[G02-1] Theory and methods

[G02-1-03] Boundary complexity in classical and variational concepts of solving geodetic boundary value problems

^○Petr Holota¹, Otakar Nesvadba² (1.Research Institute of Geodesy, Topography and Cartography, 250 66 Zdiby 98, Prague-East, Czech Republic (e-mail: [e-mail]), 2.Land Survey Office, Pod Sidlistem 9, 182 11 Prague 8, Czech Republic (e-mail: [e-mail]))

In gravity field studies the complex structure of the Earth's surface makes the solution of geodetic boundary value problems quite challenging. This equally concerns classical methods of potential theory as well as modern methods often based on a (variational or) weak solution concept. The aim of this paper is to seek a balance between the performance of an apparatus developed for the surface of the Earth smoothed up to a certain degree and an iteration procedure used to bridge the difference between the real and smoothed topography. The approach is applied to the solution of the linear gravimetric boundary value problem in geopotential determination. Within the classical concept a transformation of coordinates is used that offers a possibility to solve an alternative between the boundary complexity and the complexity of the coefficients of the partial differential equation governing the solution. Also the oblique derivative boundary condition is taken into account in this way. The use of modified spherical and also modified ellipsoidal coordinates is discussed. The complexity of the boundary is reflected in the structure of Laplace's operator. Green's function representation is applied and the structure and convergence of the iteration steps is analyzed. The weak solution concept has different features. The aim here is to simplify the bilinear form that defines the problem under consideration and to justify the use of Galerkin's matrix constructed for an approximation solution domain. The approach focuses on the use of a sphere or an ellipsoid of revolution in quality of an approximation boundary. Also in case of the weak solution the simplification is compensated by successive approximations. The related functional analytic and numerical aspects are discussed. In both the cases attention is also paid to the question concerning the possible (hidden or explicit) role of analytical continuation.