The Stokes and Vening-Meinesz functionals in a moving tangent space |
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Authors: | Erik W Grafarend Friedhelm Krumm |
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Institution: | (1) Department of Geodetic Science, University of Stuttgart, Keplerstr. 11, D-70174 Stuttgart, Germany |
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Abstract: | The regularized solution of the external sphericalStokes boundary value problem as being used for computations of geoid undulations and deflections of the vertical is based upon theGreen functions S
1( 0, 0, , ) ofBox 0.1 (R = R
0) andV
1( 0, 0, , ) ofBox 0.2 (R = R
0) which depend on theevaluation point { 0, 0} S
R0
2
and thesampling point { , } S
R0
2
ofgravity anomalies
( , ) with respect to a normal gravitational field of typegm/R ( free air anomaly ). If the evaluation point is taken as the meta-north pole of theStokes reference sphere S
R0
2
, theStokes function, and theVening-Meinesz function, respectively, takes the formS( ) ofBox 0.1, andV
2( ) ofBox 0.2, respectively, as soon as we introduce {meta-longitude (azimuth), meta-colatitude (spherical distance)}, namely {A, } ofBox 0.5. In order to deriveStokes functions andVening-Meinesz functions as well as their integrals, theStokes andVening-Meinesz functionals, in aconvolutive form we map the sampling point { , } onto the tangent plane T0S
R0
2
at { 0, 0} by means ofoblique map projections of type(i) equidistant (Riemann polar/normal coordinates),(ii) conformal and(iii) equiareal.Box 2.1.–2.4. andBox 3.1.– 3.4. are collections of the rigorously transformedconvolutive Stokes functions andStokes integrals andconvolutive Vening-Meinesz functions andVening-Meinesz integrals. The graphs of the correspondingStokes functions S
2( ),S
3(r), ,S
6(r) as well as the correspondingStokes-Helmert functions H
2( ),H
3(r), ,H
6(r) are given byFigure 4.1–4.5. In contrast, the graphs ofFigure 4.6–4.10 illustrate the correspondingVening-Meinesz functions V
2( ),V
3(r), ,V
6(r) as well as the correspondingVening-Meinesz-Helmert functions Q
2( ),Q
3(r), ,Q
6(r). The difference between theStokes functions / Vening-Meinesz functions andtheir first term (only used in the Flat Fourier Transforms of type FAST and FASZ), namelyS
2( ) – (sin /2)–1,S
3(r) – (sinr/2R
0)–1, ,S
6(r) – 2R
0/r andV
2( ) + (cos /2)/2(sin2 /2),V
3(r) + (cosr/2R
0)/2(sin2
r/2R
0), ,
illustrate the systematic errors in the flat Stokes function 2/ or flat Vening-Meinesz function –2/ 2. The newly derivedStokes functions S
3(r), ,S
6(r) ofBox 2.1–2.3, ofStokes integrals ofBox 2.4, as well asVening-Meinesz functionsV
3(r), ,V
6(r) ofBox 3.1–3.3, ofVening-Meinesz integrals ofBox 3.4 — all of convolutive type — pave the way for the rigorousFast Fourier Transform and the rigorousWavelet Transform of theStokes integral / theVening-Meinesz integral of type equidistant , conformal and equiareal . |
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Keywords: | |
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