C. Kotsakis

Estimation of the zero-height geopotential level W o LVD in a local vertical datum from inversion of co-located GPS, leveling and geoid heights: a case study in the Hellenic islands

The estimation of the zero-height geopotential level of a local vertical datum (LVD) is a key task towards the connection of isolated physical height frames and their unification into a common vertical reference system. Such an estimate resolves, in principle, the ‘ambiguity’ of a traditional crust-fixed LVD by linking it with a particular equipotential surface of Earth’s gravity field under the presence of an external geopotential model. The aim of this paper is to study the estimation scheme that can be followed for solving the aforementioned problem based on the joint inversion of co-located GPS and leveling heights in conjunction with a fixed Earth gravity field model. Several case studies with real data are also presented that provide, for the first time, precise estimates of the LVD offsets for a number of Hellenic islands across the Aegean and Ionian Sea.

Evaluation of GOCE/GRACE Global Geopotential Models over Greece with Collocated GPS/Levelling Observations and Local Gravity Data

The advent of the GOCE and GRACE missions during the last decade have brought new insights and promising results both in the static and time-variable representation of the Earth’s gravity field. The focus of this work is directed to the evaluation of most available Global Geopotential Models (GGMs) from GOCE and GRACE, both satellite only as well as combined ones. The evaluation is carried out over an extensive network of collocated GPS/Levelling benchmarks (BMs) which covers the entire part of continental Greece and with respect to the reductions the GGMs provide in existing gravity data in order to assess their performance in a scenario that a remove-compute-restore procedure would be followed for geoid determination. From the evaluation with GPS/Levelling BMs, it was concluded that the GOCE/GRACE GGMs provide an absolute accuracy at the 12–15 cm level, up to degree and order (d/o) 250, when considering the geoid omission error. This is comparable and in some cases better than the performance of EGM2008 in Greece. Moreover, the latest (Release 3) versions of the GGMs provide considerably better results compared to the earlier version by 1–5 cm. In terms of relative errors, GOCE/GRACE GGMs reach the 1 cm level for baselines between 50 and 60 km, while for longer ones, 80–90 km, their performance is analogous to the local geoid model and the ultra-high degree combined GGMs. Finally, GOCE/GRACE GGMs manage to provide the same, as EGM2008, level of reduction to the local gravity anomalies, with a std at the 26.7–27.8 mGal level, when evaluated up to d/o 250.

On the importance of intra-frame and inter-frame covariances in frame transformation theory

The coordinate frame transformation (CFT) problem in geodesy is typically solved by a stepwise approach which entails both inverse and forward treatment of the available data. The unknown transformation parameters are first estimated on the basis of common points given in both frames, and subsequently they are used for transforming the coordinates of other (new) points from their initial frame to the desired target frame. Such an approach, despite its rational reasoning, does not provide the optimal accuracy for the transformed coordinates as it overlooks the stochastic correlation (which often exists) between the common and the new points in the initial frame. In this paper we present a single-step least squares approach for the rigorous solution of the CFT problem that takes into account both the intra-frame and inter-frame coordinate covariances in the available data. The optimal estimators for the transformed coordinates are derived in closed form and they involve appropriate corrections to the standard estimators of the stepwise approach. Their practical significance is evaluated through numerical experiments with the 3D Helmert transformation model and real coordinate sets obtained from weekly combined solutions of the EUREF Permanent Network. Our results show that the difference between the standard approach and the optimal approach can become significant since the magnitude of the aforementioned corrections remains well above the statistical accuracy of the transformation results that are obtained by the standard (stepwise) solution.

Reference frame stability and nonlinear distortion in minimum-constrained network adjustment

The aim of this paper is to investigate the influence of the minimum constraints (MCs) on the reference frame parameters in a free-net solution. The non-estimable part of these parameters (which is not reduced by the available data) is analysed in terms of its stability under a numerical perturbation of the constrained datum functionals. In practice, such a perturbation can be ascribed either to hidden errors in the known coordinates/velocities that participate in the MCs or to a simple change of their a priori values due to a datum switch on a different fiducial dataset. In addition, a perturbation of this type may cause a nonlinear variation to the estimable part of the reference frame parameters, since it theoretically affects the adjusted observations that are implied by the network’s nonlinear observational model. The aforementioned effects have an impact on the quality of a terrestrial reference frame (TRF) that is established via a minimum-constrained adjustment, and our study shows that they are both controlled through a characteristic matrix which is inherently linked to the MC system (the so-called TRF stability matrix). The structure of this matrix depends on the network’s spatial configuration and the ‘geometry’ of the datum constraints, while its main role is the filtering of any MC-related errors into the least-squares adjustment results. A number of examples with different types of geodetic networks are also presented to demonstrate the theoretical findings of our study.

Estimation of the Reference Geopotential Value for the Local Vertical Datum of Continental Greece Using EGM08 and GPS/Leveling Data

Estimation of the zero-height geopotential level represented by W o LVD in a local vertical datum (LVD) is a problem of main importance for a wide range of geodetic applications related to different height frames and plays a fundamental role in the connection of traditional height reference systems into a global height system or even a modern geoid-based vertical datum. This paper aims primarily at the estimation of W o LVD for the continental part of Greece, with the use of surface gravity data and geopotential values computed from EGM08 in conjunction with GPS and orthometric heights over an extensive network which covers sufficiently the test area. The method used focuses on the estimation of W o LVD from a least squares adjustment scheme that is applied on the Helmert model for orthometric heights, using surface geopotential and gravity values (as obtained from EGM08 and the known 3D geocentric coordinates of each benchmark) along with the local Helmert heights over all network stations. Moreover, an attempt is made towards the modeling and removal of any height correlated errors in the available data according to this adjustment procedure. Different weighting schemes are tested, and, finally, some conclusions are drawn considering the accuracy of the obtained results.

Estimating Crustal Deformation Parameters from Geodetic Data: Review of Existing Methodologies, Open Problems and New Challenges

The study of crustal deformation using various types of geodetic data is a research topic whose practical importance needs hardly to be stressed, and its theoretical richness encompasses several scientific disciplines, including estimation theory, differential geometry, elasticity theory, geodynamics and physics. In this paper, an attempt is made to summarize the existing methodologies that are commonly applied in the geodetic practice for crustal deformation studies. Special emphasis is given on issues such as: (i) the definition and the estimability of frame-invariant quantities in timedependent geodetic networks, (ii) the separation of rigid motion effects from actual body deformation changes, (iii) the problem of spatial and/or temporal interpolation of the crustal deformation field, and (iv) the separation of the total deformation field into a “horizontal” part and a “vertical” part. An assessment of the remaining open problems that exist within the currently used geodetic methodologies for crustal deformation analysis is also given, and finally a number of new challenges that are imposed by the availability of data types which are essentially continuous in space and/or time, is listed.

Generalized inner constraints for geodetic network densification problems

The estimated coordinates from a minimum-constrained (MC) network adjustment are generally influenced by two different error sources, that is the data noise from the available measurements and the so-called datum noise due to random errors in the fiducial coordinates that are used for the datum definition with regard to an external reference frame. Although the latter source does not affect the estimable characteristics of a MC solution, it still contributes a datum-related noise to the estimated positions which reflects the uncertainty of the coordinate system itself for the adjusted network. The aim of this paper is to develop a new type of MCs which minimizes both of the aforementioned effects in the final coordinates of an adjusted network. This particular problem has not been treated in the geodetic literature and the solution which is presented herein offers an elegant unification of the classic inner constraints into a more general framework for the datum choice problem of network optimization theory. Furthermore, the findings of our study provide a useful and rigorous tool for frame densification problems by means of an optimal MC adjustment in geodetic networks.

Evaluation of EGM08 Using GPS and Leveling Heights in Greece

This paper presents an overview of the evaluation results for the new Earth Gravitational Model (EGM08) that was recently released by the US National Geospatial- Intelligence Agency, using GPS and leveled orthometric heights in the area of Greece. Various comparisons of geoid undulations obtained from the EGM08 model and other combined geopotential models, in conjunction with GPS/leveling data, have been performed in both absolute (at individual points) and relative (for baselines of varying length) sense. The test network covers the entire part of the Greek mainland and it consists of more than 1,500 ben- chmarks that belong to the Hellenic national triangulation network, with direct leveling ties to the Hellenic vertical reference frame. The spatial positions of these benchmarks have been recently determined at cm-level accuracy (with respect to ITRF2000) through an extensive national GPS campaign that was organized in the frame of the HEPOS project. Our results suggest that EGM08 offers a major improvement (more than 50%) in the agreement level among geoidal, ellipsoidal and orthometric heights over the mainland part of Greece, compared to the performance of previous global geopotential models for the same area.

A Study on the Effects of Data Accuracy and Datum Inconsistencies on Relative GPS Levelling

Numerical investigations into the effects of data accuracy and datum inconsistencies on relative GPS levelling are presented. Specifically, the variance/covariance information of (i) relative GPS ellipsoidal heights, (ii) geoid heights computed from a gravimetric geoid model, and (iii) orthometric heights obtained from spirit levelling methods, is used for an accuracy analysis in a combined 1D multi-data test network of GPS levelling benchmarks. Different parametric models are employed to describe the datum inconsistencies and systematic distortions inherent among the various height data sets. The a-posteriori accuracy of the adjusted parameters in the corrector surface model, along with the internal accuracy of the GPS and geoid heights, are finally used to infer the achievable accuracy of GPS levelling on baselines within the test network area.

Estimation of Variance Components Through a Combined Adjustment of GPS, Geoid and Levelling Data

The method of GPS-levelling for obtaining orthometric heights is not a new concept. In fact, many studies have proven its usefulness and the question of whether GPS-levelling can provide a viable alternative to traditional techniques is no longer an issue. An important question, however, that has yet to be satisfactorily solved is, ‘What accuracy level can be achieved using this approach?’ Over the past decade, numerous advances have been made which have placed us in a position where we can begin to address the issue with more confidence, namely (i) improved mathematical models/techniques for dealing with GPS and geoid data, (ii) increased data availability for gravimetric geoid models, and (iii) improved data processing capabilities. In this paper a statistical approach for estimating the variance components of heterogeneous groups of observations is used in the combined adjustment of GPS, geoid and levelling data. Specifically, the iterative minimum norm quadratic unbiased estimation algorithm is employed to determine the individual variance components for each of the three height types. The challenges encountered when implementing this well-known algorithm in practice with real data are discussed. The analysis provides some indication into the practicality and effectiveness of estimating variance components in mixed vertical networks. Notably, the estimation of realistic variance components provides us with important insight regarding the GPS-levelling problem in addition to other uses of combined GPS, geoid and levelling data, such as assessing the accuracy of a gravimetric geoid model.