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The Stokes and Vening-Meinesz functionals in a moving tangent space

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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 10, Φ0, Λ, Φ) ofBox 0.1 (R = R 0) andV 10, Φ0, Λ, Φ) ofBox 0.2 (R = R 0) which depend on theevaluation point0, Φ0} ∈ S R0 2 and thesampling point {Λ, Φ} ∈ S R0 2 ofgravity anomalies Δ γ (Λ, Φ) 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(Ψ) ofBox 0.1, andV 2(Ψ) ofBox 0.2, respectively, as soon as we introduce {meta-longitude (azimuth), meta-colatitude (spherical distance)}, namely {A, Ψ} 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 {Λ, Φ} onto the tangent plane T0S R0 2 at {Λ0, Φ0} 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(Ψ),S 3(r),⋯,S 6(r) as well as the correspondingStokes-Helmert functions H 2(Ψ),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(Ψ),V 3(r),⋯,V 6(r) as well as the correspondingVening-Meinesz-Helmert functions Q 2(Ψ),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(Ψ) − (sin Ψ/2)−1,S 3(r) − (sinr/2R 0)−1,⋯,S 6(r) − 2R 0/r andV 2(Ψ) + (cos Ψ/2)/2(sin2 Ψ/2),V 3(r) + (cosr/2R 0)/2(sin2 r/2R 0),⋯,\(V_6 (r) + {{(R_0 \sqrt {4R_0^2 - r^2 } )} \mathord{\left/ {\vphantom {{(R_0 \sqrt {4R_0^2 - r^2 } )} {r^2 }}} \right. \kern-\nulldelimiterspace} {r^2 }}\) illustrate the systematic errors in the”flat” Stokes function 2/Ψ or ”flat”Vening-Meinesz function −2/Ψ2. 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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Correspondence to Erik W. Grafarend.

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Grafarend, E.W., Krumm, F. The Stokes and Vening-Meinesz functionals in a moving tangent space. Journal of Geodesy 70, 696–713 (1996).

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  • Fourier Transform
  • Sampling Point
  • Systematic Error
  • Azimuth
  • Tangent Space