----------------------------------------------------------------------------- IERS Conventions Workshop -- Discussion Paper for Theme 3 ----------------------------------------------------------------------------- The realization of the origin of the terrestrial reference system G. Petit, Bureau International des Poids et Mesures J. Ries, Center for Space Research Z. Altamimi, Institut Geographique National C. Boucher, Ministère de la Recherche J. Ray, National Geodetic Survey, NOAA (Version 14 September 2007) 1) Introduction --------------- Chapter 4 of the IERS Conventions 2003 provides a conceptual and mathematical formalism for the construction of the International Terrestrial References System (ITRS) and its realization, the International Terrestrial Reference Frame (ITRF). While necessary, these detailed specifications may obscure some functional aspects of the ITRS and especially the ITRF that are important for users. First and foremost, these constructs are literally a global "system (or frame) of reference". Many users would probably like that ITRF can describe the time-varying positions of all its constituent points with utmost accuracy. The ITRF does provide time-varying positions of a network of points, but they represent the actual instantaneous positions only to some level of approximation. It is not possible to provide the utmost accuracy that can be obtained by space geodesy techniques (~ 1 mm) for a host of reasons. The ITRF provides a regularized polygon of positions for a large number of points on the Earth's surface that are known to be moving with respect to one another. This set closely approximates the actual observed positions of those points, integrated over the longest periods available and within the bounds of the IERS Conventions models that are prescribed to account for mostly tidal motions of terrestrial points. A comprehensive set of models or empirical formalisms do not exist to account for the total range of motions for all ITRF points, so at any given time the ITRF positions (plus Conventional models) will differ from the instantaneous observed positions by some non-negligible amount. As the accuracy and resolution (temporal and spatial) of the geodetic data continuously improves, expectations and demands for the ITRF also grow. In particular, an improved representation of the time-varying internal characteristics of the ITRF polygon is generally sought. But it will probably always remain beyond the scope of ITRF to provide a full and accurate description of the frequently complex local motions of every point on the Earth's surface no matter what level of effort is applied. Nevertheless, one can reasonably ask how the ITRF might be elaborated to account for a fuller range, albeit still incomplete, of internal motions. In a spherical harmonic sense, we consider in particular the lowest degree contributions. For instance, the degree one terms correspond to a global shift of the crust-fixed ITRF with respect to the center of mass of the total Earth system. For at least a decade, satellite laser ranging (SLR) data analyses have shown that the coordinate frame of tracking stations attached to the Earth's crust moves detectably relative to the planetary center of mass. This translational motion is known as "geocenter motion" and is caused by the mass movement of planetary fluids, primarily the atmosphere and oceans. At the same time, the motions of the fluid masses cause variable loadings of the Earth's surface and thus small but significant displacements of ITRF points. In principle, these effects could be modeled in the higher degrees and orders of the same expansion (e.g., Blewitt et al., 2001) or via a set of time-varying empirical grids, depending on the type and quality of data available. As stressed by Blewitt and his colleagues, observations of the degree one geocenter motion are not independent of the higher-order surface deformations. So, a major issue confronting ITRF is the extent to which such internal motions should be dealt with directly within the ITRF framework and how best to do so. In the past years, analysis centers of the IERS have implemented approaches to deal with some effects associated with geocenter motion. At the observation level, it was recognized that the tidal loading model should be used in a consistent way so that the station displacement is obtained in the appropriate frame (see below in section 3.3). At the analysis level, it was recognized that the series of individual "instantaneous" reference frame solutions should be combined appropriately in order to realize the intended global reference frame. (see below in section 4.4). While the Analysis centers have, by necessity, adopted conventional procedures to deal with the effects of geocenter motion, these did not appear in the IERS Conventions (2003) (McCarthy and Petit, 2004), which actually more reflected the situation of the year 2000. In the updates published since 2004 the situation is evolving as the opportunity of new sections allows e.g. the rewrite of section 7.1.1 on displacement due to ocean loading (see http://tai.bipm.org/iers/convupdt/convupdt_c7.html) explicitly considers the type of frame in which the displacement is computed. The present paper intends 1. to give directions for definitions and notations related to the geocenter (section 2) 2. to describe the present situation in the IERS Conventions (2003) (section 3) 3. to describe, in a consistent manner, the practical implementation of the conventional procedures used by Analysis and Product centers, as much as is needed in the IERS Conventions (section 4) 2) Definitions and notations ---------------------------- As a preamble to this section, it is recognized that many concepts and quantities which are used below do not have an accepted or widely recognized definition, or, when definitions exist, do not have an accepted or widely recognized meaning. This concerns, among others, the notions of Terrestrial reference system and Terrestrial reference frame, ITRF itself, the qualifications of a frame to be "center of mass", "center of figure", etc... ====================================================================== Therefore it is recommended that a working group on nomenclature be created in the frame of the IAG. ====================================================================== 2.1) Terminology In the literature on modeling geophysical loading, one defines [Blewitt et al., 2001; Blewitt, 2003 and references therein]: * ``center of mass'' (CM) frame: Refers to a frame which origin is the center of mass of the total Earth system, which includes the solid Earth and the atmosphere and oceans, or more generally the fluid masses. This is sensed by, and appropriate to describe, satellite dynamics. * ``crust-fixed'' frame: Refers generically to a frame which is defined by the coordinates of stations fixed to the crust. The frame having its origin at the center of mass of the solid Earth (CE frame) is appropriate for modeling solid Earth deformations due to loading. * The ``center of figure'' frame refers to a frame having no net translation with respect to the displacement field (of the points or stations used to realize it). For what concerns the origin, these last two frames can be considered identical [Blewitt 2003] and we call them CF frame. In the following, we therefore consider either CM or CF frames. These frames are to be considered 'instantaneous' (function of time), and are used to model physical phenomena. On the other hand, for the realization of a TRF, e.g. ITRF, one uses series of network solutions, each of which, by construction, realizes a CM frame over a given averaging time e.g. one week. They are noted TRF(t) where t has the indicated resolution e.g. one week. An analysis procedure (see section 4.4 for ITRF) then intends to realize a 'long term average' of the TRF(t). It is not the purpose of this paper to examine the relation between this realized 'long term average' and the above-mentioned CF frame. Rather, we consider below the relations between TRF(t) and the 'long term average'. 2.2) Geocenter motion The basic principle: When the solid Earth and fluid masses are considered as a system without any external forces acting upon it, the position of the common center of mass remains fixed in space. When a phenomenon, such as the ocean tides, causes displacements of fluid masses, the center of mass of the fluid masses moves periodically and must be compensated by an opposite motion of the center of mass of the solid Earth. The stations, being fixed to the solid Earth, are subject to this counter-motion. Considering the geophysical modeling, ``geocenter motion'' refers to the translation between the origin of the CM frame and another geometrical center of figure (see e.g. Dong et al, 1997; Greff-Lefftz, 2000; Blewitt, 2003). Although there is variation in the literature considering the orientation of this vector, Blewitt (2003) states preferable to consider the translation vector from the origin of the CM frame to the origin of the CF frame, i.e. a vector that represents the motion of the CF in the CM frame. Noting OG this vector, the coordinates of a point in the CM frame X_{CM} and in a CF frame X_{CF}, at time t, are then related by X_{CF}(t) = X_{CM}(t) - OG(t) (EQU Ia) Similarly, considering the realization of terrestrial frames such as ITRF (Altamimi et al. 2007), it has been chosen to take into account the effects of geocenter motion by considering the vector from the origin of TRF(t) (the "instantaneous" center of mass) to the origin of ITRF, so that (still noting this vector as OG (t)) X_{ITRF}(t) = X_{TRF(t)} - OG(t) (EQU Ib) Therefore, in both cases, the quantity of interest is the vector (here noted OG) with origin at the instantaneous center of mass. Although the geocenter has often been defined and used as a geometric concept (see e.g. Dong et al, 1997; Greff-Lefftz, 2000; Blewitt, 2003) and not as the center of mass, there is a disparate use of the name 'geocenter'. In order to avoid confusion, it is desirable not to name the above-defined vector the 'geocenter motion'. Instead it is proposed to use 'origin translation', leaving 'geocenter motion' to name the effect itself. ====================================================================== In dealing with geocenter motion (translation between frames, comparison of models to observed effects etc…), it is recommended to use the vector from the instantaneous center of mass to the center of the geometric frame of interest and to name this vector the "origin translation". ====================================================================== In practice, the notion of geocenter motion covers several effects, so that a generic term may become ambiguous. We distinguish * Tidal geocenter motion (with associated origin translation noted OG_t), which is mostly (but not exclusively) in the diurnal and semi-diurnal bands. In the frame of the IERS Conventions, this is defined as corresponding to those tidal effects which are modeled in the Conventions. * (Non-tidal) seasonal geocenter motion (with associated origin translation noted OG_s), which is mostly in the yearly and semi-yearly bands. * Long-term (secular) geocenter motion (with associated origin translation noted OG_l), which may be approximated by linear variation. The distincton between seasonal and secular is somewhat arbitrary and not clear-cut (e.g. one might wish to have all periodic phenomena out of the secular part, but 'quasi periodic' and long-period phenomena might exist). However, periods of 0.5 and 1 year should be correctly identified with some 10 years of good data now available. The distinction between seasonal and secular is introduced because these two classes are treated differently in the conventional analysis used to derive the IERS products. In the present practice of ITRF realization, long-term effects of geocenter motion contribute to the average variation of the regularized positions. Because these effects have their origin in several physical phenomena (plate tectonics, deglaciation, etc..) which depend on not well known physical properties of the Earth, they cannot be modeled adequately. Seasonal effects, on the other hand, are observable and must be accounted for in the analysis of series of network solutions. The procedure currently used is described in 4.4. When progress in the modeling of these effects allows, it may be necessary to reconsider this approach (see Plag et al, 2007). 3) Situation in the IERS Conventions (2003), including updates until June 2007 ------------------------------------------------------------------------------ The IERS Conventions should provide the definition of the terrestrial reference system (ITRS), and the necessary procedures used for its realization (ITRF). However, in the Conventions (2003), the global presentation is found to be not fully consistent, neither internally, nor with the definitions in the IUGG recommendations, nor with the actual procedures used for the realization of ITRF, as shown below. 3.1) Consistency in the definition of ITRS and ITRF The short summary is that ITRS is indeed defined as CM, while its realization ITRF is not consistently treated. This pertains to Chapter 4, notably to the following sections: Section 4.1.4: It is reminded that the IERS wants to provide a realization of ITRS following the IUGG Resolution No. 2, 1991 (see its text e.g. in Appendix of IERS Standards 1992) which states that the origin of the terrestrial reference system be "the geocenter of the Earth's masses including oceans and atmosphere". Section 4.2.4: It is stated that the ITRF "should be considered as a figure origin related to the crust. In order to obtain a truly geocentric position, following the ITRS definition, one must apply the geocenter motion correction dX_G". Note that this geocenter motion correction vector is the opposite of the vector OG in our equation (I) (i.e. one may be seen as the "effect", the other as the "correction"). Section 4.1.3, in describing ITRF regularized coordinates, it is stated that the instantaneous vector position X(t) of a point on the Earth's surface can be expressed in the ITRF as X(t) = X_R(t) + Sum{ delta X_i(t) } where X_R(t) are regularized coordinates and where delta X_i(t) are site-specific conventional corrections to be presented in Chapter 7 for solid Earth tides, ocean tidal loading, pole tide, atmosphere loading and geocenter motion. However in Chapter 7, conventional expressions for the last two cannot be found. These last two sections imply that ITRF is a 'CF' frame and ITRS is 'CM', with the inclusion of the geocenter motion correction in section 4.2.4. This is not in line with the notion that ITRF is a realization of ITRS. 3.2) Interpretation of EOPs In Chapter 5 "Transformation Between the Celestial and Terrestrial Systems", the coordinate transformation to be used to transform from the terrestrial reference system (TRS) to the celestial reference system (CRS) at the epoch t of the observation is specified (Equation 1) as: CRS = Q(t)R(t)W(t) TRS where Q(t), R(t) and W(t) are the transformation matrices arising from the motion of the celestial pole in the celestial system, from the rotation of the Earth around the axis of the pole, and from polar motion respectively. Section 5.10 states that Equation (1) provides the transformation from ITRS to GCRS It implies that the EOPs in the matrix W are referenced to the CM, and that the same convention should be used when determining EOPs. This is formally consistent with the writing in Chapter 4, where ITRS is defined as 'CM'. The application is unclear, as it may be understood and applied using ITRF instead of ITRS and because the definition of ITRF is ambiguous (see 3.1 above). 3.3) Conventional corrections to station coordinates In Chapter 7 "Displacement of Reference Points" of the Conventions (2003), no mention is made of geocenter. It only appears in the update (25 November 2006) of section 7.1.1 "Ocean loading". In this section, a "Note on motion of the center of mass of the solid Earth" recalls the basic principle (see above section 2): "When the solid Earth and fluid masses are considered as a system without any external forces acting upon it, the position of the common center of mass remains fixed in space. When a phenomenon, such as the ocean tides, causes displacements of fluid masses, the center of mass of the fluid masses moves periodically and must be compensated by an opposite motion of the center of mass of the solid Earth. The stations, being fixed to the solid Earth, are subject to this counter-motion." and reminds the two kinds of applications: "For observing techniques that rely upon the dynamical motions of satellites, which respond to the center of mass of the total Earth system, the modeled motions of crust-fixed stations should include the “geocenter motion” contributions that counterbalance the effects of the fluid components. For other observing techniques, such as VLBI, neglect of geocenter motion should have no observable consequences." i.e. in the first case, the coordinates obtained from / used in the analysis are 'CM' in the second case, the coordinates are 'CF'. When generating tables of amplitudes and phases using the ocean loading service, in order to compute the conventional coordinate correction due to loading, users are required to consider in which frame their regularized coordinates are expressed: one has to answer the question ``Do you want to correct your loading values for the [geocenter] motion?'' Answering ``No'' means that the coefficients do not include the large-scale effect of the geocenter motion caused by the ocean tide. This is appropriate for station coordinates given in a ``crust-fixed'' frame that is not sensitive to the Earth's center of mass. Answering ``Yes'' means that the coefficients include the large-scale effect of the geocenter motion caused by the ocean tide. This is consistent with data analyses that realize a near-instantaneous ``center of mass'' frame using observations of satellite dynamics. Regularized coordinates are 'CF' in the first case, 'CM' in the second case. Finally, the necessary formula to obtain the vector between the 'CF' coordinate correction and the 'CM' coordinate correction (Center of mass correction) is provided: If necessary, the crust-frame translation (geocenter motion) due to the ocean tidal mass, dX(t), dY(t), and dZ(t), may be computed according to the method given by Scherneck at http://www.oso.chalmers.se/~loading/cmc.html, e.g. for dX(t) as dX(t)=\sum_{k=1}^{11}X_{in}(k)\cos(\chi_k(t)) + X_{cr}(k)\sin(\chi_k(t)) where the in-phase (_{in}) and cross-phase (_{cr}) amplitudes (in meters) are tabulated for the various ocean models. Similarly for dY(t) and dZ(t). This correction should be applied, for instance, in the transformation of GPS orbits from the center-of-mass to the crust-fixed frame expected in sp3 format: X_CF = X_CM - dX, i.e. the translation vector should be substracted when going from center-of-mass to sp3. Note that this translation vector has the same definition (orientation) as the vector OG in our equation (I) in section 2.2. 4) Discussion and proposed directions ------------------------------------- In section 4.0, we recall the proposed conventional definitions and notations. In section 4.1 to 4.3, we review the three items discussed in sectons 3.1 to 3.3 above. In section 4.4, we propose that the analysis procedure used to generate regularized coordinates (ITRF, IGS) be described as a conventional procedure, i.e. provides a conventional treatment of the seasonal geocenter effects. 4.0) Conventional definitions and notations Geocenter motion literally refers to the motion of the geocenter in a frame of reference. Because geocenter is used either as the origin of a geometric frame or to name the center of mass, this can lead to confusion. In IERS applications dealing with geocenter motion it is recommended to use the vector from the instantaneous center of mass (CM) to the center of the geometric frame of interest (CF) and to name this vector "origin translation", here noted OG. The coordinates of a point in the CM frame X_{CM} and in the CF frame X_{CF} are related by X_{CF} = X_{CM} - OG (EQU I) In practice, we distinguish * Tidal geocenter motion (with associated origin translation OG_t), which is mostly (but not exclusively) in the diurnal and semi-diurnal bands. In the frame of the IERS Conventions, this is defined as corresponding to those tidal effects which are modeled in the Conventions. * (Non-tidal) seasonal geocenter motion (with associated origin translation OG_s), which is mostly in the yearly and semi-yearly bands. These effects are not modelled yet. * Long-term (secular) geocenter motion (with associated origin translation OG_l), which may be approximated by linear variation. 4.1) Consistency in the definition of ITRS and ITRF (chapter 4 of the Conventions) ITRS is defined as CM, and so is its realization ITRF. Therefore * No term for geocenter motion should appear in section 4.2.4 of chapter 4; * Geocenter motion effects should be explicitly accounted for in section 4.1.3 of chapter 4: Tidal effects OG_t(t) should be explicitly mentioned in the 'conventional corrections to station coordinates' (see 4.3 below). Seasonal effects OG_s(t) should be dealt with in a conventional analysis procedure (see 4.4 below) It is a long-term goal of the IERS to be able to model seasonal effects OG_s(t). 4.2) Interpretation of EOPs (chapter 5 of the Conventions) Equation (1) of chapter 5 should be explicitly stated as: GCRS = Q(t)R(t)W(t) ITRS ====================================================================== To Be Written ====================================================================== 4.3) Conventional corrections to station coordinates (chapter 7 of the Conventions) This concerns the handling of geocenter motion in chapter 7 (which could adequately be renamed as 'Conventional corrections to station coordinates') In 3.3) is described the treatment of geocenter motion in the frame of the ocean loading. This treatment is in line with the recommendations in PP1: "The compensating counter motions of the solid Earth due to all the fluid loading effects ("geocenter motion" of the observing networks relative to the ITRF origin) should generally be included in the modeled station displacements described by the regularization corrections, at least for those techniques that observe the dynamical motions of near-Earth satellites and respond to the center of mass of the total Earth system." This should be made as a general statement, valid for all loading corrections. ====================================================================== The "Tidal" component of the origin translation associated to all modeled loading effects should be modeled at the observation level, following the procedure used for Ocean loading in the update 25/11/2006 of Conventions. ====================================================================== 4.4) Conventional treatment of the seasonal geocenter effects 'Instantaneous' (observed) coordinates differ from 'modeled' coordinates by seasonal effects. Because the seasonal origin translation OG_s(t) cannot be modelled adequately yet, it has to be considered at the analysis level in some conventional way. A conventional procedure has been chosen for the realization of the IGS frame (Ray et al., 2004; Ferland et al., 2000) and of ITRF2005. In case of IGS frame the following procedure is applied: * unconstrain the 'instantaneous' (weekly) solutions, * align them to the initial IGS reference frame using 7-parameter Helmert transformation, * combine the aligned weekly inputs to form the final solution. In case of ITRF2005, the ILRS SLR weekly (loosely constrained) solutions were treated using the following procedure (Altamimi et al., 2007): * inner/minimal constraints were applied to each weekly solution * the obtained minimally constrained weekly solutions were then rigorously stacked to form a long-term solution (positions and velocities) where the origin and the scale are defined by the usage of intrinsic constraints [Altamimi et al., 2007]. In order to define the ITRF2005 origin, the 3 translation components and their respective rates between the ITRF2005 and the SLR long-term solution were set to zero in the final ITRF2005 combination. * During the stacking of the SLR weekly solutions, time series of the 7 weekly transformation parameters are estimated with respect to the long-term solution. The time series of the weekly translation components are considered to represent the apparent geocenter motion. Therefore, considering the CATREF combination model, the relation between each 'instantaneous' weekly SLR solution and the ITRF2005 could be written as: X_SLR(t) = X_ITRF2005(t) + T(t) + D(t)X_ITRF2005(t) + R(t).X_ITRF2005(t) where X_ designates the point coordinate vector, T(t) is the translation vector (equivalent to the OG_s(t) notation above), D(t) is the scale factor and R(t) is the rotation matrix. Notes: Phases of validation / outlier detection in the above scheme are omitted here. This describes the current conventional analysis procedure based on Helmert transformation. ====================================================================== The "Seasonal" component of the origin translation is to be dealt in the elaboration of ITRF. Chapter 4 of next edition should describe the current conventional procedure. ====================================================================== 5) Summary of propositions -------------------------- ** It is recommended that a working group on nomenclature be created in the frame of the IAG. Recommendations for the IERS ** In dealing with geocenter motion (translation between frames, comparison of models to observed effects etc…), it is recommended to use the vector from the instantaneous center of mass to the center of the geometric frame of interest; name this vector "origin translation". ** The "Tidal" component of the origin translation associated to all modeled loading effects should be modeled at the observation level, following the procedure used for Ocean loading in the update 25/11/2006 of Conventions. ** EOP formulation: To be written ** The "Seasonal" component of the origin translation is to be dealt in the elaboration of ITRF. Chapter 4 of next edition should describe the current conventional procedure. -------------------------------------------------------------------------- References Altamimi Z., X. Collilieux, J. Legrand, B. Garayt and C. 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