Length-of-day and space-geodetic determination of the Earth’s variable gravity field G. Bourda To cite this version: G. Bourda. Length-of-day and space-geodetic determination of the Earth’s variable gravity field. Journal of Geodesy, Springer Verlag, 2008, 82 (4-5), pp.295-305. 10.1007/S00190-007-0180-Y. hal- 00176445 HAL Id: hal-00176445 https://hal.archives-ouvertes.fr/hal-00176445 Submitted on 3 Oct 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Journal of Geodesy manuscript No. (will be inserted by the editor) G. Bourda Length-of-day and space-geodetic determination of the Earth’s variable gravity field Received: 17/02/2007 / Accepted: 27/07/2007 Abstract The temporal variations of the Earth’s gravity field, of the Earth (i.e., atmosphere, oceans, hydrosphere, mantle, nowadays routinely determined from Satellite Laser Rang- core; Barnes et al. 1983; Gross et al. 2004). ing (SLR) and GRACE, are related to changes in the Earth’s Each of the geophysical contributions to the LOD varia- rotation rate through the Earth’s inertia tensor. We study this tions is traditionally divided into (i) a motion term, and (ii) a connection from actual data by comparing the traditional mass term (Barnes et al. 1983; Eubanks 1993). This comes length-of-day (LOD) measurements provided by the Interna- from the Euler-Liouville equations for the angular momen- tional Earth Rotation and reference systems Service (IERS) tum conservation (Munk and McDonald 1960). For exam- to the variations of the degree-2 and order-0 Stokes coeffi- ple, the atmosphere acts on the LOD variations through the cient of the gravity field determined from fitting the orbits effect of the winds (motion term) and the effect of the at- of the LAGEOS 1 and 2 satellites since 1985. The two se- mospheric pressure onto the crust (mass term). The motion ries show a good correlation (0.62) and similar annual and contribution to the LOD variations may be modeled from semi-annual signals, indicating that the gravity-field-derived the data gathered by the Special Bureau for the Atmosphere LOD is valuable. Our analysis also provides evidence for ad- (SBA) and the Special Bureau for the Oceans (SBO), de- ditional signals common to both series, especially at a period pending on the Global Geophysical Fluid Center (GGFC) near 120 days, which could be due to hydrological effects. from the International Earth Rotation and reference systems Service (IERS). · · · Keywords Earth rotation length-of-day gravity field The mass contribution is due to dynamical processes in · · C20 satellite laser ranging LAGEOS the Earth system that affect the mass distribution (it is equiv- alent to gravitational effects). The Earth’s mass distribution, described by the inertia tensor, also acts directly on the Earth’s gravity field, therefore implying a relationship be- 1 Introduction tween the LOD and gravity field variations (Lambeck 1988; Chao 1994; Gross 2001). Studies of the Earth’s rotation are aimed at modeling its vari- The purpose of the study presented here is the investi- ations as precisely as possible, on the basis of geodetic mea- gation of this relationship by comparison of standard LOD surements and geophysical models. Offsets with respect to a measurements with gravity field data, in order to determine uniform rotation around a fixed axis or an analytical model if the latter could be useful to supplement classical LOD es- are described by the Earth’s Orientation Parameters (EOPs). timates. The motivation also originates in the requirement These parameters consist of the polar motion coordinates, to prepare for the ambitious project GGOS (Global Geode- the celestial pole offsets and the Earth’s rotation rate. tic Observing System) of the IAG (International Association The Length-of-day (LOD) is used to characterize the vari- of Geodesy), which ultimate goal is a consistent and inte- ability of the Earth’s rotation rate. The LOD variations are grated treatment of the three pillars of geodesy (Rummel et due to gravitational effects from external bodies (i.e., luni- al. 2005): (1) the geometry and deformation of the Earth’s solar tides; Defraigne and Smith 1999), but also originate surface, (2) the Earth’s rotation and orientation, and (3) the from geophysical deformations occuring in various layers Earth’s gravity field and its temporal variations. G. Bourda LOD data used in geodesy, geophysics or astronomy Centre National de Recherche Scientifique (CNRS), Observatoire Aquitain des Sciences de l’Univers (OASU), come from the combination of the results from various geode- Unit´eMixte de Recherche 5804, Universit´eBordeaux 1, tic observation techniques. Very Long Baseline Interferome- 33270 Floirac, France try (VLBI) is the primary technique, which permits determi- E-mail: [email protected] nation of all the EOPs (Schuh and Schmitz-H¨ubsch 2000; 2 G. Bourda Schl¨uter and Behrend 2007). Global Positionning System (personal communication from Lemoine 2004). In addition, (GPS) and Satellite Laser Ranging (SLR) allow to obtain po- SLR data cover a much longer period than GRACE data lar motion coordinates and LOD (Lichten at al. 1992; Tapley (20 years for SLR against 5 years for GRACE), which is et al. 1985), while Doppler Orbitography by Radioposition- more interesting for comparison of LOD measurements. ing Integrated on Satellite (DORIS) and Lunar Laser Rang- We compare the LOD mass terms obtained from both se- ing (LLR) may be selectively used for such determinations ries by determining the correlation coefficient between the (Gambis 2006; Dickey and Williams 1983). two series and by estimating annual and semi-annual signals Gravity field data, represented in the form of Stokes co- with a Fourier analysis. We also investigate the intraseasonal efficients, are nowadays routinely obtained by precise or- terms that remain in each of the series after this Fourier anal- bitography of geodetic satellites (for example based on SLR ysis. Finally, we discuss the consistency of the results and measurements; Nerem et al. 1993; Tapley et al. 1993; Bian- draw further prospects about using gravity field data to sup- cale et al. 2000) or dedicated space gravimetric missions like plement current LOD measurements. CHAMP (CHAllenging Minisatellite Payload; Reigber et al. 2002) or GRACE (Gravity Recovery And Climate Experi- ment; Reigber et al. 2005). 2 Theory Several studies have already investigated (i) the long- term variation (Yoder et al. 1983; Rubincam 1984; Cheng 2.1 Earth’s tensor of inertia et al. 1997; Cox and Chao 2002; Dickey et al. 2002) and (ii) the seasonal variations (Chao and Au 1991; Chao and The Earth’s tensor of inertia I, characterizing the mass dis- Eanes 1995; Chao and Gross 1987; Gegout and Cazenave tribution in the Earth system (i.e., solid Earth, atmosphere, 1993; Cheng and Tapley 1999) of the Stokes coefficients, oceans, hydrosphere), is defined in the terrestrial frame (Oxyz) by: especially the degree-2 and order-0 coefficient C20, which characterizes the gravitational oblateness of the Earth. I11 I12 I13 Variations of the gravity field have also been investigated I(t)= I12 I22 I23 (t) (1) by using EOP data (Chen et al. 2000; Chen and Wilson 2003; I13 I23 I33 Chen et al. 2005). Bourda (2004, 2005) and Yan et al. (2006) with: instead derived LOD and polar motion from temporal series 2 2 of the C20, C21 and S21 Stokes coefficients. I11 = (y + z ) dMI12 = xy dM %y %z −%x %y 2 2 This study is focused on the LOD parameter. For this I22 = (x + z ) dMI23 = yz dM (2) %x %z %y %z purpose, we compare the LOD series derived from the C20 − variations (hereafter referred to as geodetic data) to the stan- 2 2 I33 = (x + y ) dMI13 = xz dM dard LOD series derived mainly from VLBI, GPS and SLR %x %y −%x %z (hereafter referred to as astrometric data). These two types where M is the mass of the Earth. of data are not entirely equivalent: (1) the astrometric data In practice, the Earth is not only considered as an el- are relative to the crust of the Earth, whereas the geodetic lipsoid of rotation but also as a deformable body. For this data are relative to the entire Earth measured from space reason, its tensor of inertia is not diagonal. Then, I may be (Chao 2005); (2) the astrometric data are sensitive to all written as: geophysical processes inducing variations of the LOD (i.e., A 0 0 c c c the mass and motion terms), whereas the geodetic data are 11 12 13 I(t)= 0 B 0 + c c c (t) (3) only sensitive to gravitational effects into the Earth system 12 22 23 0 0 C c13 c23 c33 (i.e., the mass terms). These differences will be accounted for when comparing the two LOD series. where A, B and C are the mean principal moments of iner- In the following, we first detail the equations linking the tia (representing the part of the tensor that is constant with LOD and the C20 Stokes coefficient (Lambeck 1988; Gross time) and cij (for i, j = 1,2,3) are the products of inertia 2001; Chen et al. 2005). Then, we present the data used for (representing the shift to the mean ellipsoidal solid Earth) our study and the processing to determine the corresponding such that each of the cij/C is very small.