Efficient forward modelling using the self-consistent impedance method for electromagnetic surface impedance*
Hugo G. Espinosa 1 2 David V. Thiel 11 School of Engineering, Griffith University, 170 Kessels Road, Nathan, Qld 4111, Australia.
2 Corresponding author. Email: h.espinosa@griffith.edu.au
Exploration Geophysics 45(3) 201-207 https://doi.org/10.1071/EG12072
Submitted: 12 November 2012 Accepted: 26 May 2014 Published: 14 July 2014
Abstract
The two-dimensional self-consistent impedance method is used to calculate the electromagnetic surface impedance above subsurface structures at very low frequencies. The method was derived from Faraday’s and Ampere’s Laws and results in a linear matrix equation where the right hand side of the equation corresponds to the source field introduced into the model as a fixed magnetic value. An air layer above the earth’s surface is included to allow the scattered magnetic field to be calculated at the surface. The source field is applied above the earth’s surface as a Dirichlet boundary condition, and a Neumann boundary condition is applied to all other boundaries in the solution space. The left hand side of the linear equation corresponds to the impedance matrix determined by discretising the solution space into two-dimensional rectangular pixels or cells bounded by lumped impedance elements, with values determined by the electromagnetic properties of the local media and the size of the pixel in the model. The resulting sparse matrix offers the flexibility of cells of any shape or size. Due to the large matrix dimensions, an iterative solver with a preconditioning technique was used to improve the speed, size and convergence of the solution. The efficient forward modelling has been applied to the analysis of a coal seam with various structural anomalies and line of oxidation along a line defined by 500 m with 0.5 m resolution. This improved technique allows in-field inverse modelling of surface impedance data. This paper reports several likely coal-seam scenarios relevant to surface mining operations.
Key words: coal seam, electromagnetic geophysics, forward modelling, impedance method, iterative solvers, surface impedance.
References
Benzi, M., 2002, Preconditioning techniques for large linear systems: a survey: Journal of Computational Physics, 182, 418–477| Preconditioning techniques for large linear systems: a survey:Crossref | GoogleScholarGoogle Scholar |
Cagniard, L., 1953, Basic theory of the magnetotelluric method of geophysical prospecting: Geophysics, 18, 605–635
| Basic theory of the magnetotelluric method of geophysical prospecting:Crossref | GoogleScholarGoogle Scholar |
Chan, T. F., and Van der Vorst, H. A., 1997, Approximate and incomplete factorizations, in D. E. Keyes, A. Sameh, and V. Venkatakrishnan, eds., Parallel Numerical Algorithms: Springer, ICASE/LaRC Interdisciplinary Series in Science and Engineering 4, 167–202.
Collet, L. S., and Jensen, O. G., 1982, Geophysical applications of surface wave impedance measurements: Geological Survey Canada.
Elman, H. C., 1986, A stability analysis of incomplete LU factorizations: Mathematics of Computation, 47, 191–217
Espinosa, H. G., and Thiel, D. V., 2012, Efficient forward modelling of electromagnetic surface impedance for coal seam assessment: Proceedings of the 22nd International Geophysical Conference and Exhibition, Australian Society of Exploration Geophysicists, Brisbane, 1–4.
Espinosa, H. G., Heldring, A., Tamayo, J. M., Rius, J. M., and Mosig, J. R., 2006, Multilevel field interpolation algorithm for large PEC objects: EuCAP Antennas and Propagation, , 1–5
Gandhi, O. P., DeFord, J. F., and Kanai, H., 1984, Impedance method for calculation of power deposition patterns in magnetically induced hyperthermia: IEEE Transactions on Bio-Medical Engineering, BME-31, 644–651
| Impedance method for calculation of power deposition patterns in magnetically induced hyperthermia:Crossref | GoogleScholarGoogle Scholar |
James, D. A., and Thiel, D. V., 1997, Modelling eddy currents in unbounded structures using the impedance method: Applied Computational Electromagnetics Society Journal, 12, 43–49
James, D. A., O’Keefe, S. G., and Thiel, D. V., 1999, Eddy current modeling using the impedance method for surface impedance profiling: IEEE Transactions on Magnetics, 35, 1107–1110
| Eddy current modeling using the impedance method for surface impedance profiling:Crossref | GoogleScholarGoogle Scholar |
Mogensen, G. T., Espinosa, H. G., and Thiel, D. V., 2014, Surface impedance mapping using sferics: IEEE Transactions on Geoscience and Remote Sensing, 52, 2074–2080
| Surface impedance mapping using sferics:Crossref | GoogleScholarGoogle Scholar |
Porstendorfer, G., 1975, Principles of magnetotelluric prospecting: Gebruder Borntraeger.
Rankin, D., 1962, The magnetotelluric effect on a dike: Geophysics, 27, 666–676
| The magnetotelluric effect on a dike:Crossref | GoogleScholarGoogle Scholar |
Saad, Y., 2003, Iterative methods for sparse linear systems (2nd edition): Siam.
Saad, Y., and Schultz, M., 1986, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems: SIAM Journal on Scientific and Statistical Computing, 7, 856–869
| GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems:Crossref | GoogleScholarGoogle Scholar |
Silvester, P., and Haslam, C. R. S., 1972, Magnetotelluric modeling by the finite element method: Geophysical Prospecting, 20, 872–891
| Magnetotelluric modeling by the finite element method:Crossref | GoogleScholarGoogle Scholar |
Thiel, D. V., 1988, A surface impedance mapping technique based on radiation from discrete lightning strokes: Geoexploration, 25, 163–172
| A surface impedance mapping technique based on radiation from discrete lightning strokes:Crossref | GoogleScholarGoogle Scholar |
Thiel, D. V., 1990, Surface-impedance changes in the vicinity of an abrupt lateral boundary at the earth’s surface: IEEE Transactions on Geoscience and Remote Sensing, 28, 500–502
| Surface-impedance changes in the vicinity of an abrupt lateral boundary at the earth’s surface:Crossref | GoogleScholarGoogle Scholar |
Thiel, D. V., and Mittra, R., 1997, Surface impedance modeling using the finite-difference time-domain method: IEEE Transactions on Geoscience and Remote Sensing, 35, 1350–1356
| Surface impedance modeling using the finite-difference time-domain method:Crossref | GoogleScholarGoogle Scholar |
Thiel, D. V., and Mittra, R., 2001, A self-consistent method for electromagnetic surface impedance modeling: Radio Science, 36, 31–43
| A self-consistent method for electromagnetic surface impedance modeling:Crossref | GoogleScholarGoogle Scholar |
Tikhonov, A. N., 1950, On determining electrical characteristics of the deep layers of the earth’s crust: Soviet Mathematics Doklady, 2, 295–297
Ting, S. C., and Hohmann, G. W., 1981, Integral equation modeling of three-dimensional magnetotelluric response: Geophysics, 46, 182–197
| Integral equation modeling of three-dimensional magnetotelluric response:Crossref | GoogleScholarGoogle Scholar |
Wait, J. R., 1970, Electromagnetic waves in stratified media (2nd edition): Pergamon