Forward Modeling of Frequency-domain Helicopter-borne Electromagnetic Data using an Improved Finite Difference Method | ||
| Journal of Mining and Environment | ||
| مقاله 10، دوره 13، شماره 4، دی 2022، صفحه 1067-1089 اصل مقاله (4.64 M) | ||
| نوع مقاله: Original Research Paper | ||
| شناسه دیجیتال (DOI): 10.22044/jme.2022.11837.2173 | ||
| نویسندگان | ||
| Saeed Nazari؛ Alireza Arab Amiri* ؛ Abolghasem Kamkar Rouhani؛ Fereydoun Sharifi | ||
| School of Mining, Petroleum and Geophysics Engineering, Shahrood University of Technology, Shahrood, Iran | ||
| چکیده | ||
| In this work, we simulate the frequency-domain helicopter-borne electromagnetic (HEM) data over the two-dimensional (2D) and three-dimensional (3D) earth models. In order to achieve this aim, the vector Helmholtz equation is used to avoid the convergence problems in Maxwell’s equations, and the corresponding fields are divided into primary and secondary components. We use the finite difference method on a staggered grid to discretize the equations, which can be performed in two ways including the conventional and improved finite difference methods. The former is very complex in terms of programming, which causes errors. Furthermore, it requires different programming loops over each point of the grid, which increases the program’s running time. The latter is the improved finite difference method (IFDM), in which pre-made derivative matrices can be used. These pre-made derivative matrices can be incorporated into the derivative equations and convert them directly from the derivative form to the matrix form. After having the matrix form system of linear equations, Ax = b is solved by the quasi-minimal residual (QMR). IFDM does not have the complexities of the conventional method, and requires much less execution time to form a stiffness or coefficient matrix. Moreover, its programing process is simple. Our code uses parallel computing, which gives us the ability to calculate the fields for all transmitter positions at the same time, and because we use sparse matrices thorough the code memory space, requires to store the files is less than 100 MB compared with normal matrices that require more than 15 GB space in the same grid size. We implement IFDM to simulate the earth’s responses. In order to validate, we compare our results with various models including the 3D and 2D models, and anisotropic conductivity. The results show a good fit in comparison with the FDM solution of Newman and the appropriate fit integral equations solution of Avdeev that is because of the different solution methods. | ||
| کلیدواژهها | ||
| Helicopter-borne electromagnetic؛ Frequency-domain؛ Forward modeling؛ Finite difference method | ||
| مراجع | ||
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[1]. Peltoniemi, M. (1998). Depth of penetration of frequency-domain airborne electromagnetics in resistive terrains. Exploration Geophysics. 29 (2): 12-15.
[2]. Tan, K., Munday, T., Halas, L., and Cahill, K. (2009). Utilizing airborne electromagnetic data to map groundwater salinity and salt store at Chowilla, SA. ASEG Extended Abstracts. 2009 (1): 1-6.
[3]. Siemon, B., Steuer, A., Ullmann, A., Vasterling, M., and Voß, W. (2011). Application of frequency-domain helicopter-borne electromagnetics for groundwater exploration in urban areas. Physics and Chemistry of the Earth, Parts A/B/C. 36 (16): 1373-1385.
[4]. Baranwal, V. C., Brönner, M., Rønning, J. S., Elvebakk, H., and Dalsegg, E. (2020). 3D interpretation of helicopter-borne frequency-domain electromagnetic (HEM) data from Ramså Basin and adjacent areas at Andøya, Norway. Earth, Planets and Space. 72 (1): 1-14.
[5]. Huang, H. and Fraser, D.C. (1996). The differential parameter method for multi-frequency airborne resistivity mapping. Geophysics 61 (1): 100–109.
[6]. Farquharson, C.G., Oldenburg, D.W., and Routh, P.S. (2003). Simultaneous 1D inversion of loop–loop electromagnetic data for magnetic susceptibility and electrical conductivity. Geophysics. 68 (6): 1857-1869.
[7]. Yin, C. and Hodges, G. (2007). Simulated annealing for airborne EM inversion. Geophysics 72
(4): F189–F195.
[8]. Arab-Amiri, A.R., Moradzadeh, A., Fathianpour, N., and Siemon, B. (2011). Inverse modeling of
HEM data using a new inversion algorithm. JME. 1 (2): 9–20.
[9]. Siemon, B. (2012). Accurate 1D forward and inverse modeling of high-frequency
helicopter-borne electromagnetic data. Geophysics .77 (4): WB71–WB87.
[10]. Lin, C., Fiandaca, G., Auken, E., Couto, M. A., and Christiansen, A.V. (2019). A discussion of 2D induced polarization effects in airborne electromagnetic and inversion with a robust 1D laterally constrained inversion scheme. Geophysics. 84 (2): E75-E88.
[11]. Sharifi, F., Arab-Amiri, A.R., Kamkar-Rouhani, A., and Börner, R.U. (2019). One-Dimensional Modeling of Helicopter-Borne Electromagnetic Data Using Marquardt-Levenberg Including Backtracking-Armijo Line Search Strategy. International Journal of Mining and Geo-Engineering. 53 (2): 143-150.
[12]. Sharifi, F., Arab-Amiri, A.R., Kamkar-Rouhani, A., and Börner, R.U. (2020). Development of a novel approach for recovering SIP effects from 1-D inversion of HEM data: Case study from the Alut area, northwest of Iran. Journal of Applied Geophysics, 174, 103962.
[13]. Mitsuhata, Y. (2000). 2-D electromagnetic modeling by finite-element method with a dipole source and topography. Geophysics. 65 (2): 465-475.
[14]. Mitsuhata, Y., Uchida, T., and Amano, H. (2002). 2.5-D inversion of frequency-domain electromagnetic data generated by a grounded-wire source. Geophysics. 67 (6): 1753-1768.
[15]. Abubakar, A., Habashy, T.M., Druskin, V.L., Knizhnerman, L., and Alumbaugh, D. (2008). 2.5 D forward and inverse modeling for interpreting low-frequency electromagnetic measurements. Geophysics. 73 (4): F165-F177.
[16]. Ramananjaona, C. and MacGregor, L. (2010, November). 2.5 D inversion of CSEM data in a vertically anisotropic earth. In Journal of Physics: Conference Series (Vol. 255, No. 1, p. 012004). IOP Publishing.
[17]. Key, K. and Ovall, J. (2011). A parallel goal-oriented adaptive finite element method for 2.5-D electromagnetic modelling. Geophysical Journal International. 186 (1): 137-154.
[18]. Streich, R., Becken, M., and Ritter, O. (2011). 2.5 D controlled-source EM modeling with general 3D source geometries. Geophysics. 76 (6): F387-F393.
[19]. Li, W.B., Zeng, Z.F., Li, J., Chen, X., Wang, K., and Xia, Z. (2016). 2.5 D forward modeling and inversion of frequency-domain airborne electromagnetic data. Applied Geophysics. 13 (1): 37-47.
[20]. Boesen, T., Auken, E., Christiansen, A. V., Fiandaca, G., Kirkegaard, C., Aspmo Pfaffhuber, A., and Vöge, M. (2018). An efficient 2D inversion scheme for airborne frequency-domain data. Geophysics, 83(4): E189-E201.
[21]. Cheng, J., Xue, J., Zhou, J., Dong, Y., and Wen, L. (2019). 2.5-D inversion of advanced detection transient electromagnetic method in full space. IEEE Access, 8, 4972-4979.
[22]. Ghari, H.A., Voge, M., Bastani, M., Pfaffhuber, A.A., and Oskooi, B. (2020). Comparing resistivity models from 2D and 1D inversion of frequency domain HEM data over rough terrains: cases study from Iran and Norway. Exploration Geophysics. 51 (1): 45-65.
[23]. Idesman, A. and Dey, B. (2020). A new numerical approach to the solution of the 2-D Helmholtz equation with optimal accuracy on irregular domains and Cartesian meshes. Computational Mechanics. 65 (4): 1189-1204.
[24]. Li, G., Duan, S., Cai, H., Han, B., and Ye, Y. (2020). Improved interpolation scheme at receiver positions for 2.5 D frequency-domain marine CSEM forward modelling. In EGU General Assembly Conference Abstracts (p. 20994).
[25]. Avdeev, D.B. (2005). Three-dimensional electromagnetic modelling and inversion from theory to application: Surveys in Geophysics, 26, 767–799, doi: 10.1007/s10712-005-1836-x.
[26]. Zhdanov, M.S. (2009). Geophysical electromagnetic theory and methods: Elsevier.
[27]. Börner, R.U. (2010). Numerical modeling in geo-electromagnetics: Advances and challenges: Surveys in Geophysics, 31, 225–245, doi: 10.1007/s10712-009-9087-x.
[28]. Koldan, J. (2013). Numerical solution of 3-D electromagnetic problems in exploration geophysics and its implementation on massively parallel computers.
[29]. Scheunert, M. (2015). 3-D inversion of helicopter-borne electromagnetic data. PhD dissertation, TU Bergakademie Freiberg.
[30]. Dunham, M.W., Ansari, S., and Farquharson, C.G. (2018). Application of 3D marine controlled-source electromagnetic finite-element forward modeling to hydrocarbon exploration in the Flemish Pass Basin offshore Newfoundland, Canada. Geophysics. 83 (2): WB33-WB49.
[31]. Coggon, J.H. (1971). Electromagnetic and electrical modeling by the finite element method. Geophysics, 36(1): 132-155.
[32]. Xiong, Z. (1992). Electromagnetic modeling of 3-D structures by the method of system iteration using integral equations. Geophysics. 57 (12): 1556-1561.
[33]. Newman, Gregory A., and David L. Alumbaugh (1995). “Frequency-domain Modelling of Airborne Electromagnetic Responses using Staggered Finite Differences”. In: Geophysical Prospecting 43, lO21–1O42.
[34]. Habashy, T.M., Groom, R.W., and Spies, B.R. (1993). Beyond the Born and Rytov approximations: A non-linear approach to electromagnetic scattering. Journal of Geophysical Research: Solid Earth. 98 (B2): 1759-1775.
[36]. Avdeev, D.B., Kuvshinov, A.V., Pankratov, O.V., and Newman, G.A. (1998). Three-dimensional frequency-domain modeling of airborne electromagnetic responses. Exploration Geophysics. 29 (1-2): 111-119.
[37]. Zhdanov, M. and Hursan, G. (2000). 3D electromagnetic inversion based on quasi-analytical approximation. Inverse Problems. 16 (5): 1297.
[38]. Pfaffhuber, A.A., Hendricks, S., and Kvistedal, Y.A. (2012). Progressing from 1D to 2D and 3D near-surface airborne electromagnetic mapping with a multisensor, airborne sea-ice explorer. Geophysics. 77 (4): WB109-WB117.
[40]. Scheunert, M., Ullmann, A., Afanasjew, M., Börner, R.U., Siemon, B., and Spitzer, K. (2016). 3D Inversion of Helicopter-borne Electromagnetic Data-A Cut-&-Paste Strategy. In 78th EAGE Conference and Exhibition 2016 (Vol. 2016, No. 1, pp. 1-5). European Association of Geoscientists & Engineers.
[41]. Jang, H., Cho, S.O., Kim, B., Kim, H. J., and Nam, M.J. (2021). Three-dimensional finite-difference modeling of time-domain electromagnetic responses for a large-loop source. Geosciences Journal. 25 (5): 675-684.
[42]. Yin, C., Zhang, B., Liu, Y., and Cai, J. (2016). A goal-oriented adaptive finite-element method for 3D scattered airborne electromagnetic method modeling. Geophysics. 81 (5): E337-E346.
[43]. Li, J., Farquharson, C. G., and Hu, X. (2017). 3D vector finite-element electromagnetic forward modeling for large loop sources using a total-field algorithm and unstructured tetrahedral grids. Geophysics. 82 (1): E1-E16.
[44]. Castillo Reyes, O. (2017). Edge-elements formulation of 3D CSEM in geophysics: a parallel approach.
[45]. Huo, Z., Zeng, Z., Li, J., Li, W., Zhang, L., and Wang, K. (2019). 3D non-linear conjugate gradient inversion for frequency-domain airborne EM based on vector finite element method. In SEG Technical Program Expanded Abstracts 2019 (pp. 2928-2932). Society of Exploration Geophysicists.
[46]. Haber, E. (2014). Computational methods in geophysical electromagnetics. Society for Industrial and Applied Mathematics.
[47]. Jahandari, H. and Farquharson, C.G. (2015). Finite-volume modelling of geophysical electromagnetic data on unstructured grids using potentials. Geophysical Journal International. 202 (3): 1859-1876.
[48]. Jahandari, H., Ansari, S., and Farquharson, C.G. (2017). Comparison between staggered grid finite–volume and edge–based finite–element modelling of geophysical electromagnetic data on unstructured grids. Journal of Applied Geophysics, 138, 185-197.
[49]. Lu, X. and Farquharson, C.G. (2020). 3D finite-volume time-domain modeling of geophysical electromagnetic data on unstructured grids using potentials. Geophysics. 85 (6): E221-E240.
[50]. Jahandari, H. and Bihlo, A. (2021). Forward modelling of geophysical electromagnetic data on unstructured grids using an adaptive mimetic finite-difference method. Computational Geosciences. 25 (3): 1083-1104.
[51]. Farquharson, C.G. and Oldenburg, D. (2002). Chapter 1 An integral equation solution to the geophysical electromagnetic forward-modelling problem. Methods in geochemistry and geophysics, 35, 3-19.
[53]. Yoon, D., Zhdanov, M.S., Mattsson, J., Cai, H., and Gribenko, A. (2016). A hybrid finite-difference and integral-equation method for modeling and inversion of marine controlled-source electromagnetic data. Geophysics. 81 (5): E323-E336.
[54]. Sarakorn, W., and Vachiratienchai, C. (2018). Hybrid finite difference–finite element method to incorporate topography and bathymetry for two-dimensional magnetotelluric modeling. Earth, Planets and Space. 70 (1): 103.
[55]. Varilsuha, D. and Candansayar, M.E. (2018). 3D magnetotelluric modeling by using finite-difference method: Comparison study of different forward modeling approaches. Geophysics. 83 (2): WB51-WB60.
[56]. van’t Hof, B. and Vuik, M.J. (2019). Symmetry-preserving finite-difference discretizations of arbitrary order on structured curvilinear staggered grids. Journal of Computational Science, 36, 101008.
[57]. Yee, K. (1966). Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media. IEEE Transactions on antennas and propagation. 14 (3): 302-307.
[58]. Gilles, L., Hagness, S.C., and Vázquez, L. (2000). Comparison between staggered and unstaggered finite-difference time-domain grids for few-cycle temporal optical soliton propagation. Journal of Computational Physics. 161 (2): 379-400.
[59]. Rumpf, R.C. (2012). Simple implementation of arbitrarily shaped total-field/scattered-field regions in finite-difference frequency-domain. Progress in Electromagnetics Research, 36, 221-248.
[60]. Rumpf, R. C., Garcia, C. R., Berry, E. A., and Barton, J. H. (2014). Finite-difference frequency-domain algorithm for modeling electromagnetic scattering from general anisotropic objects. Progress in Electromagnetics Research B, 61, 55-67.
[61]. Ward, Stanley H. and Gerald W. Hohmann (1988). “Electromagnetic Theory for Geophysical Applications”. In: Electromagnetic Methods in Applied Geophysics. Ed. by M. N. Nabighian. Vol. 1, Theory. Investigations in Geophysics 3. Society of Exploration Geophysicists, pp. 130–311.
[62]. Lowry, T., M. B. Allen, and P. N. Shive (1989). “Singularity Removal: A Refinement of Resistivity Modeling Techniques”. In: Geophysics 54, pp. 766–774.
[63] Sarkar TK. (1987) On the application of the generalized biconjugate gradient method, Journal of Electromagnetic Waves and Applications, 1, 223-242.
[64]. Smith CF, Peterson AF, and Mittra R, (1990): The biconjugate gradient method for electromagnetic scattering, IEEE Trans. Antennas Propagat., Vol. 38, pp. 938–940.
[65]. Wang, C.F. and Jin, J.M. (1998): Simple and efficient computation of electromagnetic fields in arbitrarily shaped inhomogeneous dielectric bodies using transpose-free QMR and FFT, IEEE Transactions on Microwave Theory and Techniques, vol.46, no.5, pp.553-558.
[66]. De Gersem H., Lahaye D, Vandewalle S, and Hameyer K (1999). Comparison of quasi minimal residual and bi-conjugate gradient iterative methods to solve complex symmetric systems arising from time-harmonic simulations, COMPEL, 18, 3, 298-310.
[67]. Alumbaugh, D. L., G.A. Newman, L. Prevost, and J.N. Shadid (1996). Three-dimensional wideband electromagnetic modeling on massively parallel computers, Radio Science, 31, 1-23.
[68]. Siemon, Bernhard, Anders Vest Christiansen, and Esben Auken (2009). “A Review of Helicopter-borne Electromagnetic Methods for Groundwater Exploration”. In: Near Surface Geophysics 7, pp. 629–646. | ||
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