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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)