Divergence correction schemes in finite difference method for 3D tensor CSAMT in axial anisotropic media
Kunpeng Wang 1 Handong Tan 1 2 3 Zhiyong Zhang 1 Zhiqiang Li 1 Meng Cao 1
+ Author Affiliations
- Author Affiliations
1 School of Geophysics and Information Technology, China University of Geosciences (Beijing), Beijing 100083, China.
2 Key Laboratory of Geo-detection (China University of Geosciences), Ministry of Education, Beijing 100083, China.
3 Corresponding author. Email: thd@cugb.edu.cn
Exploration Geophysics 48(4) 363-373 https://doi.org/10.1071/EG15074
Submitted: 6 August 2015 Accepted: 18 April 2016 Published: 12 May 2016
Abstract
Resistivity anisotropy and full-tensor controlled-source audio-frequency magnetotellurics (CSAMT) have gradually become hot research topics. However, much of the current anisotropy research for tensor CSAMT only focuses on the one-dimensional (1D) solution. As the subsurface is rarely 1D, it is necessary to study three-dimensional (3D) model response. The staggered-grid finite difference method is an effective simulation method for 3D electromagnetic forward modelling. Previous studies have suggested using the divergence correction to constrain the iterative process when using a staggered-grid finite difference model so as to accelerate the 3D forward speed and enhance the computational accuracy. However, the traditional divergence correction method was developed assuming an isotropic medium. This paper improves the traditional isotropic divergence correction method and derivation process to meet the tensor CSAMT requirements for anisotropy using the volume integral of the divergence equation. This method is more intuitive, enabling a simple derivation of a discrete equation and then calculation of coefficients related to the anisotropic divergence correction equation. We validate the result of our 3D computational results by comparing them to the results computed using an anisotropic, controlled-source 2.5D program. The 3D resistivity anisotropy model allows us to evaluate the consequences of using the divergence correction at different frequencies and for two orthogonal finite length sources. Our results show that the divergence correction plays an important role in 3D tensor CSAMT resistivity anisotropy research and offers a solid foundation for inversion of CSAMT data collected over an anisotropic body.
Key words: divergence correction, finite difference, resistivity anisotropy, tensor CSAMT.
References
Bachrach, R., 2011, Elastic and resistivity anisotropy of shale during compaction and diagenesis: Joint effective medium modeling and field observations: Geophysics, 76, E175–E186| Elastic and resistivity anisotropy of shale during compaction and diagenesis: Joint effective medium modeling and field observations:Crossref | GoogleScholarGoogle Scholar |
Boerner, D. E., Wright, J. A., Thurlow, J. G., and Reed, L. E., 1993, Tensor CSAMT studies at the Buchans Mine in Central Newfoundland: Geophysics, 58, 12–19
| Tensor CSAMT studies at the Buchans Mine in Central Newfoundland:Crossref | GoogleScholarGoogle Scholar |
Caldwell, G. T., Bibby, H. M., and Brown, C., 2002, Controlled source apparent resistivity tensors and their relationship to the magnetotelluric impedance tensor: Geophysical Journal International, 151, 755–770
| Controlled source apparent resistivity tensors and their relationship to the magnetotelluric impedance tensor:Crossref | GoogleScholarGoogle Scholar |
Chen, H., Deng, J. Z., Tan, H. D., and Yang, H. Y., 2011, Study on divergence correction method in three-dimensional magnetotelluric modeling with staggered-grid finite difference method: Chinese Journal of Geophysics, 54, 1649–1659
Constable, S., 2010, Ten years of marine CSEM for hydrocarbon exploration: Geophysics, 75, 75A67–75A81
| Ten years of marine CSEM for hydrocarbon exploration:Crossref | GoogleScholarGoogle Scholar |
da Silva, N. V., Morgan, J. V., MacGregor, L., and Warner, M., 2012, A finite element multifrontal method for 3D CSEM modeling in the frequency domain: Geophysics, 77, E101–E115
| A finite element multifrontal method for 3D CSEM modeling in the frequency domain:Crossref | GoogleScholarGoogle Scholar |
Freund, R. W., and Nachtigal, N. M., 1991, QMR: A quasi-minimial residual method for non-Hermitian linear systems: Numerische Mathematik, 60, 315–339
| QMR: A quasi-minimial residual method for non-Hermitian linear systems:Crossref | GoogleScholarGoogle Scholar |
Garcia, X., Boerner, D., and Pedersen, L. B., 2003, Electric and magnetic galvanic distortion decomposition of tensor CSAMT data. Application to data from the Buchans Mine (Newfoundland, Canada): Geophysical Journal International, 154, 957–969
| Electric and magnetic galvanic distortion decomposition of tensor CSAMT data. Application to data from the Buchans Mine (Newfoundland, Canada):Crossref | GoogleScholarGoogle Scholar |
Hagiwara, T., 2012, Determination of dip and anisotropy from transient triaxial induction measurements: Geophysics, 77, D105–D112
| Determination of dip and anisotropy from transient triaxial induction measurements:Crossref | GoogleScholarGoogle Scholar |
He, Z. X., He, Z. H., Wang, X. B., and Kong, F. S., 2002, Tendency of non-seismic techniques of hydrocarbon prospecting: Progress in Geophysics, 17, 473–479
Hu, Y. C., Li, T. L., Fan, C. S., Wang, D. Y., and Li, J. P., 2015, Three-dimensional tensor controlled-source electromagnetic modeling based on the vector finite-element method: Applied Geophysics, 12, 35–46
| Three-dimensional tensor controlled-source electromagnetic modeling based on the vector finite-element method:Crossref | GoogleScholarGoogle Scholar |
Jahandari, H., and Farquharson, C. G., 2014, A finite-volume solution to the geophysical electromagnetic forward problem using unstructured grids: Geophysics, 79, E287–E302
| A finite-volume solution to the geophysical electromagnetic forward problem using unstructured grids:Crossref | GoogleScholarGoogle Scholar |
Key, K., 2012, Marine EM inversion using unstructured grids: a 2D parallel adaptive finite element algorithm: SEG Technical Program, Expanded Abstracts, 1–5.
Li, Y., and Dai, S., 2011, Finite element modelling of marine controlled-source electromagnetic responses in two-dimensional dipping anisotropic conductivity structures: Geophysical Journal International, 185, 622–636
| Finite element modelling of marine controlled-source electromagnetic responses in two-dimensional dipping anisotropic conductivity structures:Crossref | GoogleScholarGoogle Scholar |
Li, Y., and Key, K., 2007, 2D marine controlled-source electromagnetic modeling: Part I - An adaptive finite-element algorithm: Geophysics, 72, WA51–WA62
Li, X., and Pedersen, L. B., 1991, Controlled source tensor magnetotellurics: Geophysics, 56, 1456–1461
| Controlled source tensor magnetotellurics:Crossref | GoogleScholarGoogle Scholar |
Li, X., and Pedersen, L. B., 1992, Controlled-source tensor magnetotelluric responses of a layered earth with azimuthal anisotropy: Geophysical Journal International, 111, 91–103
| Controlled-source tensor magnetotelluric responses of a layered earth with azimuthal anisotropy:Crossref | GoogleScholarGoogle Scholar |
Li, Y., and Pek, J., 2008, Adaptive finite element modelling of two-dimensional magnetotelluric fields in general anisotropic media: Geophysical Journal International, 175, 942–954
| Adaptive finite element modelling of two-dimensional magnetotelluric fields in general anisotropic media:Crossref | GoogleScholarGoogle Scholar |
Li, X., Oskooi, B., and Pedersen, L. B., 2000, Inversion of controlled-source tensor magnetotelluric data for a layered earth with azimuthal anisotropy: Geophysics, 38, 1241–1245
Liu, Y. H., and Yin, C. C., 2013, Electromagnetic divergence correction for 3D anisotropic EM modeling: Journal of Applied Geophysics, 96, 19–27
| Electromagnetic divergence correction for 3D anisotropic EM modeling:Crossref | GoogleScholarGoogle Scholar |
Mackie, R. L., Madden, T. R., and Wannamaker, P. E., 1993, Three dimensional magnetotelluric modeling using difference equations-Theory and comparisons to integral equation solutions: Geophysics, 58, 215–226
| Three dimensional magnetotelluric modeling using difference equations-Theory and comparisons to integral equation solutions:Crossref | GoogleScholarGoogle Scholar |
Newman, G. A., and Alumbaugh, D. L., 1995, Frequency-domain modeling of airborne electromagnetic responses using staggered finite differences: Geophysical Prospecting, 43, 1021–1042
| Frequency-domain modeling of airborne electromagnetic responses using staggered finite differences:Crossref | GoogleScholarGoogle Scholar |
Newman, G. A., and Alumbaugh, D. L., 2000, Three-dimensional magnetotelluric inversion using non-linear conjugate gradients: Geophysical Journal International, 140, 410–424
| Three-dimensional magnetotelluric inversion using non-linear conjugate gradients:Crossref | GoogleScholarGoogle Scholar |
Newman, G. A., Commer, M., and Carazzone, J. J., 2010, Imaging CSEM data in the presence of electrical anisotropy: Geophysics, 75, F51–F61
| Imaging CSEM data in the presence of electrical anisotropy:Crossref | GoogleScholarGoogle Scholar |
Routh, P. S., and Oldenburg, D. W., 1999, Inversion of controlled-source audio-frequency magnetotellurics data for a horizontally layered earth: Geophysics, 64, 1689–1697
| Inversion of controlled-source audio-frequency magnetotellurics data for a horizontally layered earth:Crossref | GoogleScholarGoogle Scholar |
Siripunvaraporn, W., Egbert, G., and Lenbury, Y., 2002, Numerical accuracy of magnetotelluric modeling: a comparison of finite difference approximations: Earth, Planets, and Space, 54, 721–725
| Numerical accuracy of magnetotelluric modeling: a comparison of finite difference approximations:Crossref | GoogleScholarGoogle Scholar |
Siripunvaraporn, W., Egbert, G., Lenbury, Y., and Uyeshima, M., 2005, Three-dimensional magnetotelluric inversion: data-space method: Physics of the Earth and Planetary Interiors, 150, 3–14
| Three-dimensional magnetotelluric inversion: data-space method:Crossref | GoogleScholarGoogle Scholar |
Smith, J. T., 1996, Conservative modeling of 3D electromagnetic fields, Part II: biconjugate gradient solution and an accelerator: Geophysics, 61, 1319–1324
| Conservative modeling of 3D electromagnetic fields, Part II: biconjugate gradient solution and an accelerator:Crossref | GoogleScholarGoogle Scholar |
Streich, R., 2009, 3D finite-difference frequency-domain modeling of controlled-source electromagnetic data: direct solution and optimization for high accuracy: Geophysics, 74, F95–F105
| 3D finite-difference frequency-domain modeling of controlled-source electromagnetic data: direct solution and optimization for high accuracy:Crossref | GoogleScholarGoogle Scholar |
Wannamaker, P. E., 1991, Advances in three dimensional magnetotelluric modeling using integral equations: Geophysics, 56, 1716–1728
| Advances in three dimensional magnetotelluric modeling using integral equations:Crossref | GoogleScholarGoogle Scholar |
Weiss, C. J., and Constable, S., 2006, Mapping thin resistors and hydrocarbons with marine EM methods. Part II - Modeling and analysis in 3D: Geophysics, 71, G321–G332
| Mapping thin resistors and hydrocarbons with marine EM methods. Part II - Modeling and analysis in 3D:Crossref | GoogleScholarGoogle Scholar |
Yin, B., and Wang, G., 2001, Canonical decomposition of magnetotelluric impedance tensor and geological meaning of the parameters: Chinese Journal of Geophysics, 44, 275–284
| Canonical decomposition of magnetotelluric impedance tensor and geological meaning of the parameters:Crossref | GoogleScholarGoogle Scholar |
Zhou, J., Wang, J., Shang, Q., Wang, H., and Yin, C., 2014, Regularized inversion of controlled source audio-frequency magnetotelluric data in horizontally layered transversely isotropic media: Journal of Geophysics and Engineering, 11, 025003
| Regularized inversion of controlled source audio-frequency magnetotelluric data in horizontally layered transversely isotropic media:Crossref | GoogleScholarGoogle Scholar |
Zonge, K. L., and Hughes, L. J., 1991, Controlled source audio-frequency magnetotellurics, in M. S. Nabighian, ed., Electromagnetic methods in applied geophysics: Society of Exploration Geophysicists, 713–810.