On the Linearized Boundary Value Problems of Physical Geodesy
Loading...
Date
1991-02
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Ohio State University. Division of Geodetic Science
Abstract
Three versions of the Geodetic Boundary Value Problem (GBVP) have to be distinguished, namely the fixed problem involving the assumption of a completely known boundary surface, and the vectorial and scalar formulations of the free GBVP where either the spatial or the vertical components of the boundary position vector are unknown. These originally non-linear problems can be linearized using an adopted reference potential; in addition the vectorial and scalar free versions require the definition of a reference surface (telluroid). Considering first-order effects induced by the quadrapole momentum and the centrifugal term of the earth's gravity field as well as the ellipticity and the topography of the (reference) boundary surface the boundary conditions and the resulting expressions for the position correction vector (height anomaly) are derived in a consistent way for the three formulations of the GBVP. The corresponding expansions which hold in the framework of a first-order theory for arbitrary reference potentials are performed in spherical, geographical-ellipsoidal (geodetic) and elliptical (spheroidal) coordinates. Based upon these expansions the exact meaning of the concepts of spherical and constant radius approximation is explained. It is shown that the scalar and vectorial free GBVPs imply different boundary conditions which only coincide on the level of spherical approximation - in contrast to presentation in some publications mixing both concepts. A general theory for the solution of the (linear) GBVPs is set up, based on a perturbation of the corresponding spherical problems resulting from spherical and constant radius approximation. The solutions of these problems showing the classical spherical structure can be represented in analytical form and describe the "unperturbed" state. It is pointed out that most approaches proposed in literature for solving the (non-spherical) GBVPs can be considered in the framework of the general perturbation theory. Explicit solutions of the linearized GBVPs are derived, referring to a description of the boundary operators in spherical coordinates and taking into account all first-order effects induced by the quadrupole momentum and the centrifugal term of the earth's gravity field as well as the ellipticity of the earth's surface. Representing the disturbing potential in a spherical harmonic series a coupled system of algebraical equations for the unknown potential coefficients is produced which can be solved e.g. by an iteration process. It proves that in the case of a general, non-isotropic reference field the first degree harmonics are estimable quantities - in contrast to the properties of the free GBVP in sperical approximation! While the whole set of first-degree terms is estimable in the case of the scalar free GBVP, one degree of freedom (zonal term) is left in the vectorial problem. The impact of the first-degree terms on the solution of the GBVPs is discussed in detail. As a practical result, "ellipsoidal corrections" with respect to the solutions of the problems in spherical and constant radius approximation are derived. [Full text of abstract available in document.]