CG) method is shown to be an appropriate algorithm to solve this Tikhonov cost function (gravity inversion equation). The new, faster method is applied on Gaussian noise-contaminated synthetic data to demonstrate its suitability for 3D gravity inversion. To demonstrate the performance of the new algorithm on actual gravity data, we provide a case study that includes ground-based measurement of residual Bouguer gravity anomalies over the Humble salt dome near Houston, Gulf Coast Basin, off the shore of Louisiana. A 3D distribution of salt rock concentration is used to evaluate the inversion results recovered by the new SSOR iterative method. In the test model, the density values in the constructed model coincide with the known location and depth of the salt dome.

Key words: conjugate gradient, depth of the weighting function, gravity inversion, symmetric successive over-relaxation.


References

Abdelrahman, E. M., Bayoumi, A. I., and El-Araby, H. M., 1991, A least-squares minimization approach to invert gravity data: Geophysics, 56, 115–118
A least-squares minimization approach to invert gravity data:CrossRef |

Abdelrahman, E. S. M., El-Araby, T. M., El-Araby, H. M., and Abo-Ezz, E. R., 2001a, A new method for shape and depth determinations from gravity data: Geophysics, 66, 1774–1780
A new method for shape and depth determinations from gravity data:CrossRef |

Abdelrahman, E. M., El-Araby, H. M., El-Araby, T. M., and Abo-Ezz, E. R., 2001b, Three least-squares minimization approaches to depth, shape, and amplitude coefficient determination from gravity data: Geophysics, 66, 1105–1109
Three least-squares minimization approaches to depth, shape, and amplitude coefficient determination from gravity data:CrossRef |

Abedi, M., Gholami, A., Norouzi, G. H., and Fathianpour, N., 2013, Fast inversion of magnetic data using Lanczos bidiagonalization method: Journal of Applied Geophysics, 90, 126–137
Fast inversion of magnetic data using Lanczos bidiagonalization method:CrossRef |

Bear, G. W., Al-Shukri, H. J., and Rudman, A. J., 1995, Linear inversion of gravity data for 3-D density distributions: Geophysics, 60, 1354–1364
Linear inversion of gravity data for 3-D density distributions:CrossRef |

Bosch, M, and McGaughey, J, 2001, Joint inversion of gravity and magnetic data under lithologic constraints: The Leading Edge, 20, 877–881

Botros, Y. Y., and Volakis, J. L., 1999, Preconditioned generalized minimal residual iterative scheme for perfectly matched layer terminated applications: IEEE Microwave and Guided Wave Letters, 9, 45–47
Preconditioned generalized minimal residual iterative scheme for perfectly matched layer terminated applications:CrossRef |

Braile, L. W., Keller, G. R., and Peeples, W. J., 1974, Inversion of gravity data for two-dimensional density distributions: Journal of Geophysical Research, 79, 2017–2021
Inversion of gravity data for two-dimensional density distributions:CrossRef |

Canning, F. X., and Scholl, J. F., 1996, Diagonal preconditioners for the EFIE using a wavelet basis: IEEE Transactions on Antennas and Propagation, 44, 1239–1246
Diagonal preconditioners for the EFIE using a wavelet basis:CrossRef |

Caratori Tontini, F, Cocchi, L, and Carmisciano, C, 2006, Depth-to-the-bottom optimization for magnetic data inversion: magnetic structure of the Latium volcanic region, Italy: Journal of Geophysical Research: Solid Earth, 111, B11104

Chasseriau, P, and Chouteau, M, 2003, 3D gravity inversion using a model of parameter covariance: Journal of Applied Geophysics, 52, 59–74

Chen, R. S., Yung, E. K., Chan, C. H., and Fang, D. G., 2000, Application of preconditioned CG–FFT technique to method of lines for analysis of the infinite-plane metallic grating: Microwave and Optical Technology Letters, 24, 170–175
Application of preconditioned CG–FFT technique to method of lines for analysis of the infinite-plane metallic grating:CrossRef |

Chen, R. S., Yung, E. K. N., Chan, C. H., Wang, D. X., and Fang, D. G., 2002, Application of the SSOR preconditioned CG algorithm to the vector FEM for 3D full-wave analysis of electromagnetic-field boundary-value problems: IEEE Transactions on Microwave Theory and Techniques, 50, 1165–1172
Application of the SSOR preconditioned CG algorithm to the vector FEM for 3D full-wave analysis of electromagnetic-field boundary-value problems:CrossRef |

Chen, X., 2005, Preconditioners for iterative solutions of large-scale linear systems arising from Biot’s consolidation equations: Ph.D. thesis, National University of Singapore.

Čuma, M., and Zhdanov, M. S., 2014, Massively parallel regularized 3D inversion of potential fields on CPUs and GPUs: Computers & Geosciences, 62, 80–87
Massively parallel regularized 3D inversion of potential fields on CPUs and GPUs:CrossRef |

Ellis, R. G., and Oldenburg, D. W., 1994, The pole-pole 3-D Dc-resistivity inverse problem: a conjugate gradient approach: Geophysical Journal International, 119, 187–194
The pole-pole 3-D Dc-resistivity inverse problem: a conjugate gradient approach:CrossRef |

Essa, K. S., 2007, Gravity data interpretation using the s-curves method: Journal of Geophysics and Engineering, 4, 204–213
Gravity data interpretation using the s-curves method:CrossRef |

Fisher, N. J., and Howard, L. E., 1980, Gravity interpretation with the aid of quadratic programming: Geophysics, 45, 403–419
Gravity interpretation with the aid of quadratic programming:CrossRef |

Golub, G. H., and Van Loan, C. F., 1996, Matrix computations (3rd edition): Johns Hopkins University Press.

Haáz, I. B., 1953, Relationship between the potential of the attraction of the mass contained in a finite rectangular prism and its first and second derivatives: Geophysical Transactions, II, 57–66

Hinze, W. J., 1990, The role of gravity and magnetic methods in engineering and environmental studies: Geotechnical and Environmental Geophysics, 1, 75–126

Jackson, D. D., 1979, The use of a priori data to resolve non-uniqueness in linear inversion: Geophysical Journal International, 57, 137–157
The use of a priori data to resolve non-uniqueness in linear inversion:CrossRef |

Li, X., and Chouteau, M., 1998, Three-dimensional gravity modeling in all space: Surveys in Geophysics, 19, 339–368
Three-dimensional gravity modeling in all space:CrossRef |

Li, Y., and Oldenburg, D. W., 1996, 3-D inversion of magnetic data: Geophysics, 61, 394–408
3-D inversion of magnetic data:CrossRef |

Li, Y., and Oldenburg, D. W., 1998, 3-D inversion of gravity data: Geophysics, 63, 109–119
3-D inversion of gravity data:CrossRef |

Mackie, R. L., and Madden, T. R., 1993, Conjugate direction relaxation solutions for 3-D magnetotelluric modeling: Geophysics, 58, 1052–1057
Conjugate direction relaxation solutions for 3-D magnetotelluric modeling:CrossRef |

Malehmir, A., Thunehed, H., and Tryggvason, A., 2009, Case history: the Paleoproterozoic Kristineberg mining area, northern Sweden: results from integrated 3D geophysical and geologic modeling, and implications for targeting ore deposits: Geophysics, 74, B9–B22
Case history: the Paleoproterozoic Kristineberg mining area, northern Sweden: results from integrated 3D geophysical and geologic modeling, and implications for targeting ore deposits:CrossRef |

Mareschal, J. C., 1985, Inversion of potential field data in Fourier transform domain: Geophysics, 50, 685–691
Inversion of potential field data in Fourier transform domain:CrossRef |

Mohan, N. L., Anandababu, L., and Rao, S. S., 1986, Gravity interpretation using the Mellin transform: Geophysics, 51, 114–122
Gravity interpretation using the Mellin transform:CrossRef |

Najafi, H. S., and Edalatpanah, S. A., 2014, A new modified SSOR iteration method for solving augmented linear systems: International Journal of Computer Mathematics, 91, 539–552
A new modified SSOR iteration method for solving augmented linear systems:CrossRef |

Nakatsuka, T., 1995, Minimum norm inversion of magnetic anomalies with application to aeromagnetic data in the Tanna area, central Japan: Journal of Geomagnetism and Geoelectricity, 47, 295–311
Minimum norm inversion of magnetic anomalies with application to aeromagnetic data in the Tanna area, central Japan:CrossRef |

Namaki, L., Gholami, A., and Hafizi, M. A., 2011, Edge-preserved 2-D inversion of magnetic data: an application to the Makran arc-trench complex: Geophysical Journal International, 184, 1058–1068
Edge-preserved 2-D inversion of magnetic data: an application to the Makran arc-trench complex:CrossRef |

Nettleton, L. L., 1962, Gravity and magnetics for geologists and seismologists: AAPG Bulletin, 46, 1815–1838

Nettleton, L. L., 1976, Gravity and magnetics in oil prospecting: McGraw-Hill Companies.

Nocedal, J., and Wright, S., 2006, Numerical optimization: Springer Science and Business Media.

Nolet, G., 1985, Solving or resolving inadequate and noisy tomographic systems: Journal of Computational Physics, 61, 463–482
Solving or resolving inadequate and noisy tomographic systems:CrossRef |

Nolet, G., 1993, Solving large linearized tomographic problems, in H. M. Iyer, and K. Hirahara, eds., Seismic tomography: theory and practice: Chapman and Hall, 227–247.

Oldenburg, D. W., and Li, Y., 1994, Inversion of induced polarization data: Geophysics, 59, 1327–1341
Inversion of induced polarization data:CrossRef |

Oruç, B., 2010, Depth estimation of simple causative sources from gravity gradient tensor invariants and vertical component: Pure and Applied Geophysics, 167, 1259–1272
Depth estimation of simple causative sources from gravity gradient tensor invariants and vertical component:CrossRef |

Paterson, N. R., and Reeves, C. V., 1985, Applications of gravity and magnetic surveys: the state-of-the-art in 1985: Geophysics, 50, 2558–2594
Applications of gravity and magnetic surveys: the state-of-the-art in 1985:CrossRef |

Pignatelli, A., Nicolosi, I., and Chiappini, M., 2006, An alternative 3D source inversion method for magnetic anomalies with depth resolution: Annals of Geophysics, 49, 1021–1027

Pilkington, M., 1997, 3-D magnetic imaging using conjugate gradients: Geophysics, 62, 1132–1142
3-D magnetic imaging using conjugate gradients:CrossRef |

Portniaguine, O., and Zhdanov, M. S., 1999, Focusing geophysical inversion images: Geophysics, 64, 874–887
Focusing geophysical inversion images:CrossRef |

Portniaguine, O., and Zhdanov, M. S., 2002, 3-D magnetic inversion with data compression and image focusing: Geophysics, 67, 1532–1541
3-D magnetic inversion with data compression and image focusing:CrossRef |

Saad, Y., 2003, Iterative methods for sparse linear systems: SIAM.

Safon, C, Vasseur, G, and Cuer, M, 1977, Some applications of linear programming to the inverse gravity problem: Geophysics, 42, 1215–1229

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, 783–788
Linearized least-squares method for interpretation of potential-field data from sources of simple geometry:CrossRef |

Sarkar, T, and Arvas, E, 1985, On a class of finite step iterative methods (conjugate directions) for the solution of an operator equation arising in electromagnetics: IEEE Transactions on Antennas and Propagation, 33, 1058–1066

Scales, J. A., 1987, Tomographic inversion via the conjugate gradient method: Geophysics, 52, 179–185

Shamsipour, P, Marcotte, D, Chouteau, M, and Keating, P, 2010, 3D stochastic inversion of gravity data using cokriging and cosimulation: Geophysics, 75, I1–I10

Shamsipour, P., Marcotte, D., Chouteau, M., Rivest, M., and Bouchedda, A., 2013, 3D stochastic gravity inversion using nonstationary covariances: Geophysics, 78, G15–G24
3D stochastic gravity inversion using nonstationary covariances:CrossRef |

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

Shi, Y. F., Chen, R. S., and Xia, M. Y., 2008, SSOR preconditioner accelerated time domain finite element boundary integral method, in Asia-Pacific Microwave Conference 2008: IEEE, 1–4.

Smith, G. D., 1985, Numerical solution of partial differential equations: finite difference methods: Oxford University Press.

Tikhonov, A. N., and Arsenin, V. I., 1977, Solutions of ill-posed problems: Winston.

VanDecar, J. C., and Snieder, R., 1994, Obtaining smooth solutions to large, linear, inverse problems: Geophysics, 59, 818–829
Obtaining smooth solutions to large, linear, inverse problems:CrossRef |

Ward, S. H., 1990, Geotechnical and environmental geophysics: SEG.

Zhang, J., Mackie, R. L., and Madden, T. R., 1995, 3-D resistivity forward modeling and inversion using conjugate gradients: Geophysics, 60, 1313–1325
3-D resistivity forward modeling and inversion using conjugate gradients:CrossRef |

Zhdanov, M. S., 1988, Integral transforms in geophysics: Springer Verlag.

Zhdanov, M. S., 2002, Geophysical inverse theory and regularization problems: Elsevier.

Zhdanov, M. S., 2009, Geophysical electromagnetic theory and methods: Elsevier.

Zhdanov, M. S., and Wannamaker, P. E., 2002, Three-dimensional electromagnetics: Elsevier.



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