Local geoid determination with in situ geopotential data obtained from satellite-to-satellite tracking C. Jekeli, R. Garcia Civil and Environmental Engineering and Geodetic Science Ohio State University, 2070 Neil Ave., Columbus, OH 43210 Abstract. A new satellite mission to map the “drag-free” and non-gravitational accelerations Earth’s gravitational field (GRACE) is based on must be measured independently using on-board inter-satellite tracking using micro-wave ranging accelerometers. Also, the altitude of the GRACE which can be used to solve for parameters of a satellites is significantly higher (450 km) than that global spherical harmonic model of the proposed for GRM (160 km). The principal gravitational field using traditional satellite orbital differences, however, lie in the objectives of perturbation analysis. An alternative algorithm GRACE; the latter are oriented primarily to the has been developed to use the inter-satellite range- short-term Earth dynamics and the implications on rate, supplemented by GPS baseline vector climatic changes (Wahr et al., 1998), derivable velocities and accelerometer data, to estimate in from the time-varying gravity field and associated situ geopotential differences. The collection of with global ocean and atmospheric currents and these data, treated as a set of boundary values in mass/thermal transport, and the large-scale potential theory, should yield higher geopotential hydrology on continents (including mass balance resolution in the polar latitudes than provided by a of ice sheets and glaciers). truncated spherical harmonic model because of the On the other hand, the static gravity field, high concentration of data in these areas. On the averaged over several years, is also a product of other hand, in situ data must be downward the mission, advertised as a global spherical continued for local geoid determination, whereas harmonic model up to degree and order 150 this is already incorporated in the spherical (resolution of 130 km). By the nature of the harmonic models. Simulations confirm that GRACE orbits, being nearly polar (the inclination although spherical harmonic estimations also yield is 89°), there is considerable densification of higher accuracy near the poles, local geoid measurements and increase of track cross-over estimation from in situ geopotential difference angles in the polar regions, as well as abrupt lack measurements is even more accurate at the poles, of data in small (200 km diameter) caps centered by a factor of two or higher for the case of on the poles. The higher density and directional nineteen days' worth of GRACE data. heterogeneity of the data present an opportunity for more local, high-resolution, gravity modeling, Keywords. Satellite-to-satellite tracking, geoid while the discontinuity presents a challenge in the determination, downward continuation, spherical combination of GRACE data with existing harmonic model regional gravity data. The fundamental GRACE measurement is the near-instantaneous range between the two 1. Introduction satellites. The ranging is accomplished with a microwave (K-band) satellite link that provides A satellite mission dedicated to the improvement two one-way ranges, each obtained by comparing of our knowledge of the Earth’s gravitational field an on-board generated phase to the received by the use of a direct measurement system has phase. Both the transmitted and the on-board now been approved and is expected to be realized phases are generated by the same ultra-stable in 2001: GRACE, the Gravity Recovery And oscillator (USO). In addition, accelerometers Climate Experiment (Tapley and Reigber, 1998). measure non-gravitational accelerations of the GRACE is a variant of the erstwhile GRAVSAT center of mass of each satellite, and star cameras and GRM mission concepts (Keating et al., 1986) ensure accuracy in the orientation of the in that two low-altitude satellites will track each transmitting and receiving K-band horns, as well other as they circle the Earth in identical near- as the accelerometers. The range values are to be polar orbits. Unlike GRM, the satellites are not obtained at a sampling rate of 10 Hz. These will - 1 - be decimated and filtered to produce range-rates system, while the term with the explicit and range-rate-rates (line-of-sight (LOS) dependence of V on time accounts for the time- accelerations) at a sampling rate of 0.2 Hz. The varying nature of the potential, primarily due to predicted accuracy in the decimated, filtered Earth’s rotation. These two terms and the range-rate is 10±6 m/s . A second band (Ka-band) constant can be calculated from the accelerometer will be used to calibrate first-order ionospheric and GPS position/velocity data and from initial ∆ delays in the signal. conditions; they are combined as one term, V . In addition, each satellite is equipped with a The difference of gravitational potentials GPS antenna and receiver, running off the same between two satellites and referenced to a given USO. The GPS data will be used for orbit low-degree spherical harmonic model is then determination and atmospheric sounding on the given by (the details of the derivation are found in basis of GPS satellite occultations. The GPS data (Jekeli, 1999)): rate is nominally 1 Hz, but greater for the δρ occultation measurements. The K-band ranges T12 = x01 12 can be used with orbit determination algorithms to estimate the spherical harmonic coefficients T δ (spectral components) of the Earth’s gravitational ± x02 ± x01 e12 x12 field (Yuan et al., 1988; Cui and Lelgemann, 2000). (2) δ δ T Instead of the conventional global (spherical ± x1 ± x01 e12 x012 harmonic) gravity modeling approach we propose to derive local gravity field parameters from the in T 1 2 situ data products of GRACE. In situ data ± x δx ± δx ± ∆V , 1 12 2 12 12 products are defined as those data observed directly by the on-board instruments and subjected to minimal preprocessing, where the K-band where the appended 0 signifies a reference field ranging system is considered one of the “in situ” quantity and δ the corresponding residual, a instruments. These products include the range- single subscript 1 identifies the quantity as rate and line-of-sight (LOS) accelerations derived belonging to the first satellite, a double subscript from the inter-satellite ranges, accelerometer and 12 implies a difference of corresponding star tracker measurements on both satellites, GPS quantities at the two satellite points, and e is a phase data obtained on both satellites, as well as GPS position and velocity vectors of both unit vector. The significance of (2) lies in the fact satellites. that the disturbing potential difference, T12 , is δρ related to the residual range rate, 12 , between the satellites, plus correction terms that can be 2. Observation Models computed from other data. Simpler models can be derived for the line-of-sight acceleration, but we The models that relate the measurements to local do not consider these here. gravitational quantities, such as at-altitude Clearly, the accuracy of the in situ disturbing geopotential differences and gravitational potential difference depends on the accuracy with acceleration differences, have been developed by which the correction terms can be computed. A Jekeli (1999). We start with the energy equation: cursory analysis of (2) yields a quantification of the sensitivities of the correction terms to orbit t t error (either absolute or relative between the 1 2 ∂V V= x ±FΣ k xk dt + dt ±E0 , (1) satellites). From Table 1 we see that the 2 k ∂t t0 t0 correction terms are most sensitive to the relative position vector between the satellites, which, if accurate to 1 mm, contributes an error of about for the gravitational potential, V, in inertial space, 2 2 T 0.008 m /s to the disturbing potential difference. where x =x , x , x is the velocity vector of 1 2 3 This is commensurate with the accuracy the satellite, Fk is a component of specific force anticipated from the range-rate, since the average acting on the satellite (such as atmospheric drag), t speed of the satellite is 7700 m/s . is time, and E0 is a constant. The specific-force The registration of the measurement within a term represents the dissipative energy of the particular coordinate frame may contribute an - 2 - error as well. However, if the orbits of the two imλ satellites are highly correlated (as they would be if χ θ θ θ nm =Pnm cos e , (5) determined with relative GPS positioning), then the potential difference is not sensitive to and where λθ is the longitude of a point on a registration errors. single revolution of a Keplerian, circular orbit whose co-latitude is θ (see Figure 1). Table 1: Sensitivity of T12 correction terms to orbit error. Given observations T =T + ε corrupted 12 12 T12 orbit quantity sensitivity 2 by white-noise errors with variance σ , we can absolute position 0.02 m2/s2 per m form the following least-squares estimate of a 2 2 baseline vector 8m /s per m finite number of geopotetnial coefficients (say up 2 2 to degree n ): absolute velocity 0.2 m /s per mm/s max 2 2 baseline velocity 0.5 m /s per mm/s ±1 γ * * nm =AA A T12 , (6) For our simulations, we assume that disturbing potential differences have been obtained by where the values of the elements of the matrix, A, suitably processing the measurements from the * suite of GRACE instruments, principally the K- can be inferred from (4) and A is its Hermitian band ranging assembly, the accelerometers, and (Jekeli, 1996; Lemoine et al., 1998, chapter 8). GPS according to models (1) and (2). One may The geoid undulation is related to the disturbing now proceed along two different approaches potential according to Bruns’ formula: toward the computation of the geoid undulation.