Abstract
Next we present two novel exploration methods, thereby using the structure of a methodological circuit (as presented in Sect. 2.6), respectively. We start with inverse gravimetry, which becomes an increasing importance, e.g., in geothermal research. Then we go over to a standard technique in geoexploration, namely reflection seismics, for which a “mollifier inversion procedure” similar to the approach in gravimetry will be developed.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Achenbach, J.D.: Wave Propagation in Elastic Solids. North Holland Publishing Company, New York (1973)
Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)
Baysal, E., Kosloff, D.D., Sherwood, J.W.C.: A two-way nonreflecting wave equation. Geophysics 49, 132–141 (1984)
Beylkin, G., Monzón, L.: On approximation of functions by exponential sums. Appl. Comput. Harmon. Anal. 19, 17–48 (2005)
Beylkin, G., Monzón, L.: Approximation of functions by exponential sums revisited. Appl. Comput. Harmon. Anal. 28, 131–149 (2010)
Blick, C., Freeden, W., Nutz, H.: Gravimetry and Exploration. In: Freeden, W., Nashed, M.Z. (eds.) Handbook of Mathematical Geodesy. Geosystems Mathematics, pp. 687–752. Birkhäuser/Springer, Basel/New-York/Heidelberg (2018)
Burschäpers, H.C.: Local modeling of gravitational data. Master Thesis, University of Kaiserslautern, Mathematics Department, Geomathematics Group (2013)
Cheng, H., Greengard, L., Rokhlin, V.: A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys. 155, 468–498 (1999)
Claerbout, J.: Basic Earth Imaging. Standford University, Standford (2009)
Davis, P.J.: Interpolation and Approximation. Blaisdell, New York (1963)
Evans, L.D.: Partial Differential Equation, Third Printing. American Mathematical Society, Providence (2002)
Freeden, W.: Geomathematics: Its Role, Its Aim, and Its Potential. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, vol. 1, 2nd edn, pp. 3–78. Springer, Heidelberg (2015)
Freeden, W., Blick, C.: Signal decorrelation by means of multiscale methods. World Min. 65, 1–15 (2013)
Freeden, W., Gerhards, C.: Geomathematically Oriented Potential Theory. Chapman and Hall/CRC Press, Boca Raton/London (2013)
Freeden, W., Nashed, M.Z.: Inverse gravimetry: Background material and multiscale mollifier approaches. GEM Int. J. Geomath. 9, 199–264 (2018)
Freeden, W., Nashed, M.Z.: Ill-Posed Problems: Operator Methodologies of Resolution and Regularization. In: Freeden, W., Nashed, M.Z. (eds.) Handbook of Mathematical Geodesy. Geosystems Mathematics, pp. 201–314. Springer, Basel (2018)
Freeden, W., Nashed, M.Z.: Gravimetry As an Ill-Posed Problem in Mathematical Geodesy. In: Freeden, W., Nashed, M.Z. (eds.) Handbook of Mathematical Geodesy. Geosystems Mathematics, pp. 641–686. Springer, Basel (2018)
Freeden, W., Nutz, H.: Mathematische Methoden. In: Bauer, M., Freeden, W., Jacobi, H., Neu, T. (Herausgeber) Handbuch Tiefe Geothermie. Springer, Heidelberg (2014)
Freeden, W., Nutz, H.: Mathematik als Schlüsseltechnologie zum Verständnis des Systems “Tiefe Geothermie”. Jahresber. Deutsch. Math. Vereinigung (DMV) 117, 45–84 (2015)
Freeden, W., Sansó, F.: Geodesy and Mathematics: Interactions, Acquisitions, and Open Problems. In: International Association of Geodesy Symposia (IAGS), IX Hotine-Marussi Symposium Rome. Springer, Heidelberg (submitted, 2019). Preprint (2019)
Freeden, W., Schreiner, M.: Spherical Functions of Mathematical Geosciences – A Scalar, Vectorial, and Tensorial Setup. Springer, Heidelberg (2009)
Freeden, W., Schreiner, M.: Mathematical Geodesy: Its Role, Its Potential and Its Perspective. In: Freeden, W., Rummel, R. (eds.) Handbuch der Geodäsie. Springer Reference Naturwissenschaften. Springer, Cham (2019). https://doi.org/10.1007/978-3-662-46900-2_91_1
Freeden, W., Witte, B.: A combined (spline-) interpolation and smoothing method for the determination of the gravitational potential from heterogeneous data. Bull. Géod. 56, 53–62 (1982)
Freeden, W., Sonar, T., Witte, B.: Gauss as Scientific Mediator Between Mathematics and Geodesy from the Past to the Present. In: Freeden, W., Nashed, M.Z. (eds.) Handbook of Mathematical Geodesy, pp. 1–164. Geosystems Mathematics. Springer, Basel (2018)
Grafarend, E.W.: Six Lectures on Geodesy and Global Geodynamics. In: Moritz, H., Sünkel, H. (eds.) Proceedings of the Third International Summer School in the Mountains, pp. 531–685 (1982)
Greengard, L., Rokhlin, V.: A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numer. 6, 229–269 (1997)
Groten, E.: Geodesy and the Earth’s Gravity Field I + II. Dümmler, Bonn (1979)
Gutting, M.: Fast multipole methods for oblique derivative problems. Ph.D. thesis, University of Kaiserslautern, Mathematics Department, Geomathematics Group (2007)
Gutting, M.: Fast Spherical/Harmonic Spline Modeling. In: Freeden, W., Nashed, Z., Sonar, T. (eds.) Handbook of Geomathematics, vol. 3, 2nd edn., pp. 2711–2746. Springer, New York (2015)
Hackbusch, W.: Entwicklungen nach Exponentialsummen. Technical Report. Max-Planck-Institut für Mahematik in den Naturwissenschaften, Leipzig (2010)
Hackbusch, W., Khoromoskij, B.N., Klaus, A.: Approximation of functions by exponential sums based on the Newton-type optimisation. Technical Report, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig (2005)
Hadamard, J.: Sur les problèmes aux dérivées partielles et leur signification physique. Princet. Univ. Bull. 13, 49–52 (1902)
Heiskanen, W.A., Moritz, H.: Physical Geodesy. Freeman, San Francisco (1967)
Helmert, F.: Die Mathematischen und Physikalischen Theorien der Höheren Geodäsie, I, II. B.G. Teubner, Leipzig (1884)
Hille, E.: Introduction to the general theory of reproducing kernels. Rocky Mountain J. Math. 2, 321–368 (1972)
Hofmann-Wellenhof, B., Moritz, H.: Physical Geodesy. Springer, Wien (2005)
Ilyasov, M.: A tree algorithm for Helmholtz potential wavelets on non-smooth surfaces: theoretical background and application to seismic data processing. Ph.D. thesis, Geomathematics Group, University of Kaiserslautern (2011)
Jakobs, F., Meyer, H.: Geophysik – Signale aus der Erde. Teubner, Leipzig (1992)
Listing, J.B.: Über unsere jetzige Kenntnis der Gestalt und Größe der Erde. Dietrichsche Verlagsbuchhandlung, Göttingen (1873)
Marks, D.L.: A family of approximations spanning the Born and Rytov scattering series. Opt. Express 14, 8837–8848 (2013)
Martin, G.S., Marfurt, K.J., Larsen, S.: Marmousi-2: An updated model for the investigation of AVO in structurally complex areas. In: Proceedings, SEG Annual Meeting, Salt Lake City (2002)
Martin, G.S., Wiley, R., Marfurt, K.J.: Marmousi2: An elastic upgrade for marmousi. Lead. Edge 25, 156–166 (2006)
Marussi, A.: Intrinsic Geodesy. Springer, Berlin (1985)
Meissl, P.A.: Hilbert spaces and their applications to geodetic least squares problems. Boll. Geod. Sci. Aff. 1, 181–210 (1976)
Michel, V.: A multiscale method for the gravimetry problem: theoretical and numerical aspects of harmonic and anharmonic modelling. Ph.D. thesis, University of Kaiserslautern, Mathematics Department, Geomathematics Group, Shaker, Aachen (1999)
Michel, V.: Lectures on Constructive Approximation. Applied and Numerical Harmonic Analysis. Birkhäuser, New York (2013)
Michel, V., Fokas, A.S.: A unified approach to various techniques for the non-uniqueness of the inverse gravimetric problem and wavelet-based methods. Inverse Prob. 24 (2008). https://doi.org/10.1088/0266-5611/24/4/045019
Möhringer, S.: Decorrelation of gravimetric data. Ph.D. thesis, University of Kaiserslautern, Mathematics Department, Geomathematics Group (2014)
Moritz, H.: Advanced Physical Geodesy. Herbert Wichmann Verlag/Abacus Press, Karlsruhe/Tunbridge (1980)
Moritz, H.: The Figure of the Earth. Theoretical Geodesy of the Earth’s Interior. Wichmann Verlag, Karlsruhe (1990)
Müller, C.: Foundations of the Mathematical Theory of Electromagnetic Waves. Springer, Berlin (1969)
Nolet, G.: Seismic Tomography: Imaging the Interior of the Earth and Sun. Cambridge University Press, Cambridge (2008)
Popov, M.M., Semtchenok, N.M., Popov, P. M., Verdel, A.R.: Gaussian beam migration of multi-valued zero-offset data. In: Proceedings, International Conference, Days on Diffraction, St Petersburg, Russia, pp. 225–234 (2006)
Popov, M.M., Semtchenok, N.M., Popov, P.M., Verdel, A.R.L.: Reverse time migration with gaussian beams and velocity analysis applications. In: Extended Abstracts, 70th EAGE Conference & Exhibitions, Rome, F048 (2008)
Rummel, R.: Geodesy. In: Encyclopedia of Earth System Science, vol. 2, pp. 253–262. Academic, New York (1992)
Saitoh, S.: Theory of Reproducing Kernels and Its Applications. Longman, New York (1988)
Skudrzyk, E.: The Foundations of Acoustics. Springer, Heidelberg (1972)
Snieder, R.: The Perturbation Method in Elastic Wave Scattering and Inverse Scattering in Pure and Applied Science. General Theory of Elastic Waves, pp. 528–542. Academic, San Diego (2002)
Symes, W.W.: The Rice Inversion Project, Department of Computational and Applied Mathematics, Rice University, Houston, Texas, USA. http://www.trip.caam.rice.edu/downloads/downloads.html. Accessed 12 Sept 2016
Torge, W.: Gravimetry. de Gruyter, Berlin (1989)
Torge, W.: Geodesy. de Gruyter, Berlin (1991)
Weck, N.: Zwei inverse Probleme in der Potentialtheorie. Mitt. Inst. Theor. Geodäsie, Universität Bonn 4, 27–36 (1972)
Yilmas, O.: Seismic Data Analysis: Processing, Inversion and Interpretation of Seismic Data. Society of Exploration Geophysicists, Tulsa (1987)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Freeden, W., Heine, C., Nashed, M.Z. (2019). Exemplary Applications: Novel Exploration Methods. In: An Invitation to Geomathematics. Lecture Notes in Geosystems Mathematics and Computing. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-13054-1_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-13054-1_3
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-13053-4
Online ISBN: 978-3-030-13054-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)