Skip to main content
SpringerLink
Log in

Inverse Magnetometry

  • Chapter
  • First Online:
Inverse Magnetometry

Part of the book series: Lecture Notes in Geosystems Mathematics and Computing ((LNGMC))

  • 258 Accesses

Abstract

This chapter discusses inverse magnetometry as an ill-posed problem in dipole reflected nomenclature. It is mentioned that all criteria of Hadamard’s classification (existence, uniqueness, and stability) are violated for terrestrial data. Consequently, inverse magnetometry is considered to be “too ill-posed” in order to use regularization techniques other than mollifier regularization.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alberto, P., Oliveira, O., Pais, M.A.: On the non-uniqueness of main geomagnetic field determined by surface intensity measurements: The Backus Problem. Geophys. J. Int. 159, 548–554 (2004)

    Article  Google Scholar 

  2. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  3. Backus, G.E.: Non-uniqueness of the external geomagnetic field determined by surface intensity measurements. J. Geophys. Res. 75, 6339–6341 (1970)

    Article  Google Scholar 

  4. Backus, G.E., Gilbert, F.: Numerical applications of a formalism for geophysical inverse problems. Geophys. J. R. Astron. Soc. 13, 247–276 (1967)

    Article  Google Scholar 

  5. Backus, G.E., Gilbert, F.: The resolving power of gross Earth data. Geophys. J. R. Astron. Soc. 16, 169–205 (1968)

    Article  MATH  Google Scholar 

  6. Backus, G.E., Gilbert, F.: Uniqueness of the inversion of inaccurate gross Earth data. Philos. Trans. R. Soc. Lond. 226, 123–197 (1970)

    MathSciNet  Google Scholar 

  7. Blakely, R.J.: Potential Theory in Gravity and Magnetic Application. Cambridge University, Cambridge (1996)

    Google Scholar 

  8. Bullard E.C.: The magnetic field within the Earth. Proc. Camb. Phil. Soc. A197, 438–453 (1949)

    Google Scholar 

  9. Eggermont, P.N., LaRiccia, V., Nashed, M.Z.: Noise models for ill-posed problems. In: W. Freeden, Z. Nashed, T. Sonar (eds.) Handbook of Geomathematics, 2nd edn., vol. 2, pp. 1633–1658. Springer, New York (2015)

    Chapter  Google Scholar 

  10. Engl, H.: Integralgleichungen. Springer Lehrbuch Mathematik, Wien (1997)

    Google Scholar 

  11. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publisher, Dordrecht (1996)

    Book  MATH  Google Scholar 

  12. Engl, H., Louis, A.K., Rundell, W. (eds.) Inverse problems in geophysical applications. SIAM, Philadelphia (1997)

    Google Scholar 

  13. Freeden, W.: On approximation by harmonic splines. Manuscr. Geodaet. 6, 193–244 (1981)

    MATH  Google Scholar 

  14. Freeden, W.: Multiscale Modeling of Spaceborne Geodata. Teubner, Stuttgart (1999)

    MATH  Google Scholar 

  15. Freeden, W.: Geomathematics: Its Role, its Aim, and its Potential. In: Freeden, W., Nashed, Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn., vol. 1, pp. 3–78 Springer, New York (2015)

    Google Scholar 

  16. Freeden, W.: Decorrelative Mollifier Gravimetry–Basics, Concepts, Examples and Perspectives. Geosystems Mathematics, Birkhäuser (2021)

    Google Scholar 

  17. Freeden, W., Bauer, M.: Dekorrelative Gravimetrie—Ein innovativer Zugang in Exploration und Geowissenschaften. Springer Spektrum, Berlin (2020)

    Book  Google Scholar 

  18. Freeden, W., Blick, C.: Signal decorrelation by means of multiscale methods. World Min. 65, 304–317 (2013)

    Google Scholar 

  19. Freeden, W., Gerhards, C.: Geomathematically Oriented Potential Theory. CRC Press/Taylor and Francis, Boca Raton (2013)

    MATH  Google Scholar 

  20. Freeden, W., Gutting, M.: Integration and Cubature Methods. Chapman and Hall/CRC Press, Boca Raton/New York (2018)

    MATH  Google Scholar 

  21. Freeden, W., Michel, V.: Multiscale Potential Theory (with Applications to Geoscience). Birkhäuser, Boston (2004)

    Book  MATH  Google Scholar 

  22. Freeden, W., Nashed, M.Z.: Ill-posed problems: operator methodologies of resolution and regularization approaches. In: Freeden, W. and Nashed, M.Z. (eds.) Handbook of Mathematical Geodesy. Geosystems Mathematics, pp. 201–314. Springer/Birkhäuser, Basel/New York (2018a)

    Chapter  Google Scholar 

  23. Freeden, W., Nashed, M.Z.: Inverse gravimetry: background material and multiscale mollifier approaches. GEM Int. J. Geomath. 9, 199–264 (2018c)

    Article  MathSciNet  MATH  Google Scholar 

  24. Freeden, W., Nashed, M.Z.: Operator-theoretic and regularization approaches to ill-posed problems. GEM Int. J. Geomath. 9, 1–115 (2018d)

    Article  MathSciNet  MATH  Google Scholar 

  25. Freeden, W., Nashed, M.Z.: Inverse gravimetry: density signatures from gravitational potential data. In: Freeden, W., Rummel, R. (eds.) Handbuch der Geodäsie, Mathematische Geodäsie/Mathematical Geodesy, pp. 969–1052. Springer Spektrum, Heidelberg (2020)

    Chapter  Google Scholar 

  26. Freeden, W., Nashed, M.Z., Schreiner, M.: Spherical Sampling. Geosystems Mathematics, Springer, Basel (2018)

    Google Scholar 

  27. Freeden, W., Nutz, H.: Inverse Probleme der Geodäsie: Ein Abriss mathematischer Lösungsstrategien. In: Freeden, W., Rummel, R. (eds.) Handbuch der Geodäsie, Mathematische Geodäsie/Mathematical Geodesy, pp. 65–90. Springer Spektrum, Heidelberg (2020)

    Chapter  Google Scholar 

  28. Freeden, W., Schreiner, M.: Local multiscale modeling of geoid undulations from deflections of the vertical. J. Geodesy 79, 641–651 (2006)

    Article  MATH  Google Scholar 

  29. Freeden, W., Heine, C., Nashed M.Z.: An Invitation to Geomathematics. Lecture Notes in Geosystem Mathematics and Computing (2019)

    Google Scholar 

  30. Freeden W., Nutz H., Rummel R., Schreiner M.: Satellite gravity gradiometry (SGG): Methodological foundation and geomathematical advances. In: Freeden, W., Rummel, R. (eds.) Handbuch der Geodäsie, Mathematische Geodäsie/Mathematical Geodesy, pp. 1185–1256. Springer Spektrum, Heidelberg (2020)

    Google Scholar 

  31. Hadamard, J.: Sur les problèmes aux dérivés partielles et leur signification physique. Princeton Univ. Bull. 13, 49–52 (1902)

    MathSciNet  Google Scholar 

  32. Hadamard, J.: Lectures on the Cauchy-problem in linear partial differential equations. Yale University, New Haven (1923)

    MATH  Google Scholar 

  33. Jacobs, F., Meyer, H.: Geophysik–Signale aus der Erde. Teubner, Leipzig (1992)

    Book  Google Scholar 

  34. Leweke, S., Michel, V., Telschow, R.: On the uniqueness of gravitational and magnetic field data inversion. In: Freeden, W., Nashed, M.Z. (eds.): Handbook of Mathematical Geodesy. Geosystems Mathematics, pp. 883–920. Springer International Publishing, Birkhäuser, New York (2018)

    Google Scholar 

  35. Lima, E.A., Irimia, A., Wikswo, J.P.: The magnetic inverse problem. In: Clarke, J., and Braginski, A.E. (eds.) The SQUID Handbook, vol. II. WILEY-VCH, Weinheim (2006)

    Google Scholar 

  36. Louis, A.K., Maass, P.: A mollifier method for linear equations of the first kind. Inverse Prob. 6, 427–440 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mayer, C.: Wavelet modelling of the spherical inverse source problem with application to geomagnetism. Inverse Prob. 20, 1713–1728 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  38. Michel, V.: A multiscale approximation for operator equations in separable Hilbert spaces—case study: reconstruction and description of the Earth’s interior. University of Kaiserslautern, Geomathematics Group, Habilitation Thesis (2002)

    Google Scholar 

  39. Nashed, M.Z.: Generalized inverses, normal solvability and iteration for singular operator equations. In: Rall, L.B. (ed.) Nonlinear Functional Analysis and Applications, pp. 311–359. Academic Press, New York (1971)

    Chapter  Google Scholar 

  40. Nashed, M.Z.: Aspects of generalized inverses in analysis and regularization. In: Generalized Inverses and Applications, pp. 193–244. Academic Press, New York (1976)

    Google Scholar 

  41. Nashed, M.Z.: Operator-theoretic and computational approaches to ill-posed problems with applications to antenna theory. IEEE Trans. Antennas Propag. 29, 220–231 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  42. Nashed, M.Z.: A new approach to classification and regularization of ill-posed operator equations, inverse and ill-posed problems. In: Engl, H.W. and Groetsch, C.W. (eds.) Notes and Reports in Mathematics in Science and Engineering, vol. 4. Academic Press, New York (1987)

    Google Scholar 

  43. Nashed, M.Z., Votruba, F.G.: A unified operator theory of generalized inverses. In: Nashed, M.Z. (ed.) Generalized Inverses and Applications, pp. 1–109. Academic Press, New York (1976)

    Google Scholar 

  44. Nashed, M.Z., Walter, G.G.: General sampling theorems for functions in reproducing kernel Hilbert space. Math. Control Signals Syst. 4, 363–390 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  45. Nashed, M.Z., Walter, G.G.: Reproducing kernel Hilbert space from sampling expansions. Contemp. Math. 190, 221–226 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  46. Parker, R.L.: The inverse problem of electromagnetic induction: Existence and construction of solutions based on incomplete data. J. Geophys. Res. 85, 3321–3328 (1980)

    Google Scholar 

  47. Saltus, R.W., Blakely, R.J.: Unique geologic insights from “non-unique” gravity and magnetic interpretation. GSA Today 21, 4–11 (2011)

    Article  Google Scholar 

  48. Shure, L., Parker, R.L., Backus, G.E.: Harmonic splines for geomagnetic modelling. Phys. Earth Planet. Inter. 28, 215–229 (1982)

    Article  Google Scholar 

  49. Shure, I., Parker, R.L., Langel, R.A.: A preliminary harmonic spline model from Magsat data. J. Geophys. Res. 90, 11505–11512 (1985)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Business und Management mbH, CBM - Gesellschaft für Consulting, Bexbach, Saarland, Germany

    Christian Blick & Helga Nutz

  2. Mathematics Department, University of Kaiserslautern, Kaiserslautern, Rheinland-Pfalz, Germany

    Willi Freeden

  3. Mathematics Department, University of Central Florida, Orlando, FL, USA

    M. Zuhair Nashed

  4. OST, Institute for Computational Engineering, Buchs, Switzerland

    Michael Schreiner

Authors
  1. Christian Blick
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Willi Freeden
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Zuhair Nashed
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Helga Nutz
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Michael Schreiner
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Blick, C., Freeden, W., Nashed, M.Z., Nutz, H., Schreiner, M. (2021). Inverse Magnetometry. In: Inverse Magnetometry. Lecture Notes in Geosystems Mathematics and Computing. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-79508-5_4

Download citation

Publish with us

Policies and ethics

Search

Navigation