In this work the mathematical and probabilistic background of standard linear estimation techniques used in geodesy is clarified, and their interrelationship is revealed with the help of best approximation theory and the normal equations. Emphasis is given to the separation of the deterministic solution to the approximation problem from the probabilistic justification of the metric of the approximation. Least squares prediction has been related to deterministic (exact) collocation, and minimum error bound has been identified as a prediction optimality criterion in the latter. Criteria for the optimal choice of norm in Hilbert space collocation are proposed for gravimetric geodesy problems.

A representation of the geodetic boundary value problem linearized with respect to the Somigliana-Pizzetti field is given in terms of ellipsoidal coordinates on the telluroid. In addition, the rigorous boundary condition for the Stokes problem is presented in terms of ellipsoidal, geodetic or spherical coordinates, on the reference ellipsoid.

The problem of the minimization of the cost for a given trilateration network design by optimizing the planning and execution of fieldwork is studied with the help of graph theory tools. The case with one instrument and one reflector is shown to be a problem in graph theory which has some of the characteristics of the travelling salesman problem but is much more complicated. Since the determination of the optimal solution requires enormous computational time, a heuristic algorithm which leads to a sub-optimal solution is developed, along the lines of a similar heuristic algorithm for the much simpler travelling salesman problem. The algorithm is elaborated with the help of a computer program written for this purpose. Finally the problems where a) instrument and reflector have the same transportation cost, b) each distance is observed twice, c) instrument and reflector have a common means of transportation and d) observations extend over more than one day with returns to home-base, are shortly discussed in the light of graph theory.

The differential geometric characteristics of the gravity potential of the earth are utilized for the derivation of rigorous boundary conditions for three types of geodetic boundary value problems: the vectorial free, the scalar free and the fixed problem. The conditions are first derived for any type of normal field and are further simplified with the consideration of rotationally symmetric ones. For the free problems the conditions are given on the telluroid (Molodensky’s problem) and on the reference ellipsoid (Stokes’ problem). All boundary conditions are analytically expressed for three coordinate systems: geodetic, spherical and ellipsoidal. General expressions are also given for the second order terms of the residuals neglected in the linearization.

A simple comparison of the Kriging estimation techniques with the method of collocation widely used in geodesy is carried out. Similarities and differences between the two methods are pointed out and explained on the basis of their corresponding underlying stochastic assumptions.

Die Verwendung geodätischer Daten für Studien der Krustendeformation führt zu zwei möglichen Fällen: (a) dem Vergleich der Ausprägung zweier Epochen, für die Stationskoordinaten verfügbar sind, um in jeder gewünschten Genauigkeit invariante planare Deformationsparameter zu berechnen, (b) der Nutzanwendung von Koordinaten und Geschwindigkeiten, um während einer bestimmten Epoche Zeitableitungen der Deformationsparameter zu bestimmen. Der vorliegende Beitrag widmet sich der klassischen approximativen ,,infinitesimalen“ Theorie ebenso wie der weit verbreiteten Finite-Element-Methode mittels triangulärer Elemente zur Interpolation von Stationsverschiebungen und Geschwindigkeiten. Eine strenge Theorie der Erweiterung planarer Deformation auf den dreidimensionalen Fall wird aufgewiesen.

After a short introduction on the basics of reference system theory and its application for the description of earth rotation, the problem of establishing a reference system for the discrete stations of a geodetic network is studied, from both a theoretical and a practical – implementation point of view.First the case of rigid networks is examined, which covers also the case of deformable networks with data collected within a time span, small enough for the network shape to remain practically unaltered. The problem of how to analyze observations, which are invariant under particular changes of the reference system, is examined within the framework of least squares estimation theory, with a rank deficiency in the design matrix. The complete theory is presented, including all necessary proofs. Not only of the usual statistical results for the rank deficient linear Gauss-Markov model, but also those of the rich geodetic theory are presented, based on the fact that the physical cause of the rank deficiency is known to be the lack of definition of the reference system. The additional geodetic results are based on the fact that one can easily construct a matrix with columns that are a basis of the null space of the design matrix. Insights are presented into the geometric characteristics of the problem and its relation to the theory of generalized inverses. Passing into deformable networks, a deterministic mathematical model is presented, based of the concept of geodesic lines which are the shortest between linear shape manifolds, associated with the network shape at each instant. Reference system optimality for a discrete network is related to the relevant ideas of Tisserand, developed for the continuum of the earth masses.The practical problem of choosing a reference system for a coordinate time series is examined, for the case where a linear-in-time model is adopted for the temporal variation of coordinates. The choice of reference system is related to the choice of minimal constraints for obtaining one out of the infinitely many least squares solutions, corresponding to descriptions in different reference systems of the same sequence of network shapes. The a-posteriori change of the reference system is examined, where one moves from one least squares solution to another one, satisfying particular minimal constraints. Kinematic minimal constraints are also introduced, leading to coordinates that demonstrate the minimum coordinate variation and are thus connected to the ideas of Tisserand for reference system optimality. It is also shown how to convert a reference system of a geodetic network to one for the whole continuous earth, or at least the lithosphere, utilizing additional geophysical information.The last item is the combination of data from four space techniques (VLBI, SLR, GPS, DORIS) in order to establish a global reference system realized though a number of parameters that constitute the International Terrestrial Reference Frame. After a theoretical exposition of the basics of data combination, the various methods of spatial data combination are presented, for both coordinate and Earth Orientation Parameter time series, while alternatives are presented for the choice of the origin (geocenter) and the network scale from the scale of VLBI and SLR. Finally, existing and new methodologies are presented for building post linear models, describing the temporal variation of station coordinates.

A new approach to the determination of least-squares solutions for the rank deficient model is presented, which, in place of the classical minimal constraints, utilizes the expression of the excess parameters as a function of a subset of unknown parameters describing the model without ambiguities. The role of the lack of reference system definition in the resulting rank deficiency is explained, first for the original non-linear model case, and is specialized next to the linearized model. The whole set of solutions is parameterized in terms of two matrices defining the linearized relation from the model describing parameters to the remaining ones. Particular values are obtained for the case of solution with minimal trace of its covariance matrix, as well for the solution for minimum norm. Finally, the connection with the existing classical approach is established, while the approach is further elaborated in terms of the full-rank decompositions of the model design matrix.

The aim of this research was to investigate the feasibility of merging the benefits offered by low-cost GNSS and MEMS accelerometers technology, in order to promote the diffusion of low-cost monitoring solutions.A merging approach was set up at the level of the combination of kinematic results (velocities and displacements) coming from the two kinds of sensors, whose observations were separately processed, following to the so called loose integration, which sounds much more simple and flexible thinking about the possibility of an easy change of the combined sensors.At first, the issues related to the difference in reference systems, time systems and measurement rate and epochs for the two sensors were faced with. An approach was designed and tested to transform into unique reference and time systems the outcomes from GPS and MEMS and to interpolate the usually (much) more dense MEMS observation to common (GPS) epochs. The proposed approach was limited to time-independent (constant) orientation of the MEMS reference system with respect to the GPS one.Then, a data fusion approach based on the use of Discrete Fourier Transform and cubic splines interpolation was proposed both for velocities and displacements: MEMS and GPS derived solutions are firstly separated by a rectangular filter in spectral domain, and secondly back-transformed and combined through a cubic spline interpolation.Accuracies around 5 mm for slow and fast displacements and better than 2 mm/s for velocities were assessed. The obtained solution paves the way to a powerful and appealing use of low-cost single frequency GNSS receivers and MEMS accelerometers for structural and ground monitoring applications.Some additional remarks and prospects for future investigations complete the paper. (C) 2017 COSPAR. Published by Elsevier Ltd. All rights reserved.

New tools in the form of angular deficiency indices are developed for the physical interpretation of numerically detected rank deficiencies or weaknesses, in terms of defects or weaknesses in the definition of the reference system characteristics (origin, orientation and scale) to which coordinates and other unknown parameters refer. Within a least squares estimation procedure, the numerical detection is based on the subspace of the parameter space spanned by eigenvectors of the coefficient matrix of the normal equations, which correspond to zero or very small eigenvalues. The physical interpretation is based on the well-known theoretically derived basis of the null space of the design matrix, provided by the columns of the matrix appearing in the so-called inner constraints. This null space basis is elaborated for single-epoch (weekly) solutions of GNSS networks, for deformable networks with constant station velocities, as well as for the stacking problem of coordinate time series. The applicability of these new tools and their implementation strategy are demonstrated by simple numerical examples.

In this review paper theoretical aspects of global reference systems are critically discussed in relation to their practical implementation through reference frames. These include the problem of themathematical modeling of a spatiotemporal reference system for the deforming Earth, the relation of geodetic discrete-network reference systems to geophysical ones for the Earth mass continuum, the contribution of the various geodetic space techniques, the estimation issues related to the combination of the various data types, and issues relating to the compatibility of earth rotation representation. Finally issues related to future development of the International Terrestrial Reference Frame are discussed, concerning the addition of non-linear quasi-periodic terms in coordinate variation and their proper geophysical interpretation.

By considering a deformable geodetic network, deforming in a linear-in-time mode, according to a coordinate-invariant model, it becomes possible to get an insight into the rank deficiency of the stacking procedure, which is the standard method for estimating initial station coordinates and constant velocities, from coordinate time series. Comparing any two out of the infinitely many least squares estimates of stacking unknowns (initial station coordinates, velocity components and transformation parameters for the reference system in each data epoch), it is proven that the two solutions differ only by a linear-in-time trend in the transformation parameters. These pass over to the initial coordinates (the constant term) and to the velocity estimates (the time coefficient part). While the difference in initial coordinates is equivalent to a change of the reference system at the initial epoch, the differences in velocity components do not comply with those predicted by the same change of reference system for all epochs. Consequently, the different velocity component estimates, obtained by introducing different sets of minimal constraints, correspond to physically different station velocities, which are therefore non-estimable quantities. The theoretical findings are numerically verified for a global, a regional and a local network, by obtaining solutions based on four different types of minimal constraints, three usual algebraic ones (inner or partial inner) and the lately introduced kinematic constraints. Finally, by resorting to the basic ideas of Felix Tisserand, it is explained why the station velocities are non-estimable quantities in a very natural way. The problem of the optimal choice of minimal constraints and, hence, of the corresponding spatio-temporal reference system is shortly discussed.

Algorithms for the determination of the transformation parameters from one reference system to the other are presented for the case where the data points lie on the same surface or curve, while there exist no common data points. Iterative solutions are proposed based on localized surface or curve interpolation and least squares best fitting. Three application examples are presented. The first is the matching of two overlapping digital terrain models where horizontally gridded height data are referring to two different horizontal and vertical reference systems. The second is the matching of two point clouds with coordinates obtained from two independent scannings of overlapping surfaces, which refer to different three-dimensional reference systems. The last example is the matching of velocities derived from GNSS and accelerometers, with different reference and time systems, where the data are velocity component time series, i.e. points on a three-dimensional timedependent curve.

Coodinate time series from a regional GNSS network in Greece covering a period of 5 years are used in order to study the effect of the choice of the reference system on their linear, nonlinear and spectral characteristics. The standard solution where the reference system is defined by removing the translational rank defect in GNSS data by partial inner constraints is compared with two different solutions. The first is obtained by a posteriori aligning the network at each epoch to the IGS08 reference system through the coordinates of common points by means of a similarity (Helmert) transformation. The second solution is achieved by a regional stacking where the original standard coordinate time series are best fit to a linear in time coordinate model and the derived similarity transformation parameters are used to convert the standard solution into coordinate time series where variations due to reference system instability are removed. The analysis shows that the stacking solution leads to better results more suitable for regional geodynamic studies free from effects, which are not reflecting actual temporal variations in the shape of the network.

We will also compare two different traditions where the linear model is applied: The first is the classical one of applications in physical sciences, in particular astronomy and geodesy. It starts with the least squares method of Gauss and Legendre and it culminates in the statistical justification of the choice of weight matrix as a scalar multiple of the inverse of the error covariance matrix. According to the celebrated Gauss-Markov theorem with this choice the least squares solution provides Best Linear Uniformly Unbiased Estimates (BLUUE or simply BLUE) for all linear functions of the parameters, which are estimable, i.e. functions of the observables only.

Ideally, the origin of the Terrestrial Reference Frame (TRF) is defined as the center of mass of the whole Earth system, the time evolution of its orientation is such that no global net rotation of the whole Earth’s surface is possible and the TRF scale is specified through the adoption of some physical constants and time-scale. These parameters need to be accurately determined since their choice has an influence on many Earth’s science applications. The aim of the task force “External evaluation of the Terrestrial Reference Frame” is to review all the applications for which the TRF accuracy is of fundamental importance. As the TRF choice has an influence on the interpretation of the results in these specific applications, we investigate if some evaluation procedures could be established. We classified the methods that allow evaluation of the TRF using ground, geodetic data or models that have not been used in the TRF construction, based on their expected contributions. Some of these methods have been applied to the latest International Terrestrial Reference System realizations and the results are presented here. Although further analysis will be necessary to deliver a more precise error budget, our findings demonstrate that the most recent realizations of the ITRS are more accurate than the previous in terms of origin and scale rate definition. The current level of ITRF2008 accuracy is likely to be at the level of 0.5 mm/year along each origin component and better than 0.3 mm/year in the scale rate according to the most recent studies.

The one-step approach for ITRF formulation is compared with the two-step technique where the ITRF parameters are first separately estimated for each space technique separately (stacking per technique) and are then combined into the final ITRF estimates (combination step). The comparison is achieved by splitting the one-step approach into two equivalent steps such that the first one is identical to the first step of the two-step approach. It is shown that the two approaches give equivalent results when the input covariance matrices in the combination step of the two-step approach are the normal equation coefficient matrices formulated in the corresponding stacking per technique first step. Furthermore it is shown that the model of the combination step can be significantly simplified by ignoring the parameters of reference system transformation from the per technique estimates to those of the final ITRF parameter estimates.

The problem of choosing an optimal reference system for the International Terrestrial Reference Frame (ITRF) is studied for both the rigorous solution which is a simultaneous stacking (removal of the reference system at each data epoch and implementation of a linear in time coordinate model) for all techniques, as well as for the usual numerically convenient separation into a set of individual stackings one for each technique and a final combination step for the derived initial coordinates and velocities. Two approaches are followed, an algebraic and a kinematic one. The algebraic approach implements the inner constraints, which minimize the sum of squares of the unknown parameters, as well as partial inner constraints, which minimize the sum of squares of a subset of the unknown parameters. In the kinematical approach the optimal minimal constraints are derived by requiring the minimization of the apparent coordinate variations: (a) with respect to the origin by imposing constant coordinates for the network barycenter, (b) with respect to orientation by imposing zero relative angular momentum for the network points conceived as mass points with equal mass and (c) with respect to the scale by imposing constant mean quadratic size (involving the distances of stations from their barycenter).