Boundary Value Problem Of Physical Geodesy Research Articles

Spatial and Spectral Representations of the Geoid-to-Quasigeoid Correction

In geodesy, the geoid and the quasigeoid are used as a reference surface for heights. Despite some similarities between these two concepts, the differences between the geoid and the quasigeoid (i.e. the geoid-to-quasigeoid correction) have to be taken into consideration in some specific applications which require a high accuracy. Over the world’s oceans and marginal seas, the quasigeoid and the geoid are identical. Over the continents, however, the geoid-to-quasigeoid correction could reach up to several metres especially in the mountainous, polar and geologically complex regions. Various methods have been developed and applied to compute this correction regionally in the spatial domain using detailed gravity, terrain and crustal density data. These methods utilize the gravimetric forward modelling of the topographic density structure and the direct/inverse solutions to the boundary-value problems in physical geodesy. In this article, we provide a brief summary of existing theoretical and numerical studies on the geoid-to-quasigeoid correction. We then compare these methods with the newly developed procedure and discuss some numerical and practical aspects of computing this correction. In global applications, the geoid-to-quasigeoid correction can conveniently be computed in the spectral domain. For this purpose, we derive and present also the spectral expressions for computing this correction based on applying methods for a spherical harmonic analysis and synthesis of global gravity, terrain and crustal structure models. We argue that the newly developed procedure for the regional gravity-to-potential conversion, applied for computing the geoid-to-quasigeoid correction in the spatial domain, is numerically more stable than the existing inverse models which utilize the gravity downward continuation. Moreover, compared to existing spectral expressions, our definition in the spectral domain takes not only the terrain geometry but also the mass density heterogeneities within the whole Earth into consideration. In this way, the geoid-to-quasigeoid correction and the respective geoid model could be determined more accurately.

IV: GEODESY: Remarks on the Role of Height Datum in Altimetry-Gravimetry Boundary-Value Problems

The problem of global geoid determination is usually solved using satellite altimetry data on the oceans, together with an oceanographic model of sea surface topography, and gravity anomaly data on the continents. Such data, however, enable to obtain only potential differences with respect to a reference surface whose absolute potential is unknown. This situation suggests to modify the classical mixed boundary-value problem of physical geodesy by inserting into the boundary conditions an unknown additive constant, that must be determined by imposing a suitable additional constraint. Yet, such formulation of the boundary-value problem, from the point of view of its mathematical properties, is not unconditionally well-posed, and, furthermore, does not reflect faithfully the available physical model, as the present knowledge of ocean circulation does not allow to connect along coastlines the reference surfaces defined on the oceans and on the continents. The introduction of two different unknown additive constants, one for the oceans and one for the earth, to be determined by imposing two additional constraints, gives rise to a more faithful picture of the present physical knowledge, and, at the same time, to a new well-posed formulation of the boundary-value problem.

The boundary-value problem of physical geodesy and its solution

Краевая задача физической геодезин не являеетор задачей тиоа Неймана. Она формуларуется следующим образом: На поверхности Земли пусть даны наблюденные краевые значения Коши и по ним требуется определить иеометрию зтой поверхности. Окаоалооь, что формулы, раньше данные автором, практически решают эту проблему. Члены, пренебреженные в зтом решении, имеют порядок наклонов тела. Исходя из этой строгой основы, было найдено, что все остальные редукшионные методы могут применяться как строгие ъе, в которых работается со значениями на поверхности, несмотря на компликации т. наз. новой теории гравиметрической чеодезии.