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...esfera terrestre dos áreas de integración, la zona cercana y la lejana, empleándose observaciones puntuales en la primera y un modelo geopotencial en la segunda. En este documento se muestra la formulación matemática para evaluar la contribución de la zona lejana en el kernel esférico de Stokes minimizando los coeficientes de truncación de Molodenskij, tomando como área de aplicación el territorio mexicano.
Abstract
The solution of the geodetic boundary value problem requires the evaluation of the Stokes integral all over the Earth. Since the distribution of gravity observations on the surface of our planet is not homogenous, the terrestrial sphere is divided into two areas of integration, the near zone and the far zone. Point observations are used in the first zone and a geopotencial model in spectral form in the second. In this paper the mathematical formulation for evaluating the contribution of the far zone with Stokes's spherical kernel is shown, considering Mexican territory as the application area.
Introduction
Stokes's classical approach is based on the solution of the external boundary value problem for the disturbing potential T. The famous Stokes integral reads (Vanícek and Krakiwsky, 1986, eq. 22.16):
(1) T([OMEGA]) [[integral].sub.[OMEGA]'] [DELTA]g([OMEGA]) S([psi])d[OMEGA]',
where:
T([OMEGA]) is the disturbing potential at W
[DELTA]g([OMEGA]) is the gravity anomaly at W
S([psi]) is the spherical Stokes' function
[OMEGA] is the pair of angular spherical coordinates
[psi] is the angular distance between two points
R is the mean Earth radius
The spherical Stokes kernel (function) is an isotropic and homogeneous function. This means that the function depends neither on direction nor on the position of the integration point. Its value is only a function of the spherical distance between the integration point and the dummy point.
The integration expressed in equation (1) must be carried out over the whole earth (sphere), and the approximate equality sign in this equation is because the expression is correct only to the order of e2 (the square of eccentricity of the reference ellipsoid).
The Stokes kernel may be represented in spatial form (Heiskanen and Moritz, 1967, eq. 2-164) as:
(2) S([psi]) = 1 + 1/sin[[psi]/2] - 6 sin [[psi]/2] - 5cos [psi] - 3cos[psi](sin[[psi]/2] + [sin.sup.2][[psi]/2])
or in spectral form (Vanícek and Krakiwsky, 1986, eq. 22.15) as:
(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where:
[P.sub.j]: are the Legendre's functions
In Figure 1 the shape of the Stokes...
NOTE: All illustrations and photos
have been removed from this article.
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