Stokes’ Formula

2020 ◽  
pp. 147-157
Author(s):  
Kenji Fukaya ◽  
Yong-Geun Oh ◽  
Hiroshi Ohta ◽  
Kaoru Ono
Keyword(s):  
Stokes Formula

scholarly journals Unbiased least-squares modification of Stokes’ formula

2020 ◽  
Vol 94 (9) ◽  
Author(s):  
Lars E. Sjöberg
Keyword(s):  
Gravity Data ◽  
Harmonic Series ◽  
Local Error ◽  
Data Sets ◽  
Spherical Cap ◽  
Local Variance ◽  
Error Covariance ◽  
Stokes Formula ◽  
Gravity Signal ◽  
High Degree

Abstract As the KTH method for geoid determination by combining Stokes integration of gravity data in a spherical cap around the computation point and a series of spherical harmonics suffers from a bias due to truncation of the data sets, this method is based on minimizing the global mean square error (MSE) of the estimator. However, if the harmonic series is increased to a sufficiently high degree, the truncation error can be considered as negligible, and the optimization based on the local variance of the geoid estimator makes fair sense. Such unbiased types of estimators, derived in this article, have the advantage to the MSE solutions not to rely on the imperfectly known gravity signal degree variances, but only the local error covariance matrices of the observables come to play. Obviously, the geoid solution defined by the local least variance is generally superior to the solution based on the global MSE. It is also shown, at least theoretically, that the unbiased geoid solutions based on the KTH method and remove–compute–restore technique with modification of Stokes formula are the same.

scholarly journals Comparison of remove-compute-restore and least squares modification of Stokes' formula techniques to quasi-geoid determination over the Auvergne test area

2012 ◽  
Vol 2 (1) ◽  
pp. 53-64 ◽  
Author(s):  
H. Yildiz ◽  
R. Forsberg ◽  
J. Ågren ◽  
C. Tscherning ◽  
L. Sjöberg
Keyword(s):  
Least Squares ◽  
Gravity Data ◽  
Test Area ◽  
Geoid Determination ◽  
Geopotential Model ◽  
Low Degree ◽  
Residual Terrain Modelling ◽  
The Alps ◽  
Stokes Formula ◽  
Regional Geoid Determination

Comparison of remove-compute-restore and least squares modification of Stokes' formula techniques to quasi-geoid determination over the Auvergne test areaThe remove-compute-restore (RCR) technique for regional geoid determination implies that both topography and low-degree global geopotential model signals are removed before computation and restored after Stokes' integration or Least Squares Collocation (LSC) solution. The Least Squares Modification of Stokes' Formula (LSMS) technique not requiring gravity reductions is implemented here with a Residual Terrain Modelling based interpolation of gravity data. The 2-D Spherical Fast Fourier Transform (FFT) and the LSC methods applying the RCR technique and the LSMS method are tested over the Auvergne test area. All methods showed a reasonable agreement with GPS-levelling data, in the order of a 3-3.5 cm in the central region having relatively smooth topography, which is consistent with the accuracies of GPS and levelling. When a 1-parameter fit is used, the FFT method using kernel modification performs best with 3.0 cm r.m.s difference with GPS-levelling while the LSMS method gives the best agreement with GPS-levelling with 2.4 cm r.m.s after a 4-parameter fit is used. However, the quasi-geoid models derived using two techniques differed from each other up to 33 cm in the high mountains near the Alps. Comparison of quasi-geoid models with EGM2008 showed that the LSMS method agreed best in term of r.m.s.

scholarly journals On the topographic effects by Stokes’ formula

2014 ◽  
Vol 4 (1) ◽  
Author(s):  
L.E. Sjöberg
Keyword(s):  
Gravity Anomaly ◽  
Bouguer Anomaly ◽  
Gravity Anomalies ◽  
Geoid Height ◽  
Topographic Effects ◽  
Gravimetric Geoid ◽  
Bouguer Gravity ◽  
The Third ◽  
Stokes Formula ◽  
The Difference

AbstractTraditional gravimetric geoid determination relies on Stokes’ formula with removal and restoration of the topographic effects. It is shown that this solution is in error of the order of the quasigeoid-to-geoid difference, which is mainly due to incomplete downward continuation (dwc) of gravity from the Earth’s surface to the geoid. A slightly improved estimator, based on the surface Bouguer gravity anomaly, is also biased due to the imperfect harmonic dwc the Bouguer anomaly. Only the third estimator,which uses the (harmonic) surface no-topography gravity anomaly, is consistent with the boundary condition and Stokes’ formula, providing a theoretically correct geoid height. The difference between the Bouguer and no-topography gravity anomalies (on the geoid or in space) is the “secondary indirect topographic effect”, which is a necessary correction in removing all topographic signals.

On Stokes' formula for the force of resistance

1990 ◽  
Vol 45 (5) ◽  
pp. 232-234
Author(s):  
T Yu Krasulina
Keyword(s):  
Stokes Formula

scholarly journals A semi-canonical reduction for periods of Kontsevich–Zagier

2020 ◽  
pp. 1-28
Author(s):  
Juan Viu-Sos
Keyword(s):  
Integral Representation ◽  
Rational Functions ◽  
Finite Sequence ◽  
Complex Numbers ◽  
Effective Algorithm ◽  
Algebraic Set ◽  
Change Of Variables ◽  
Stokes Formula ◽  
Algebraic Domains

The [Formula: see text]-algebra of periods was introduced by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of [Formula: see text]-rational functions over [Formula: see text]-semi-algebraic domains in [Formula: see text]. The Kontsevich–Zagier period conjecture affirms that any two different integral expressions of a given period are related by a finite sequence of transformations only using three rules respecting the rationality of the functions and domains: additions of integrals by integrands or domains, change of variables and Stokes formula. In this paper, we prove that every non-zero real period can be represented as the volume of a compact [Formula: see text]-semi-algebraic set obtained from any integral representation by an effective algorithm satisfying the rules allowed by the Kontsevich–Zagier period conjecture.

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