The Torus Approach in Spaceborne Gravimetry C. Xu, M.G. Sideris Department of Geomatics Engineering, University of Calgary, 2500 University Dr., NW, Calgary, Canada T2N 1N4, e-mail: {xuc, sideris}@ucalgary.ca N. Sneeuw Geod¨ atisches Institut, Stuttgart Universit¨ at, Stuttgart, Germany, D-70147, e-mail: sneeuw@gis.uni-stuttgart.de Abstract. Direct global gravity field recovery from the dedicated gravity field satellite missions CHAMP, GRACE and GOCE is a very demanding task. This has led to so-called semi-analytical schemes that are usually classified as either space-wise or time- wise under certain approximations and assumptions. Both approaches are efficient in that they provide a block-diagonal structure of the normal matrix for the least squares adjustment. At the same time, both approaches suffer from limitations implied by these approximations and assumptions. In this paper we will focus mainly on the so- called torus approach, that combines the strengths of both time-wise and space-wise approaches, in space- borne gravimetry. The semi-analytical algorithm will be addressed by comparing the different projection domains: periodic orbit, sphere and torus. Lumped coefficients are obtained from a two-dimensional Fourier analysis of the geopotential functionals on a nominal torus. Subsequently, block-diagonal least squares inversion provides the gravity field spherical harmonic coefficients as output. In addition, important issues, such as interpola- tion, regularization and optimal weighting will be discussed through the processing of two-year real CHAMP data. Keywords. Torus approach, spaceborne gravime- try, block-diagonality, semi-analytical formulation, spherical harmonic coefficients 1 Introduction Traditional techniques of gravity field determination, e.g. mean gravity anomalies from terrestrial and ship-borne gravimetry, satellite altimetry for ocean areas and satellite orbit analysis, have reached their intrinsic limits. Any advances must rely on space techniques only because they provide global, regular and dense data sets of high and homogeneous quality, cf. (ESA, 1999; Rummel et al., 2002). The dedicated gravity field satellite missions, CHAMP,GRACE and GOCE represent a new frontier in studies of the Earth and its fluid envelope, such as ocean dynamics and heat flux, ice mass balance and sea level, solid Earth, and geodesy, cf. (NRC, 1997). Spaceborne gravime- try missions provide millions of measurements during their life time. Consequently, direct global gravity field recovery up to a certain maximum spherical harmonic degree from these observations is a very demanding task. For instance, the number of unknowns for GOCE will be close to 100 000 when the maximum degree reaches L = 300. The conventional numerical methods (brute force approach), which are based on the orbit pertur- bation theory, are unable to solve the huge nor- mal equation system in the least squares inversion because it demands enormous computational time and high memory. This has led to so-called semi- analytical schemes that are usually classified as either space-wise or time-wise approach, cf. (Rum- mel et al., 1993). As a boundary value problem approach to physical geodesy, the former approach treats gravitational observable as a function of spa- tial coordinates, usually leading to a spherical projec- tion. Rooted in celestial mechanics, the latter treats the measurements as an along-orbit time-series, lead- ing to a one-dimensional Fourier analysis under a repeat orbit assumption, cf. (Sneeuw, 2000a). Both approaches are efficient in that they provide a block- diagonal structure of the normal matrix for the least squares adjustment under certain approximations and assumptions, cf. Figure 1 and (Koop, 1993). How- ever, both approaches have their intrinsic limitations. The space-wise approach lacks for instance any link to the dynamic orbit configurations, making it very difficult to implement a stochastic model from an error power spectral density (PSD). The time-wise approach, on the other hand, is very sensitive to data gaps, and the repeat orbit requirement is not always realistic, cf. (Rummel et al., 1993; Sneeuw, 2000a). In this paper we will focus on the so-called torus approach, which combines the strengths of both space-wise and time-wise approaches, 23