Athanasios Dermanis

The Choice of Reference System in ITRF Formulation

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).

Establishing Global Reference Frames. Nonlinear, Temporal, Geophysical and Stochastic Aspects

The problem of the optimal definition of a Global Reference Frame based on a geodetic network of continuously observing stations is analyzed from various points of view. The non-linear datum definition problem is extended to the time domain and the optimal reference frame for a de-formable network is obtained as a geodesic line on a curved manifold in the vector space of all network coordinates, which is the union of instantaneous shape manifolds. The original ideas of Meissl are extended from the linear to the non-linear case and from the space to the space-time domain. The resulting definition of a Meissl Reference Frame is shown to be a geodesic frame as well as a Tisser-and-type frame. The difference between the operational Meissl-Tisserand geodetic network frame and the theoretical geophysical Tisserand earth frame is emphasized and it is shown how a connection between the two can be established by incorporating geophysical hypotheses, such as plate tectonics. Finally the stochastic problem of the optimal combination of estimated network frames is examined from both a non-linear and an approximate linearized solution point of view.

Data analysis methods in geodesy

“Geodesy” is a term coined by the Greeks in order to replace the original term “geometry”, which had meanwhile lost its original meaning of “earth or land measuring” (surveying) and acquired the new meaning of an abstract “theory of shapes”. Aristotle tells us in his “Metaphysics” that the two terms differ only in this respect: “Geodesy refers to things that can be sensed, while geometry to things that they cannot”. Many centuries afterwards the word geodesy was set in use anew, to denote the determination of the shape of initially parts of the earth surface and eventually, with the advent of space methods, the shape of the whole earth. Thus it remained an applied science, while facing at the same time significant and challenging theoretical problems, in both physical modeling and data analysis methodology.

The role of frame definitions in the geodetic determination of crustal deformation parameters

A study of the role of coordinate frame definitions in the determination of crustal deformation parameters is first carried out for the theoretical case where displacement information between two discrete time epochs is continuously available for all area points. The obtained results are next applied to the realistic case where the required continuous information is derived by means of an interpolation of the known coordinate variations at the points of a horizontal geodetic network.The problem is different from the usual one of frame-invariant interpolation, since not only the domain of definition, but also the interpolated quantities, depend on independent choices of coordinate frames.Specific necessary and sufficient conditions for the invariance of derived crustal deformation parameters are given for linear type of interpolations of either the coordinates at the second epoch or of the displacements.With the help of the above conditions the invariance characteristics of two commonly used types of linear interpolations are finally derived, in order to illustrate the practical significance of the 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.

The rank deficiency in estimation theory and the definition of reference systems

The concept of reference frame is examined from the viewpoint of both geophysical and geodetic applications. The concept of parameter estimability in linear models is related to the deterministic concept of determinability in linear or nonlinear improper models without full rank. The geometry of such models is investigated in its linear and nonlinear aspects with emphasis on the common invariance characteristics of observable and estimable parameters and is applied to the choice of datum problem in geodetic networks. The time evolution of the reference frame is investigated and optimal choices are presented from different equivalent points of view. The transformation of a global geodetic network into an estimate of a geocentric Tisserand frame for the whole earth is investigated and a solution is given for the rotational part. The translation to a geocentric frame poses the problem of the estimability of the geocenter coordinates and the more general problem of estimability of coefficients of an unknown function of position, having as domain the frame-dependent coordinates.

The ITRF Beyond the “Linear” Model. Choices and Challenges

The current solution to the choice of a reference system for the coordinates of a global geodetic network is based on a linear model for the time evolution of station coordinates. The advantages and disadvantages between a mathematical approach and a physical approach to the optimal definition of a reference system for the International Terrestrial Reference Frame (ITRF) are examined. The optimality conditions are derived for a general class of models, consisting of linear combinations of a system of base functions which is closed under differentiation and multiplication. The general results are then applied and elaborated for polynomial and Fourier Series models. Finally the problem of how these conditions should be implemented in practice is investigated.

A coordinate-invariant model for deforming geodetic networks: understanding rank deficiencies, non-estimability of parameters, and the effect of the choice of minimal constraints

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.

A Study on the Impact of Reference Frame Implementation Strategy on GNSS Time Series for Regional Network Analysis

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.

The Treatment of Time Continuous GPS Observations for the Determination of Regional Deformation Parameters

The calculation of invariant deformation parameters, entering in the constitutional equations of crustal dynamics, requires information on the geometry of the crust in the study region, which is continuous in both the time and spatial domain. The use of continuously observing GPS stations provides geodetic data which are practically timecontinuous and must be only spatially interpolated. A strategy is developed for the treatment of dense series of horizontal coordinates from a regional GPS network, which are typically exhibiting a timelinear behavior. The role of the choice of reference system is examined for the removal of trend before the spatial interpolation as well as the determination of the motion of the region as whole with respect to the ITRF or of the relative motion of tectonically homogeneous sub regions. Rigorous formulas are presented for various horizontal deformation parameters and their intrinsic time derivatives, without the usual infinitesimal approximations. Finally the problem of quality assessment for the derived parameters is investigated completely ignoring the questionable formal statistical characteristic of the original geodetic data. A realistic numerical example demonstrates the suggested techniques, involving spatial interpolation by the classical finiteelement method. A software package in standard C language has been developed in order to implement the proposed algorithms.