On Combining the Directional Solutions of the Gravitational Curvature Boundary-Value Problem

Martin Pitoňák, Pavel Novák, Michal Šprlák, Robert Tenzer

Research output: Chapter in book / Conference proceedingConference article published in proceeding or bookAcademic researchpeer-review


In global studies, the Earth’s gravitational field is conveniently described in terms of spherical harmonics. Four integral-based solutions to a gravitational curvature boundary-value problem can formally be formulated for the vertical-vertical-vertical, vertical-vertical-horizontal, vertical-horizontal-horizontal and horizontal-horizontal-horizontal components of the third-order gravitational tensor. Each integral equation provides an independent set of spherical harmonic coefficients because each component of the third-order gravitational tensor is sensitive to gravitational changes in the different directions. In this contribution, estimations of spherical harmonic coefficients of the gravitational potential are carried out by combining four solutions of the gravitational curvature boundary-value problem using three methods, namely an arithmetic mean, a weighted mean and a conditional adjustment model. Since the third-order gradients of the gravitational potential are not yet observed by satellite sensors, we synthesise them at the satellite altitude of 250 km from a global gravitational model up to the degree 360 while adding a Gaussian noise with the standard deviation of 6.3 × 10−19 m−1 s−2. Results of the numerical analysis reveal that the arithmetic mean model provides the best solution in terms of the RMS fit between predicted and reference values. We explain this result by the facts that the conditions only create additional stochastic bindings between estimated parameters and that more complex numerical schemes for the error propagation are unnecessary in the presence of only a random noise.

Original languageEnglish
Title of host publication9th Hotine-Marussi Symposium on Mathematical Geodesy - Proceedings of the Symposium in Rome, 2018
EditorsPavel Novák, Mattia Crespi, Nico Sneeuw, Fernando Sansò
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages7
ISBN (Print)9783030542665
Publication statusPublished - May 2019
Event9th Hotine-Marussi Symposium on Mathematical Geodesy, 2018 - Rome, Italy
Duration: 18 Jun 201822 Jun 2018

Publication series

NameInternational Association of Geodesy Symposia
ISSN (Print)0939-9585
ISSN (Electronic)2197-9359


Conference9th Hotine-Marussi Symposium on Mathematical Geodesy, 2018


  • Conditional adjustment
  • Gravitational curvature
  • Spherical harmonics

ASJC Scopus subject areas

  • Computers in Earth Sciences
  • Geophysics


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