(Note: The following text is mainly taken from Plag, 2006). The ITRS is realized through a reference frame specifying a set of coordinates for a network of stations. These coordinates are given as Cartesian equatorial coordinates triples $x_i \equiv (X, Y, Z)$ by preference. The IERS Conventions suggest that if geographical coordinates are needed, the GRS80 ellipsoid should be used. Realizations of the ITRS are determined by the IERS in a nearly annual sequence and denoted as ITRF\_nn, where nn identifies the epoch of the frame \cite[see, for example][]{boucher++99}. These realizations are determined through combination of results from individual techniques. They are based on results provided by the different IERS analysis centers. The realization consists of lists of coordinates and velocities for a selection of IERS sites, which may be tracking stations or related ground markers. The station coordinates are expressed through \be \label{e-smm} x_i(t) = x_i^0 + v_i^0(t-t_0) + \sum_{j=1}^k \delta x_i^j(t), \; \; i=1,2,3 \ee where $x_i^0$ and $v_i^0$ are the position and velocity at epoch $t=t_0$ and $\delta x_i^k$ are corrections due to the $k$-th process inducing time variable contributions to the coordinates. Such processes are, for example, solid Earth tide displacements, ocean loading, atmospheric loading, and postglacial rebound. The first two terms in eq.~(\ref{e-smm}), that is \be \tilde{x}_i(t) = x_i^0 + v_i^0(t-t_0), \ee are denoted as regularized coordinates \cite[]{mccarthy+pet2003}. These regularized coordinates are listed in the ITRF publications. They depend on the selection of processes included in eq.~(\ref{e-smm}), that is the geophysical model used in the modeling of \be \Delta_i (t) = \sum_{j=1}^k \delta x_i^j (t). \ee The agreed upon processes to be included and the underlying geophysical models are specified in the IERS Conventions (for a critical discussion of the regularized coordinates, see Section \ref{ss-brief-itrs-itrf} on Page \pageref{ss-brief-itrs-itrf}). For different realizations of the ITRS, transformations are given to convert coordinates from one ITRF to another. The basic transformation formula is a seven parameter similarity transformation, often denoted as Helmert Transformation. This is given by \be \label{e-helmert-1} x'_i = s R_{ij}x_j + t_i, \; i=1,2,3 \ee where $x_i$ and $x'_i$ are the coordinate vectors of the point in the unprimed and primed frame, respectively, and $t_i$ is the vector describing the offset of the origin between the primed and unprimed system measured in scale units of the primed system. $s$ is the scale change of the primed frame with respect to the unprimed frame, and $R_{ij}$ is a rotation matrix \be R_{ij} = R^{(1)}_{ik}(\epsilon)R^{(2)}_{kl}(\psi)R^{(3)}_{lj}(\omega). \ee Here, the $(R^{(n)}_{ij}(\alpha))$ are rotation matrices describing a rotation around the $n$-th axis. For $j = n {\rm (modulo 3)} + 1; \; k= j {\rm (modulo 3)} + 1$, we have \begin{eqnarray} R_{nn} & = & 1 \\ \nonumber R_{nj} & = & R_{jn}=R_{nk} = R_{kn} = 0 \\ \nonumber R_{jj} & = & R_{kk} = \cos \alpha \\ \label{e-rotation-matrix} R_{jk} & = & \sin \alpha \\ \nonumber R_{kj} & = & -\sin \alpha \end{eqnarray} For infinitesimal rotations, (\ref{e-helmert-1}) can be written as \be x'_i = (1+\delta s) \tilde{R}_{ij} x_j + t_i \ee where $\delta s$ is the incremental scale change and with \be (\tilde{R}_{ij}) = \left(\begin{array}{ccc} 1 & \omega &-\psi \\ -\omega & 1 & \epsilon \\ \psi & -\epsilon & 1 \end{array} \right) \ee where $\epsilon, \psi$, and $\omega$ are given in radian. Then, \be \label{e-7-helmert} x'_i = x_j + \delta s \tilde{R}_{ij} x_j + t_i \ee which is the form given in the IERS Conventions \cite[]{mccarthy+pet2003}. For later versions of the ITRF (from 1993 onwards), not only the transformation parameters are give but also rates of changes of these parameters. In this case, for a given transformation parameter $q$ valid at the epoch $t_0$, its value at time $t$ is given by \be \label{e-14-helmert} q(t) = q(t_0) + \dot{q} (t-t_0). \ee The coordinate transformation described by equations~(\ref{e-7-helmert}) and (\ref{e-14-helmert}) are rigid and thus not able to account for any deformations of two reference frames. This is a significant deficiency of the transformation. With increasing time, the subsequent ITRFs are based on a growing number of techniques (including VLBI, SLR, GPS, DORIS, and, for ITRF2000, also GLONASS), improved global networks, improved information on the local ties between individual techniques at sites where techniques are collocated, improved analyses procedures for the single techniques, and improved methodologies for the combination of the single-technique reference results \cite[see][for a recent example]{boucher++99}. This leads inevitably to deformations of two subsequent reference frames and renders the transformations inaccurate. For many applications, it is necessary to compare coordinates of a point determined at different epochs or to refer coordinates to a reference epoch different from the central epoch of observations. Within the same reference frame, this can be achieved by \be \label{e-timeshift} x_i(t_r) = x_i(t_c) + (t_r-t_c) \cdot v_i \ee where $t_r$ and $t_c$ are the reference epoch and the central epoch of measurement, respectively, and $x_i$ and $v_i$ are the position and velocity vectors given in the relevant ITRF. If positions given for different epochs in different versions of the ITRF are to be compared, then equations (\ref{e-7-helmert} to (\ref{e-timeshift}) should be, in principle, sufficient. However, due to the deformation of the reference frames with respect to each other, these equations are not accurate and the error increases with time. It should be mentioned here, that some criticism has been articulated concerning appropriateness of the combination of results from single techniques in order to form the ITRF. It has been claimed that a combination on the observation level will lead to a more stable and more accurate realization of the ITRS \cite[]{andersen97,andersen2000}. However, the superiority of the combination at the observational level advocated, for example, by \cite[]{andersen97} has not been demonstrated up to now. In fact, the current combination (at the normal equation level), when done properly, should be equivalent to the observation level combinations. Today, the accuracy of the ITRF is estimated to be of the order of 10-20 mm in station coordinates. This accuracy refers to the regularized coordinates. In order to understand the effect of periodic point motion (see table~\ref{t-factors-displ} for an overview of the periodic motions) on coordinate accuracy, it is necessary to take into account the the definition of point coordinates given above. ITRF does not only consist of the coordinates and velocities given for the ITRF points but also of the IERS Conventions which give many details on what models are to be used when processing geodetic observations from, for example, VLBI, SLR, and GNSS in order of get ITRF coordinates. The conventions are to a very large extent followed by the IERS Analysis Centers contributing to the determination of ITRF. Violations of the conventions by a user of the ITRF will lead to non-ITRF or at least biased ITRF coordinates. It needs to be mentioned here that highly accurate geodetic coordinates determined with daily or longer datasets are free of period movements due to tides and polar motion: these are taken into account in the station motion model used in the different softwares to model the point motion over time. Thus, we expect time series of daily coordinates to be "tide-free" and to show basically linear trends and some long-period movement due to, for example, surface loading. This also implies that "accuracy in ITRF" refers to "tide-free" coordinates. Consequently, in order to provide reference coordinates that are of a certain accuracy in ITRF, these coordinates have to be as far as possible free of periodic movements. However, for a global frame, the secular motion has to be part of the temporal dependency of the coordinates. Considering higher temporal resolution than one day for geodetic GPS analyses, one may choose to leave the tides in the time series or one may still choose to model them. Nevertheless, when considering ``reference coordinates,'' it is implicitly assumed that these are ``tide-free.'' Moreover, for many national reference system such as the EUREF89 realizations in the different countries, the reference coordinates are not only "tide-free" but they are considered to be free of regional secular motion. In most national reference systems, the coordinates are kept fixed over long time intervals (hopefully several decades) as long as intra-plate deformation is very small (except for the vertical). -------------------------------------------------------------- \begin{figure} \vspace*{-1.8cm} \bc \includegraphics[width=13.2cm]{figures/figure2.3.eps} \ec \vspace{-9.0cm} \caption[The “three pillars of geodesy” and their techniques]{\label{f-geodesy-cont}The “three pillars of geodesy” and their techniques. Today, the space-geodetic techniques and dedicated satellite missions are crucial in the determination and monitoring of geokinematics, Earth's rotation and the gravity field. Together, these observations provide the basis to determine the geodetic reference frames with high accuracy, spatial resolution and temporal stability. From \cite{plag2006b}, modified from \cite{rummel2000}. For acronyms, see the list in Appendix~\ref{a-acronyms}.} \end{figure} \begin{table} \caption[The Global Geodetic Observing System (GGOS)]{\label{t-ggos-1}The Global Geodetic Observing System (GGOS). For acronyms, see the list in Appendix~\ref{a-acronyms}.} \bc \begin{tabular}{p{2.4cm}p{3.3cm}p{2.8cm}p{2.8cm}}\hline Component & Objective & Techniques & Responsible \\\hline\hline I. Geokinematics (size, shape, kinematics, deformation) & Shape and temporal variations of land/ice/ocean surface (plates, intra-plates, volcanos, earthquakes, glaciers, ocean variability, sea level) & Altimetry, InSAR, GNSS-cluster, VLBI, SLR, DORIS, imaging techniques, leveling, tide gauges & International and national projects, space missions, IGS, IAS, future InSAR service \\ \hline II. Earth Rotation (nutation, precession, polar motion, variations in LOD) & Integrated effect of changes in angular momentum and moment of inertia tensor (mass changes in atmosphere, cryosphere, oceans, solid Earth, core/mantle; momentum exchange between Earth system components) & Classical astronomy, VLBI, LLR, SLR, GNSS, DORIS, under development: terrestrial gyroscopes & International geodetic and astronomical community (IERS, IGS, IVS, ILRS, IDS) \\ \hline III. Gravity field & Geoid, Earth's static gravitational potential, temporal variations induced by solid Earth processes and mass transport in the global water cycle.& Terrestrial gravimetry (absolute and relative), airborne gravimetry, satellite orbits, dedicated satellite missions (CHAMP, GRACE, GOCE) & International geophysical and geodetic community (GGP, IGFS, IGeS, BGI) \\ \hline IV. Terrestrial Frame & Global cluster of fiducial point, determined at mm to cm level & VLBI, GNSS, SLR, LLR, DORIS, time keeping/transfer, absolute gravimetry, gravity recording & International geodetic community (IERS with support of IVS, ILRS, IGS, and IDS) \\ \hline \end{tabular} \ec \end{table} Today, the toolbox of \index{geodesy!toolbox of}geodesy comprises a number of space-geodetic and terrestrial techniques, which together allow for detailed observations of the “three pillars of geodesy” on a wide range of spatial and temporal scales (Figure~\ref{f-geodesy-cont}). With a mix of terrestrial, airborne, and spaceborne techniques, geodesy today determines and monitors changes in Earth's shape, gravitational field and rotation with unprecedented accuracy, resolution (temporal as well as spatial), and long-term stability (Table~\ref{t-ggos-1}). At the same time, geodetic observation technologies are in constant development with new technologies extending the observation capabilities almost continuously in terms of accuracy, spatial and temporal coverage and resolution, parameters observed, latency and quality. Together, these observations provide the basis to determine and monitor the ITRF and ICRF as the \index{Earth observations!metrological basis of}metrological basis for all Earth observations. Equally important, the observations themselves are directly related to mass transport and dynamics in the Earth system. Thus, the geodetic measurements form the basis for Earth system observations in the true meaning of these words. \cite{beutler++99b} suggested a development towards an interdisciplinary service in support of Earth sciences for the IGS. With the establishment of GGOS, IAG has extended this concept of an observing system and service for Earth system sciences to the whole of geodesy. From the discussion of the reference systems and frames in the previous section it is obvious that there is an intimate relationship between the three pillars of geodesy and the reference systems and frames (Figure \ref{f-geodesy-cont}). For geokinematics and Earth rotation, the relationship works both ways: The reference systems are required for positioning purposes (terrestrial and celestial) and for studying Earth rotation, and monitoring through the space geodetic techniques is necessary to realize the two frames and the (time-dependent) transformation between them. The ICRF, the ITRF, and the EOPs are needed to derive a gravity field, which is consistent with the ICRF, the ITRF, and the corresponding EOPs. Therefore, one might think at first that the gravity field is not necessary to define and realize the geometric reference systems. However, in order to realize the ITRF, observations made by the satellite geodetic techniques (SLR, GNSS, DORIS) are needed. For these techniques, a gravitational reference system and frame (including a gravity field representation and the parameters associated with it, and the geoid, the mean equipotential surface “near sea level”, which may be derived from the gravity field representation) is required as well and cannot be separately determined from the geometrical frames. The problems are obviously inseparable when dealing with the definition in the geometry and gravity domains (origin, orientation, scale of the geometric networks, low degree and order terms of the Earth's gravity field). This \index{consistency}consistency between geometric and gravitational products is important today, it will be of greatest relevance in the future for the understanding of the mass transport and the exchange of angular momentum between the Earth's constituents, in particular between solid Earth, atmosphere, and oceans. The aspect of consistency is also of greatest importance for all studies related to global change, sea level variation, and to the monitoring of ocean currents. Only if consistency on the \power{10}{-9} level or better between all reference frames it achieved, will it be possible to perform meaningful research in the areas mentioned. In the narrowest possible sense, geodesy has the \index{geodesy!tasks of}tasks to define the geometric and gravitational reference systems, and to establish the celestial, terrestrial, and gravitational reference frames. Moreover geodesy has to provide the transformation between the terrestrial and celestial reference frames. These key tasks would be relatively simple to accomplish on a rigid Earth without hydrosphere and atmosphere. However, in the real Earth environment already the definition of the terrestrial and gravitational reference systems is a challenge. The corresponding reference frames can only be established by permanent monitoring based on a polyhedron of terrestrial geodetic observing sites, and of space missions. This ambitious and expensive geodetic monitoring is necessary and its result, properly time-tagged and mutually consistent, is a stringent requirement in a broad field of scientific and societal applications. There is strong science justification for these geodetic products as a prerequisite (see Chapter~\ref{s-urscience}). Also, some tasks of societal relevance may only be addressed if this permanent geodetic monitoring is available (see Chapter~\ref{s-ursocietal}), and monitoring of the Earth system, including for example sea level and ice sheet variations, would not be possible without it (see Chapter~\ref{s-ureos}). The following three sections give an overview of the current status of the global geodetic observing system relevant to the three pillars. Many (but not all) items or activities, which will be mentioned in these section below, are coordinated by entities working under the auspices of IAG. IAG has been in the “monitoring business” since the late \cent{19}{th} century, when the \ac{ILS} was created to monitor polar motion. More recently the IAG created technique-specific Services to coordinate observation and analysis for the new space geodetic techniques. Also, on the level of IUGG and IAU the IERS was given the charter mentioned above and is coordinating related activities. These Services, which will be mentioned below, are important building blocks of the GGOS.