Modification of the least-squares collocation method for non-stationary gravity field modelling
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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.
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Darbeheshti, Neda; Featherstone, Will (2009)Standard least-squares collocation (LSC) assumes 2D stationarity and 3D isotropy, and relies on a covariance function to account for spatial dependence in the ob-served data. However, the assumption that the spatial ...
Darbeheshti, Neda; Featherstone, Will (2010)This paper reviews a field that is herein termed spatial ?non-stationarity?, which is specifically concerned with non-stationarity in the geodetic theory of least-squares collocation (LSC). In practice, many geodesists ...
Carli, F.; Ferrante, Augusto; Pavon, M.; Picci, G. (2011)Stationary reciprocal processes defined on a finite interval of the integer line can be seen as a special class of Markov random fields restricted to one dimension. Nonstationary reciprocal processes have been extensively ...