SETUP OF OBSERVATIONAL FUNCTIONALS IN GRAVITY SPACE AS WELL AS IN GEOMETRY SPACE

The determination of the terrestrial gravitational field is traditiona lly based on the following procedure: Observables on the Earth's bound ary surface like gravitational potential differences, the spherical co ordinates of the gravitational vector (length of the gravitational vec tor, astronomical latitude, astronomical longitude) as well as higher order derivatives of the gravitational potential are used to formulate a particular boundary value problem of the Laplace - Poisson equation . In case the boundary surface as a star-shaped orientable smooth surf ace is perfectly known, we are led to an overdetermined boundary value problem which is solved by adjustment procedures in function spaces, namely in the Earth's external space and on its complicated boundary s urface. As soon as observables of different types are only known parti ally on the ocean (observation of type one), partially on the continen t (observation of type two) we are led to mixed boundary value problem s. In contrast, if we know nothing about the surface geometry, but bou ndary observational functionals are available we are confronted with t he task to solve nonlinear free boundary value problems. In between of perfect information and no information about the boundary surface geo metry is the case of a weakly known boundary geometry. Such a situatio n is motivated by the fact that the coordinates of a parameterized sur face of the Earth or of artificial satellites is derived from noisy ge odetic observations, e.g. by Local Positioning Systems (LPS) or Global Positioning Systems (GPS). The weak boundary information of the Earth 's surface or the surface of artificial satellite orbits (Maupertuis m anifold of all orbits for a given gravitational field) is given by sec ond order statistics by means of first order moments (mean values mu(( $) over cap x)), possibly subject to bias, and by second order moments (variance-covariance matrix Sigma(($) over cap x)). In addition, the boundary observational functionals are characterized by means of first order moments (mean values mu(Y)), possibly subject to bias, and by s econd order moments (variance-covariance matrix Sigma(Y)). Here we out line the linearization of the nonlinear observational functionals firs tly in paragraph one with respect to stochastic boundary information a nd secondly in paragraph two with respect to a reference gravitational field leading to the ''mixed model'' boundary value problem (1.10) wi th unknowns in gravity space as well as in geometry space. Partial eli mination either of the displacement vector Delta ($) over cap x or of the disturbing potential delta u(($) over cap x) leads to linearized b oundary operators of observational type either in gravity space or in geometry space (1.12ii) versus (1.14ii). In the centre is an extensive derivation of the boundary observational functionals in local coordin ates, namely {ellipsoidal longitude, ellipsoidal latitude, ellipsoidal height} for a ''realistic'' topographic boundary surface of the Earth . The ellipsoidal height functions are scalar functions on the Earth's reference ellipsoid of revolution and are accordingly represented by amplitude- versus phase-modulated spherical functions, being orthonorm al on the reference ellipsoid of revolution. The symbols of the metric and of connection (Christoffel symbols) are computed in order to esta blish surface covariant differentiation. The highlight are the boundar y observational functionals (1.28), (1.31), (1.32), (1.33) given in lo cal coordinates of the parameterized topographic surface of the Earth. Finally, as an example, the boundary observational functionals are gi ven in spherical approximation of gravity and geometry space.