The earth's gravity field G * at a point P in the region surrounding the earth's surface is defined as the force acting on a unit mass concentrated at P. This is a force resulting from two components: (1) G1 due to the gravitational attraction of the earth's mass, and (2) G2 due to the earth's rotation.

The determination of the figure of the Earth and its gravity field has long relied on methodologies that approximate the Earth by a sphere, but this level of accuracy is no longer adequate for many applications, due to the advent of new and advanced measurement techniques. New, practical and highly accurate methodologies for gravity field modelling that describe the Earth as an oblate ellipsoid of revolution are therefore required. The foundation for these methodologies is formed by solutions to ellipsoidal geodetic boundary-value problems. In this thesis, new solutions to the ellipsoidal Dirichlet, Neumann and second-order boundary-value problems, as well as the fixed- and free-geodetic boundary-value problems, are derived. These solutions do not rely on any spherical approximation, but are nevertheless completely based on a simple spherical harmonic expansion of the function that is to be determined. They rely on new relations among spherical harmonic base functions. In the new solutions, solid spherical harmonic coefficients of the desired function are expressed as a weighted summation over surface spherical harmonic coefficients of the data on the ellipsoidal boundary, or alternatively as a weighted summation over coefficients that are computed under the approximation that the boundary is a sphere. / Specific applications of the new solutions are the computation of geopotential coefficients from terrestrial gravimetric data and local or regional gravimetric geoid determination. Numerical closed-loop simulations have shown that the accuracy of geopotential coefficients obtained with the new methods is significantly higher than the accuracy of existing methods that use the spherical harmonic framework. The ellipsoidal corrections to a Stokesian geoid determination computed from the new solutions show strong agreement with existing solutions. In addition, the importance of the choice of the reference sphere radius in Stokes's formula and its effect on the magnitude and spectral sensitivity of the ellipsoidal corrections are pointed out.

Gravity Recovery and Climate Experiment (GRACE) mission has amplified the knowledge of both static and time-variable part of the Earth’s gravity field. Currently, GRACE maps the Earth’s gravity field with a near-global coverage and over a five year period, which makes it possible to apply statistical analysis techniques to the data. The objective of this study is to analyse the most dominant spatial and temporal variability of the Earth’s gravity field observed by GRACE using a combination of analytical and statistical methods such as Harmonic Analysis (HA) and Principal Component Analysis (PCA). The HA is used to gain general information of the variability whereas the PCA is used to find the most dominant spatial and temporal variability components without having to introduce any presetting. The latter is an important property that allows for the detection of anomalous or a-periodic behaviour that will be useful for the study of various geophysical processes such as the effect from earthquakes. The analyses are performed for the whole globe as well as for the regional areas of: Sumatra- Andaman, Australia, Africa, Antarctica, South America, Arctic, Greenland, South Asia, North America and Central Europe. On a global scale the most dominant temporal variation is an annual signal followed by a linear trend. Similar results mostly associated to changing land hydrology and/or snow cover are obtained for most regional areas except over the Arctic and Antarctic where the secular trend is the prevailing temporal variability. / Apart from these well-known signals, this contribution also demonstrates that the PCA is able to reveal longer periodic and a-periodic signal. A prominent example for the latter is the gravity signal of the Sumatra-Andaman earthquake in late 2004. In an attempt to isolate these signals, linear trend and annual signal are removed from the original data and the PCA is once again applied to the reduced data. For a complete overview of these results the most dominant PCA modes for the global and regional gravity field solutions are presented and discussed.

Orbit determination has long relied on the use of the Kalman filter, or specifically the extended Kalman filter, as a means of accurately navigating spacecraft. With the advent of cheaper, more powerful computers more accurate techniques such as the particle filter have been utilized. These Bayesian types of filters have in more recent years found their way to other applications. Dr. Furfaro and B. Gaudet have demonstrated the ability of the particle filter to accurately estimate the angular velocity, homogenous density, and rotation angle of a non-uniformly rotating ellipsoid shaped asteroid. This paper extends that work by utilizing a particle filter to accurately estimate the angular velocity and homogenous density of an ellipsoidal asteroid while simultaneously determining the location and mass of a mass concentration modeled as a point mass embedded within the asteroid. This work shows that by taking measurements in several locations around the asteroid, the asteroid's rotation state and mass distribution can be discerned.

Geodesy deals with the accurate analysis of spatial and temporal variations in the geometry and physics of the Earth at local and global scales. In geodesy, least-squares collocation (LSC) is a bridge between the physical and statistical understanding of different functionals of the gravitational field of the Earth. This thesis specifically focuses on the [incorrect] implicit LSC assumptions of isotropy and homogeneity that create limitations on the application of LSC in non-stationary gravity field modeling. In particular, the work seeks to derive expressions for local and global analytical covariance functions that account for the anisotropy and heterogeneity of the Earth's gravity field. / Standard LSC assumes 2D stationarity and 3D isotropy, and relies on a covariance function to account for spatial dependence in the observed data. However, the assumption that the spatial dependence is constant throughout the region of interest may sometimes be violated. Assuming a stationary covariance structure can result in over-smoothing, e.g., of the gravity field in mountains and under-smoothing in great plains. The kernel convolution method from spatial statistics is introduced for non-stationary covariance structures, and its advantage in dealing with non-stationarity in geodetic data is demonstrated. / Tests of the new non-stationary solutions were performed over the Darling Fault, Western Australia, where the anomalous gravity field is anisotropic and non-stationary. Stationary and non-stationary covariance functions are compared in 2D LSC to the empirical example of gravity anomaly interpolation. The results with non-stationary covariance functions are better than standard LSC in terms of formal errors and cross-validation. Both non-stationarity of mean and covariance are considered in planar geoid determination by LSC to test how differently non-stationarity of mean and covariance affects the LSC result compared with GPS-levelling points in this area. Non-stationarity of the mean was not very considerable in this case, but non-stationary covariances were very effective when optimising the gravimetric quasigeoid to agree with the geometric quasigeoid. / In addition, the importance of the choice of the parameters of the non-stationary covariance functions within a Bayesian framework and the improvement of the new method for different functionals on the globe are pointed out.

Title: Forward and Inverse Modeling of Planetary Gravity and Topography Author: Martin Pauer Department/Institute: Department of Geophysics MFF UK Supervisor of the doctoral thesis: Doc. RNDr. Ondřej Čadek, CSc., Department of Geophysics MFF UK Abstract: The aim of this work was to investigate various mechanisms compensating the observed planetary topography - crustal isostasy, elastic support and dynamic support caused by mantle flow. The investigated models were applied to three different planetary problems. Firstly we applied dynamic compensation model to explain today large-scale gravity and topography fields of Venus and investigate its mantle viscosity structure. The results seem to support not only models with constant viscosity structure but also a model with a stiff lithosphere and a gradual increase of viscosity toward a core. In the second paper several crust compensation models were employed to estimate the density of the Martian southern highlands crust. Since the used methods depends differently on crustal density changes, we were able to provide some constraints on the maximum density of the studied region. In the third application, the strength of a possible ocean floor gravity signal of Jupiter's moon Europa was studied. It turned out that if the long wavelength topography reaches height at...

One of the applications of the Satellite Laser Ranging (SLR) technique is the derivation of gravity field models; these models have various geophysical and geodynamical applications. Gravity field modelling has reached a new era where the latest satellite missions (CHAMP, GRACE and GOCE) are thought to provide significant improvement of global gravity field information in terms of quality and spatial resolution. In particular, the recent satellite missions carry on-board Global Navigation Satellite System (GNSS) receivers, accelerometers, K/Kaband microwave system (e.g. in GRACE) and gradiometers (e.g. in GOCE) allowing measurements of gravity field with unprecedented accuracy in contrast to the unsteady and fragmented orbit tracking by unevenly distributed SLR ground stations. Numerous gravity field models have been derived based on the newly available data sets by various research groups globally. Due to the availability of high quality SLR and satellite data, some of the older gravity field models are being updated as new models with higher degree and order are developed. Notwithstanding the significant progress in gravity field modelling, research focusing on assessing the accuracy and precision of the existing gravity field models has largely remained insufficient. The difference between the observed and computed satellite orbit (which is often expressed as the O-C range residuals) is used as a parameter for Precise Orbit Determination (POD) of satellites. Furthermore, O-C range residuals computed during SLR analysis are used as proxy parameters for evaluating the accuracy of gravity field models. The work presented in this thesis firstly reviewed and evaluated the accuracy of gravity field models released between 1990 and 2008. The accuracy of the gravity field models was examined by analysing the O-C residuals computed from LAGEOS 1 and 2 data analysis based on a set of twelve gravity field models. The results demonstrated that in general, there has been an improvement in the accuracy of gravity field models released between 1990 and 2008 by a factor of 2 based on improvements in the O-C residuals. Additionally, the influence of SLR tide parameterization (the IERS 2010 solid Earth and pole tide models) on the O-C residuals across five gravity field models has been assessed and results illustrate that the solid Earth and pole tides parameterization influence on the O-C residuals is dependent on the type of gravity field model. In order to ascertain the significance of mean differences in the Standard Deviations (SD) of O-C residuals based on the tide parameterization options, the student’s t-test was used. Results suggest that in general the O-C residuals derived from SLR LAGEOS 1 data have insignificant mean SD differences across the tide parameterizations. On the other hand analysis of SLR observations of LAGEOS 2 resulted in statistically significant mean SD differences in the O-C based on EIGEN-CG03C, EGM2008 and AIUB-GRACE01S gravity field models. The J2 coefficient forms part of the SLR Data Analysis Software (SDAS) package output products and was investigated in this thesis due to its role in understanding mass-redistribution within the Earth system (i.e. the equatorial bulge due to centrifugal force and rotation). In particular, the J 2 coefficient computed from SLR analysis of LAGEOS 1 and 2 data sets and based on the four selected gravity field models were compared with a priori J2 coefficients from the four models and those published in the literature. The results indicated that the J2 coefficients computed from the SDAS package were in agreement with the published coefficients. For geophysical applications, the relationship between the J2 parameter and LOD and AAM was investigated by use of data adaptive analysis methodology (the empirical mode decomposition). The results demonstrated that some degree of synchronization exists between the signal components of J2 and LOD and J2 and AAM. / Thesis (PhD)--University of Pretoria, 2013. / Geography, Geoinformatics and Meteorology / Unrestricted

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