The exterior Airy/Heiskanen topographic-isostatic gravity potential, anomaly and the effect of analytical continuation in Stokes' formula

1998 ◽  
Vol 72 (11) ◽  
pp. 654-662 ◽  
Author(s):  
L. E. Sjöberg
Keyword(s):  
Analytical Continuation ◽  
Gravity Potential ◽  
Isostatic Gravity ◽  
Stokes Formula

scholarly journals The geoid or quasigeoid – which reference surface should be preferred for a national height system?

2013 ◽  
Vol 3 (2) ◽  
pp. 103-109 ◽  
Author(s):  
L. E. Sjöberg
Keyword(s):  
Density Distribution ◽  
Analytical Continuation ◽  
Reference Surface ◽  
Numerical Instability ◽  
Convergence Problem ◽  
The World ◽  
Height System ◽  
Normal Heights ◽  
Integration Problem ◽  
Stokes Formula

Abstract Most European states use M. S. Molodensky’s concept of normal heights for their height systems with a quasigeoid model as the reference surface, while the rest of the world rely on orthometric heights with the geoid as the zero-level. Considering the advances in data caption and theory for geoid and quasigeoid determinations, the question is which system is the best choice for the future. It is reasonable to assume that the latter concept, in contrast to the former, will always suffer from some uncertainty in the topographic density distribution, while Molodensky’s approach to quasigeoid determination has a convergence problem. On the contrary, geoid and quasigeoid models computed by analytical continuation (e.g., rcr technique or KTH method) have no integration problem, and the quasigeoid can always be determined at least as accurate as the geoid. As the numerical instability of the analytical continuation is better controlled in the KTH method vs. the rcr method, we propose that any future height system be based on normal heights with a quasigeoid model computed similar to or directly based on the KTH method (Least squares modification of Stokes formula with additive corrections).

scholarly journals The topographic bias in Stokes’ formula vs. the error of analytical continuation by an Earth Gravitational Model - are they the same?

2015 ◽  
Vol 5 (1) ◽  
Author(s):  
L. E. Sjöberg
Keyword(s):  
Gravity Anomaly ◽  
Spherical Harmonics ◽  
Analytical Continuation ◽  
Geoid Determination ◽  
Disturbing Potential ◽  
Solid Spherical Harmonics ◽  
Topographic Bias ◽  
Gravitational Model ◽  
Stokes Formula ◽  
Original Formula

AbstractGeoid determination below the topographic surface in continental areas using analytical continuation of gravity anomaly and/or an external type of solid spherical harmonics determined by an Earth GravitationalModel (EGM) inevitably leads to a topographic bias, as the true disturbing potential at the geoid is not harmonic in contrast to its estimates. We show that this bias differs for the geoid heights represented by Stokes’ formula, an EGMand for the modified Stokes formula. The differences are due to the fact that the EGM suffers from truncation and divergence errors in addition to the topographic bias in Stokes’ original formula.

Isostatic gravity map of the Las Vegas 30 x 60 minute quadrangle, California and Nevada

1999 ◽  
Author(s):  
V.E. Langenheim ◽  
R.I. Morin ◽  
J.G. Davidson ◽  
K.M. Schmidt ◽  
H.R. Blank
Keyword(s):  
Las Vegas ◽  
Isostatic Gravity ◽  
Gravity Map

scholarly journals The surface layer integral method for the modelling of the radial gravity gradient component over sea surface

2019 ◽  
Vol 9 (1) ◽  
pp. 127-132
Author(s):  
D. Zhao ◽  
Z. Gong ◽  
J. Feng
Keyword(s):  
Surface Layer ◽  
Integral Method ◽  
Integral Formula ◽  
Radial Component ◽  
Second Order ◽  
Sea Surface ◽  
Gravity Potential ◽  
Gravity Gradient ◽  
Disturbing Potential ◽  
Gradient Component

Abstract For the modelling and determination of the Earth’s external gravity potential as well as its second-order radial derivatives in the space near sea surface, the surface layer integral method was discussed in the paper. The reasons for the applicability of the method over sea surface were discussed. From the original integral formula of disturbing potential based on the surface layer method, the expression of the radial component of the gravity gradient tensor was derived. Furthermore, an identity relation was introduced to modify the formula in order to reduce the singularity problem. Numerical experiments carried out over the marine area of China show that, the modi-fied surface layer integral method effectively improves the accuracy and reliability of the calculation of the second-order radial gradient component of the disturbing potential near sea surface.

scholarly journals Analytical Continuation within the Freundlich Adsorption Model. Comment on Edet, U.A.; Ifelebuegu, A.O. Kinetics, Isotherms, and Thermodynamic Modeling of the Adsorption of Phosphates from Model Wastewater Using Recycled Brick Waste. Processes 2020, 8, 665

Processes ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 1251
Author(s):  
Michael Vigdorowitsch ◽  
Alexander N. Pchelintsev ◽  
Liudmila E. Tsygankova
Keyword(s):  
Experimental Data ◽  
Power Function ◽  
Thermodynamic Modeling ◽  
Mathematical Theory ◽  
Surface Adsorption ◽  
Analytical Continuation ◽  
Adsorption Model ◽  
Saturation Regime

Using experimental data for the adsorption of phosphates out of wastewater on waste recycled bricks, published independently in MDPI Processes before (2020), this message re-visits the mathematical theory of the Freundlich adsorption model. It demonstrates how experimental data are to be deeper treated to model the saturation regime and to bridge a chasm between those areas where the data fit the Freundlich power function and where a saturation of surface adsorption centers occurs.

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 Review of Environmental and Geological Microgravity Applications and Feasibility of Its Employment at Archaeological Sites in Israel

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Lev V. Eppelbaum
Keyword(s):  
Gravity Anomalies ◽  
Gravity Potential ◽  
Archaeological Sites ◽  
Useful Signal ◽  
Geometric Size ◽  
Archaeological Models ◽  
Vertical Gravity ◽  
Transformation Methods ◽  
Derivatives Of ◽  
Archaeological Objects

Microgravity investigations are widely applied at present for solving various environmental and geological problems. Unfortunately, microgravity survey is comparatively rarely used for searching for hidden ancient targets. It is caused mainly by small geometric size of the desired archaeological objects and various types of noise complicating the observed useful signal. At the same time, development of modern generation of field gravimetric equipment allows to register promptly and digitally microGal (10-8 m/s2) anomalies that offer a new challenge in this direction. An advanced methodology of gravity anomalies analysis and modern 3D modeling, intended for ancient targets delineation, is briefly presented. It is supposed to apply in archaeological microgravity the developed original methods for the surrounding terrain relief computing. Calculating second and third derivatives of gravity potential are useful for revealing some closed peculiarities of the different Physical-Archaeological Models (PAMs). It is underlined that physical measurement of vertical gravity derivatives in archaeological studying has a significant importance and cannot be replaced by any transformation methods. Archaeological targets in Israel have been ranged by their density/geometrical characteristics in several groups. The performed model computations indicate that microgravity investigations might be successfully applied at least in 20–25% of archaeological sites in Israel.

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