Skip to main content
SpringerLink
Log in

Graphical Representation and Explanation of the Conductivity Tensor of Anisotropic Media

  • Published:
Surveys in Geophysics Aims and scope Submit manuscript

Abstract

Electrical anisotropy is a property of the Earth materials that can be studied through electromagnetic geophysical methods, such as magnetotellurics. It consists of the electrical conductivity changing with the orientation and being mathematically characterized by the conductivity tensor. In order to better understand the conductivity tensor and provide more effective tools for quantitatively analyzing the conductivity tensor of anisotropic structures, three graphical representations for symmetric tensors using ellipsoids, Mohr circles and geometric forms are presented. The ellipsoid representation can be applied to indicate the strength of the anisotropy in different directions. The Mohr circle provides a graphic representation of a tensor as a function of the rotation of the coordinate system. For the geometric forms, one-dimensional (1-D), two-dimensional (2-D) and three-dimensional (3-D) sheet models with given parameters (sizes and resistivities of the constituent prisms), the macroscopic anisotropic conductivity may be calculated using the closed-form mathematical formulas. These three graphical representations have different abilities for revealing information on the conductivity tensor. Four synthetic examples involving uniaxial or biaxial anisotropic conductivity structures are examined in the principal axis coordinate system to investigate the advantages and disadvantages of the graphical displays.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  • Adetunji AQ, Ferguson IJ, Jones AG (2015) Reexamination of magnetotelluric responses and electrical anisotropy of the lithospheric mantle in the Grenville Province, Canada. J Geophys Res Solid Earth 120(3):1890–1908

    Google Scholar 

  • Atkinson G (1967) An approximate flow equation for geomagnetic flux tubes and its application to polar substorms. J Geophys Res 72(21):5373–5382

    Google Scholar 

  • Baba K, Chave AD, Evans RL, Hirth G, Mackie RL (2006) Mantle dynamics beneath the East Pacific Rise at 17 degrees S: Insights from the mantle electromagnetic and tomography (MELT) experiment. J Geophys Res-Solid Earth. https://doi.org/10.1029/2004jb003598

    Article  Google Scholar 

  • Bahr K (1997) Electrical anisotropy and conductivity distribution functions of fractal random networks and of the crust: the scale effect of connectivity. Geophys J Int 130(3):649–660. https://doi.org/10.1111/j.1365-246X.1997.tb01859.x

    Article  Google Scholar 

  • Bahr K, Duba A (2000) Is the asthenosphere electrically anisotropic? Earth Planet Sci Lett 178(1):87–95. https://doi.org/10.1016/S0012-821X(00)00070-4

    Article  Google Scholar 

  • Bahr K, Smirnov M, Steveling E (2002) A gelation analogy of crustal formation derived from fractal conductive structures. J Geophys Res: Solid Earth 107(B11):ECV-18

    Google Scholar 

  • Baker WG, Martyn D (1953) The electric current in the ionosphere, part 1, the conductivity. Phil.trans.r.soc.lond.a, 246

  • Bigalke J (1999) Investigation of the conductivity of random networks. Phys A 272(3–4):281–293

    Google Scholar 

  • Bigalke J (2000a) Derivation of an equation to calculate the average conductivity of random networks. Phys A 285(3–4):295–305

    Google Scholar 

  • Bigalke J (2000b) A study concerning the conductivity of porous rock. Phys Chem Earth Part A 25(2):189–194

    Google Scholar 

  • Börner JH, Girault F, Bhattarai M, Adhikari LB, Deldicque D, Perrier F et al (2018) Anomalous complex electrical conductivity of a graphitic black schist from the Himalayas of central Nepal. Geophys Res Lett 45(9):3984–3993

    Google Scholar 

  • Chapagain NP (2016) Ionospheric plasma drift and neutral winds modeling. Res J Phys Sci 4(7):5–10

    Google Scholar 

  • Cowling TG (1945) The electrical conductivity of an ionized gas in a magnetic field, with applications to the solar atmosphere and the ionosphere. Proc R Soc A Math Phys Eng Sci 183(183):453–479

    Google Scholar 

  • Culmann C (1866) Die graphische statik (vol 1). Meyer & Zeller

  • Davydycheva S, Wang T (2011) Modeling of electromagnetic logs in a layered, biaxially anisotropic medium. In SEG technical program expanded abstracts 2011. Society of Exploration Geophysicists, pp 494–498

  • Dekker DL, Hastie LM (1980) Magneto-telluric impedances of an anisotropic layered Earth model. Geophys J R Astron Soc 61(1):11–20

    Google Scholar 

  • Du Frane WL, Roberts JJ, Toffelmier DA, Tyburczy JA (2005) Anisotropy of electrical conductivity in dry olivine. Geophys Res Lett 32(24)

  • Eisel M, Haak V (1999) Macro-anisotropy of the electrical conductivity of the crust: a magnetotelluric study of the German Continental Deep Drilling site (KTB). Geophys J Int 136(1):109–122. https://doi.org/10.1046/j.1365-246X.1999.00707.x

    Article  Google Scholar 

  • Everett ME, Constable S (1999) Electric dipole fields over an anisotropic seafloor: theory and application to the structure of 40 Ma Pacific Ocean lithosphere. Geophys J Int 136(1):41–56. https://doi.org/10.1046/j.1365-246X.1999.00725.x

    Article  Google Scholar 

  • Falae PO, Kanungo D, Chauhan P, Dash RK (2019) Recent trends in Application of electrical resistivity tomography for landslide study. In: Renewable energy and its innovative technologies. Springer, Berlin, pp 195–204

    Google Scholar 

  • Ferdinand PB, Johnson ER (1992) Mechanics of materials. McGraw-Hill, New York

    Google Scholar 

  • Feynman RP, Leighton RB, Sands M, Hafner EM (1965) The Feynman lectures on physics, vol II. Addison-Wesley Publishing, Reading

    Google Scholar 

  • Gololobov DV, Malevich IY (2005). Physical and electrochemical processes in the medium above the hydrocarbon deposit, Reports of the Belarusian State University of Informatics and Radioelectronics, (1 (9), (In Russian). (Vol. 1(9))

  • Grant FS, West GF (1965) Interpretation theory in applied geophysics. McGraw-Hill, New York

    Google Scholar 

  • Greenhalgh S, Zhou B, Greenhalgh M, Marescot L, Wiese T (2009) Explicit expressions for the Fréchet derivatives in 3D anisotropic resistivity inversion. Geophysics 74(3):F31–F43

    Google Scholar 

  • Griffiths D (1990) Failure criteria interpretation based on Mohr-Coulomb friction. J Geotech Eng 116(6):986–999

    Google Scholar 

  • Heise W, Caldwell T, Bibby H, Brown C (2006) Anisotropy and phase splits in magnetotellurics. Phys Earth Planet Int 158(2–4):107–121

    Google Scholar 

  • Herwanger J, Pain C, Binley A, De Oliveira C, Worthington M (2004) Anisotropic resistivity tomography. Geophys J Int 158(2):409–425

    Google Scholar 

  • Holder DS (2004) Electrical impedance tomography: methods, history and applications. CRC Press, Boca Raton

    Google Scholar 

  • Hoversten GM, Myer D, Key K, Alumbaugh D, Hermann O, Hobbet R (2015) Field test of sub-basalt hydrocarbon exploration with marine controlled source electromagnetic and magnetotelluric data. Geophys Prospect 63(5):1284–1310

    Google Scholar 

  • Jaeger J (1969) Elasticity, fracture and flow (3rd ed.). Methuen

  • Jeong WC, Sajib SZ, Katoch N, Kim HJ, Kwon OI, Woo EJ (2017) Anisotropic conductivity tensor imaging of in vivo canine brain using DT-MREIT. IEEE Trans Med Imaging 36(1):124–131

    Google Scholar 

  • Jolly R, Sanderson D (1997) A Mohr circle construction for the opening of a pre-existing fracture. J Struct Geol 19(6):887–892

    Google Scholar 

  • Kirkby A, Heinson G (2015) Linking electrical and hydraulic conductivity through models of random resistor networks. ASEG Ext Abstr 2015(1):1–5

    Google Scholar 

  • Kirkby A, Heinson G (2017) Three-dimensional resistor network modeling of the resistivity and permeability of fractured rocks. J Geophys Res: Solid Earth 122(4):2653–2669

    Google Scholar 

  • Kirkby A, Heinson G, Krieger L (2016a) Relating electrical resistivity to permeability using resistor networks. ASEG Ext Abstr 2016(1):1–7

    Google Scholar 

  • Kirkby A, Heinson G, Krieger L (2016b) Relating permeability and electrical resistivity in fractures using random resistor network models. J Geophys Res: Solid Earth 121(3):1546–1564

    Google Scholar 

  • Kittel C (2005) Introduction to solid state physics, 8th edn. Wiley, New York

    Google Scholar 

  • Kozlovskaya E, Hjelt S-E (2000) Modeling of elastic and electrical properties of solid-liquid rock system with fractal microstructure. Phys Chem Earth Part A 25(2):195–200

    Google Scholar 

  • Kwon OI, Jeong WC, Sajib SZ, Kim HJ, Woo EJ (2014) Anisotropic conductivity tensor imaging in MREIT using directional diffusion rate of water molecules. Phys Med Biol 59(12):2955

    Google Scholar 

  • Labuz JF, Zang A (2012) Mohr-Coulomb failure criterion. Rock Mech Rock Eng 45(6):975–979

    Google Scholar 

  • Le G, Slavin J, Strangeway R (2010) Space technology 5 observations of the imbalance of regions 1 and 2 field‐aligned currents and its implication to the cross‐polar cap Pedersen currents. J Geophys Res: Space Phys 115(A07202)

  • Li YG (2002) A finite-element algorithm for electromagnetic induction in two-dimensional anisotropic conductivity structures. Geophys J Int 148(3):389–401. https://doi.org/10.1046/j.1365-246x.2002.01570.x

    Article  Google Scholar 

  • Lilley FEM (1976) Short note: diagrams for magnetotelluric data. Geophysics 41(4):766–770

    Google Scholar 

  • Lilley FEM (1993a) Magnetotelluric analysis using Mohr circles. Geophysics 58(10):1498–1506

    Google Scholar 

  • Lilley FEM (1993b) Mohr circles in magnetotelluric interpretation (i) simple static shift; (ii) Bahr’s analysis. J Geomagn Geoelectr 45(9):833–839

    Google Scholar 

  • Lilley FEM (1998a) Magnetotelluric tensor decomposition: Part II. Examples of a basic procedure. Geophysics 63(6):1898–1907

    Google Scholar 

  • Lilley FEM (1998b) Magnetotelluric tensor decomposition: Part I, Theory for a basic procedure. Geophysics 63(6):1885–1897

    Google Scholar 

  • Lilley FEM (2012) Magnetotelluric tensor decomposition: insights from linear algebra and Mohr diagrams. In: Lim H-S (ed) New achievements in geoscience. InTech, London

    Google Scholar 

  • Lilley FE (2016) The distortion tensor of magnetotellurics: a tutorial on some properties. Explor Geophys 47(2):85–99

    Google Scholar 

  • Lilley FE (2018) The magnetotelluric tensor: improved invariants for its decomposition, especially ‘the 7th’. Explor Geophys 49(5):622–636

    Google Scholar 

  • Løseth LO (2007) Modelling of controlled source electromagnetic data. Norwegian University of Science and Technology, Trondheim

    Google Scholar 

  • Löwer A, Junge A (2017) Magnetotelluric transfer functions: phase tensor and tipper vector above a simple anisotropic three-dimensional conductivity anomaly and implications for 3D isotropic inversion. Pure Appl Geophys 174(5):2089–2101

    Google Scholar 

  • Madden TR (1976) Random networks and mixing laws. Geophysics 41(6):1104–1125

    Google Scholar 

  • Maillet R (1947) The fundamental equations of electrical prospecting. Geophysics 12(4):529–556

    Google Scholar 

  • Makris J, Bogris N, Eftaxias K (1999) A new approach in the determination of characteristic directions of the geoelectric structure using Mohr circles. Earth Planets Space 51(10):1059–1065

    Google Scholar 

  • Mandolesi E (2013) Inversion of magnetotelluric data in an anisotropic domain. National University of Ireland, Galway

    Google Scholar 

  • Mandolesi E, Jones AG (2012) Magnetotelluric inversion in a 2D anisotropic environment. In: EGU general assembly conference abstracts, 2012 (vol 14, p 13561)

  • Martí A (2014) The role of electrical anisotropy in magnetotelluric responses: from modelling and dimensionality analysis to inversion and interpretation. Surv Geophys 35(1):179–218

    Google Scholar 

  • Martí A, Queralt P, Ledo J, Farquharson C (2010) Dimensionality imprint of electrical anisotropy in magnetotelluric responses. Phys Earth Planet Int 182(3–4):139–151. https://doi.org/10.1016/j.pepi.2010.07.007

    Article  Google Scholar 

  • Masahiko Takeda (1991) Role of Hall conductivity in the ionospheric dynamo. J Geophys Res Space Phys 96(A6):9755–9759

    Google Scholar 

  • McKeagney C, Boulter C, Jolly R, Foster R (2004) 3-D Mohr circle analysis of vein opening, Indarama lode-gold deposit, Zimbabwe: implications for exploration. J Struct Geol 26(6–7):1275–1291

    Google Scholar 

  • Meju, M. A., Saleh, A. S., Mackie, R. L., Miorelli, F., Miller, R. V., & Mansor, N. K. S. (2018). Workflow for improvement of 3D anisotropic CSEM resistivity inversion and integration with seismic using cross-gradient constraint to reduce exploration risk in a complex fold-thrust belt in offshore northwest Borneo. Interpretation, 6(3), SG49–SG57. doi:10.1190/int-2017-0233.1

    Google Scholar 

  • Miensopust MP, Jones AG (2011) Artefacts of isotropic inversion applied to magnetotelluric data from an anisotropic Earth. Geophys J Int 187(2):677–689

    Google Scholar 

  • Mogilatov V, Bespalov A (2009) Biaxial anisotropy in geoelectric prospecting. Izvestiya Phys Solid Earth 45(9):822–828

    Google Scholar 

  • Mohr O (1882) Ueber die Darstellung des Spannungszustandes und des Deformationszustandes eines Korperelementes und uber die Anwendung derselben in der Festigkeitslehre. Civilengenieur 28:113–156

    Google Scholar 

  • Mollison, R., Schon, J., Fanini, O., Kreigshauser, B., Meyer, W., & Gupta, P. (1999) A model for hydrocarbon saturation determination from an orthogonal tensor relationship in thinly laminated anisotropic reservoirs. In: SPWLA 40th Annual Logging Symposium, 1999: Society of Petrophysicists and Well-Log Analysts

  • Montgomery H (1971) Method for measuring electrical resistivity of anisotropic materials. J Appl Phys 42(7):2971–2975

    Google Scholar 

  • Moran J, Gianzero S (1979) Effects of formation anisotropy on resistivity-logging measurements. Geophysics 44(7):1266–1286

    Google Scholar 

  • Nekut AG (1994) Anisotropy induction logging. Geophysics 59(3):345–350

    Google Scholar 

  • Nye JF (1985) Physical properties of crystals: their representation by tensors and matrices. Oxford University Press, Oxford

    Google Scholar 

  • Okazaki T, Oshiman N, Yoshimura R (2016) Analytical investigations of the magnetotelluric directionality estimation in 1-D anisotropic layered media. Phys Earth Planet Inter 260:25–31

    Google Scholar 

  • Onwumechilli A (1967) Geomagnetic variations in the equatorial zone. In: International geophysics, vol 11. Elsevier, Amsterdam, pp 425–507

    Google Scholar 

  • Pain CC, Herwanger JV, Saunders JH, Worthington MH, de Oliveira CR (2003) Anisotropic resistivity inversion. Inverse Prob 19(5):1081

    Google Scholar 

  • Pek J, Santos FAM (2006) Magnetotelluric inversion for anisotropic conductivities in layered media. Phys Earth Planet Inter 158:139–158

    Google Scholar 

  • Pek J, Verner T (1997) Finite-difference modelling of magnetotelluric fields in two-dimensional anisotropic media. Geophys J Int 128:505–521. https://doi.org/10.1111/j.1365-246X.1997.tb05314.x

    Article  Google Scholar 

  • Pervukhina M, Kuwahara Y, Ito H (2005) Fractal network and mixture models for elastic and electrical properties of porous rock. In: Fractal behaviour of the earth system. Springer, Berlin, pp 97-118

  • Plotkin VV (2017) Method for determining the contribution of the Hall effect in magnetotelluric sounding. In: Interexpo GEO-Siberia-2017. XIII Intern. scientific Kongr (In Russian)

  • Pommier A (2014) Interpretation of magnetotelluric results using laboratory measurements. Surv Geophys 35(1):41–84

    Google Scholar 

  • Qin L, Yang C (2016) Analytic magnetotelluric responses to a two-segment model with axially anisotropic conductivity structures overlying a perfect conductor. Geophys J Int 205(3):1729–1739

    Google Scholar 

  • Qin L, Yang C, Chen K (2013) Quasi-analytic solution of 2-D magnetotelluric fields on an axially anisotropic infinite fault. Geophys J Int 192(1):67–74

    Google Scholar 

  • Rastogi RG, Chandra H (2006) Conductivity, electric field and electron drift velocity within the equatorial electrojet. Earth, Planets and Space 58(8):1071–1077

    Google Scholar 

  • Regis C, Rijo L (1997) 1-D inversion of anisotropic magnetotelluric data. In: Extended abstracts book from the 50th congresso internacional da sociedade Brasileira de Geofisica, Brasil, vol 2, pp 673–674

  • Regis C, Rijo L (2000) Approximate equality constraints in the inversion of anisotropic MT data. In: Abstracts book, 15th workshop on electromagnetic induction in the earth, Cabo Frio, Brazil, p 47

  • Rokityansky II (1982) Geoelectromagnetic fields. In: Geoelectromagnetic investigation of the Earth’s crust and mantle. Springer, Berlin, pp 1–26

    Google Scholar 

  • Santos FAM, Mendes-Victor LA (2000) ID Anisotropic Versus 2D Isotropic Media In Magnetotellurics. Acta Geodaetica et Geophysica Hungarica 35(1):49–61

    Google Scholar 

  • Sarma S (1975) Seismic stability of earth dams and embankments. Geotechnique 25(4):743–761

    Google Scholar 

  • Sasaki Y (1994) 3-D resistivity inversion using the finite-element method. Geophysics 59(12):1839–1848

    Google Scholar 

  • Schmeling H (1986) Numerical models on the influence of partial melt on elastic, anelastic and electrical properties of rocks. Part II: electrical conductivity. Phys Earth Planet Interiors 43(2):123–136

    Google Scholar 

  • Schön JH (2015) Physical properties of rocks: fundamentals and principles of petrophysics (vol 65). Elsevier, Amsterdam

    Google Scholar 

  • Schön JH, Mollison R, Georgi D (1999) Macroscopic electrical anisotropy of laminated reservoirs: a tensor resistivity saturation model. In: SPE annual technical conference and exhibition. Society of Petroleum Engineers

  • Schön JH, Georgi D, Fanini O (2000) Anisotropic reservoir characterization (laminated sands) using orthogonal resistivity, NMR, and formation test data. In: EAGE conference on exploring the synergies between surface and borehole geoscience-petrophysics meets geophysics

  • Secor DT (1965) Role of fluid pressure in jointing. Am J Sci 263(8):633–646

    Google Scholar 

  • Semeriyanov F, Saphiannikova M, Heinrich G (2009) Anisotropic generalization of Stinchcombe’s solution for the conductivity of random resistor networks on a Bethe lattice. J Phys A: Math Theor 42(46):465001

    Google Scholar 

  • Sen AK, Torquato S (1989) Effective conductivity of anisotropic two-phase composite media. Phys Rev B 39(7):4504

    Google Scholar 

  • Seo JK, Pyo HC, Park C, Kwon O, Woo EJ (2004) Image reconstruction of anisotropic conductivity tensor distribution in MREIT: computer simulation study. Phys Med Biol 49(18):4371

    Google Scholar 

  • Ugural AC, Fenster SK (2011) Advanced mechanics of materials and applied elasticity. Pearson Education

    Google Scholar 

  • Wang T, Fang S (2001) 3-D electromagnetic anisotropy modeling using finite differences. Geophysics 66(5):1386–1398

    Google Scholar 

  • Wannamaker PE (2005) Anisotropy versus heterogeneity in continental solid earth electromagnetic studies: fundamental response characteristics and implications for physicochemical state. Surv Geophys 26(6):733–765

    Google Scholar 

  • Weaver JT (2004) The use of Mohr circles in the interpretation of magnetotelluric data. ASEG Extended Abstracts 2004(1):1–4

    Google Scholar 

  • Weaver JT, Lilley F (2004) Using Mohr circles to identify regional dimensionality and strike angle from distorted magnetotelluric data. Explor Geophys 35(4):251–254

    Google Scholar 

  • Weidelt P, Oristaglio M, Spies B (1999) 3-D conductivity models: implications of electrical anisotropy. In: Oristaglio M, Spies B, Cooper MR (eds) Three-dimensional electromagnetics. Society of Exploration Geophysicists, Tulsa, pp 119–137

  • Weiss CJ, Newman GA (2002) Electromagnetic induction in a fully 3-D anisotropic earth. Geophysics 67(4):1104–1114

    Google Scholar 

  • Wiese T, Greenhalgh S, Zhou B, Greenhalgh M, Marescot L (2015) Resistivity inversion in 2-D anisotropic media: numerical experiments. Geophys J Int 201(1):247–266

    Google Scholar 

  • Yang C (1997) MT numerical simulation of symmetrically 2D Anisotropic media based on the finite element method (in Chinese). Northwest Seismol J 19(2):27–33

    Google Scholar 

  • Yin C (2000) Geoelectrical inversion for 1D anisotropic models and inherent non-uniqueness. Geophys J Int 140:11–23

    Google Scholar 

  • Yin C (2003) Inherent nonuniqueness in magnetotelluric inversion for 1D anisotropic models. Geophysics 68(1):138–146

    Google Scholar 

  • Yin C, Weidelt P (1999) Geoelectrical fields in a layered earth with arbitrary anisotropy. Geophysics 64(2):426–434

    Google Scholar 

  • Yu L, Evans R, Edwards R (1997) Transient electromagnetic responses in seafloor with triaxial anisotropy. Geophys J Int 129(2):292–304

    Google Scholar 

  • Yuan N, Nie XC, Liu R, Qiu CW (2010) Simulation of full responses of a triaxial induction tool in a homogeneous biaxial anisotropic formation. Geophysics 75(2):E101–E114

    Google Scholar 

Download references

Acknowledgements

The research was supported by the National Natural Science Foundation of China (Grant Nos. 41474054, 40774035 and 41776079). The authors thank Dr. Pilar Queralt (Editor) and two reviewers Dr. Ian Ferguson and Dr. Anna Martí for their thorough reading of the manuscript and for their insightful comments and constructive suggestions which certainly improved the quality of this paper.

Author information

Authors and Affiliations

  1. School of Earth Sciences, Zhejiang University, Hangzhou, 310027, China

    Chang-fu Yang

  2. Key Laboratory of Submarine Geosciences, Ministry of Natural Resources, Hangzhou, 310012, China

    Lin-jiang Qin

  3. Second Institute of Oceanography, Ministry of Natural Resources, Hangzhou, 310012, China

    Lin-jiang Qin

Authors
  1. Chang-fu Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lin-jiang Qin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chang-fu Yang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, Cf., Qin, Lj. Graphical Representation and Explanation of the Conductivity Tensor of Anisotropic Media. Surv Geophys 41, 249–281 (2020). https://doi.org/10.1007/s10712-019-09581-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10712-019-09581-5

Keywords

Search

Navigation