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An evaluation of accuracy of stokes' series and of some attempts to improve his theory

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Bulletin Géodésique (1946-1975)

Abstract

Four different variants of expanding the perturbing potential in terms of spherical functions were carried out for an Earth model in the shape of a sphere girded along the equator with a tor, the said tor being covered, on both sides with “deckings” (the greatest inclination angle of the model surface is near 10o):

  1. 1)

    On the basis of a given distribution of anomalous masses correct values of the coefficients (A) were calculated

  2. 2)

    By formally referring to the sphere of reference the quasigeoidal heights at points on the model surface (coefficients Aζe)

  3. 3)

    By formally referring to the sphere the gravity anomalies and by formally applying the Stokes series (coefficients Aζ),

  4. 4)

    By introducing corrections ΔAζ to the coefficients Aζ of the third variant given above, to account for corrections G1 of the first order obtainable by solving numerically the integral equation of Molodensky.

A marked discrepancy between the first and the third variants was brought to light (see Tables I and II). A conclusion was drawn on unfitness of the current practical method of calculating coefficients of the perturbing potential on the base of gravity measurements. To obtain an acceptable accuracy it is necessary to make use of the theory of investigating the external gravitational field and the shape of Earth's physical surface.

In connection with the results of the described computations some attempts by Gromov and Hirvonen of improving accuracy of the Stokes theory are investigated. It is show that the way proposed by the said authors cannot lead to an improvement of the accuracy of the Stokes theory, whereas the proposal made by Bjerhammar in relation to determining Green's function can only give the same approximation as the Stokes formula. Taking instead of the real Earth its model, as proposed by de Graaff-Hunter, we are faced with the same difficulties as in the case of application of the Stokes theory to the real Earth. The integral equation for the perturbing potential developed by Levallois and Bjerhammar is affected by an error of order of the slope of Earth's physical surface.

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Authors
  1. M. S. Molodensky
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  2. V. F. Yeremeyev
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  3. M. I. Yurkina
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Molodensky, M.S., Yeremeyev, V.F. & Yurkina, M.I. An evaluation of accuracy of stokes' series and of some attempts to improve his theory. Bull. Geodesique 63, 19–37 (1962). https://doi.org/10.1007/BF02528170

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