1887
Volume 67, Issue 8
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

We present a concept of the hybrid finite volume–integral equation technique for solving Maxwell's equation in a quasi‐static form. The divergence correction was incorporated to improve the convergence and stability of the governing linear system equations which pose a challenge on the discretization of the curl–curl Helmholtz equation. A staggered finite volume approach is applied for discretizing the system of equations on a structured mesh and solved in a secondary field technique. The bi‐conjugate gradient stabilizer was utilized with block incomplete lower‐upper factorization preconditioner to solve the system of equation. To obtain the electric and magnetic fields at the receivers, we use the integral Green tensor scheme. We verify the strength of our hybrid technique with benchmark models relative to other numerical algorithms. Importantly, from the tested models, our scheme was in close agreement with the semi‐analytical solution. It also revealed that the use of a quasi‐analytical boundary condition helps to minimize the runtime for the linear system equation. Furthermore, the integral Green tensor approach to compute at the receivers demonstrates better accuracy compared with the conventional interpolation method. This adopted technique can be applied efficiently to the inversion procedure.

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2019-09-10
2024-04-25
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References

  1. AlumbaughD., NewmanG., PrevostL. and ShadidJ.1996. Three‐dimensional wide‐band electromagnetic modelling on massively parallel computer. Radio Science31, 1–23.
    [Google Scholar]
  2. AvdeevD.2005. Three‐dimensional electromagnetic modeling and inversion from theory to application. Surveys in Geophysics26, 767–799.
    [Google Scholar]
  3. AvdeevD. and KnizhnikS.2009. 3D integral equation modeling with a linear dependence on dimensions. Geophysics74, F89–F94.
    [Google Scholar]
  4. AvdeevD., KuvshinovA., PankratovO. and NewmanG.1997. High‐performance three‐dimensional electromagnetic modeling using modified Neumann series. Wide‐band numerical solution and examples. Journal of Geomagnetism and geoelectricity, 49, 1519–1539.
    [Google Scholar]
  5. BerdichevskiiM. and DmitrievV.2008. Models and Methods of Magnetotellurics. Springer.
    [Google Scholar]
  6. BestM., DuncanP., JacobsF. and ScheenW.1985. Numerical modelling of the electromagnetic response of three‐dimensional conductors in layered earth. Geophysics50, 665–676.
    [Google Scholar]
  7. BörnerR.2010. Numerical modelling in geo‐electromagnetics: advances and challenges. Surveys in Geophysics31, 225–245.
    [Google Scholar]
  8. ChaveA. and JonesA.2012. The Magnetotelluric Method – Theory and Practice. Cambridge University Press.
    [Google Scholar]
  9. CoxL.H. and ZhdanovM.2014. 3D airborne electromagnetic inversion using a hybrid edge‐based FE‐IE method with moving sensitivity domain. 84th Annual International Meeting, SEG, Expanded Abstracts, 739–744.
  10. De Groot‐HedlinC.2006. Finite‐difference modeling of magnetotelluric fields: error estimates for uniform and nonuniform grids. Geophysics71, 3, G97–G106.
    [Google Scholar]
  11. EgbertG. and KelbertA.2012. Computational recipes for electromagnetic inverse problems. Geophysical Journal International189, 251–267.
    [Google Scholar]
  12. EndoM., ČumaM. and ZhdanovM.2008. A multigrid integral equation method for large‐scale models with inhomogeneous backgrounds. Journal of Geophysics and Engineering5, 438–447.
    [Google Scholar]
  13. FarquharsonC. and MiensopustM.2011. Three‐dimensional finite element modelling of magnetotelluric data with a divergence correction. Journal of Applied Geophysics75, 699–710.
    [Google Scholar]
  14. FomenkoE. and MogiT.2002. A new computation method for a staggered grid of 3‐D EM field conservative modelling. Earth, Planets and Space54, 499–509.
    [Google Scholar]
  15. GuptaP., BennettjL. and HaichejA.1987. Hybrid calculations of the three‐dimensional electromagnetic response of buried conductors. Geophysics52, 301–306.
    [Google Scholar]
  16. HabashyT., GroomR. and SpiesB.1993. Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering. Journal of Geophysical Research98, 1759.
    [Google Scholar]
  17. HaberE., AscherU., AruliahD. and OldenburgD.2000. Fast simulation of 3D electromagnetic problems using potentials. Journal of Computational Physics163, 150–171.
    [Google Scholar]
  18. HaberE. and RuthottoL.2014. A multiscale finite volume method for Maxwell's equations at low frequencies. Geophysical Journal International199, 1268–1277.
    [Google Scholar]
  19. HursánG. and ZhdanovM.2002. Contraction integral equation method in three‐dimensional electromagnetic modelling. Radio Science37, 1‐1–1‐13.
    [Google Scholar]
  20. HymanJ. and ShashkovM.1997. The adjoint operators for natural discretizations for the divergence, gradient and curl on logically rectangular grids. IMACS Journal of Applied Numerical Mathematics25, 413–442.
    [Google Scholar]
  21. KoyamaT., UtadaH. and AvdeevD.2008. Fast and memory‐saved 3‐D forward modeling code for magnetotelluric by using integral equation method. Abstract Book, 19th Workshop on Electromagnetic Induction in the Earth, China.
  22. KruglyakovM., GeraskinA. and KuvshinovA.2016. Novel accurate and scalable 3‐D MT forward solver based on a contracting integral equation method. Computers & Geosciences96, 208–217.
    [Google Scholar]
  23. KruglyakovM. and KuvshinovA.2018. Using high‐order polynomial basis in 3‐D EM forward modelling based on volume integral equation method. Geophysical Journal International213, 1387–1401.
    [Google Scholar]
  24. LeeK., PridmoreD. and MorrisonH.1981. A hybrid 3D electromagnetic modelling scheme. Geophysics46, 796–805.
    [Google Scholar]
  25. MackieR., SmithJ. and MaddenT.1994. Three‐dimensional electromagnetic modeling using finite difference equations. The magnetotelluric example. Radio Science29, 923–935.
    [Google Scholar]
  26. MiensopustM., QueraltP. and JonesA., et al. 2013. Magnetotelluric 3‐D inversion—a review of two successful workshops on forward and inversion code testing and comparison. Geophysical Journal International193, 1216–1238.
    [Google Scholar]
  27. MitsuhataY. and UchidaT.2004. 3D magnetotelluric modeling using the T‐Ω finite‐element method. Geophysics69, 108–119.
    [Google Scholar]
  28. NewmanG.2014. A review of high‐performance computational strategies for modeling and imaging of electromagnetic induction data. Surveys in Geophysics35, 85–100.
    [Google Scholar]
  29. PankratovO. and KuvshinovA.2016. Applied mathematics in EM studies with special emphasis on an uncertainty quantification and 3‐D IE modelling. Surveys in Geophysics37, 109–147.
    [Google Scholar]
  30. RaicheA.1974. An integral equation approach to three‐dimensional modelling. Geophysical Journal of the Royal Astronomical Society36, 363–376.
    [Google Scholar]
  31. RenZ., KalscheuerT., GreenhalghS. and MaurerH.2014. A hybrid boundary element finite element approach to modeling plane wave 3D electromagnetic induction responses in the Earth. Journal of Computational Physics258, 705–17.
    [Google Scholar]
  32. SimpsonF. and BahrK.2005. Practical Magnetotellurics. Cambridge University Press.
    [Google Scholar]
  33. SiripunvarapornW. and EgbertG., and LenburyY.2002. Numerical accuracy of magnetotelluric modelling. A comparison of finite difference approximations. Earth, Planets and Space54, 721–725.
    [Google Scholar]
  34. SmithJ.1996a. Conservative modeling of 3‐D electromagnetic fields, part I: properties and error analysis. Geophysics61, 1308–1318.
    [Google Scholar]
  35. SmithJ.1996b. Conservative modeling of 3‐D electromagnetic fields, Part II: biconjugate gradient solution and an accelerator. Geophysics61, 1319–1324.
    [Google Scholar]
  36. TingS. and HohmannG.1981. Integral equation modeling of three‐dimensional magnetotelluric response. Geophysics46, 182–197.
    [Google Scholar]
  37. TongX., LiuJ., XieW., XuL., GuoR. and ChengY.2009. Three‐dimensional forward modeling for magnetotelluric sounding by finite element method. Journal of Central South University of Technology16, 136–142.
    [Google Scholar]
  38. Van der VorstH.1992. BI‐CGSTAB: a fast and smoothly convergent variant of BI‐CG for the solution of nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing13, 631–644.
    [Google Scholar]
  39. WannamakerP.1991. Advances in three dimensional magnetotelluric modeling using integral equations. Geophysics56, 1716–1728.
    [Google Scholar]
  40. WardS. and HohmannG.1987. Electromagnetic theory for geophysical applications. In: Electromagnetic Methods in Applied Geophysics 1: Investigations in Geophysics (ed. M.Nabighian), pp. 130–311. Society of Exploration Geophysicists.
    [Google Scholar]
  41. WeideltP.1975. Electromagnetic induction in three‐dimensional structures. J. Geophysics41, 85–109.
    [Google Scholar]
  42. WeissC.2013. Project APhiD: a Lorenz‐gauged A‐Φ decomposition for parallelized computation of ultra‐broadband electromagnetic induction in a fully heterogeneous Earth. Computers and Geosciences58, 40–52.
    [Google Scholar]
  43. WeissC. and ConstableS.2006. Mapping thin resistor and hydrocarbons with marine EM methods: modeling and analysis in 3D. Geophysics71, G321–G332.
    [Google Scholar]
  44. YoonD., ZhdanovM., MattssonJ., CaiH. and GribenkoA.2016. A hybrid finite‐difference and integral‐equation method for modeling and inversion of marine controlled‐source electromagnetic data. Geophysics81, E323–36.
    [Google Scholar]
  45. ZaslavskyM., DruskinV., DavydychevaS., KnizhnermanL., AbubakarA. and HabashyT. 2011. Hybrid finite‐difference integral equation solver for 3D frequency domain anisotropic electromagnetic problems. Geophysics76, 2, F123–F137.
    [Google Scholar]
  46. ZhdanovM.2002. Geophysical Inverse Theory and Regularization Problems. Elsevier.
    [Google Scholar]
  47. ZhdanovM.2009. Geophysical Electromagnetic Theory and Methods. Elsevier.
    [Google Scholar]
  48. ZhdanovM.2010. Electromagnetic geophysics: notes from the past and the road ahead. Geophysics75, 75A49–75A66.
    [Google Scholar]
  49. ZhdanovM., DmitrievV I., FangS. and HursanG.2000. Quasi‐analytical approximations and series in electromagnetic modeling, Geophysics65, 1746.
    [Google Scholar]
  50. ZhdanovM., LeeS. and YoshiokaK.2006. Integral equation method for 3D modelling of electromagnetic fields in complex structures with inhomogeneous background conductivity. Geophysics71, G333–G345.
    [Google Scholar]
  51. ZhdanovM., VarentovI., WeaverJ., GolubevN. and KrylovbV.1997. Methods for modeling electromagnetic fields: results from COMMEMI—the international project on the comparison of modelling methods for electromagnetic induction. Journal of Applied Geophysics37, 133–271.
    [Google Scholar]
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  • Article Type: Research Article
Keyword(s): 3D; Iterative solver; Magnetotellurics; Modelling

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