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Up and Down Through the Gravity Field

  • F. SansóEmail author
  • M. Capponi
  • D. Sampietro
Living reference work entry
Part of the Springer Reference Naturwissenschaften book series (SRN)

Zusammenfassung

Die Kenntnis des Schwerefeldes hat weitreichende Anwendungen in den Geo- wissenschaften, insbesondere in Geodsie und Geophysik. Unser Anliegen in diesem Beitrag ist die Beschreibung von Eigenschaften zur Fortpflanzung des Potentials oder seiner relevanten Funktionale nach oben und nach unten. Die Fortpflanzung des Potentials nach oben (“upward continuation”) ist stets ein wohlgestelltes Problem. “Downward Continuation” ist stets ein schlechtgestelltes Problem, nicht nur wegen der numerischen Instabilitten, sondern vor allem wegen der Nichteindeutigkeit der Bestimmung von Massenschichtung aus Potentialwerten.

Als Konsequenz fokussiert sich der Beitrag auf neuere Resultate aus dem Bereich geodätischer Randwertprobleme und zum anderen auf das inverse Gravimetrieproblem. Dabei machen wir den Versuch, die Bedeutung von mathematischer Theorie für numerische Anwendungen heraus zu streichen.

Das Paper ist vom mathematischem Anspruch her schlicht gehalten. Dazu bedienen wir uns oftmals der Rückführung auf sphärische Beispiele. Der größte Teil des Materials ist bereits in der Literatur vorhanden, bis auf Teile für die globalen Modelle und das inverse Gravimetrieproblem für Schichtungen.

Keywords

Upward continuation Downward continuation Geodetic boundary value problem Inverse gravimetric problem 

Abstract

The knowledge of the gravity field has widespread applications in geosciences, in particular in Geodesy and Geophysics. The point of view of the paper is to describe the properties of the propagation of the potential, or of its relevant functionals, while moving upward or downward. The upward propagation is always a properly posed problem, in fact a smoothing and somehow related to the Newton integral and to the solution of boundary value problems (BVP). The downward propagation is always improperly posed, not only due to its intrinsic numerical instability but also because of the non-uniqueness that is created as soon as we penetrate layers of unknown mass density.

So the paper focuses on recent results on the Geodetic Boundary Value Problems on the one side and on the inverse gravimetric problem on the other, trying to highlight the significance of mathematical theory to numerical applications. Hence, on the one hand we examine the application of the BVP theory to the construction of global gravity models, on the other hand the inverse gravimetric problem is studied for layers together with proper regularization techniques.

The level of the mathematics employed in the paper is willingly kept at medium level, often recursing to spherical examples in support to the theory. Most of the material is already present in literature but for a few parts concerning global models and the inverse gravimetric problem for layers.

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Copyright information

© Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringPolitecnico di MilanoMilanItaly
  2. 2.Department of Civil, Constructional and Environmental EngineeringUniversità di Roma La SapienzaRomeItaly
  3. 3.Department of Civil and Environmental EngineeringPolitecnico di MilanoMilanItaly
  4. 4.Geomatics Research & Development s.r.l.ComoItaly

Section editors and affiliations

  • Willi Freeden
    • 1
  1. 1.Geomathematics GroupUniversity of KaiserslauternKaiserslauternGermany

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