Geodetic Methods for Calibration of GRACE and GOCE

It is beyond doubt that calibration and validation are essential tools in the process of reaching the goals of gravity missions like GRACE and GOCE and to obtain results of the highest possible quality. Both tools, although general and obvious instruments for any mission, have specific features for gravity missions. Therefore, it is necessary to define exactly what is expected (and what cannot be expected) from calibration and what from validation and how these tools should work in our case. The general calibration and validation schemes for GRACE and GOCE are outlined. Calibration will be linked directly to the instrument and the measurements whereas validation will be linked to data derived from the original measurements. Calibration includes on-ground, internal, and external calibration as well as error assessment. The calibration phase results in corrected measurements along with an a posteriori error model. Validation of e.g. calibrated measurements or geoid heights means checking against independent data to assess whether there are no systematic errors left and/or whether the error model describes the true error reasonably well. Geodetic methods for calibration typically refer to external calibration and error assessment, and will be illustrated with an example. This revised version was published online in August 2006 with corrections to the Cover Date.     

Space-Wise,Time-Wise,Torus and Rosborough Representations in Gravity Field Modelling

The decade of the geopotentials started July 2000 with the launch of the German high-low SST mission CHAMP. Together with the joint NASA-DLR low-low SST mission GRACE and the ESA gradiometry mission GOCE an unprecedented wealth of geopotential data becomes available over the next few years. Due to the sheer number of unknown gravity field parameters (up to 100 000) and of observations (millions), especially the latter two missions are highly demanding in terms of computational requirements. In this paper several modelling strategies are presented that are based on a semi-analytical approach. In this approach the set of normal equations becomes block-diagonal with maximum block-sizes smaller than the spherical harmonic degree of resolution. The block-diagonality leads to a rapid and powerful gravity field analysis tool. Beyond the more-or-less conventional space-wise and time-wise formulations, the torus approach and Rosborough's representation are discussed. A trade-off between pros and cons of each of the modelling strategies will be given. This revised version was published online in August 2006 with corrections to the Cover Date.     

Possible Future Use of Laser Gravity Gradiometers

With the GRACE mission under way and the GOCE mission well along in the design process, detailed questions concerning the type of future mission that may follow them have arisen. It is generally agreed that determining the time variations in the Earth's gravity field with as high spatial and temporal resolution as is feasible will be the main driver for such a mission. The possible use of laser heterodyne measurements between separate satellites in such a mission has been discussed by a number of people. The first suggestion of emphasizing time variation measurements in a laser mission was the TIDES concept presented in 1992 by Colombo and Chao. Then, in 2000, a GRACE Follow-On mission using laser measurements between two drag-free satellites was discussed by Watkins el al. (2000). More recently, the possibility of utilizing laser measurements between more than two satellites in order to determine two or more components of the gravity gradient tensor simultaneously has been proposed by Balmino. This approach may be desirable in order to reduce the aliasing of time variations between geopotential terms of different degree and order, as well as to improve the resolution in longitude, despite the cost of the additional satellites. In this paper, we discuss specific possible mission geometries for measuring the two diagonal in-plane components of the gravity gradient tensor simultaneously. This could be done, for example, by laser heterodyne measurements between two pairs of satellites in coplanar and nearly polar orbits. This revised version was published online in August 2006 with corrections to the Cover Date.     

Combined Use of Altimetry and In Situ Gravity Data for Coastal Dynamics Studies

Accurate local geoids derived from in situ gravity data will be valuable in the validation of GOCE results. In addition it will be a challenge to use GOCE data in an optimal way, in combination with in situ gravity, to produce better local geoid solutions. This paper discusses the derivation of a new geoid over the NW European shelf, and its comparison with both tide gauge and altimetric sea level data, and with data from ocean models. It is hoped that over the next few years local geoid methods such as these can be extended to cover larger areas and to incorporate both in situ and satellite measured gravity data. This revised version was published online in August 2006 with corrections to the Cover Date.     

Microscope Instrument Development,Lessons for GOCE

Two space missions are presently under development with payload based on ultra-sensitive electrostatic accelerometers. The GOCE mission takes advantage of a three axis gradiometer accommodated in a very stable thermal case on board a drag-free satellite orbiting at a very low altitude of 250 km. This ESA mission will perform the very highly accurate mapping of the Earth gravity field with a geographical resolution of 100 km. The MICROSCOPE mission is devoted to the test of the “Universality of free fall” in view of the verification of the Einstein Equivalence Principle (EP) and of the search of a new interaction. The MICROSCOPE instrument is composed of two pairs of differential electrostatic accelerometers and the accelerometer proof-masses are the bodies of the EP test. The satellite is also a drag-free satellite exhibiting a fine attitude control and in a certain way, each differential accelerometer is a one axis gradiometer with an arm of quite null length. The development of this instrument much interests the definition and the evaluation of the sensor cores of the gradiometer. The in flight calibration process of both instruments is also very similar. Lessons form these parallel developments are presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

V: SEA LEVEL: Benefits of GRACE and GOCE to sea level studies

The recently published Third Assessment Reports of the Intergovernmental Panel on Climate Change have underlined the scientific interest in, and practical importance of past and potential future sea level changes. Space gravity missions will provide major benefits to the understanding of the past, and, thereby, in the prediction of future, sea level changes in many ways. The proposal for the GOCE mission described well the improvements to be expected from improved gravity field and geoid models in oceanography (for example, in the measurement of the time-averaged, or ‘steady state’, ocean surface circulation and better estimation of ocean transports), in geophysics (in the improvement of geodynamic models for vertical land movements), in geodesy (in positioning of tide gauge data into the same reference frame as altimeter data, and in improvement of altimeter satellite orbits), and possibly in glaciology (in improved knowledge of bedrock topography and ice sheet mass fluxes). GRACE will make many important steps towards these ‘steady state’ aims. However, its main purpose is the provision of oceanographic (and hydrological and meteorological) temporally-varying gravity information, and should in effect function as a global ‘bottom pressure recorder’, providing further insight into the 3-D temporal variation of the ocean circulation, and of the global water budget in general. This paper summaries several of these issues, pointing the way towards improved accuracy of prediction of future sea level change. This revised version was published online in August 2006 with corrections to the Cover Date.     

How to Climb the Gravity Wall

Space Science Reviews - What type of gravity satellite mission is required for the time after GRACE and GOCE? Essentially, the variables at our disposal are experiment altitude, compensation of...     

Aiming at a 1-cm Orbit for Low Earth Orbiters: Reduced-Dynamic and Kinematic Precise Orbit Determination

The computation of high-accuracy orbits is a prerequisite for the success of Low Earth Orbiter (LEO) missions such as CHAMP, GRACE and GOCE. The mission objectives of these satellites cannot be reached without computing orbits with an accuracy at the few cm level. Such a level of accuracy might be achieved with the techniques of reduced-dynamic and kinematic precise orbit determination (POD) assuming continuous Satellite-to-Satellite Tracking (SST) by the Global Positioning System (GPS). Both techniques have reached a high level of maturity and have been successfully applied to missions in the past, for example to TOPEX/POSEIDON (T/P), leading to (sub-)decimeter orbit accuracy. New LEO gravity missions are (to be) equipped with advanced GPS receivers promising to provide very high quality SST observations thereby opening the possibility for computing cm-level accuracy orbits. The computation of orbits at this accuracy level does not only require high-quality GPS receivers, but also advanced and demanding observation preprocessing and correction algorithms. Moreover, sophisticated parameter estimation schemes need to be adapted and extended to allow the computation of such orbits. Finally, reliable methods need to be employed for assessing the orbit quality and providing feedback to the different processing steps in the orbit computation process. This revised version was published online in August 2006 with corrections to the Cover Date.     

VI: FUTURE CONCEPTS: Attitude and Drag Control: An Application to the GOCE Satellite

The Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite, currently planned to he launched in the course of 2006, will require a precise drag compensation and a fine attitude control along the Local Orbiting Reference Frame (LORF) of a polar Sun-synchronous low orbit, allowing the Earth gravity field to be recovered with unprecedented accuracy by post-processing the scientific telemetry. To this aim, the spectral density of the spacecraft linear and angular accelerations must be limited below 0.025 respectively, in the frequency range from 5 mHz to 0.1 Hz, the gradiometer measurement bandwidth. In the same range, the orientation errors of the spacecraft in the LORF and of the LORF in the inertial frame must be kept below 10 . The Drag-Free Mode, encharged of drag-free and attitude control (DFAC) during measurement phases, determines the spacecraft state vector using a very precise gradiometer, one large Field-of-View Star Tracker and a Satellite-to-Satellite Tracking Instrument. Force and torque commands are actuated by two assemblies of thrusters: a single ion-thruster acting along the orbital direction, a set of eight micro-thrusters acting along the other five degrees of freedom. To cover every mission scenario, other control modes have been studied and designed: the Coarse Pointing Mode dedicated to rate damping and Sun acquisition, the Fine Pointing Mode handling the transition to Drag-Free Mode and the Ultimate Safe Mode, a survival operative mode to improve mission reliability. Results presented in this paper give a positive perspective on the solidity of the current DFAC design. This revised version was published online in August 2006 with corrections to the Cover Date.     

Resolution Needed for an Adequate Determination of the Mean Ocean Circulation from Altimetry and an Improved Geoid

The sea surface topography observed by satellite altimetry is a combination of the geoid and of the ocean dynamic topography. Satellite altimetry has thus the potential to supply quasi-global maps of mean sea surface heights from which the mean geostrophic surface ocean currents can be derived, provided that the geoid is known with a sufficient absolute accuracy. At present, however, given the limited accuracy of the best available geoid, altimetric mean sea surface topographies have been derived only up to degree 15 or so, i.e. for wavelengths of approximately 2000 km and larger. CHAMP, GRACE, and the future GOCE missions are dedicated to the improvement of the Earth's gravity field from space. Several studies have recently investigated the impact of these improvements for oceanography, concluding to reductions of uncertainties on the oceanic flux estimates as large as a factor of 2 in the regions of intense an narrow currents. The aim of this paper is to focus on what are the typical horizontal scales of the mean dynamic topography of the ocean, and to compare their characteristics to the error estimates expected from altimetry and these future geoids. It gives also an illustration of the oceanic features that will be resolved by the combination of altimetry and the GRACE and GOCE geoids. It further reassesses the very demanding requirements in term of accuracy and resolution agreed in the design of these new gravity missions for ocean science applications. The present study relies on recent very high-resolution numerical Ocean General Circulation Model simulations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Benefits from GOCE Within Solid Earth Geophysics

This contribution focuses on the results of a series of mathematical models which simulate the most important geophysical processes responsible for the deformation at crustal, lithospheric and mantle level. Post-Glacial Rebound, continent-continent collision and oceanic subduction are modelled by means of analytical and finite element models in order to emphasize the benefits in the usage of the GOCE data within Solid Earth Geophysics. Our results indicate that the new image of the Earth interior as provided by GOCE will contribute to better constrain, with respect to the present status, the physical parameters defining the outermost portion of out planet and its dynamics. This revised version was published online in August 2006 with corrections to the Cover Date.     

Deep and Shallow Solid-Earth Structures Reconstructed with Sequential Integrated Inversion (SII) of Seismic and Gravity Data

In this paper, the possibility of using simultaneously seismic and gravity data, for the reconstruction of solid-Earth structures, has been investigated through the use of an algorithm which allows joint efficient and reliable optimisation of compressional velocity and mass density parameters. We view the measured data as a realisation of a stochastic process generated by the physical parameters to be sought and we construct a “probability density function” which includes three kinds of information: information derived from gravity measurements; information derived from seismic travel time inversion and information on the physical correlation among density and velocity parameters. We show that combining data has a beneficial effect on the inversion since: it makes the problem more stable and as a consequence, providing that the quality of data is sufficiently high, enables more accurate and reliable reconstruction of the unknown parameters. In this context, we look forward the GOCE mission, which promises high spatial resolution (100–200 km) and accurate (1–2 mGals) gravity data. We show results obtained from data sets calculated for a lateral inhomogeneous earth synthetic model and from seismic and gravity field data analysed: — in the framework of TOMOVES (TOMOgraphy of Mt. VESuvius) experiment, an European project aiming at reconstructing the 3-D image of Mt. Vesuvius volcano and the crust underneath. using high resolution seismic tomography techniques and other geophysical methods; — for a profile inserted in a project aiming at reconstructing the crustal structure between Corsica and the Northern Appennines which crosses the Ligurian Sea and cuts the Ligurian Appennines W of La Spezia, extending up to Parma. This revised version was published online in August 2006 with corrections to the Cover Date.     

Gravity Field Recovery from GRACE: Unique Aspects of the High Precision Inter-Satellite Data and Analysis Methods

The very high accuracy of the Doppler and range measurements between the two low-flying and co-orbiting spacecraft of the GRACE mission, which will be at the μm/sec and ≈10 μm levels respectively, requires that special procedures be applied in the processing of these data. Parts of the existing orbit determination and gravity field parameters retrieval methods and software must be modified in order to fully benefit from the capabilities of this mission. This is being done in the following areas: (i) numerical integration of the equations of motion (summed form, accuracy of the predictor-corrector loop, Encke's formulation): (ii) special inter-satellite dynamical parameterization for very short arcs; (iii) accurate solution of large least-squares problems (normal equations vs. orthogonal decomposition of observation equations); (iv) handling the observation equations with high accuracy. Theoretical concepts and first tests of some of the newly implemented algorithms are presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

Ocean Tides in GRACE Monthly Averaged Gravity Fields

The GRACE mission will map the Earth's gravity fields and its variations with unprecedented accuracy during its 5-year lifetime. Unless ocean tide signals and their load upon the solid earth are removed from the GRACE data, their long period aliases obscure more subtle climate signals which GRACE aims at. In this analysis the results of Knudsen and Andersen (2002) have been verified using actual post-launch orbit parameter of the GRACE mission. The current ocean tide models are not accurate enough to correct GRACE data at harmonic degrees lower than 47. The accumulated tidal errors may affect the GRACE data up to harmonic degree 60. A study of the revised alias frequencies confirm that the ocean tide errors will not cancel in the GRACE monthly averaged temporal gravity fields. The S₂ and the K₂ terms have alias frequencies much longer than 30 days, so they remain almost unreduced in the monthly averages. Those results have been verified using a simulated 30 days GRACE orbit. The results show that the magnitudes of the monthly averaged values are slightly higher than the previous values. This may be caused by insufficient sampling to fully resolve and reduce the tidal signals at short wavelengths and close to the poles. This revised version was published online in August 2006 with corrections to the Cover Date.     

What Might GRACE Contribute to Studies of Post Glacial Rebound?

The NASA/DLR satellite gravity mission GRACE, launched in March, 2002, will map the Earth's gravity field at scales of a few hundred km and greater, every 30 days for five years. These data can be used to solve for time-variations in the gravity field with unprecedented accuracy and resolution. One of the many scientific problems that can be addressed with these time-variable gravity estimates, is post glacial rebound (PGR): the viscous adjustment of the solid Earth in response to the deglaciation of the Earth's surface following the last ice age. In this paper we examine the expected sensitivity of the GRACE measurements to the PGR signal, and explore the accuracy with which the PGR signal can be separated from other secular gravity signals. We do this by constructing synthetic GRACE data that include contributions from a PGR model as well as from a number of other geophysical processes, and then looking to see how well the PGR model can be recovered from those synthetic data. We conclude that the availability of GRACE data should result in improved estimates of the Earth's viscosity profile. This revised version was published online in August 2006 with corrections to the Cover Date.     

Gravity Recovery and Interior Laboratory (GRAIL): Mapping the Lunar Interior from Crust to Core

The Gravity Recovery and Interior Laboratory (GRAIL) is a spacecraft-to-spacecraft tracking mission that was developed to map the structure of the lunar interior by producing a detailed map of the gravity field. The resulting model of the interior will be used to address outstanding questions regarding the Moon’s thermal evolution, and will be applicable more generally to the evolution of all terrestrial planets. Each GRAIL orbiter contains a Lunar Gravity Ranging System instrument that conducts dual-one-way ranging measurements to measure precisely the relative motion between them, which in turn are used to develop the lunar gravity field map. Each orbiter also carries an Education/Public Outreach payload, Moon Knowledge Acquired by Middle-School Students (MoonKAM), in which middle school students target images of the Moon for subsequent classroom analysis. Subsequent to a successful launch on September 10, 2011, the twin GRAIL orbiters embarked on independent trajectories on a 3.5-month-long cruise to the Moon via the EL-1 Lagrange point. The spacecraft were inserted into polar orbits on December 31, 2011 and January 1, 2012. After a succession of 19 maneuvers the two orbiters settled into precision formation to begin science operations in March 1, 2012 with an average altitude of 55 km. The Primary Mission, which consisted of three 27.3-day mapping cycles, was successfully completed in June 2012. The extended mission will permit a second three-month mapping phase at an average altitude of 23 km. This paper provides an overview of the mission: science objectives and measurements, spacecraft and instruments, mission development and design, and data flow and data products.     

How Operational Oceanography Can Benefit from Dynamic Topography Estimates as Derived from Altimetry and Improved Geoid

With a precise geoid, GOCE will allow an estimation of absolute dynamic topography from altimetry. The projected benefits to operational oceanography and its applications are analyzed herein. After a brief overview of operational oceanography, we explain how the new geoids will be used in the future to improve real time altimeter products and to better constrain modelling and data assimilation systems. A significant impact is expected both for mesoscale (e.g. better estimations and forecasts of currents for pollution monitoring, marine safety, offshore industry) and climate (better initialization of coupled ocean/atmosphere models) applications. This revised version was published online in August 2006 with corrections to the Cover Date.     

Tidal Models in a New Era of Satellite Gravimetry

The high precision gravity measurements to be made by recently launched (and recently approved) satellites place new demands on models of Earth, atmospheric, and oceanic tides. The latter is the most problematic. The ocean tides induce variations in the Earth's geoid by amounts that far exceed the new satellite sensitivities, and tidal models must be used to correct for this. Two methods are used here to determine the standard errors in current ocean tide models. At long wavelengths these errors exceed the sensitivity of the GRACE mission. Tidal errors will not prevent the new satellite missions from improving our knowledge of the geopotential by orders of magnitude, but the errors may well contaminate GRACE estimates of temporal variations in gravity. Solar tides are especially problematic because of their long alias periods. The satellite data may be used to improve tidal models once a sufficiently long time series is obtained. Improvements in the long-wavelength components of lunar tides are especially promising. This revised version was published online in August 2006 with corrections to the Cover Date.     

New Horizons Mission Design

In the first mission to Pluto, the New Horizons spacecraft was launched on January 19, 2006, and flew by Jupiter on February 28, 2007, gaining a significant speed boost from Jupiter’s gravity assist. After a 9.5-year journey, the spacecraft will encounter Pluto on July 14, 2015, followed by an extended mission to the Kuiper Belt objects for the first time. The mission design for New Horizons went through more than five years of numerous revisions and updates, as various mission scenarios regarding routes to Pluto and launch opportunities were investigated in order to meet the New Horizons mission’s objectives, requirements, and goals. Great efforts have been made to optimize the mission design under various constraints in each of the key aspects, including launch window, interplanetary trajectory, Jupiter gravity-assist flyby, Pluto–Charon encounter with science measurement requirements, and extended mission to the Kuiper Belt and beyond. Favorable encounter geometry, flyby trajectory, and arrival time for the Pluto–Charon encounter were found in the baseline design to enable all of the desired science measurements for the mission. The New Horizons mission trajectory was designed as a ballistic flight from Earth to Pluto, and all energy and the associated orbit state required for arriving at Pluto at the desired time and encounter geometry were computed and specified in the launch targets. The spacecraft’s flight thus far has been extremely efficient, with the actual trajectory error correction ΔV being much less than the budgeted amount.     

Needs and Tools for Future Gravity Measuring Missions

This paper compares the requirements that can be expected of gravity measuring missions with respect to the status of the instrumentation and satellite technologies. The error sources of gravity gradiometry and satellite-to-satellite tracking are analysed and the elements limiting the accuracy are identified. Proposed and approved future missions that will fly technologies of interest for gravity sensing are recalled. Areas of technical development of interest are reviewed. The article finishes with two possible conceptual missions presented as examples and with a chapter of conclusions. This revised version was published online in August 2006 with corrections to the Cover Date.