POSITIONING

DETERMINATION OF THE GRAVITY FIELD

The project UNIGRACE applied three ballistic absolute gravity meters based on the free fall and two based on rise and fall method for unification of gravity systems of 11 countries in Central and Eastern Europe. One absolute observation campaign was performed in 1998 and the second one will be in 2000 (Reinhart et al. 1998). The sites Wettzell and Jozefoslav will be used as to intercompare the absolute gravimeters used in the project. Each absolute site will be connected to the national fundamental gravity networks to correct the gravity datum and gravity scale by relative gravity meters. Each absolute site is supplemented by a gravimetric micronetwork in order to monitor local ground deformations and gravity variations caused by local ground water level and atmospheric variations.

Gravimetric model of Slovak quasigeoid

The gravimetric model of Slovak quasigeoid was determined from gravimetric data, discrete (4-6 points/km² ) and mean 5´ x 7,5´ grid by remove-restore technique. The gravimetric quasigeoid was fitted by 12 GPS/leveling points (Mojzeš 1996). Polynomial model was used for fitting. After using three degree polynomial model with seven coefficients and 40 GPS/leveling points, the maximum residual was 0.114 m, the minimum residual was -0.068 m and standard deviation of residuals was 0.046 m (Mojzeš & Janák 1998).

GENERAL THEORY AND METHODOLOGY

Truncated geoid and gravity inversion

The research was concerned with the investigation of the possibility to use the truncated geoid in the inverse problem of gravimetry.
     The truncated geoid, defined by the truncated Stokes integral transform, an integral convolution of gravity anomalies with the Stokes function on a spherical cap, is often used as a mathematical tool in geoid via Stokes´ integral computations to overcome computational difficulties, particularly the need to integrate over the entire boundary spheroid. The first objective is to demonstrate, that the truncated geoid does, besides having mathematical applications, have physical interpretation, and thus may be used in gravity inversion. For the sake of such demonstration a very simple model of one point mass anomaly was chosen and a method for inverting its synthetic gravity field with the use of the truncated geoid was presented in Vajda & Vaníček (1996). The method of inverting the synthetic field generated by one point mass anomaly has become fundamental for the inversion studies for sets of point mass anomalies, which are published in Vajda & Vaníček (1997). More general applications are currently under investigation. Since it was not developed an inversion technique for physically meaningful mass distributions based on the truncated geoid yet, this work was not related to any of the existing gravity inversion techniques. The inversion for one point mass is based on the onset of the so called „dimple event", which occurs in the sequence of surfaces (or profiles) of the first derivative of the truncated geoid with respect to the truncation parameter (radius of the integration cap), its only free parameter. Computing the truncated geoid at various values of the truncation parameter may be understood as spatial filtering of surface gravity data, sort of a weighted spherical windowing method. Studying the change of the truncated geoid represented by its first derivative may be understood as a data enhancement method. The instant of the dimple onset is practically independent of mass of the point anomaly and linearly dependent on its depth.
     In Vajda & Vaníček (1997), the change of the truncated geoid in response to the change of the truncation parameter was studied in the context of the mass density that generates the gravity field. This study was limited to synthetic densities represented by point mass anomalies imbedded in a massive sphere. Computer simulation was chosen as the basic technique for this study. It demonstrated a clear sensitivity of the truncated geoid to the depths of point mass anomalies generating the synthetic surface gravity.
     In Vajda & Vaníček (1998a) the numerical aspects of the evaluation of the truncated geoid were investigated. The truncated geoid can be computed from gravity anomalies either in spectral form, providing the geopotential is given in a spectral form, or directly by numerical integration over mean gravity anomalies. Alternatively, the truncated geoid may be evaluated analytically from geoidal undulations via another integral transform, namely the so called „Altimetry Integral". This becomes particularly useful when interpreting the truncated geoid on sea, where the geoidal undulations of global coverage and good quality can be obtained after processing the readily available satellite altimetry data. However, the numerical evaluation of the „Altimetry Integral" poses a challenge.
     The deploying spectral filtering in the inversion method by means of the truncated geoidal heights was investigated. The combination of spectral filtering with truncation is tackled in the following two papers. In Vajda & Vaníček (1998b), producing a frequency window (wavelength bandwidth) part of the truncated geoid is understood under the term „spectral filtering of the truncated geoid". Furthermore the high frequency (short wavelength) part of the truncated geoid was studied. Three kinds of the high frequency part of the truncated geoid are introduced herein. All three are defined via integral transforms and referred to as high degree truncated geoids. Spectral form expressions are derived for these transforms. It is shown, that the three transforms differ from each other, and represent slightly different physical quantities.
     In Vajda & Vaníček (1998c), the incorporating spectral filtering into the truncation technique for one point mass anomaly was studied, the objective being demonstrating such concept on a trivial case. The relationship between the instant of the dimple onset in the sequence of the derivative of the high degree (high-pass filtered) truncated geoid and the depth of the point mass is derived.

Gravimetric inverse problem

A formula for the computation of the gravity field of a polyhedral body with linearly increasing density was derived (Pohánka 1998).This formula has the following properties: (i) it is the simplest possible; (ii) it is valid for every point of space, and (iii) it needs no special attention for the points near to or on the surface of the body. Hewever, in contrast to the formula for a polyhedral body of uniform density, for the points very far from the body, a substantial numerical error can arise. Therefore, for numerical calculation, the higher-precision expression of real numbers must be used.
     New method for solving the inverse problem of gravimetry for a spherical planetary body using the decomposition of the interior potential into a series of polyharmonic functions is presented in Pohánka (1997). The interior potential of a spherical planetary body is expressed as the sum of two parts, the first given uniquely by the external gravity field and the surface density and the second depending on an arbitrary function. From the general solution a single solution is then chosen; this is done by imposing certain natural conditions on it, among others that this particular solution is an n-harmonic function for n as small as possible. In the case we are interested only in a local inverse problem (where the curvature of the surface of the body can be ignored), it is advantageous to use the planar version of this solution, which can be obtained by fixing some point of the surface of the body and letting the radius of the body tend to infinity (Pohánka 1998) In this limit the surface of the body becomes a plane dividing the halfspace containing the matter (the interior domain) from the other (exterior) one. As we intend to treat only the local inverse problem, the density of matter and the gravity field (generated by this density) will be the anomalous one. The inverse problem of gravimetry for a planetary body of arbitrary smooth shape is solved by a suitable expression of the gravity potential in the interior of the body (Pohánka 1997), provided the solution of the interior Dirichlet and exterior Neumann problem for the Laplace equation for the domain representing the interior of the body is known.

A new model for polynomial transformation between 3D reference frames if one of them is locally deformed is proposed by Hefty & Frohmann (1998b). The algorithms using the quadratic, cubic and bi-quadratic forms enable removal of local distortions and efficient transformation of non-identical points. The method is demonstrated on transformation between terrestrial and satellite GPS networks. Application of polynomial transformation for studies of the regional deformation of ITRF is presented in Hefty (1998). The transformation of 3D geocentric coordinates into the local planar system minimising the projection errors are presented by Melicher & Flassik (1997) and Melicher & Galgonová (1998). The algorithms are well suited for limited local networks.

GEODYNAMICS

The computation of simple analytical models of surface displacements and gravity changes in layered elastic-gravitational medium and in an elastic halfspace with point source of heat is presented in Brimich et al. (1996).The comparison of the radial and vertical components of the displacement and gravity changes indicates that the horizontal changes of these quantities are smaller for the thermoelastic model than for elastic-gravitational.
     The magmatic intrusion in the Earth’s crust will cause a series of effects related to its mass as well as to pressurization of the chamber due to everfilling or temperature changes. In deformation modelling and prediction the most interesting effects are those which in principle could be detected on the surface before the eruption, including surface gravity changes and deformation. In consequence, the theoretical problem lies in calculating the gravitational potential and gravity changes, and the deformation produced on the Earth’s surface by a magmatic intrusion in the crust. The thermo-viscoelastic models of the deformations and gravity changes due to the anomalous sources of heat of prismatic shape are presented in Brimich (in press).

Earth tides research

The Earth’s tides research was aimed at the study of the extensometric measurements at the tidal station of the Geophysical Institute of the Slovak Academy of Sciences in Vyhne (Brimich 1998). The tidal measurements are affected by local effects and the study of these effects was our second topic in Earth’s tides investigation. Tidal forces generate periodic, low-frequency loading of the Earth’s interior. Evaluation of tidal observations requires the knowledge of the cavity effect due to buried gallery of the tidal station. The problem of the cavity effect can be solved by various methods. Estimation of this effect using the boundary integral method is presented in Brimich & Hvoždara (1997). The finite element approximatin of the estimation of the cavity, topographic and geologic effects is given in Kostecký & Kohút (1998). The comparison of the boundary integral and finite element method is presented in Brimich et al. (1997).

Rotation of the Earth

The study of the Earth’s rotation focuses on analysis of the short-term oscillations. The combination of diurnal polar motion coordinates obtained from the GPS using the variance component estimation is described by Hefty (1995). The short-period UT1 variations determined by single-baseline (Westford - Wettzell) VLBI are analysed in Hefty & Gontier (1997). A new VLBI geometric delay model including the adjustment of corrections to atmospheric modelling and terrestrial and celestial reference frames enable removing of biases due to phenomena mentioned. The UT1 quarterly steps due to observing schedule alterations were significantly reduced using the algorithm proposed. The subdaily Earth rotation variations observed by GPS are analysed by Rothacher et al. (1998). The main driving force are the ocean tides, the consistency of observed and predicted tidal constituents are at 10 m as level. The role of atmospheric excitation of subdaily variations is discussed by Hefty et al. (1997). It is shown, that the GPS does not observe the retrograde diurnal atmospheric polar motion variations even the expected amplitude is over the observation noise.

Intraplate tectonics

Slovak Republic actively participated in the Central European Regional Geodynamics Project (CERGOP) devoted to study of long-term geo-kinematics in the Central Europe using the GPS technique. There are three epoch sites on the territory of Slovakia included in the CEGRN - Central European Regional Network - Modra-Piesok in Small Carpathian (from 1996 the observations are on the permanent basis), Skalnaté Pleso in High Tatra and Strážna hora in Krupina Mountains. The epoch GPS network CEGRN consisting of more than 30 stations observed yearly from 1994 has been analysed by Hefty & Gerhátová (1996, 1997).
     The reference frame for geo-kinematic interpretation of repeated epoch observations in CERGOP has been proposed by Hefty (1997a). It is based on the ITRF94 coordinates and velocities of the long-term observed IGS sites conserving the modelled motion of Eurasian tectonic plate. The analysis of the expected accuracy is in Hefty (1997a) and the results of first three years of the project are presented in Hefty (1997b). The proposed reference frame named as CETRF was officially adopted for interpretation of CERGOP. The coordinate evolution of 31 CERGOP sites is given in Rogowski & Hefty (1998). The results show the horizontal repeatability within 10 mm and the observed trends are below 5mm/year. The effect of introducing the ITRF96 for interpretation of CERGOP results is studied in Hefty (1999). The internal consistency of individual GPS epoch campaigns increased, the behaviour of individual station remained generally unchanged.
     The project CERGOP applied GPS techniques on a regional basis aiming to obtain a 3 dimensional velocity field and provide a most reference frame best suited for the region. Four measurements were carried out spanning 3 years from 1994-1997. The results of the yearly epoch measurements were the basis for a  number of studies. The epoch data processing showed that an overall site position accuracy of 2-4 mm in horizontal coordinates and 4-8 mm in vertical coordinates have been achieved Fejes et al. (1998). For the major development of the project were established CERGOP Study Groups (CSG). CSG.2 contributed to the assessment and maintenance of quality standards within the project (Levai et al. 1998). CSG 7 main activity consist in listing of possibilities to observe the absolute gravity for geodynamical purposes in Central Europe (Barlík & Mojzeš 1998). For local deformation studies in Central Europe was established project Geodynamics of Tatra mountains (Czarnecki et al. 1998).
     The mathematical model for estimate of 3D velocities on the basis of GPS epoch campaigns is presented by Hefty (1998a). The results for permanent GPS stations are consistent with ITRF96 Boucher et al. (1998), estimates of velocities for 20 new epoch sites are obtained with accuracy 2mm/year. The differential velocity field in Central Europe estimated in Hefty (1998a) shows that there are slight horizontal movements in the region not exceeding few mm per year.

REFERENCES