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 satellites are highly correlated (as they would be if χ θ θ imλθ 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 baseline vector 8m 2/s2 per m finite number of geopotetnial coefficients (say up absolute velocity 0.2 m2/s2 per mm/s to degree nmax ): baseline velocity 0.5 m2/s2 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. T The first is based on the estimation of coefficients N= , (7) in the spherical harmonic series for the disturbing γ potential, that in spherical coordinates (r,θ ,λ) is where γ is normal gravity; and, thus, with ∞ ∞ n+1 estimates of the geopotential coefficients, it can be R λ T(r,θ,λ)= ∑ ∑ γ P (cosθ)eim , computed (with respect to the reference model): r nm nm m=±∞ n= m (3) nmax nmax n+1 1 ∑ ∑ R γ θ imλ N = γ nm Pnm(cos )e , rG m=±nmax n= m where Pnm is the fully normalized associated γ (8) Legendre function of the first kind, nm is a geopotential coefficient of degree n and order m, where rG is the radius to a point on the geoid. If and R is a mean Earth radius. The set γ nm this point is inside the crust, a correction must be presumably excludes coefficients of the low- applied since the spherical harmonic expansion is degree reference model. valid only in free space (Rapp, 1997). Our model of the observable T12 makes the The second approach makes use of the Poisson simplifying assumption that the points, upward continuation integral for the disturbing representing the ends of a baseline for the potential (Heiskanen and Moritz, 1967): difference, have the same geocentric radius and are regularly distributed in longitude. Then we r2 ±R2 T(R,θ',λ') T(r,θ,λ)= dS , can write: 4π R 3 (9) S ∞ ∞ R n+1 T = ∑ ∑ 12 r m=±∞ n= m where S denotes the a sphere of radius R and is (r,θ ,λ) (4) the distance between the evaluation point, , ∆λ (R, θ',λ') γ χ θ ± χ θ eimk , and the integration point, . Following nm nm 2 nm 1 ideas expounded by Bjerhammar (1987), the Poisson integral may be interpreted as an analytic where continuation operator and discretized (using

- 3 - impulse functions), where the sources, in general, The geopotential values on the orbital grid were need not lie on a sphere. Considering the obtained from the EGM96 model restricted to difference of T at altitude, we have degrees 41 through 360 (all quantities are simulated and estimated with respect to a 40- degree reference field) and corrupted by simulated ≈ 1 1 2 2 1 1 ∆ T12 ΣΣ r ±rG ± T S . (10) 2 2 2 2 4π rG 3 3 white noise with variance, σ = 0.01 m /s S 2 1 (equivalent to the anticipated range noise of 1 µ m/s ; Jekeli, 1999). Inverting this for T on the basis of the observations, T12 , and using (7), we write the estimate for the geoid undulation as

T α ±1 T N =B B+ I B T12 , (11) where a Tikhonov regularization factor, α , is introduced because the downward continuation is known to be unstable (Bouman 1998).

3. Simulation

The estimation formulas developed in the previous section were tested with simulated potential values on an idealized set of “orbits”. The ground tracks of these orbits are depicted in Figure 1 and are generated from a single parent orbit that is one circular revolution of the satellites at a certain Figure 1: Geometry of simulated orbital grid inclination. The observables are registered along (shown with more illustrative parameters). this orbit according to a particular sampling interval. This defines the set of co-latitudes for γ The geopotential coefficients, nm , up to one satellite; the co-latitudes for the other satellite are defined by its separation from the first satellite degree nmax = 150 were estimated from the global along the same orbit. Each co-latitude defines a set of observations, while the downward longitude according to the position of the satellite continuation operator was applied to observations along the great circle. The resulting set of point in global bands delimited by parallels of co- θ , λθ , θ , λθ latitude. (In this sense, the geoid estimation is pairs 1 1 2 2 for one revolution is local only with respect to latitude, not longitude; then rotated in longitude by the angle k ∆λ , but serves the purpose of the investigation.) In k = 1, ..., M±1 , thus simulating Earth’s rotation, both cases the geoid was estimated at a spherical where the number, M, is defined by the mission surface, rG = R = 6371 km . duration. Table 2 gives all relevant parameters for Figure 2 shows the accuracy of geoid estimation the simulated orbits. via the estimated geopotential coefficients and as a function of latitude (root-mean-square (rms) Table 2: Parameters of the simulation. over all longitudes). The improved accuracy at orbit: high (absolute) latitudes is attributable to the inclination 89° increased density of observations resulting from eccentricity 0° the orbital geometry (Figure 1). The relatively gravitational field central term only large error (of the order of 1.5 m around the altitude 400 km equator) compared to the predicted GRACE satellite separation 200 km accuracy (20 cm) is due to the smaller number of observation grid: orbits (300 vs ~470 for 30 days) and because the sampling interval 10 s estimated geoid is compared to the 360-degree number of orbits 300 (~ 19 days) field (rather than the 120-degree field).

- 4 - 2.0 Finally, Table 3 also shows the value of the regularization factor, α , that was chosen in the local estimation to achieve the best result over 1.5 each zonal region. Its indicated variability implies that the downward continuation is not robust. Adapting the best regularization to a particular 1.0

error [m] region is one of the essential problems of the local estimation approach (see also Ilk, 2000).

0.5

4. Summary 0.0 -50 0 50 latitude [deg] A comparison of the two methods of geoid Figure 2: Accuracy of geoid calculated from estimation from in situ GRACE observations of estimated spherical harmonic coefficients (rms the disturbing potential may be summarized as over all longitudes). follows. The spherical harmonic model approach used here relies on a regular grid of data (at Table 3 divides the southern hemisphere into constant radial distance) in order to make the three zonal regions with indicated latitude 2 problem of estimating the nmax +1 boundaries. The last column lists the rms errors of geopotential coefficients reasonably tractable. the estimated geoid from spherical harmonics,

Table 3: Geoid accuracy from inverse-Poisson downward continuation (rms errors). Estimation region RMS Error RMS Error Along Central α × 1036 RMS Error from 0° < λ < 360° (excluding edges) Meridian Parallel Spherical Harmonics -20° < φ < 18° 1.28 m 0.21 m 1.19 m 0.1 1.50 m -63° < φ < -24° 0.47 m 0.74 m 0.47 m 0.3 1.42 m -89° < φ < -52° 0.43 m 0.42 m 0.28 m 2 1.02 m

showing the decrease in error from 1.50 m near This is nearly satisfied by the geometry of the the equator to 1.02 m near the pole. A more orbital ground-tracks for the actual GRACE dramatic decrease to 0.43 m is obtained with the mission, but some interpolation or extrapolation, local inverse-Poisson downward continuation especially in radius, will be required. [We note using (11), as indicated in the second row of Table that the approach based on orbital perturbation 3. In this case, however, the edges of each zonal analysis, a priori, does not require this region are excluded, and the errors refer only to simplification, which makes it computationally the middle third. In addition, we see significant burdensome; but we assume that the spherical dependence of the error on the ground-track harmonic estimation is largely independent of the geometry. Near the equator, where the tracks are observations, as well as the method of processing almost parallel, the geoid error computed along them.] meridians is very much smaller than along The local, inverse-Poisson, downward parallels, implying that the estimation suffers from continuation involves fewer unknowns and thus the directionality of the observed differences in allows greater flexibility in the data distribution. disturbing potential. In mid-latitudes and near the On the other hand, being local in implementation, pole, no such clear distinction can be inferred as it suffers from the usual edge effects along the the ascending and descending tracks begin to borders of the estimation area. For the same cross with more significant angles. In both cases, reason, there is no chance to estimate the long- the estimation errors, as computed along the wavelength components of the geoid; whereas, all central parallel is somewhat smaller than along the but the zero-degree coefficient can be estimated central meridian. from a global distribution of T12 observations. Moreover, while the local downward continuation

- 5 - definitely requires some form of regularization, using a spherical harmonic representation of the the estimated spherical harmonic model includes a height anomaly/geoid undulation difference. J. relatively stable downward continuation through Geodesy, 71(5), 282-289. the factor R/r n+1 . Tapley, B.D. and C. Reigber (1998): GRACE: A Despite the shortcomings of the local procedure, satellite-to-satellite tracking geopotential the results of the simulation seem to confirm the mapping mission. Proceedings of Second Joint principal hypothesis of the investigation, that the Meeting of the Int. Gravity Commission and the greater density of observations at higher latitudes Int. Geoid Commission, 7-12 September 1998, yields improved geoid determination through local Trieste. estimation. The spherical harmonic approach also Wahr, J., M. Molenaar, F. Bryan (1998): Time shows an improvement in polar latitudes, but it is variability of the Earth’s gravity field: less pronounced than in the case of local hydrological and oceanic effects and their downward continuation of in situ data, which possible detection using GRACE. J. Geophys. yields substantially better results. Res., 103(B12), 30205-30229. Yuan, D.N., C.K. Shum, and B.D. Tapley (1988): Gravity field determination and error References assessment techniques. Proceedings of the Chapman Conference for Progress in the Bjerhammar, A. (1987): Discrete physical Determination of the Earth’s Gravity Field, geodesy. Report no.380, Department of pp.15-18, 13-16 September 1988, Ft. Geodetic Science and Surveying, Ohio State Lauderdale, Florida. University, Columbus, Ohio. Bouman, J. (1998): Quality of regularization methods. Report no.98.2, DEOS, Delft University Press, The Netherlands. C. Cui, D. Lelgemann (2000): On non-linear low- low SST observation equations for the determination of the geopotential based on an analytical solution. Journal of Geodesy, 74(5), 431-440. Heiskanen, W.A. and H. Moritz, 1967: Physical Geodesy. Freeman and Co., San Francisco. Ilk, K.H., J. Kusche, and S. Rudolph (2000): A contribution to data combination in ill-posed downward continuation problems. Submitted to Journal of Geodynamics Research. Jekeli, C. (1996): Spherical harmonic analysis, aliasing, and filtering. Journal of Geodesy, 70(4), 214-223. Jekeli, C. (1999): The determination of gravitational potential differences from satellite- to-satellite tracking. Celestial Mechanics and Dynamical Astronomy, 75(2), 85-100. Keating, T., P. Taylor, W. Kahn, F. Lerch (1986): Geopotential Research Mission, Science, Engineering, and Program Summary. NASA Tech. Memo. 86240. Lemoine, F.G. et al. (1998): The development of the joint NASA GSFC and the National Imagery Mapping Agency (NIMA) geopotential model EGM96. NASA Technical Report NASA/TP-1998-206861, Goddard Space Flight Center, Greenbelt, Maryland. Rapp, R.H. (1997): Use of potential coefficient models for geoid undulation determinations

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