ISSN:
1432-1394
Source:
Springer Online Journal Archives 1860-2000
Topics:
Architecture, Civil Engineering, Surveying
Notes:
Abstract. The regularized solution of the external spherical Stokes boundary value problem as being used for computations of geoid undulations and deflections of the vertical is based upon the Green functions S 1 (Λ0, Φ0, Λ, Φ) of Box 0.1 (R = R 0 and V 1(Λ0, Φ0, Λ, Φ) of Box 0.2 (R = R 0) which depend on the evaluation point{Λ0, Φ0}∈ S2 R0 and the sampling point{Λ, Φ}∈ S2 R0 of gravity anomaliesΔγ(Λ, Φ) with respect to a normal gravitational field of type gm/R ("free air anomaly"). If the evaluation point is taken as the meta-north pole of the Stokes reference sphere S2 R0, the Stokes function, and the Vening-Meinesz function, respectively, takes the form S 2(Ψ) of Box 0.1, and V 2(Ψ) of Box 0.2, respectively, as soon as we introduce {meta-longitude (azimuth), meta-colatitude (spherical distance)}, namely {A, Ψ} of Box 0.5. In order to derive Stokes functions and Vening-Meinesz functions as well as their integrals, the Stokes and Vening-Meinesz functionals, in a convolutive form we map the sampling point {Λ, Φ} onto the tangent plane T0S2 R0 at {Λ0, Φ0} by means of oblique map projections of type (i) equidistant (Riemann polar/normal coordinates), (ii) conformal and (iii) equiareal. Box 2.1–2.4. and Box 3.1–3.4 are collections of the rigorously transformed convolutive Stokes functions and Stokes integrals and convolutive Vening-Meinesz functions and Vening-Meinesz integrals. The graphs of the corresponding Stokes functions S 2(Ψ), S 3(r),...,S 6(r) as well as the corresponding Stokes-Helmert functions H 2(Ψ), H 3(r),...,H 6(r) are given by Figure 4.1–4.5. In contrast, the graphs of Figure 4.6–4.10 illustrate the corresponding Vening-Meinesz functions V 2(Ψ), V 3(r),...,V 6(r) as well as the corresponding Vening-Meinesz-Helmert functions Q 2(Ψ), Q 3(r),...,Q 6(r). The difference between the Stokes functions / Vening-Meinesz functions and their first term (only used in the Flat Fourier Transforms of type FAST and FASZ), namely S 2(Ψ) – (sin Ψ/2)–1, S 3(r) – (sin r / 2R 0)–1,...,S 6(r) – 2R 0/r and V 2(Ψ) + (cos Ψ/2)/2(sin2Ψ/2), V 3(r) + (cos r/2R 0)/2(sin2 r/2R 0,...,V 6(r)+(R 0√4R 2 0–r 2)/r 2 illustrate the systematic errors in the "flat" Stokes function 2/Ψ or "flat" Vening-Meinesz function–2/Ψ2. The newly derived Stokes functions S 3(r),...,S 6(r) of Box 2.1–2.3, of Stokes integrals of Box 2.4, as well as Vening-Meinesz functions V 3(r),...,V 6(r) of Box 3.1–3.3, of Vening-Meinesz integrals of Box 3.4— all of convolutive type — pave the way for the rigorous Fast Fourier Transform and the rigorous Wavelet Transform of the Stokes integral / the Vening-Meinesz integral of type "equidistant". "conformal" and "equiareal".
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00867148
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