Gravity recovery capability of four generic satellite formations

Transcription

1 Gravity recovery capability of four generic satellite formations M.A. Sharifi, N. Sneeuw, W. Keller Institute of Geodesy, Universität Stuttgart, Geschwister-Scholl-Str. 24D, D-7174 Stuttgart, Germany Abstract. Of all potential satellite formations a GRACE-type in-plane leader-follower configuration is one of the weakest choices in terms of gravitational signal in the satellite-to-satellite tracking (SST) observable. In this paper we simulate four basic types of formation flying (FF) missions and analyse them in terms of sampling geometry (baseline orientation), gravitational signal content and gravity field recovery capability. The four FF types are, respectively a GRACE-type SST misison, a SWARM-like pendulum configuration, a Cartwheel formation on a 2:1 relative ellipse and a LISA-type formation. All formations have comparable orbit characteristics: near polar, near eccentric, and short baselines of typically 1 2 km length. The baseline orientations in the latter three FFtypes contain additional cross-track and/or radial information. They cover a larger azimuth-elevation sampling space. As a result, compared to the GRACEtype formation, all three other formations are superior in terms of more gravitational signal content, though not equally. The qualitative difference in the respective FF- SST observable is reflected in the recovered gravity fields. Comparison between the input and output fields demonstrates that the recovery process using an observable with radial and/or cross-track information results in drastically improved accuracy and isotropy. The recovered solutions from the latter three formation types possess a lower error spectrum with a more homogeneous structure. Keywords. Observation equation, formation flight, gravity field recovery, satellite-to-satellite tracking 1 Introduction The key GRACE-type SST observables are the intersatellite distance and scalar relative velocity in a leader-follower configuration at near-polar inclination. This type of observable inherently suffers from the weakness that it is mainly sensitive along the lineof-sight, i.e. in North-South direction. Consequently, typical streaks emerge along the meridians in the monthly GRACE solutions (e.g. Tapley et al., 24). Sneeuw et al. (26) have shown that the weakness can be overcome if the SST observation contains radial and/or cross-track gravitational signal. To that end, a next generation gravity mission should preferably be a FF with a rotating baseline in the satellites local frame. To support and quantify this assertion a gravity recovery experiment is set up that makes use of the concept of formation flying. Four generic types of Low-Earth Formations (LEF) are simulated: a GRACE-type leader-follower configuration, with along-track orientation; a pendulum scenario, which adds a cross-track component, but stays in the local horizon; a Cartwheel formation that performs 2:1-relative elliptical motion in the orbital plane, i.e. stays in the local vertical plane; and a LISA-type formation that performs circular relative motion including out-of-plane and radial relative motion. All four LEF will have a typical baseline length of around 1 2 km. The names GRACE, Pendulum, Cartwheel and LISA will be used in this paper as generic for these basic formation types and should not be mistaken for actual missions. The dynamics of such formations are easily understood in the framework of homogeneous Hill equations. Thus, the first part of the paper is concerned with initial conditions and modelling the relative motion of the configurations in the local Hill frame. To show the differences in the signal content of the configurations, the second part of the paper is dedicated to their comparative analysis at the observation level. It shows how the projection of the additional components into the observable amplify the signal. Eventually, the recovered solutions are compared from different perspectives both in the time and frequency domains. 2 Equations of relative motion Let us adopt the following formation flying convention. All formations consist of a chief satellite and one or more deputy satellites. We assume a local orbital reference frame (or Hill frame) with its origin

2 in this chief satellite and oriented in along-track (x), cross-track (y) and radial (z) direction. The relative motion of the deputy in the local orbital reference frame is described in general by Schaub and Junkins (23): ( ẍ + z θ + 2ż θ x θ 2 µ ) rc 3 = a x (1a) ÿ + µ rc 3 y = a y (1b) ( z z θ µ ) rc 3 x θ 2ẋ θ = a z (1c) where µ is the gravitational constant and (a x,a y,a z ) are non-keplerian forces acting on the deputy satellite. They could be due to atmospheric drag, Earth oblateness effects or control thrusters. Here we consider only formations where the chief motion is essentially circular. In this case the chief rate θ is constant and equal to the mean orbit rate n = µ/r 3 c. The equations of motion simplify to the well-known linearized Hill equations (Hill, 1878), see also (Clohessy and Wiltshire, 196): ẍ + 2nż = a x ÿ + n 2 y = a y z 2nẋ 3n 2 z = a z (2a) (2b) (2c) We will refer to them as HE in the sequel. Equation (2) has been used extensively in spacecraft formation flying mission analysis and control research. They are reasonable as long as (x,y,z) are small compared to the chief orbital radius. Since they are linear, the HE can be solved analytically. Assuming no perturbations or thrusting is present (a x = a y = a z = ), all possible deputy relative motions can be expressed in closed form (Schaub and Junkins, 23): x(t) = 2A sin(nt + α) 3 2 ntz off + x off y(t) = B cos(nt + β) z(t) = A cos(nt + α) + z off (3a) (3b) (3c) Note that the out-of-plane motion is decoupled from the in-plane motion. The integration constants can be expressed in terms of initial conditions through: A = 1 ż 2 n + (2ẋ + 3nz ) 2 (4a) B = 1 ẏ 2 n + (ny ) 2 (4b) ( ) ż α = arctan (4c) 3nz + 2ẋ ( ) ẏ β = arctan ny (4d) z off = 2 n (ẋ + 2nz ) (4e) x off = x 2ż n (4f) The solution of the homogeneous HE allows to analyze the motion of the aforementioned four basic formation types in terms of the parameters in (4). For the relative motion to be bounded, we must require the drift term to vanish, i.e. z off = for all missions. The four generic FF types are now characterized by: GRACE is purely along-track. All periodic terms are zero and the variable x off determines the baseline length. The pendulum scenario also has a constant alongtrack term x off, but additionally a non-zero crosstrack amplitude B. The relative motion takes place in the xy-plane, i.e. the local horizontal plane. The intersatellite baseline is variable. Only its component in along-track direction is constant. The Cartwheel configuration has a non-zero A value. Without cross-track motion (B = ) this results in an in-plane elliptical relative motion (Massonnet, 21). The maximum along-track separation is twice as large as the maximum radial separation. Hence a 2:1 relative ellipse. The LISA-type mission achieves a relative circular motion by setting B = 3A and matching the phases α and β. Within the approximation of the HE, the baseline is constant. The necessary condition for achieving these configurations and the corresponding differential elements in the Hill frame are summarized in table 1. These differential elements can be converted to inertial orbital elements for integration purposes (Alfriend et al., 2). In table 1, relative position and velocity vectors are expressed in terms of the chief satellite s Kepler elements (a,e,u) and the baseline length ρ. The latter two configurations are defined by setting the free parameter (z ). The homogeneous HE are a helpful tool for first order formation design. However, the solution in (3) is no longer valid if the chief motion is not circular. Even small amounts of eccentricity can produce modelling errors comparable to those produced by J 2 gravitational perturbations or atmospheric drag. Alternatively, the formations can be designed directly using differential Kepler elements. This ap-

3 Table 1. Initial conditions and state vectors of different formation types formation initial conditions initial position (ρ ) initial velocity ( ρ ) GRACE A = B = z off = (ρ,, ) (,, ) Pendulum A = z off = (ρ x, ρ y cos u, ) (, nρ y sin u, ) Cartwheel B = x off = z off = (±2 4a 2 e 2 z 2,, z 2ae) ( 2nz,, ±n 4a 2 e 2 z 2 ) LISA x off = z off = ; α = β B = 3A = 3ρ/2 (± ρ 2 4z 2, ± 3z, z ρ/2) ( 2nz, ± 3n ρ 2 /4 z 2, ±n ρ2 /4 z 2 ) proach, in contrast, can be used for designing a configuration even with an elliptical chief motion in J 2 gravitational field (Sneeuw et al., 26). 3 Observation equation of FF LL-SST As for the GRACE mission, let us assume range-rate as the basic observable of a future LEF mission. The range rate ρ between two satellites is the projection of the relative vectorial velocity ρ on the line-of-sight unit vector e, e.g. (Keller and Sharifi, 25). The scalar range acceleration is derived by time differentiation and involves additional centrifugal terms: ρ = ρ e (5a) ρ = ρ e + 1 ( ρ ρ ρ 2 ). ρ (5b) Using Newton s equations, the vectorial acceleration difference ρ equals the difference in gravitational attraction V 1,2 between the two satellites 1 and 2. For practical applications, the scalar range acceleration ρ can be obtained from the observed range rate by numerical differentiation. To extract the gravitational information, one should further correct for the relative velocity terms at the right of (5b). In the absence of the nuisance forces, the vectorial gradient difference V1,2 LOS is parameterized in terms of the unknown spherical harmonic coefficients. To set up the mathematical model for the recovery of the field, Newton s equation is employed and (5b) is recast into V1,2 LOS = ρ + ρ2 ρ ρ 2. (6) ρ The left-hand side represents the gravitational attraction difference between the two satellites projected along the Line Of Sight (LOS). As such, it is justified to speak of spaceborne gravimetry, though in a differential sense. The right-hand side consists of the HL- and LL-SST measurements. Depending on the formation type, each term on the observation side has different magnitude and pattern and consequently different contribution to the total observable. Furthermore, the recovered solution s quality explicitly depends on the gravity signal captured by the formation type. Therefore, the accuracy and the resolution of the recovered field is a configuration specific aspect. To underline this point, the signal decomposition according to (6) is visualized in 1 both for GRACE and LISA. The top row represents the quantity at the left-hand side of (6), i.e. the quantity that is used to extract gravity field information. The left column is a visualization of one orbital revolution in the time domain, whereas the right column shows the power spectral density of the signal in the measurement bandwidth. L [mgal] c 1 c 3 c time domain GRACE LISA time [min] = + power [mgal/ Hz] frequency domain = frequency [mhz] Figure 1. Observation decomposition in time and space domains (L = ρ e, c 1 = ρ, c 2 = ρ 2 /ρ and c 3 = ρ 2 /ρ) As can be clearly seen in the time domain representation, the total gravitational signal of LISA and its individual components are significantly larger than those of GRACE. Comparing the missions spectra shows that LISA captures more information than GRACE. This richer gravitational signal content is due to the the radial and cross-track components in the relative motion. The GRACE mission, in contrast, only contains the along-track component which is a from a gravity gradiometry viewpoint a relatively weak observ-

4 able (Sneeuw and Schaub, 25). Apparently the common mode motion of the satellites in a GRACEtype mission cancels a large part of the signal. It should also be noted here that the LISA s spectrum is at least one order of magnitude higher than that of the GRACE. Moreover, more peaks are clearly visible in the LISA s spectrum. Further numerical analysis (not shown here) would demonstrate that the LISA mission has the richest signal, followed by Cartwheel and Pendulum. 4 Simulation setup For a comparative study of the four mission types closed-loop simulations have been performed. All configurations have comparable orbit characteristics: near polar, near eccentric, and short baselines of typically 1 2 km length. In order to achieve a nearly stable and bounded motion, the necessary conditions, listed in table 1, were imposed on the initial values. The relative state vectors can then be transformed to the corresponding initial differential Kepler elements (Alfriend et al., 2). Alternatively, the formations can be designed directly in terms of differential Kepler elements (Sneeuw et al., 26). We have used the latter approach and the obtained initial differential Kepler elements have been listed in table 2. The differential Table 2. Differential mean orbital elements Grace pendul. cartwh. LISA a e.1 I [ ] Ω[ ] ω [ ] M [ ] elements of the GRACE and Pendulum missions are the same, except for the ascending node definition. In the latter mission a non-zero value for Ω is required to introduce cross-track baseline component. In a next step, for all missions observations of differential gravimetry type (6) were simulated. The mission duration and the sampling frequency are one month and.2 Hz, respectively. To generate these SST observations EGM96 was used up to degree and order 6 as input model. All kinematic quantities like intersatellite range, range-rate and rangeacceleration vectors are derived from the orbit integration process. As shown in (6), the observable consists of the LL-mode observations ( ρ, ρ) and the HL-mode measurements ρ. Noise-free simulated range-rate is contaminated with a white noise with σ ρ = 1µm/s, comparable to the nominal noise RMS of the real GRACE mission (Reigber et al., 25). The rangeacceleration ρ is then numerically derived from the noisy range-rate observations using spline differentiation. Furthermore, noise-free simulated relative velocity vectors are contaminated with a correlated noise sequence, that reflects the GPS-derived LEO orbit coordinates. We assumed a standard deviation of 6 mm in each coordinate direction with a correlation length of about 3 minutes. An identical noise pattern but independently simulated sequences were used for each individual component of the relative velocity vector. These noise sequences were used for all the formations. 5 Gravity field recovery To close the simulation loop gravity fields were recovered from the noisy observations for all four scenarios. Spherical harmonic coefficients and their covariance matrix were estimated using brute-force least squares inversion. degree RMS original signal GRACE error Pendulum error Cartwheel error Lisa error degree Figure 2. Dimensionless degree RMS of the recovered solutions. The achieved results are evaluated from different perspectives. A first spectral comparison in terms of error degree RMS curves shows that the introduction of cross-track and/or radial information into the observable greatly improves the quality of the solution by nearly an order of magnitude, see figure 2. As expected, the GRACE mission, with along-track information alone, yields the poorest solution whereas the LISA mission gives the best due to the contribution of all three components. It is followed by Cartwheel mission which moves only in satellites orbital plane. Compared to LISA, the Cartwheel ob-

5 servable does not carry cross-track information. Nevertheless the Cartwheel solution is just marginally worse than LISA s. In contrast, the Pendulum observable contains only along-track and cross-track, i.e. horizontal information. Therefore, it does not achieve the performance level of the Cartwheel mission. These effects, in which the radial component is the dominant source of gravity field information, is known from satellite gravity gradiometry as well, e.g. (Sneeuw, 2). Still, the Pendulum configuration outperforms a GRACE-type LEF. By lumping over the order m, the degree RMS curves are not representative of non-isotropic error spectra. Therefore we visualize the full error spectra in 3, both in terms of input-minus-output coefficient differences C lm and S lm and in terms of formal standard deviations. sectorial harmonics, the recovered spectrum from a Cartwheel-type mission is relatively homogeneous. It s error spectrum resembles a cross between V xz and V zz gradiometry. A LISA-type mission also appears to provide a homogeneous error spectrum, although the error structure resembles that of the observable (V xx V yy ), a term that is obtained by rotating gradiometers differences C lm, S lm standard deviations σ lm degree Slm < order > Clm 6 Coefficient Size [log1] Slm < order > Clm Figure 3. Spherical harmonic error spectra (from top: GRACE, Pendulum, Cartwheel and LISA). Left: actual input-minus-output coefficient differences. Right: formal standard deviations. Indeed, these full error spectra reveal the nonisotropic behaviour of some of the mission types. A GRACE-type mission performs best in the low order range. Its error spectrum resembles that of alongtrack gradiometry (V xx ). The reverse is true for the Pendulum mission, whose error spectrum resembles that of cross-track (V yy ) gradiometry. Except of Figure 4. The geoid height residuals in meter (from top: GRACE, Pendulum, Cartwheel and LISA). Note the different gray scale for GRACE. These spectral results are also reflected in the spatial domain. Differences between the geoid height of the input and recovered models show the spatial distribution of the errors. Due to preference for low orders, the familiar North-South streaks are clearly visible in GRACE geoid errors. In contrast, they ap-

6 pear in East-West direction in the Pendulum configuration. The last two solutions are almost homogeneous although a very weak diagonal pattern is observed in the LISA solution. 6 Conclusion When designing future gravity field missions, formations that involve a cross-track or radial component outperform a GRACE-type leader-follower configuration. The observable in such formations are significantly richer in gravitational content, leading to a higher S/N ratio. Observing at least one of these components, preferably the radial one, improves the results both in terms of error level and of isotropy. Employing the Pendulum mission involves the gravity gradient component in East-West direction whereas the Cartwheel scenario projects the radial gravity gradient V zz onto the SST observable. The observable in a LISA-type mission combines all the components. Thus, the inherent weakness and the non-isotropic behaviour of the conventional low-low SST observable can be solved by formation flying. If the relative orbits comprise a cross-track motion, the corresponding observables gain sensitivity in East-West direction. This may be helpful in dealiasing signals. Moreover, including the radial component s contribution leads to a nearly homogeneous results in the LISA configuration. Consequently, a LEF with sufficiently many satellites linked together in a strategic way, can observe full-tensor gravity gradiometry. References Alfriend, K.T., H. Schaub, D.W. Gim (2). Gravitational Perturbations, Nonlinearity and Circular Orbit Assumption Effects on Formation Flying Control Strategies. Proceedings of the Annual AAS Rocky Mountain Conference, Breckenridge, CO, Feb. 26, pp Clohessy W.H., R.S. Wiltshire (196). Terminal guidance system for satellite rendezvous. Journal of the Areospace Sciences, 27: Hill, G.W. (1878). Researches in the lunar theory. Am. Journal of Math., I:5 26, , Massonnet, D. (21). The interferometric Cartwheel: a constellation of passive satellite to produce radar images to be coherently combined. Int. J. Remote Sensing 22: Reigber, C., R. Schmidt, F. Flechtner, R. König, U. Meyer, K.H. Neumayer, P. Schwintzer, S.Y. Zhu (25). An Earth gravity field model complete to degree and order 15 from GRACE: EIGEN-GRACE2S. Journal of Geodynamics 39:1 1 Keller, W., M.A. Sharifi (25) Gradiometry using a satellite pair. Journal of Geodesy 78: Schaub, H., J.L. Junkins (23). Analytical mechanics of space systems. AIAA education series Reston VA Sneeuw, N. (2). A semi-analytical approach to gravity field analysis from satellite observations. Deutsche Geod ätische Kommission, Series C, Nr Sneeuw, N., H. Schaub (25). Satellite Cluster for future gravity field missions. In: Proc of IAG symposia Gravity, Geoid and space missions Jekeli C Bastos L Fernandes J (eds) Vol. 129, pp 12 17, Springer Sneeuw, N., M.A. Sharifi W. Keller (26). Gravity Recovery from Formation Flight Missions. In: Proc of IAG symposia Hotine-Marussi Symposium of Theoretical and Computational Geodesy: Challenge and Role of Modern Geodesy, Wuhan, 29 May to 2 June 26 Tapley B.D., S. Bettadpur, J.C. Ries, P.F. Thompson, M.M. Watkins (24) GRACE measurements of mass variability in the Earth system. Science 35: 53 55