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

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(Λ₀, Φ₀, Λ, Φ) ofBox 0.1 (R = R ₀) andV ₁(Λ₀, Φ₀, Λ, Φ) ofBox 0.2 (R = R ₀) which depend on theevaluation point {Λ₀, Φ₀} ∈ S ²_R0 and thesampling point {Λ, Φ} ∈ S ²_R0 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 ²_R0 , 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 ²_R0 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”.