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.     

Feasibility,and Contribution to Ocean Circulation Studies,of Ocean Bottom Pressure Determination

An assessment is presented of the probable magnitude of ocean signals causing aliasing in ocean bottom pressure measurements from the GRACE satellite mission. Even after modelling as much of the high frequency signal as possible, variability between 1 mbar (in quiet ocean regions) and 10 mbar (on some shelves) is likely to remain. Interpretation of the resulting retrievals will therefore rely on the facts that the satellite sampling will average the aliasing signal to some extent, and that the spatial patterns of aliased signal and true signal will be different. To this end, a theoretical argument is given, and supported by model diagnostics, suggesting that observable bottom pressure signals will be strongly constrained by the shape of the ocean floor. The modelled magnitudes offer the prospect of significant detectable signals and, while the model accuracy can be called into question, there are hints from Earth rotation and satellite orbit measurements that significant mass redistributions occur in the ocean. It seems certain that we will learn something new about the oceans from GRACE. This revised version was published online in August 2006 with corrections to the Cover Date.     

Error Characteristics Estimated from Champ,GRACE and GOCE Derived Geoids and from Satellite Altimetry Derived Mean Dynamic Topography

This paper presents a review of geoid error characteristics of three satellite gravity missions in view of the general problem of separating scientifically interesting signals from various noise sources. The problem is reviewed from the point of view of two proposed applications of gravity missions, one is the observation of the mean oceanic circulation whereby an improved geoid model is used as a reference surface against the long term mean sea level observed by altimetry. In this case we consider the presence of mesoscale variability during assimilation of derived surface currents in inverse models. The other experiment deals with temporal changes in the gravity field observed by GRACE in which case a proposed experiment is to monitor changes in the geoid in order to detect geophysical interesting signals such as variations in the continental hydrology and non-steric ocean processes. For this experiment we will address the problem of geophysical signal contamination and the way it potentially affects monthly geoid solutions of GRACE. 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.     

Measuring the Distribution of Ocean Mass Using GRACE

The Gravity Recovery and Climate Experiment (GRACE), which was successfully launched March 17, 2002, has the potential to create a new paradigm in satellite oceanography with an impact perhaps as large as was observed with the arrival of precision satellite altimetry via TOPEX/Poseidon (T/P) in 1992. The simulations presented here suggest that GRACE will be able to monitor non-secular changes in ocean mass on a global basis with a spatial resolution of ≈500 km and an accuracy of ≈3 mm water equivalent. It should be possible to recover global mean ocean mass variations to an accuracy of ≈1 mm, possibly much better if the atmospheric pressure modeling errors can be reduced. We have not considered the possibly significant errors that may arise due to temporal aliasing and secular gravity variations. Secular signals from glacial isostatic adjustment and the melting of polar ice mass are expected to be quite large, and will complicate the recovery of secular ocean mass variations. Nevertheless, GRACE will provide unprecedented insight into the mass components of sea level change, especially when combined with coincident satellite altimeter measurements. Progress on these issues would provide new insight into the response of sea level to climate change. 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.     

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.     

VII: CLOSING SESSION: GOCE: ESA's First Earth Explorer Core Mission

This paper introduces the first ESA Core Earth Explorer mission, GOCE, in the context of ESA's Living Planet programme. GOCE will measure highly accurate, high spatial resolution differential accelerations in three dimensions along a well characterised orbit: the mission is planned for launch in early 2006. The mission objectives are to obtain gravity gradient data such that new global and regional models of the static Earth's gravity field and of the geoid can be deduced at length scales down to 100 km. These products will have broad application in the fields of geodesy, oceanography, solid-earth physics and glaciology. This revised version was published online in August 2006 with corrections to the Cover Date.     

Monitoring Changes in Continental Water Storage with GRACE

The Gravity Recovery and Climate Experiment, GRACE , will enable the recovery of monthly estimates of changes in water storage, on land and in the ocean, averaged over arbitrary regions having length scales of a few hundred km and larger. These data will allow the examination of changes in the distribution of water in the ocean, in snow and ice on polar ice sheets, and in continental water and snow storage. Extracting changes in water storage from the GRACE dataset requires the use of averaging kernels which can isolate a particular region. To estimate the accuracy to which continental water storage changes in a few representative regions may be recovered, we construct a synthetic GRACE dataset from global, gridded models of surface-mass variability. We find that regional changes in water storage can be recovered with rms error less than 1 cm of equivalent water thickness, for regions having areas of 4 × 10⁵ km² and larger. Signals in smaller regions may also be recovered; however, interpretations of such results require a careful consideration of model resolution, as well as the nature of the averaging kernel. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

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.     

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.     

The Lunar Gravity Ranging System for the Gravity Recovery and Interior Laboratory (GRAIL) Mission

The Lunar Gravity Ranging System (LGRS) flying on NASA’s Gravity Recovery and Interior Laboratory (GRAIL) mission measures fluctuations in the separation between the two GRAIL orbiters with sensitivity below 0.6 microns/Hz^1/2. GRAIL adapts the mission design and instrumentation from the Gravity Recovery and Climate Experiment (GRACE) to a make a precise gravitational map of Earth’s Moon. Phase measurements of Ka-band carrier signals transmitted between spacecraft with line-of-sight separations between 50 km to 225 km provide the primary observable. Measurements of time offsets between the orbiters, frequency calibrations, and precise orbit determination provided by the Global Positioning System on GRACE are replaced by an S-band time-transfer cross link and Deep Space Network Doppler tracking of an X-band radioscience beacon and the spacecraft telecommunications link. Lack of an atmosphere at the Moon allows use of a single-frequency link and elimination of the accelerometer compared to the GRACE instrumentation. This paper describes the implementation, testing and performance of the instrument complement flown on the two GRAIL orbiters.     

Spacecraft orbit determination using GPS navigation solutions

The orbit determination using the GPS navigation solutions for the KOMPSAT-1 spacecraft has been studied. The Cowell method of special perturbation theories was employed to develop a precision orbit propagation, and the perturbations due to geopotential, the gravity of the Sun and the Moon, solid Earth tides, ocean tides, the Earth's dynamic polar motion, solar radiation pressure, and atmospheric drag were modeled. Specifically, the satellite box-wing macro model was applied to minimize the drag errors at low altitude. The estimation scheme consisted of an extended Kalman filter and Bayesian least square method. To investigate the applicability of the method to the KOMPSAT-1 spacecraft, the orbit determination was accomplished using the GPS navigation solutions for the TOPEX/POSEIDON and TAOS satellites. The orbit determination results were compared with NASA POE generated by global laser tracking. The position and velocity accuracy was estimated about 16∼7 m and 0.0157∼0.0074 m·s⁻¹ RMS, respectively, for the two satellites in the presence of SA. These results verify that an orbit determination scheme using GPS navigation solutions can provide the static orbit information and reduce conspicuously the position and velocity errors of navigation solutions. It can be suggested that the sequential and batch orbit determination using the GPS navigation solutions be the most appropriate method in the KOMPSAT-1 type mission.     

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.     

III: OCEAN CIRCULATION: Global Ocean Data Assimilation and Geoid Measurements

Parts of geodesy and physical oceanography are about to mature into a single modeling problem involving the simultaneous estimation of the marine geoid and the general circulation. Both fields will benefit. To this end, we present an ocean state estimation (data assimilation) framework which is designed to obtain a dynamically consistent picture of the changing ocean circulation by combining global ocean data sets of arbitrary type with a general circulation model (GCM). The impact of geoid measurements on such estimates of the ocean circulation are numerous. For the mean circulation, a precise geoid describes the reference frame for dynamical signals in altimetric sea surface height observations. For the time-varying ocean signal, changing geoid information might be a valuable new information about correcting the changing flow field on time scales from a few month to a year, but the quantitative utility of such information has not yet been demonstrated. For a consistent estimate, some knowledge of the prior error covariances of all data fields is required. The final result must be consistent with prior error estimates for the data. State estimation is thus one of the few quantitative consistency checks for new geoid measurements anticipated from forthcoming space missions. Practical quantitative methods will yield a best possible estimate of the dynamical sea surface which, when combined with satellite altimetric surfaces, will produce a best-estimate marine geoid. The anticipated accuracy and precision of such estimates raises some novel modeling error issues which have not conventionally been of concern (the Boussinesq approximation, self-attraction and loading). Model skill at very high frequencies is a major concern because of the need to de-alias the data obtained by the inevitable oceanic temporal undersampling dictated by realistic satellite orbit configurations. 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.     

Spaceborne GPS receiver antenna phase center offset and variation estimation for the Shiyan 3 satellite

In determining the orbits of low Earth orbit (LEO) satellites using spaceborne GPS, the errors caused by receiver antenna phase center offset (PCO) and phase center variations (PCVs) are gradually becoming a major limiting factor for continued improvements to accuracy. Shiyan 3, a small satellite mission for space technology experimentation and climate exploration, was developed by China and launched on November 5, 2008. The dual-frequency GPS receiver payload delivers 1 Hz data and provides the basis for precise orbit determination within the range of a few centime-ters. The antenna PCO and PCV error characteristics and the principles influencing orbit determi-nation are analyzed. The feasibility of PCO and PCV estimation and compensation in different directions is demonstrated through simulation and in-flight tests. The values of receiver antenna PCO and PCVs for Gravity Recovery and Climate Experiment (GRACE) and Shiyan 3 satellites are estimated from one month of data. A large and stable antenna PCO error, reaching up to 10.34 cm in the z-direction, is found with the Shiyan 3 satellite. The PCVs on the Shiyan 3 satellite are estimated and reach up to 3.0 cm, which is slightly larger than that of GRACE satellites. Orbit validation clearly improved with independent k-band ranging (KBR) and satellite laser ranging (SLR) measurements. For GRACE satellites, the average root mean square (RMS) of KBR resid-uals improved from 1.01 cm to 0.88 cm. For the Shiyan 3 satellite, the average RMS of SLR resid-uals improved from 4.95 cm to 4.06 cm.     

Orbit Design for the THEMIS Mission

THEMIS, NASA’s fifth Medium Class Explorer (MIDEX) mission will monitor the onset and macro-scale evolution of magnetospheric substorms. It is a fleet of 5 small satellites (probes) measuring in situ the magnetospheric particles and fields while a network of 20 ground based observatories (GBOs) monitor auroral brightening over Northern America. Three inner probes (～1 day period, 10 R_E apogee) monitor current disruption and two outer probes (～2 day and ～4 day period, 20 R_E and 30 R_E apogees respectively) monitor lobe flux dissipation. In order to time and localize substorm onsets, THEMIS utilizes Sun–Earth aligned conjunctions between the probes when the ground-based observatories are on the nightside. To maintain high recurrence of conjunctions the outer orbits have to be actively adjusted during each observation season. Orbit maintenance is required to rearrange the inner probes for dayside observations and also inject the probes into their science orbits after near-simultaneous release from a common launch vehicle. We present an overview of the orbit strategy, which is primarily driven by the scientific goals of the mission but also represents a compromise between the probe thermal constraints and fuel capabilities. We outline the process of orbit design, describe the mission profile and explain how mission requirements are targeted and evaluated. Mission-specific tools, based on high-fidelity orbit prediction and common magnetospheric models, are also presented. The planning results have been verified by in-flight data from launch through the end of the first primary science seasons and have been used for mission adjustments subject to the early scientific results from the coast phase and first tail season.