Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-22T23:47:10.519Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  07 April 2022

Volker Michel
Affiliation:
Universität Siegen, Germany
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Geomathematics
Modelling and Solving Mathematical Problems in Geodesy and Geophysics
, pp. 433 - 446
Publisher: Cambridge University Press
Print publication year: 2022

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M., and Stegun, I. A. 1972. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications Inc.Google Scholar
Abrikosov, O., and Schwintzer, P. 2004. Recovery of the Earth’s gravity field from GOCE satellite gravity gradiometry: a case study. In: Lacoste, H. (ed.), GOCE. The Geoid and Oceanography, Proc. of the 2nd International Workshop, Held 8–10 March in Frascati, Italy. https://ui.adsabs.harvard.edu/abs/2004ESASP.569E..18A/abstract.Google Scholar
Agarwal, R. P., and O’Regan, D. 2009. Ordinary and Partial Differential Equations. With Special Functions, Fourier Series, and Boundary Value Problems. New York: Springer.CrossRefGoogle Scholar
Aki, K., and Richards, P. G. 2002. Quantitative Seismology. 2nd ed. Sausalito: University Science Books.Google Scholar
Akram, M. 2008. Constructive Approximation on the 3-Dimensional Ball with Focus on Locally Supported Kernels and the Helmholtz Decomposition. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group. Shaker, Aachen.Google Scholar
Akram, M., and Michel, V. 2010. Locally supported approximate identities on the unit ball. Revista Matemática Complutense, 23, 233249.CrossRefGoogle Scholar
Akram, M., Amina, I., and Michel, V. 2011. A study of differential operators for particular complete orthonormal systems on a 3D ball. International Journal of Pure and Applied Mathematics, 73, 489506.Google Scholar
Albertella, A., Sansò, F., and Sneeuw, N. 1999. Band-limited functions on a bounded spherical domain: the Slepian problem on the sphere. Journal of Geodesy, 73, 436447.Google Scholar
Amann, H., and Escher, J. 2008. Analysis III. 2nd ed. Basel: Birkhäuser.Google Scholar
Amirbekyan, A. 2007. The Application of Reproducing Kernel Based Spline Approximation to Seismic Surface and Body Wave Tomography: Theoretical Aspects and Numerical Results. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group. https://nbn-resolving.org/urn:nbn:de:hbz:386-kluedo-21039.Google Scholar
Amirbekyan, A., and Michel, V. 2008. Splines on the three-dimensional ball and their application to seismic body wave tomography. Inverse Problems, 24, 125.CrossRefGoogle Scholar
Amirbekyan, A., Michel, V., and Simons, F. J. 2008. Parameterizing surface-wave tomographic models with harmonic spherical splines. Geophysical Journal International, 174, 617628.Google Scholar
Atiyah, M., et al. 2001a. Lexikon der Mathematik, vol. 2. Heidelberg: Spektrum Akademischer Verlag.Google Scholar
Atiyah, M., et al. 2001b. Lexikon der Mathematik, vol. 3. Heidelberg: Spektrum Akademischer Verlag.Google Scholar
Atkinson, K., and Han, W. 2012. Spherical Harmonics and Approximations on the Unit Sphere: An Introduction. Berlin, Heidelberg: Springer.CrossRefGoogle Scholar
Aubin, T. 1982. Nonlinear Analysis on Manifolds, Monge–Ampère Equations. Grundlagen der mathematischen Wissenschaften, vol. 252. New York: Springer.Google Scholar
Backus, G. 1986. Poloidal and toroidal fields in geomagnetic field modeling. Reviews of Geophysics, 24, 75109.Google Scholar
Backus, G., Parker, R., and Constable, C. 1996. Foundations of Geomagnetism. Cambridge: Cambridge University Press.Google Scholar
Bagherbandi, M. 2012. A comparison of three gravity inversion methods for crustal thickness modelling in Tibet plateau. Journal of Asian Earth Sciences, 43, 8997.CrossRefGoogle Scholar
Balandin, A. L., Ono, Y., and You, S. 2012. 3D vector tomography using vector spherical harmonics decompositon. Computers and Mathematics with Applications, 63, 14331441.CrossRefGoogle Scholar
Ballani, L., Engels, J., and Grafarend, E. W. 1993. Global base functions for the mass density in the interior of a massive body (Earth). Manuscripta Geodaetica, 18, 99114.Google Scholar
Ballmann, W. 2015. Einführung in die Geometrie und Topologie. Mathematik Kompakt. Basel: Springer.Google Scholar
Barrera, R. G., Estévez, G. A., and Giraldo, J. 1985. Vector spherical harmonics and their application to magnetostatics. European Journal of Physics, 6, 287294.Google Scholar
Bauer, F., Gutting, M., and Lukas, M. A. 2015. Evaluation of parameter choice methods for regularization of ill-posed problems in geomathematics. Pages 17131774 of: Freeden, W., Nashed, M. Z., and Sonar, T. (eds.), Handbook of Geomathematics, 2nd ed. Berlin, Heidelberg: Springer.Google Scholar
Bauer, H. 1990. Maß- und Integrationstheorie. Berlin: Walter de Gruyter.Google Scholar
Baur, O., and Sneeuw, N. 2011. Assessing Greenland ice mass loss by means of point-mass modeling: a viable methodology. Journal of Geodesy, 85, 607615.CrossRefGoogle Scholar
Bayer, M. 2000. Geomagnetic Field Modelling from Satellite Data by First and Second Generation Vector Wavelets. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group. Shaker, Aachen.Google Scholar
Beals, R., and Wong, R. 2016. Special Functions and Orthogonal Polynomials. Cambridge Studies in Advanced Mathematics, vol. 153. Cambridge: Cambridge University Press.Google Scholar
Ben-Menahem, A., and Singh, S. J. 1981. Seismic Waves and Sources. New York: Springer.Google Scholar
Berkel, P. 2009. Multiscale Methods for the Combined Inversion of Normal Mode and Gravity Variations. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group. Shaker, Aachen.Google Scholar
Berkel, P., and Michel, V. 2010. On mathematical aspects of a combined inversion of gravity and normal mode variations by a spline method. Mathematical Geosciences, 42, 795816.CrossRefGoogle Scholar
Berkel, P., Fischer, D., and Michel, V. 2011. Spline multiresolution and numerical results for joint gravitation and normal mode inversion with an outlook on sparse regularisation. GEM: International Journal on Geomathematics, 1, 167204.Google Scholar
Bernstein, I. N., and Gerver, M. L. 1978. On the problem of integral geometry for a set of geodesic lines and on the inverse kinematic problem of seismology. Doklady Akademii Nauk SSSR, 243, 302305.Google Scholar
Bianco, M. J., and Gerstoft, P. 2018. Travel time tomography with adaptive dictionaries. IEEE Transactions on Computational Imaging, 4, 499511.Google Scholar
Billingham, J., and King, A. C. 2000. Wave Motion. Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press.Google Scholar
Blakely, R. J. 1996. Potential Theory in Gravity and Magnetic Applications. Cambridge: Cambridge University Press.Google Scholar
Blick, C., Freeden, W., and Nutz, H. 2017. Feature extraction of geological signatures by multiscale gravimetry. GEM: International Journal on Geomathematics, 8, 5783.Google Scholar
Bolton, S., Levin, S., and Bagenal, F. 2017. Juno’s first glimpse of Jupiter’s complexity. Geophysical Research Letters, 44, 76637667. Special Section Early Results: Juno at Jupiter.CrossRefGoogle Scholar
Boulanger, O., and Chouteau, M. 2001. Constraints in 3D gravity inversion. Geophysical Prospecting, 49, 265280.Google Scholar
Buchheim, W. 1975. Zur geophysikalischen Inversionsproblematik. Pages 305–310 of: Maaz, R. (ed.), Seismology and Solid-Earth-Physics: Proc. Int. Symp. on the Occasion of 50 Years of Seismological Research and 75 Years of Seismic Registration at Jena 1974.Google Scholar
Candès, E. J., Romberg, J., and Tao, T. 2006. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 52, 489509.CrossRefGoogle Scholar
Canuto, C., and Tabacco, A. 2010. Mathematical Analysis II. Milan: Springer.Google Scholar
Carrascal, B, Estévez, , G. A., Lee, P., and Lorenzo, V. 1991. Vector spherical harmonics and their application to classical electrodynamics. European Journal of Physics, 12, 184191.Google Scholar
Chattopadhyay, A. 2004. Wave reflection and refraction in triclinic crystalline media. Archive of Applied Mechanics, 73, 568579.CrossRefGoogle Scholar
Chen, J. L., Wilson, C. R., and Tapley, B. D. 2006. Satellite gravity measurements confirm accelerated melting of Greenland ice sheet. Science, 313, 19581960.CrossRefGoogle ScholarPubMed
Chihara, T. S. 1978. An Introduction to Orthogonal Polynomials. New York: Gordon and Breach.Google Scholar
Chui, C. K. 1992. An Introduction to Wavelets. San Diego: Academic Press.Google Scholar
Clapp, R. E., and Li, H. T. 1970. Six integral theorems for vector spherical harmonics. Journal of Mathematical Physics, 11, 49.Google Scholar
Craven, B. D. 1982. Lebesgue Measure and Integral. Boston: Pitman.Google Scholar
Dahlen, F. A., and Tromp, J. 1998. Theoretical Global Seismology. Princeton: Princeton University Press.Google Scholar
Dahlen, F. A., and Simons, F. J. 2008. Spectral estimation on a sphere in geophysics and cosmology. Geophysical Journal International, 174, 774807.CrossRefGoogle Scholar
Dai, F., and Xu, Y. 2014. The Hardy–Rellich inequality and uncertainty principle on the sphere. Constructive Approximation, 40, 141171.Google Scholar
Dang, P., Qian, T., and Chen, Q. 2017. Uncertainty principle and phase-amplitude analysis of signals on the unit sphere. Advances in Applied Clifford Algebras, 27, 29853013.Google Scholar
Davis, P. J. 1975. Interpolation and Approximation. New York: Dover Publications.Google Scholar
Dieudonné, J. 1960. Foundations of Modern Analysis. New York and London: Academic Press.Google Scholar
Donoho, D. L. 2006. Compressed sensing. IEEE Transactions on Information Theory, 52, 12891306.CrossRefGoogle Scholar
Dufour, H. M. 1977. Fonctions orthogonales dans la sphère. Résolution théorique du problème du potentiel terrestre. Bulletin Géodésique, 51, 227237.Google Scholar
Dunkl, C. F., and Xu, Y. 2001. Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and Its Applications. Cambridge: Cambridge University Press.Google Scholar
Dziewonski, A. M., and Anderson, D. L. 1981. Preliminary reference Earth model. Physics of the Earth and Planetary Interiors, 25, 297356.Google Scholar
Edmonds, A. R. 1957. Angular Momentum in Quantum Mechanics. Princeton: Princeton University Press.Google Scholar
Efthimiou, C., and Frye, C. 2014. Spherical Harmonics in p Dimensions. Singapore: World Scientific.Google Scholar
Eicker, A., Mayer-Gürr, T., and Ilk, K. H. 2005. Global gravity field solutions based on a simulation scenario of GRACE SST data and regional refinements by GOCE SGG observations. Pages 66–71 of: Jekeli, C., Bastos, L., and Fernandes, J. (eds.), Gravity, Geoid and Space Missions. International Association of Geodesy Symposia, vol. 129. Berlin, Heidelberg: Springer.Google Scholar
Engl, H. W., and Gfrerer, H. 1988. A posteriori parameter choice for general regularization methods for solving linear ill-posed problems. Applied Numerical Mathematics, 4, 395417.CrossRefGoogle Scholar
Engl, H. W., Hanke, M., and Neubauer, A. 1996. Regularization of Inverse Problems. Dordrecht, Boston, London: Kluwer Academic Publishers.Google Scholar
Eshagh, M. 2009. Spatially restricted integrals in gradiometric boundary value problems. Artificial Satellites, 44, 131148.Google Scholar
Fengler, M. J. 2005. Vector Spherical Harmonic and Vector Wavelet Based Non-Linear Galerkin Schemes for Solving the Incompressible Navier–Stokes Equation on the Sphere. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group. Shaker, Aachen.Google Scholar
Fengler, M. J., Michel, D., and Michel, V. 2006. Harmonic spline-wavelets on the 3-dimensional ball and their application to the reconstruction of the Earth’s density distribution from gravitational data at arbitrarily shaped satellite orbits. Zeitschrift für Angewandte Mathematik und Mechanik, 86, 856873.CrossRefGoogle Scholar
Fengler, M. J., Freeden, W., Kohlhaas, A., Michel, V., and Peters, T. 2007. Wavelet modelling of regional and temporal variations of the Earth’s gravitational potential observed by GRACE. Journal of Geodesy, 81, 515.Google Scholar
Ferrers, N. M. 1877. Spherical Harmonics. London: Macmillan and Co.Google Scholar
Fersch, B., Kunstmann, H., Bárdossy, A., Devaraju, B., and Sneeuw, N. 2012. Continental-scale basin water storage variation from global and dynamically downscaled atmospheric water budgets in comparison with GRACE-derived observations. Journal of Hydrometeorology, 13, 15891603.Google Scholar
Fischer, D. 2011. Sparse Regularization of a Joint Inversion of Gravitational Data and Normal Mode Anomalies. Ph.D. thesis, University of Siegen, Department of Mathematics, Geomathematics Group. Dr. Hut, Munich, https://nbn-resolving.org/urn:nbn:de:hbz:467-5448.Google Scholar
Fischer, D., and Michel, V. 2012. Sparse regularization of inverse gravimetry – case study: spatial and temporal mass variations in South America. Inverse Problems, 28, 065012 (34pp).Google Scholar
Fischer, D., and Michel, V. 2013a. Automatic best-basis selection for geophysical tomographic inverse problems. Geophysical Journal International, 193, 12911299.CrossRefGoogle Scholar
Fischer, D., and Michel, V. 2013b. Inverting GRACE gravity data for local climate effects. Journal of Geodetic Science, 3, 151162.CrossRefGoogle Scholar
Flechtner, F., Morton, P., Watkins, M., and Webb, F. 2014. Status of the GRACE Follow-On mission. Pages 117121 of: Marti, U. (ed.), Gravity, Geoid and Height Systems. International Association of Geodesy Symposia, vol. 141. Cham: Springer.Google Scholar
Folland, G. B. 1976. Introduction to Partial Differential Equations. Princeton: Princeton University Press.CrossRefGoogle Scholar
Fox, A. J., and Johnson, F. A. 1966. On finding the eigenvalues of real symmetric tridiagonal matrices. The Computer Journal, 9, 98105.Google Scholar
Freeden, W. 1981a. On approximation by harmonic splines. Manuscripta Geodaetica, 6, 193244.Google Scholar
Freeden, W. 1981b. On spherical spline interpolation and approximation. Mathematical Methods in the Applied Sciences, 3, 551575.Google Scholar
Freeden, W., Gervens, T., and Schreiner, M. 1994. Tensor spherical harmonics and tensor spherical splines. Manuscripta Geodaetica, 19, 70100.Google Scholar
Freeden, W., and Schneider, F. 1998. Regularization wavelets and multiresolution. Inverse Problems, 14, 225243.CrossRefGoogle Scholar
Freeden, W., Gervens, T., and Schreiner, M. 1998. Constructive Approximation on the Sphere with Applications to Geomathematics. Oxford: Oxford University Press.CrossRefGoogle Scholar
Freeden, W. 1999. Multiscale Modelling of Spaceborne Geodata. Stuttgart, Leipzig: B. G. Teubner.Google Scholar
Freeden, W., and Hesse, K. 2002. On the multiscale solution of satellite problems by use of locally supported kernel functions corresponding to equidistributed data on spherical orbits. Studia Scientiarum Mathematicarum Hungarica, 39, 3774.Google Scholar
Freeden, W., Michel, V., and Nutz, H. 2002. Satellite-to-satellite tracking and satellite gravity gradiometry (advanced techniques for high-resolution geopotential field determination). Journal of Engineering Mathematics, 43, 1956.Google Scholar
Freeden, W., and Michel, V. 2004. Multiscale Potential Theory (with Applications to Geoscience). Boston: Birkhäuser.CrossRefGoogle Scholar
Freeden, W., and Gutting, M. 2008. On the completeness and closure of vector and tensor spherical harmonics. Integral Transforms and Special Functions, 19, 713734.Google Scholar
Freeden, W., and Schreiner, M. 2009. Spherical Functions of Mathematical Geosciences, a Scalar, Vectorial, and Tensorial Setup. Berlin: Springer.Google Scholar
Freeden, W., and Gerhards, C. 2013. Geomathematically Oriented Potential Theory. Boca Raton: CRC Press.Google Scholar
Freeden, W., and Gutting, M. 2013. Special Functions of Mathematical (Geo-) Physics. Basel: Birkhäuser.Google Scholar
Freeden, W., Michel, V., and Simons, F. J. 2018. Spherical harmonics based special function systems and constructive approximation methods. Pages 753819 of: Freeden, W., and Nashed, M. Z. (eds.), Handbook of Mathematical Geodesy. Geosystems Mathematics. Basel: Birkhäuser.Google Scholar
Fukushima, T. 2012. Numerical computation of spherical harmonics of arbitrary degree and order by extending exponent of floating point numbers. Journal of Geodesy, 86, 271285.Google Scholar
Gilbarg, D., and Trudinger, N. S. 1977. Elliptic Partial Differential Equations of Second Order. Grundlagen der mathematischen Wissenschaften, vol. 224. Berlin: Springer.Google Scholar
Gorenflo, R., and Vessella, S. 1991. Abel Integral Equations – Analysis and Applications. Lecture Notes in Mathematics, no. 1461. Berlin: Springer.Google Scholar
Grasmair, M., and Naumova, V. 2016. Conditions on optimal support recovery in unmixing problems by means of multi-penalty regularization. Inverse Problems, 32, 104007 (16pp).Google Scholar
Grünbaum, F. A., Longhi, L., and Perlstadt, M. 1982. Differential operators commuting with finite convolution integral operators: some non-Abelian examples. SIAM Journal on Applied Mathematics, 42, 941955.Google Scholar
Gubbins, D., Ivers, D., Masterton, S. M., and Winch, D. E. 2011. Analysis of lithospheric magnetization in vector spherical harmonics. Geophysical Journal International, 187, 99117.Google Scholar
Gurtin, M. E. 1984. The linear theory of elasticity. Pages 1295 of: Truesdell, C. (ed.), Linear Theories of Elasticity and Thermoelasticity. Linear and Nonlinear Theories of Rods, Plates, and Shells. Mechanics of Solids, vol. II. Berlin: Springer.Google Scholar
Gutting, M., Kretz, B., Michel, V., and Telschow, R. 2017. Study on parameter choice methods for the RFMP with respect to downward continuation. Frontiers in Applied Mathematics and Statistics, 3, article 10.Google Scholar
Hadamard, J. 1902. Sur les problèmes aux dérivées partielles et leur signification physique. Princeton University Bulletin, 13, 4952.Google Scholar
Hanke, M. 2017. A Taste of Inverse Problems. Basic Theory and Examples. Other Titles in Applied Mathematics, no. 153. Philadelphia: Society for Industrial and Applied Mathematics (SIAM).Google Scholar
Hanke, M., and Engl, H. W. 1994. An optimal stopping rule for the ν-method for solving ill-posed problems using Christoffel functions. Journal of Approximation Theory, 79, 89108.Google Scholar
Hansen, P. C. 1992. Analysis of discrete ill-posed problems by means of the L-curve. SIAM Review, 34, 561580.Google Scholar
Hansen, P. C., and O’Leary, D. P. 1993. The use of the L-curve in the regularization of discrete ill-posed problems. SIAM Journal on Scientific Computing, 14, 14871503.Google Scholar
Hansen, P. C. 1998. Rank-Deficient and Discrete Ill-Posed Problems. Numerical Aspects of Linear Inversion. Philadelphia: SIAM.CrossRefGoogle Scholar
Hardy, G. H. 1916. Weierstrass’s non-differentiable function. Transactions of the American Mathematical Society, 17, 301325.Google Scholar
Harig, C., and Simons, F. J. 2016. Icemass loss in Greenland, the Gulf of Alaska, and the Canadian Archipelago: seasonal cycles and decadal trends. Geophysical Research Letters, 43, 31503159.Google Scholar
Heiskanen, W. A., and Moritz, H. 1981. Physical Geodesy, Reprint. Technical University Graz/Austria: Institute of Physical Geodesy.Google Scholar
Hettlich, F., and Rundell, W. 1996. Iterative methods for the reconstruction of an inverse potential problem. Inverse Problems, 12, 251266.Google Scholar
Heuser, H. 1991. Gewöhnliche Differentialgleichungen. 2nd ed. Stuttgart: B. G. Teubner.Google Scholar
Heuser, H. 1992. Funktionalanalysis. 3rd ed. Stuttgart: B. G. Teubner.Google Scholar
Heuser, H. 2009. Lehrbuch der Analysis, Teil 1. 17th ed. Mathematische Leitfäden. Wiesbaden: Vieweg + Teubner.Google Scholar
Higuchi, A. 1987. Symmetric tensor spherical harmonics on the N-sphere and their application to the Sitter group SO(N,1). Journal of Mathematical Physics, 28, 15531566.CrossRefGoogle Scholar
Hill, E. L. 1954. The theory of vector spherical harmonics. American Journal of Physics, 22, 211214.Google Scholar
Hobson, E. W. 1965. The Theory of Spherical and Ellipsoidal Harmonics. New York: Chelsea Publ. Co.Google Scholar
Hofmann, B. 1986. Regularization for Applied Inverse and Ill-Posed Problems. Teubner-Texte zur Mathematik. Leipzig: BSB Teubner.Google Scholar
Hofmann, B., Kaltenbacher, B., Pöschl, C., and Scherzer, O. 2007. A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse Problems, 23, 9871010.Google Scholar
Hofmann-Wellenhof, B., and Moritz, H. 2005. Physical Geodesy. Vienna, New York: Springer.Google Scholar
Holmes, S. A., and Featherstone, W. E. 2002. A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions. Journal of Geodesy, 76, 279299.Google Scholar
Humphreys, E., and Clayton, R. W. 1988. Adaptation of back projection tomography to seismic travel time problems. Journal of Geophysical Research: Solid Earth, 93, 10731085.Google Scholar
Iglewska-Nowak, I. 2016. Multiresolution on n-dimensional spheres. Kyushu Journal of Mathematics, 70, 353374.Google Scholar
Ilk, K. H., Feuchtinger, M., and Mayer-Gürr, T. 2005. Gravity field recovery and validation by analysis of short arcs of a satellite-to-satellite tracking experiment as CHAMP and GRACE. Pages 189194 of: Sansò, F. (ed.), A Window on the Future of Geodesy. International Association of Geodesy Symposia, vol. 128. Berlin, Heidelberg: Springer.Google Scholar
Isakov, V. 1990. Inverse Source Problems. Providence: American Mathematical Society.Google Scholar
Isakov, V. 2006. Inverse Problems for Partial Differential Equations. 2nd ed. New York: Springer.Google Scholar
Ishtiaq, A. 2018. Grid Points and Generalized Discrepancies on the d-Dimensional Ball. Ph.D. thesis, University of Siegen, Department of Mathematics, Geomathematics Group. https://nbn-resolving.org/urn:nbn:de:hbz:467-13733.Google Scholar
Ishtiaq, A., and Michel, V. 2017. Pseudo-differential operators, cubature and equidistribution on the 3D ball: an approach based on orthonormal basis systems. Numerical Functional Analysis and Optimization, 38, 891910.Google Scholar
Ishtiaq, A., Michel, V., and Scheffler, H. P. 2019. Theory of generalized discrepancies on a ball of arbitrary dimensions and algorithms for finding low-discrepancy point sets. GEM: International Journal on Geomathematics, 10, article 21.Google Scholar
Iske, A. 2018. Approximation. Berlin: Springer Spektrum.Google Scholar
Jahn, K., and Bokor, N. 2012. Vector Slepian basis functions with optimal energy concentration in high numerical aperture focusing. Optics Communications, 285, 20282038.Google Scholar
Jahn, K., and Bokor, N. 2013. Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields. Optics Communications, 288, 1316.Google Scholar
James, R. W. 1976. New tensor spherical harmonics, for application to the partial differential equations of mathematical physics. Philosophical Transactions of the Royal Society A, 281, 195221.Google Scholar
Jänich, K. 2005. Topologie. 8th ed. Berlin: Springer.Google Scholar
Jansen, M. J. F., Gunter, B. C., and Kusche, J. 2009. The impact of GRACE, GPS and OBP data on estimates of global mass redistribution. Geophysical Journal International, 177, 113.Google Scholar
Jekeli, C. 1999. The determination of gravitational potential differences from satellite-to-satellite tracking. Celestial Mechanics and Dynamical Astronomy, 75, 85101.Google Scholar
Jensen, J. L. W. V. 1906. Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Mathematica, 30, 175193.Google Scholar
Jones, F. 1993. Lebesgue Integration on Euclidean Spaces. Boston: Jones and Bartlett Publishers.Google Scholar
Jones, L. K. 1987. On a conjecture of Huber concerning the convergence of projection pursuit regression. The Annals of Statistics, 15, 880882.Google Scholar
Kaban, M. K., Flóvenz, Ó. G., and Pálmason, G. 2002. Nature of the crust-mantle transition zone and the thermal state of the upper mantle beneath Iceland from gravity modelling. Geophysical Journal International, 149, 281299.Google Scholar
Kant, I. 1786. Metaphysische Anfangsgründe der Naturwissenschaft. Riga: Johann Friedrich Hartknoch.Google Scholar
Kant, I. 1883. Kant’s Prolegomena and Metaphysical Foundations of Natural Science. London: George Bell and Sons. Translated from the original, with a biography and introduction by E. B. Bax.Google Scholar
Kazantsev, S. G., and Kardakov, V. B. 2019. Poloidal-toroidal decomposition of solenoidal vector fields in the ball. Journal of Applied and Industrial Mathematics, 13, 480499.Google Scholar
Kellogg, O. D. 1967. Foundations of Potential Theory. Berlin: Springer.Google Scholar
Khalid, Z., Kennedy, R. A., and McEwen, J. D. 2016. Slepian spatial-spectral concentration on the ball. Applied and Computational Harmonic Analysis, 40, 470504.Google Scholar
Kirsch, A. 1996. An Introduction to the Mathematical Theory of Inverse Problems. Applied Mathematical Sciences, no. 120. New York: Springer.Google Scholar
Kontak, M. 2018. Novel Algorithms of Greedy-Type for Probability Density Estimation as well as Linear and Nonlinear Inverse Problems. Ph.D. thesis, University of Siegen, Department of Mathematics, Geomathematics Group. Dr. Hut, Munich, https://nbn-resolving.org/urn:nbn:de:hbz:467-13160.Google Scholar
Kontak, M., and Michel, V. 2018. A greedy algorithm for non-linear inverse problems with an application to non-linear inverse gravimetry. GEM: International Journal on Geomathematics, 9, 167198.Google Scholar
Kontak, M., and Michel, V. 2019. The regularized weak functional matching pursuit for linear inverse problems. Journal of Inverse and Ill-Posed Problems, 27, 317340.CrossRefGoogle Scholar
Koyré, A., and Cohen, I. B. 1726. Isaac Newton’s Philosophiae Naturalis Principia Mathematica, the 3rd Edition with Variant Readings. Cambridge: Harvard University Press.Google Scholar
Landau, H. J., and Pollak, H. O. 1961. Prolate spheroidal wave functions, Fourier analysis and uncertainty – II. Bell System Technical Journal, 40, 6584.Google Scholar
Lang, S. 2001. Undergraduate Analysis. 2nd ed. New York: Springer.Google Scholar
Last, B. J., and Kubik, K. 1983. Compact gravity inversion. Geophysics, 48, 713721.Google Scholar
Laures, G., and Szymik, M. 2015. Grundkurs Topologie. 2nd ed. Berlin: Springer Spektrum.Google Scholar
Lauricella, G. 1912. Sulla distribuzione della massa nell’interno dei pianeti. Rendiconti Accademia Nazionale dei Lincei, XXI, 1826.Google Scholar
Leweke, S. 2018. The Inverse Magneto-Encephalography Problem for the Spherical Multiple-Shell Model: Theoretical Investigations and Numerical Aspects. Ph.D. thesis, University of Siegen, Department of Mathematics, Geomathematics Group. https://nbn-resolving.org/urn:nbn:de:hbz:467-13967.Google Scholar
Leweke, S., Michel, V., and Telschow, R. 2018a. On the non-uniqueness of gravitational and magnetic field data inversion (survey article). Pages 883919 of: Freeden, W., and Nashed, M. Z. (eds.), Handbook of Mathematical Geodesy. Basel: Birkhäuser.Google Scholar
Leweke, S., Michel, V., and Schneider, N. 2018b. Vectorial Slepian functions on the ball. Numerical Functional Analysis and Optimization, 39, 11201152.Google Scholar
Leweke, S., Michel, V., and Fokas, A. S. 2020. Electro-magnetoencephalography for a spherical multiple-shell model: novel integral operators with singular-value decompositions. Inverse Problems, 035003 (31p).Google Scholar
Li, Y., and Oldenburg, D. W. 1998. 3-D inversion of gravity data. Geophysics, 63, 109119.Google Scholar
Louis, A. K. 1989. Inverse und schlecht gestellte Probleme. Stuttgart: Teubner.Google Scholar
Lu, S., and Pereverzev, S. V. 2013. Regularization Theory for Ill-Posed Problems: Selected Topics. Berlin, Boston: de Gruyter.Google Scholar
MacRobert, T. M. 1927. Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. London: E. P. Dutton.Google Scholar
Magnus, W., Oberhettinger, F., and Soni, R. P. 1966. Formulas and Theorems for the Special Functions of Mathematical Physics. Berlin: Springer.Google Scholar
Mallat, S. G., and Zhang, Z. 1993. Matching pursuits with time-frequency dictionaries. IEEE Transactions on Signal Processing, 41, 33973415.Google Scholar
Maniar, H., and Mitra, P. P. 2005. Local basis expansions for MEG source localization. International Journal of Bioelectromagnetism, 7, 3033.Google Scholar
Marquering, H., Nolet, G., and Dahlen, F. A. 1998. Three-dimensional waveform sensitivity kernels. Geophysical Journal International, 132, 521534.CrossRefGoogle Scholar
Marquering, H., Dahlen, F. A., and Nolet, G. 1999. Three-dimensional sensitivity kernels for finite-frequency traveltimes: the banana-doughnut paradox. Geophysical Journal International, 137, 805815.Google Scholar
Marsden, J. E., and Hughes, T. J. R. 1994. Mathematical Foundations of Elasticity. New York: Dover Publications.Google Scholar
Martinec, Z. 2003. Green’s function solution to spherical gradiometric boundary-value problems. Journal of Geodesy, 77, 4149.Google Scholar
Marussi, A. 1980. On the density distribution in bodies of assigned outer Newtonian attraction. Bollettino di Geofisica Teorica ed Applicata, XXII, 8394.Google Scholar
Mathews, J. 1962. Gravitational multipole radiation. Journal of the Society for Industrial and Applied Mathematics, 10, 768780.Google Scholar
McShane, E. J. 1974. Integration. Princeton: Princeton University Press.Google Scholar
Michel, V. 1999. A Multiscale Method for the Gravimetry Problem: Theoretical and Numerical Aspects of Harmonic and Anharmonic Modelling. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group. Shaker, Aachen.Google Scholar
Michel, V. 2002a. A Multiscale Approximation for Operator Equations in Separable Hilbert Spaces – Case Study: Reconstruction and Description of the Earth’s Interior. Aachen: Shaker. Habilitation thesis.Google Scholar
Michel, V. 2002b. Scale continuous, scale discretized and scale discrete harmonic wavelets for the outer and the inner space of a sphere and their application to an inverse problem in geomathematics. Applied and Computational Harmonic Analysis, 12, 7799.Google Scholar
Michel, V. 2003. Theoretical aspects of a multiscale analysis of the eigenoscillations of the Earth. Revista Matemática Complutense, 16, 519554.Google Scholar
Michel, V. 2005. Regularized wavelet-based multiresolution recovery of the harmonic mass density distribution from data of the Earth’s gravitational field at satellite height. Inverse Problems, 21, 9971025.Google Scholar
Michel, V., and Fokas, A. S. 2008. A unified approach to various techniques for the non-uniqueness of the inverse gravimetric problem and wavelet-based methods. Inverse Problems, 24, 045019.Google Scholar
Michel, V., and Wolf, K. 2008. Numerical aspects of a spline-based multiresolution recovery of the harmonic mass density out of gravity functionals. Geophysical Journal International, 173, 116.Google Scholar
Michel, V. 2013. Lectures on Constructive Approximation – Fourier, Spline and Wavelet Methods on the Real Line, the Sphere, and the Ball. New York: Birkhäuser.Google Scholar
Michel, V., and Telschow, R. 2014. A non-linear approximation method on the sphere. GEM: International Journal on Geomathematics, 5, 195224.Google Scholar
Michel, V. 2015. RFMP – An iterative best basis algorithm for inverse problems in the geosciences. Pages 21212147 of: Freeden, W., Nashed, M. Z., and Sonar, T. (eds.), Handbook of Geomathematics, 2nd ed. Berlin, Heidelberg: Springer.Google Scholar
Michel, V., and Orzlowski, S. 2016. On the null space of a class of Fredholm integral equations of the first kind. Journal of Inverse and Ill-Posed Problems, 24, 687710.Google Scholar
Michel, V., and Orzlowski, S. 2017. On the convergence theorem for the Regularized Functional Matching Pursuit (RFMP) algorithm. GEM: International Journal on Geomathematics, 8, 183190.Google Scholar
Michel, V., and Simons, F. J. 2017. A general approach to regularizing inverse problems with regional data using Slepian wavelets. Inverse Problems, 33, 125016.Google Scholar
Michel, V., and Telschow, R. 2016. The regularized orthogonal functional matching pursuit for ill-posed inverse problems. SIAM Journal on Numerical Analysis, 54, 262287.Google Scholar
Michel, V., and Schneider, N. 2020. A first approach to learning a best basis for gravitational field modelling. GEM: International Journal on Geomathematics, 11, article 9.Google Scholar
Michel, V., Plattner, A., and Seibert, K. 2021. A unified approach to scalar, vector, and tensor Slepian functions on the sphere and their construction by a commuting operator. Preprint, arXiv:2103.14650, accepted for publication in: Analysis and Applications.Google Scholar
Mie, G. 1908. Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen. Annalen der Physik, 25, 377445.Google Scholar
Mikhlin, S. G. 1970. Mathematical Physics, an Advanced Course. Amsterdam: North-Holland Publishing Company.Google Scholar
Moritz, H. 1990. The Figure of the Earth. Theoretical Geodesy of the Earth’s Interior. Karlruhe: Wichmann.Google Scholar
Morozov, V. A. 1966. On the solution of functional equations by the method of regularization. Soviet Mathematics. Doklady, 7, 414417.Google Scholar
Morozov, V. A. 1967. Choice of parameter for the solution of functional equations by the regularization method. Soviet Mathematics. Doklady, 8, 10001003.Google Scholar
Morozov, V. A. 1984. Methods for Solving Incorrectly Posed Problems. New York: Springer.Google Scholar
Morse, P. M. C., and Feshbach, H. 1953a. Methods of Theoretical Physics. Vol. 1. New York: McGraw–Hill.Google Scholar
Morse, P. M. C., and Feshbach, H. 1953b. Methods of Theoretical Physics. Vol. 2. New York: McGraw–Hill.Google Scholar
Müller, C. 1966. Spherical Harmonics. Berlin: Springer.Google Scholar
Müller, C. 1969. Foundations of the Mathematical Theory of Electromagnetic Waves. Berlin: Springer.Google Scholar
Munkres, J. R. 2000. Topology. 2nd ed. Upper Saddle River: Prentice Hall.Google Scholar
Narcowich, F. J., and Ward, J. D. 1996. Nonstationary wavelets on the m-sphere for scattered data. Applied and Computational Harmonic Analysis, 3, 324336.Google Scholar
Nashed, M. Z. 1987. A new approach to classification and regularization of ill-posed operator equations. Pages 5375 of: Engl, H. W., and Groetsch, C. W. (eds.), Inverse and Ill-Posed Problems. Notes and Reports in Mathematics in Science and Engineering, vol. 4. Boston: Academic Press.Google Scholar
Newton, I. 1687. Philosophiae Naturalis Principia Mathematica. London: Royal Society.Google Scholar
Nolet, G. 2008. A Breviary of Seismic Tomography: Imaging the Interior of the Earth and Sun. Cambridge: Cambridge University Press.Google Scholar
Novikoff, P. 1938. Sur le problème inverse du potentiel. Comptes Rendus de l’Académie des Sciences de l’URSS, XVIII, 165168.Google Scholar
Pail, R., and Wermuth, M. 2003. GOCE SGG and SST quick-look gravity field analysis. Advances in Geosciences, 1, 59.Google Scholar
Pail, R., Bruinsma, S., Migliaccio, F., Förste, C, et al. 2011. First GOCE gravity field models derived by three different approaches. Journal of Geodesy, 85, 819843.Google Scholar
Panet, I., Bonvalot, S., Narteau, C., Remy, D., and Lemoine, J.-M. 2018. Migrating pattern of deformation prior to the Tohoku-Oki earthquake revealed by GRACE data. Nature Geoscience, 11, 367373.Google Scholar
Pati, Y. C., Rezaiifar, R., and Krishnaprasad, P. S. 1993. Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition. Pages 4044 of: Asilomar Conference on Signals, Systems and Computers. Los Alamitos: IEEE Computer Society.Google Scholar
Pavlis, N. K., Holmes, S. A., Kenyon, S. C., and Factor, J. K. 2012. The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). Journal of Geophysical Research: Solid Earth, 117, B04406. Erratum in Journal of Geophysical Research: Solid Earth, 118, 2633–2633.Google Scholar
Peter, S., Artina, M., and Fornasier, M. 2015. Damping noise-folding and enhanced support recovery in compressed sensing. IEEE Transactions on Signal Processing, 63, 59906002.Google Scholar
Phillips, D. L. 1962. A technique for the numerical solution of certain integral equations of the first kind. Journal of the Association for Computing Machinery, 9, 8497.Google Scholar
Pizzetti, P. 1909. Corpi equivalenti rispetto alla attrazione newtoniana esterna. Accademia dei Lincei, Rendiconti, XVIII, 211215.Google Scholar
Pizzetti, P. 1910. Intorno alle possibili distribuzioni della massa nell’interno della terra. Annali di Matematica, Serie III, XVII, 225258.Google Scholar
Plato, R. 1990. Optimal algorithms for linear ill-posed problems yield regularization methods. Numerical Functional Analysis and Optimization, 11, 111118.Google Scholar
Plattner, A., Simons, F. J., and Wei, L. 2012. Analysis of real vector fields on the sphere using Slepian functions. Pages 14 of: IEEE Statistical Signal Processing Workshop (SSP), Ann Arbor, MI, USA.Google Scholar
Plattner, A., and Simons, F. J. 2014. Spatiospectral concentration of vector fields on a sphere. Applied and Computational Harmonic Analysis, 36, 122.Google Scholar
Plattner, A., and Simons, F. J. 2017. Internal and external potential field estimation from regional vector data at varying satellite altitude. Geophysical Journal International, 211, 207238.Google Scholar
Protter, M. H., and Morrey, C. B. 1977. A First Course in Real Analysis. 2nd ed. New York: Springer.Google Scholar
Ramillien, G., Lombard, A., Cazenave, A., et al. 2006. Interannual variations of the mass balance of the Antarctica and Greenland ice sheets from GRACE. Global and Planetary Change, 53, 198208.Google Scholar
Raus, T. 1985. On the discrepancy principle for the solution of ill-posed problems with non-selfadjoint operators. Acta et Commentationes Universitatis Tartuensis de Mathematica, 715, 1220. In Russian.Google Scholar
Regge, T., and Wheeler, J. A. 1957. Stability of a Schwarzschild singularity. Physical Review, 108, 10631069.Google Scholar
Reigber, C., Schmidt, R., Flechtner, F., König, R., and Meyer, U. 2005. An Earth gravity field model complete to degree and order 150 from GRACE: EIGEN-GRACE02S. Journal of Geodynamics, 39, 110.Google Scholar
Rennhack, S. 2018. Der Regularized Weak Functional Matching Pursuit am Beispiel der Erdgravitationsfeldmodellierung. B.Sc. Thesis, University of Siegen, Department of Mathematics, Geomathematics Group.Google Scholar
Resmerita, E. 2005. Regularization of ill-posed problems in Banach spaces: convergence rates. Inverse Problems, 21, 13031314.Google Scholar
Reuter, R. 1982. Über Integralformeln der Einheitssphäre und harmonische Splinefunktionen. Ph.D. thesis, RWTH Aachen, Veröffentlichungen des Geodätischen Instituts der Rheinisch-Westfälischen Technischen Hochschule in Aachen, no. 33.Google Scholar
Rieder, A. 2003. Keine Probleme mit Inversen Problemen. Wiesbaden: Vieweg.Google Scholar
Riley, K. F., Hobson, M. P., and Bence, S. J. 2008. Mathematical Methods for Physics and Engineering. 4th ed. Cambridge: Cambridge University Press.Google Scholar
Roberts, C. E. 2010. Ordinary Differential Equations. Applications, Models, and Computing. Boca Raton: CRC Press.Google Scholar
Robin, L. 1957. Fonctions Sphérique de Legendre et Fonctions Sphéroïdale. Vol. 1. Paris: Gauthier-Villars.Google Scholar
Robin, L. 1958. Fonctions Sphérique de Legendre et Fonctions Sphéroïdale. Vol. 2. Paris: Gauthier-Villars.Google Scholar
Robin, L. 1959. Fonctions Sphérique de Legendre et Fonctions Sphéroïdale. Vol. 3. Paris: Gauthier-Villars.Google Scholar
Rubin, M. A., and Ordóñez, C. R. 1984. Eigenvalues and degeneracies for n-dimensional tensor spherical harmonics. Journal of Mathematical Physics, 25, 28882894.Google Scholar
Rubincam, D. P. 1979. Gravitational potential energy of the Earth: a spherical harmonics approach. Journal of Geophysical Research, 84, 62196225.Google Scholar
Rudin, W. 1987. Real and Complex Analysis. New York: McGraw–Hill.Google Scholar
Rummel, R., Yi, W., and Stummer, C. 2001. GOCE gravitational gradiometry. Journal of Geodesy, 85, 777790.Google Scholar
Rummel, R. 2003. How to climb the gravity wall. In: Beutler, G., Drinkwater, M. R., Rummel, R., and von Steiger, R. (eds.), Earth Gravity Field from Space – From Sensors to Earth Sciences. Space Sciences Series of ISSI, vol. 17. Dordrecht: Springer.Google Scholar
Sandberg, V. D. 1978. Tensor spherical harmonics on S2 and S3 as eigenvalue problems. Journal of Mathematical Physics, 19, 24412446.Google Scholar
Sanna, N. 2000. Vector spherical harmonics: concepts and applications to the single centre expansion method. Computer Physics Communications, 132, 6683.Google Scholar
Sasgen, I., van den Broeke, M., Bamber, J. L., et al. 2012. Timing and origin of recent regional ice-mass loss in Greenland. Earth and Planetary Science Letters, 333 –334, 293303.Google Scholar
Schmidt, R., Flechtner, F., Meyer, U., et al. 2008. Hydrological signals observed by the GRACE satellites. Surveys in Geophysics, 29, 319334.Google Scholar
Schneider, F. 1997. Inverse Problems in Satellite Geodesy and Their Approximate Solution by Splines and Wavelets. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group. Shaker, Aachen.Google Scholar
Schneider, N. 2020. Learning Dictionaries for Inverse Problems on the Sphere. Ph.D. thesis, University of Siegen, Department of Mathematics, Geomathematics Group. http://dx.doi.org/10.25819/ubsi/5431.Google Scholar
Schock, E. 1985. Approximate solution of ill-posed equations: arbitrarily slow convergence vs. superconvergence. Pages 234243 of: Hämmerlin, G., and Hoffmann, K.-H. (eds.), Constructive Methods for the Practical Treatment of Integral Equations. Proceedings of the Conference at the Mathematisches Forschungsinstitut Oberwolfach, June 24–30, 1984. International Series of Numerical Mathematics, vol. 73. Basel, Boston, Stuttgart: Birkhäuser.Google Scholar
Schöpfer, F., Louis, A. K., and Schuster, T. 2006. Nonlinear iterative methods for linear ill-posed problems in Banach spaces. Inverse Problems, 22, 311329.Google Scholar
Schreiner, M. 1994. Tensor Spherical Harmonics and Their Application in Satellite Gradiometry. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group. Shaker, Aachen.Google Scholar
Schuster, T., Kaltenbacher, B., Hofmann, B., and Kazimierski, K. S. 2012. Regularization Methods in Banach Spaces. Radon Series on Computational and Applied Mathematics, no. 10. Berlin, Boston: de Gruyter.Google Scholar
Seibert, K. 2018. Spin-Weighted Spherical Harmonics and Their Application for the Construction of Tensor Slepian Functions on the Spherical Cap. Ph.D. thesis, University of Siegen, Department of Mathematics, Geomathematics Group. Universi, Siegen, https://nbn-resolving.org/urn:nbn:de:hbz:467-14210.Google Scholar
Sigloch, K. 2008. Two-stage subduction history under North America inferred from multiple-frequency tomography. Nature Geoscience, 1, 458462.Google Scholar
Simons, F. J., and Dahlen, F. A. 2006. Spherical Slepian functions and the polar gap in geodesy. Geophysical Journal International, 166, 10391061.Google Scholar
Simons, F. J., Dahlen, F. A., and Wieczorek, M. A. 2006. Spatiospectral concentration on a sphere. SIAM Review, 48, 504536.Google Scholar
Simons, F. J. 2010. Slepian functions and their use in signal estimation and spectral analysis. Pages 891923 of: Freeden, W., Nashed, M. Z., and Sonar, T. (eds.), Handbook of Geomathematics. Heidelberg: Springer.Google Scholar
Slepian, D., and Pollak, H. O. 1961. Prolate spheroidal wave functions, Fourier analysis and uncertainty—I. Bell System Technical Journal, 40, 4363.Google Scholar
Slepian, D. 1964. Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: extensions to many dimensions; generalized prolate spheroidal functions. Bell System Technical Journal, 43, 30093057.Google Scholar
Smirnow, W. I. 1977. Lehrgang der Höheren Mathematik. Hochschulbücher für Mathematik, vol. 5. Berlin: VEB Deutscher Verlag der Wissenschaften. Translated from Russian to German by C. Berg and L. Berg.Google Scholar
Stokes, G. G. 1867. On the internal distribution of matter which shall produce a given potential at the surface of a gravitating mass. Proceedings of the Royal Society, 16 May 1867, 482–486. And Collected Works IV, pp 277–288.Google Scholar
Szegö, G. 1975. Orthogonal Polynomials. 14th ed. Vol. XXIII. Providence: AMS Colloquium Publications.Google Scholar
Tanabe, H. I. Y. F. K., and Ogata, Y. 1990. Whole mantle P-wave travel time tomography. Physics of the Earth and Planetary Interiors, 59, 294328.Google Scholar
Tapley, B. D., Bettadpur, S., Ries, J. C., Thompson, P. F., and Watkins, M. M. 2004a. GRACE measurements of mass variability in the Earth system. Science, 305, 503505.Google Scholar
Tapley, B. D., Bettadpur, S., Watkins, M., and Reigber, C. 2004b. The gravity recovery and climate experiment: mission overview and early results. Geophysical Research Letters, 31, L09607.Google Scholar
Telschow, R. 2014. An Orthogonal Matching Pursuit for the Regularization of Spherical Inverse Problems. Ph.D. thesis, University of Siegen, Department of Mathematics, Geomathematics Group. Dr. Hut, Munich.Google Scholar
Temlyakov, V. N. 2000. Weak greedy algorithms. Advances in Computational Mathematics, 12, 213227.CrossRefGoogle Scholar
Tesauro, M., Kaban, M. K., and Cloetingh, S. A. P. L. 2008. EuCRUST-07: A new reference model for the European crust. Geophysical Research Letters, 35, L05313.Google Scholar
Tian, Y., Hung, S.-H., Nolet, G., Montelli, R., and Dahlen, F. A. 2007. Dynamic ray tracing and traveltime corrections for global seismic tomography. Journal of Computational Physics, 226, 672687.Google Scholar
Tian, Y., Sigloch, K., and Nolet, G. 2009. Multiple-frequency SH-wave tomography of the western US upper mantle. Geophysical Journal International, 178, 13841402.Google Scholar
Tipler, P. A. 1982. Physics. 2nd ed. Vol. 1. New York: Worth Publishers.Google Scholar
Tomita, K. 1982. Tensor spherical and pseudo-spherical harmonics in four-dimensional spaces. Progress of Theoretical Physics, 68, 310313.Google Scholar
Tosic, I., Jovanovic, I., Frossard, P., Vetterli, M., and Duric, N. 2010. Ultrasound tomography with learned dictionaries. Pages 55025505 of: Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing.Google Scholar
Trim, D. 1993. Calculus. Scarborough: Prentice Hall.Google Scholar
Tscherning, C. C. 1996. Isotropic reproducing kernels for the inner of a sphere or spherical shell and their use as density covariance functions. Mathematical Geology, 28, 161168.Google Scholar
Velicogna, I., and Wahr, J. 2006. Acceleration of Greenland ice mass loss in spring 2004. Nature, 443, 329331.Google Scholar
Velicogna, I., and Wahr, J. 2013. Time-variable gravity observations of ice sheet mass balance: precision and limitations of the GRACE satellite data. Geophysical Research Letters, 40, 30553063.Google Scholar
Vincent, P., and Bengio, Y. 2002. Kernel matching pursuit. Machine Learning, 48, 169191.Google Scholar
Vogel, C. R. 1996. Non-convergence of the L-curve regularization parameter selection method. Inverse Problems, 12, 535547.Google Scholar
Voigt, A., and Wloka, J. 1975. Hilberträume und elliptische Differentialoperatoren. Mannheim: Bibliographisches Institut.Google Scholar
von Brecht, J. H. 2016. Localization and vector spherical harmonics. Journal of Differential Equations, 260, 16221655.Google Scholar
Wahba, G. 1981. Spline interpolation and smoothing on the sphere. SIAM Journal on Scientific and Statistical Computing, 2, 516.CrossRefGoogle Scholar
Walter, E., et al. (eds.). 2005. Cambridge Advanced Learner’s Dictionary. 2nd ed. Cambridge: Cambridge University Press.Google Scholar
Walter, W. 1971. Einführung in die Potentialtheorie. Mannheim: Bibliographisches Institut.Google Scholar
Walter, W. 1990. Analysis II. Berlin, Heidelberg: Springer.Google Scholar
Wang, T., and Xu, W. 2016. Sparsity-based approach for ocean acoustic tomography using learned dictionaries. Pages 16 of: OCEANS 2016 – Shanghai. Hoboken: IEEE. https://ieeexplore.ieee.org/abstract/document/7485626.Google Scholar
Watson, G. N. 1944. A Treatise on the Theory of Bessel functions. 2nd edn. Cambridge: Cambridge University Press.Google Scholar
Weck, N. 1972. Inverse Probleme der Potentialtheorie. Applicable Analysis, 2, 195204.Google Scholar
Weinberg, E. J. 1994. Monopole vector spherical harmonics. Physical Review D, 49, 10861092.Google Scholar
Wieczorek, M. A., and Simons, F. J. 2005. Localized spectral analysis on the sphere. Geophysical Journal International, 162, 655675.Google Scholar
Wieczorek, M. A., and Simons, F. J. 2007. Minimum-variance spectral analysis on the sphere. Journal of Fourier Analysis and Applications, 13, 665692.Google Scholar
Winter, J. 1982. Tensor spherical harmonics. Letters in Mathematical Physics, 6, 9196.Google Scholar
Wolfers, J. P. 1872. Sir Isaac Newton’s Mathematische Principien der Naturlehre. Mit Bemerkungen und Erläuterungen. Berlin: R. Oppenheim.Google Scholar
Yegorova, T. P., Kozlenko, V. G., Pavlenkova, N. I, and Starostenko, V. I. 1995. 3D density model for the lithosphere of Europe: construction method and preliminary results. Geophysical Journal International, 121, 873892.Google Scholar
Yegorova, T. P., Starostenko, V. I., Kozlenko, V. G., and Pavlenkova, N. I. 1997. Three-dimensional gravity modelling of the European Mediterranean lithosphere. Geophysical Journal International, 129, 355367.Google Scholar
Yegorova, T. P., and Starostenko, V. I. 1999. Large-scale three-dimensional gravity analysis of the lithosphere below the transition zone fromWestern Europe to the East European Platform. Tectonophysics, 314, 83100.Google Scholar
Yomogida, K. 1992. Fresnel zone inversion for lateral heterogeneities in the Earth. Pure and Applied Geophysics, 138, 391406.Google Scholar
Yosida, K. 1995. Functional Analysis. 6th ed. Classics in Mathematics. Berlin: Springer.Google Scholar
Zeidler, E. (ed.). 1996. Teubner-Taschenbuch der Mathematik, originally from I.N. Bronstein and K.A. Semendjajew. Leipzig: Teubner.Google Scholar
Zerilli, F. J. 1970a. Gravitational field of a particle falling in a Schwarzschild geometry analyzed in tensor harmonics. Physical Review D, 2, 21412160.Google Scholar
Zerilli, F. J. 1970b. Tensor harmonics in canonical form for gravitational radiation and other applications. Journal of Mathematical Physics, 11, 22032208.Google Scholar
Zhang, S., and Xin, J. 2018. Minimization of transformed L1 penalty: theory, difference of convex function algorithm, and robust application in compressed sensing. Mathematical Programming, 169, 307336.Google Scholar
Zuber, M., Smith, D., Lehman, D., Hoffmann, T., Asmar, S., and Watkins, M. 2013. Gravity recovery and interior laboratory (GRAIL): mapping the lunar interior from crust to core. Space Science Reviews, 178, 324.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • References
  • Volker Michel, Universität Siegen, Germany
  • Book: Geomathematics
  • Online publication: 07 April 2022
  • Chapter DOI: https://doi.org/10.1017/9781108297882.010
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • References
  • Volker Michel, Universität Siegen, Germany
  • Book: Geomathematics
  • Online publication: 07 April 2022
  • Chapter DOI: https://doi.org/10.1017/9781108297882.010
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Volker Michel, Universität Siegen, Germany
  • Book: Geomathematics
  • Online publication: 07 April 2022
  • Chapter DOI: https://doi.org/10.1017/9781108297882.010
Available formats
×