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Reconstruction and Decomposition of Scalar and Vectorial Potential Fields on the Sphere

A Brief Overview

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Handbuch der Geodäsie

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Abstract

We give a brief overview on approximation methods on the sphere that can be used in a variety of geophysical setups. A particular focus is on methods related to potential field problems and spatial localization, such as spherical splines, multiscale methods, and Slepian functions. Furthermore, we introduce the common Helmholtz and Hardy-Hodge decompositions of spherical vector fields together with some related recent results. The methods are illustrate for two different examples: determination of the disturbing potential from deflections of the vertical and approximation of magnetic fields induced by oceanic tides.

Zusammenfassung

Wir geben einen kurzen Überblick über Approximationsmethoden auf der Sphäre, welche Anwendung in verschiedenen geophysikalischen Fragestellungen finden. Im Speziellen geht es um Methoden mit Bezug zu Potentialfeldpro- blemen und Lokalisierung auf der Sphäre (z. B. Splines, Multiskalenmethoden und Slepian Funktionen). Des Weiteren führen wir zwei bekannte Vektorfeldzerlegungen (Helmholtz und Hardy-Hodge) ein und stellen die Verbindung zu einigen neueren Resultaten her. Abschliessend illustrieren wir unsere Ansätze an zwei geophysikalischen Beispielen: der Bestimmung des Störpotentials aus Lotabweichungen und der Approximation des Magnetfelds, welches durch Ozeangezeiten erzeugt wird.

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Notes

  1. 1.

    Typically, the order is (0,  0),  (1, −1),  (1,  0),  (1,  1),  (2, −2), … such that the pair (n, k) is at position n 2 + n + k + 1 in a row or column.

References

  1. Alfeld, P., Neamtu, M., Shumaker, L.L.: Fitting data on sphere-like surfaces using splines. J. Comput. Appl. Math. 73, 5–43 (1995)

    Article  Google Scholar 

  2. Atkinson, K.: Numerical integration on the sphere. J. Austr. Math. Soc. 23, 332–347 (1982)

    Article  Google Scholar 

  3. Backus, G., Parker, R., Constable, C.: Foundations of Geomagnetism. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  4. Baratchart, L., Gerhards, C.: On the recovery of crustal and core contributions in geomagnetic potential fields. SIAM J. Appl. Math. 77, 1756–1780 (2017)

    Article  Google Scholar 

  5. Baratchart, L., Hardin, D.P., Lima, E.A., Saff, E.B., Weiss, B.P.: Characterizing kernels of operators related to thin plate magnetizations via generalizations of Hodge decompositions. Inverse Prob. 29, 015004 (2013)

    Article  Google Scholar 

  6. Bauer, F., Gutting, M., Lukas, M.A.: Evaluation of parameter choice methods for regularization of ill-posed problems in geomathematics. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn. Springer, Berlin (2015)

    Google Scholar 

  7. Bauer, F., Reiß, M.: Regularization independent of the noise level: an analysis of quasi-optimality. Inverse Prob. 24, 055009 (2008)

    Article  Google Scholar 

  8. Berkel, P., Fischer, D., Michel, V.: Spline multiresolution and numerical results for joint gravitation and normal-mode inversion with an outlook on sparse regularisation. GEM Int. J. Geomath. 1, 167–204 (2011)

    Article  Google Scholar 

  9. Chambodut, A., Panet, I., Mandea, M., Diamet, M., Holschneider, M., Jamet, O.: Wavelet frames: an alternative to spherical harmonic representation of potential fields. Geophys. J. Int. 163, 875–899 (2005)

    Article  Google Scholar 

  10. Dahlke, S., Dahmen, W., Schmitt, E., Weinreich, I.: Multiresolution analysis and wavelets on S 2 and S 3. Num. Func. Anal. Appl. 16, 19–41 (1995)

    Article  Google Scholar 

  11. Driscoll, J.R., Healy, M.H. Jr.: Computing fourier transforms and convolutions on the 2-sphere. Adv. Appl. Math. 15, 202–250 (1994)

    Article  Google Scholar 

  12. Fehlinger, T., Freeden, W., Mayer, C., Michel, D., Schreiner, M.: Local modelling of sea surface topography from (geostrophic) ocean flow. ZAMM 87, 775–791 (2007)

    Article  Google Scholar 

  13. Fehlinger, T., Freeden, W., Mayer, C., Schreiner, M.: On the local multiscale determination of the earths disturbing potential from discrete deflections of the vertical. Comput. Geosci. 12, 473–490 (2009)

    Article  Google Scholar 

  14. Fischer, D., Michel, V.: Sparse regularization of inverse gravimetry – case study: spatial and temporal mass variations in South America. Inverse Prob. 28, 065012 (2012)

    Article  Google Scholar 

  15. Fischer, D., Michel, V.: Automatic best-basis selection for geophysical tomographic inverse problems. Geophys. J. Int. 193, 1291–1299 (2013)

    Article  Google Scholar 

  16. Freeden, W.: On integral formulas of the (unit) sphere and their application to numerical computation of integrals. Computing 25, 131–146 (1980)

    Article  Google Scholar 

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

    Google Scholar 

  18. Freeden, W.: Multiscale Modelling of Spaceborne Geodata. Teubner, Leipzig (1999)

    Google Scholar 

  19. Freeden, W., Gerhards, C.: Poloidal and toroidal field modeling in terms of locally supported vector wavelets. Math. Geosc. 42, 818–838 (2010)

    Article  Google Scholar 

  20. Freeden, W., Gerhards, C.: Geomathematically Oriented Potential Theory. Pure and Applied Mathematics. Chapman & Hall/CRC, Boca Raton (2012)

    Book  Google Scholar 

  21. Freeden, W., Gerhards, C.: Romberg extrapolation for Euler summation-based cubature on regular regions. GEM Int. J. Geomath. 8, 169–182 (2017)

    Article  Google Scholar 

  22. Freeden, W., Gerhards, C., Schreiner, M.: Disturbing potential from deflections of the vertical: from globally reflected surface gradient equation to locally oriented multiscale modeling. In: Grafarend, E. (ed.) Encyclopedia of Geodesy. Springer International Publishing (2015)

    Google Scholar 

  23. Freeden, W., Gervens, T.: Vector spherical spline interpolation – basic theory and computational aspects. Math. Methods Appl. Sci. 16, 151–183 (1993)

    Article  Google Scholar 

  24. Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere (with Applications to Geomathematics). Oxford Science Publications. Clarendon Press, Oxford (1998)

    Google Scholar 

  25. Freeden, W., Gutting, M.:Special Functions of Mathematical (Geo-)Physics. Applied and Numerical Harmonic Analysis. Springer, Basel (2013)

    Book  Google Scholar 

  26. Freeden, W., Hesse, K.: On the multiscale solution of satellite problems by use of locally supported kernel functions corresponding to equidistributed data on spherical orbits. Stud. Sci. Math. Hungar. 39, 37–74 (2002)

    Google Scholar 

  27. Freeden, W., Michel, V.: Constructive approximation and numerical methods- in geodetic research today – an attempt at a categorization based on an uncertainty principle. J. Geod. 73, 452–465 (1999)

    Article  Google Scholar 

  28. Freeden, W., Schneider, F.: Regularization wavelets and multiresolution. Inverse Prob. 14, 225–243 (1998)

    Article  Google Scholar 

  29. Freeden, W., Schreiner, M.: Local multiscale modeling of geoidal undulations from deflections of the vertical. J. Geod. 78, 641–651 (2006)

    Article  Google Scholar 

  30. Freeden, W., Schreiner, M.: Spherical Functions of Mathematical Geosciences. Springer, Berlin/Heidelberg (2009)

    Google Scholar 

  31. Freeden, W., Windheuser, U.: Combined spherical harmonics and wavelet expansion – a future concept in Earth’s gravitational potential determination. Appl. Comput. Harm. Anal. 4, 1–37 (1997)

    Article  Google Scholar 

  32. Gemmrich, S., Nigam, N., Steinbach, O.: Boundary integral equations for the Laplace-Beltrami operator. In: Munthe-Kaas, H., Owren, B. (eds.) Mathematics and Computation, a Contemporary View. Proceedings of the Abel Symposium 2006. Springer, Berlin (2008)

    Google Scholar 

  33. Gerhards, C.: Spherical decompositions in a global and local framework: theory and an application to geomagnetic modeling. GEM Int. J. Geomath. 1, 205–256 (2011)

    Article  Google Scholar 

  34. Gerhards, C.: Locally supported wavelets for the separation of spherical vector fields with respect to their sources. Int. J. Wavel. Multires. Inf. Process. 10, 1250034 (2012)

    Article  Google Scholar 

  35. Gerhards, C.: A combination of downward continuation and local approximation for harmonic potentials. Inverse Prob. 30, 085004 (2014)

    Article  Google Scholar 

  36. Gerhards, C.: Multiscale modeling of the geomagnetic field and ionospheric currents. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn. Springer, Berlin (2015)

    Google Scholar 

  37. Gerhards, C.: On the unique reconstruction of induced spherical magnetizations. Inverse Prob. 32, 015002 (2016)

    Article  Google Scholar 

  38. Gerhards, C.: On the reconstruction of inducing dipole directions and susceptibilities from knowledge of the magnetic field on a sphere. Inv. Probl. Sci. Engin. https://doi.org/10.1080/17415977.2018.1438426, to appear.

    Article  Google Scholar 

  39. Gerhards, C.: Spherical potential theory: tools and applications. In: Freeden, W., Nashed, M.Z. (eds.) Handbook of mathematical geodesy – functional analytic and potential theoretic methods, Birkhäuser, Basel (2018)

    Google Scholar 

  40. Gerhards, C., Pereverzyev, S. Jr., Tkachenko, P.: A parameter choice strategy for the inversion of multiple observations. Adv. Comp. Math. 43, 101–112 (2017)

    Article  Google Scholar 

  41. Gerhards, C., Pereverzyev, S. Jr., Tkachenko, P.: Joint inversion of multiple observations. In: Freeden, W., Nashed, M.Z. (eds.) Handbook of mathematical geodesy – functional analytic and potential theoretic methods, Birkhäuser, Basel (2018)

    Google Scholar 

  42. Gubbins, D., Ivers, D., Masterton, S.M., Winch, D.E.: Analysis of lithospheric magnetization in vector spherical harmonics. Geophys. J. Int. 187, 99–117 (2011)

    Article  Google Scholar 

  43. Gutkin, E., Newton, K.P.: The method of images and green’s function for spherical domains. J. Phys. A: Math. Gen. 37, 11989–12003 (2004)

    Article  Google Scholar 

  44. Gutting, M.: Fast spherical/harmonic spline modeling. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn. Springer, Berlin (2015)

    Google Scholar 

  45. Gutting, M., Kretz, B., Michel, V., Telschow, R.: Study on parameter choice methods for the RFMP with respect to downward continuation. Front. Appl. Math. Stat. 3, 10 (2017)

    Article  Google Scholar 

  46. Haar, A.: Zur Theorie der orthogonalen Funktionensysteme. Math. Ann. 69, 331–371 (1910)

    Article  Google Scholar 

  47. Hesse, K., Sloan, I., Womersley, R.: Numerical integration on the sphere. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn. Springer, Berlin (2015)

    Google Scholar 

  48. Hofmann-Wellenhof, B., Moritz, H.: Physical Geodesy, 2nd edn. Springer, Vienna (2006)

    Google Scholar 

  49. Holschneider, M.: Continuous wavelet transforms on the sphere. J. Math. Phys. 37, 4156–4165 (1996)

    Article  Google Scholar 

  50. Hubbert, S., LeGia, Q.T., Morton, T.: Spherical Radial Basis Functions, Theory and Applications. Springer International Publishing (2015)

    Google Scholar 

  51. Kamman, P., Michel, V.: Time-dependent Cauchy-Navier splines and their application to seismic wave front propagation. ZAMM J. Appl. Math. Mech. 88, 155–178 (2008)

    Article  Google Scholar 

  52. Kidambi, R., Newton, K.P.: Motion of three point vortices on a sphere. Phys. D 116, 143–175 (1998)

    Article  Google Scholar 

  53. Kidambi, R., Newton, K.P.: Point vortex motion on a sphere with solid boundaries. Phys. Fluids 12, 581–588 (2000)

    Article  Google Scholar 

  54. Kuvshinov, A.V.: 3-D global induction in the ocean and solid earth: recent progress in modeling magnetic and electric fields from sources of magnetospheric, ionospheric, and ococean origin. Surv. Geophys. 29, 139–186 (2008)

    Article  Google Scholar 

  55. LeGia, Q.T., Sloan, I., Wendland, H.: Multiscale analysis on sobolev spaces on the sphere. SIAM J. Num. Anal. 48, 2065–2090 (2010)

    Article  Google Scholar 

  56. Lima, E.A., Weiss, B.P., Baratchart, L., Hardin, D.P., Saff, E.B.: Fast inversion of magnetic field maps of unidirectional planar geological magnetization. J. Geophys. Res. Solid Earth 118, 1–30 (2013)

    Article  Google Scholar 

  57. Mallat, S., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41, 3397–3415 (1993)

    Article  Google Scholar 

  58. Masterton, S., Gubbins, D., Müller, R.D., Singh, K.H.: Forward modelling of oceanic lithospheric magnetization. Geophys. J. Int. 192, 951–962 (2013)

    Article  Google Scholar 

  59. Mayer, C., Maier, T.: Separating inner and outer Earth’s magnetic field from CHAMP satellite measurements by means of vector scaling functions and wavelets. Geophys. J. Int. 167, 1188–1203 (2006)

    Article  Google Scholar 

  60. Michel, V.: Lectures on Constructive Approximation – Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball. Birkhäuser, Boston (2013)

    Google Scholar 

  61. Michel, V., Simons, F.: A general approach to regularizing inverse problems with regional data using Slepian wavelets. Inverser Prob. 33, 125016 (2018)

    Article  Google Scholar 

  62. Michel, V., Telschow, R.: A non-linear approximation method on the sphere. GEM Int. J. Geomath. 5, 195–224 (2014)

    Article  Google Scholar 

  63. Michel, V., Telschow, R.: The regularized orthogonal functional matching pursuit for ill-posed inverse problems. SIAM J. Num. Anal. 54, 262–287 (2016)

    Article  Google Scholar 

  64. Michel, V., Wolf, K.: Numerical aspects of a spline-based multiresolution recovery of the harmonic mass density out of gravity functionals. Geophys. J. Int. 173, 1–16 (2008)

    Article  Google Scholar 

  65. Müller, C.: Spherical Harmonics. Springer, New York (1966)

    Book  Google Scholar 

  66. Olsen, N., Glassmeier, K-H., Jia, X.: Separation of the magnetic field into external and internal parts. Space Sci. Rev. 152, 135–157 (2010)

    Article  Google Scholar 

  67. Olsen, N., Lühr, H., Finlay, C.C., Sabaka, T.J., Michaelis, I., Rauberg, J., Tøffner-Clausen, L.: The CHAOS-4 geomagnetic field model. Geophys. J. Int. 197, 815–827 (2014)

    Article  Google Scholar 

  68. Plattner, A., Simons, F.J.: slepian_golf version 1.0.0. https://doi.org/10.5281/zenodo.583627

  69. Plattner, A., Simons, F.J.: Spatiospectral concentration of vector fields on a sphere. Appl. Comp. Harm. Anal. 36, 1–22 (2014)

    Article  Google Scholar 

  70. Plattner, A., Simons, F.J.: Potential-field estimation from satellite data using scalar and vector Slepian functions. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn. Springer, Berlin (2015)

    Google Scholar 

  71. Plattner, A., Simons, F.J.: Internal and external potential-field estimation from regional vector data at varying satellite altitude. Geophys. J. Int. 211, 207–238 (2017)

    Google Scholar 

  72. Sabaka, T., Tyler, R., Olsen, N.: Extracting ocean-generated tidalmagnetic signals from Swarm data through satellite gradiometry. Geophys. Res. Lett. 43, 3237–3245 (2016)

    Article  Google Scholar 

  73. Sabaka, T.J., Olsen, N., Tyler, R.H., Kuvshinov, A.: CM5, a pre-Swarm comprehensive geomagnetic field model derived from over 12 years of CHAMP, ørsted, SAC-C and observatory data. Geophys. J. Int. 200, 1596–1626 (2015)

    Google Scholar 

  74. Schreiner, M. Locally supported kernels for spherical spline interpolation. J. Approx. Theory 89, 172–194 (1997)

    Article  Google Scholar 

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

    Article  Google Scholar 

  76. Simons, F.J., Dahlen, F.A., Wieczorek, M.A.: Spatiospectral localization on a sphere. SIAM Rev. 48, 505–536 (2006)

    Article  Google Scholar 

  77. Simons, F.J., Plattner, A.: Scalar and vector slepian functions, spherical signal estimation and spectral analysis. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn. Springer, Berlin (2015)

    Google Scholar 

  78. Sloan, I., Womersley, R.: Filtered hyperinterpolation: a constructive polynomial approximation on the sphere. GEM Int. J. Geomath. 3, 95–117 (2012)

    Article  Google Scholar 

  79. Telschow, R.: An Orthogonal Matching Pursuit for the Regularization of Spherical Inverse Problems. PhD thesis, University of Siegen (2014)

    Google Scholar 

  80. Tyler, R., Maus, S., Lühr, H.: Satellite observations of magnetic fields due to ocean tidal flow. Science 299, 239–240 (2003)

    Article  Google Scholar 

  81. Vervelidou, F, Lesur, V., Morschhauser, A.,Grott, M., Thomas, P.: On the accuracy of paleopole estimations from magnetic field measurements. Geophys. J. Int. 211, 1669–1678 (2017)

    Article  Google Scholar 

  82. Wahba, G.: Spline inteprolation and smoothing on the sphere. SIAM J. Sci. Stat. Comput. 2, 5–16 (1981)

    Article  Google Scholar 

  83. Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4, 389–396 (1995)

    Article  Google Scholar 

  84. Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)

    Google Scholar 

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Acknowledgements

This work was partly supported by DFG grant GE 2781/1-1.

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Authors and Affiliations

  1. Computational Science Center, University of Vienna, Vienna, Austria

    Christian Gerhards & Roger Telschow

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  1. Christian Gerhards
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  2. Roger Telschow
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Correspondence to Christian Gerhards .

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Editors and Affiliations

  1. Lehr- und Forschungsgebiet "Geomathemati, TU Kaiserslautern, Kaiserslautern, Germany

    Willi Freeden

  2. Physikalische Geodäsie, TU München Inst. Astronomische und, München, Germany

    Reiner Rummel

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  1. Geomathematics Group, University of Kaiserslautern, Kaiserslautern, Germany

    Willi Freeden

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Gerhards, C., Telschow, R. (2018). Reconstruction and Decomposition of Scalar and Vectorial Potential Fields on the Sphere. In: Freeden, W., Rummel, R. (eds) Handbuch der Geodäsie. Springer Reference Naturwissenschaften . Springer Spektrum, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46900-2_103-1

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