Anno Löcher

The Use of Energy Balance Relations for Validation of Gravity Field Models and Orbit Determination Results

There is a need for a proper validation procedure of gravity field solutions, especially for those high precise ones which are derived from the dedicated gravity field missions as CHAMP, GRACE and in future GOCE. In this paper the balance equations for the energy and for the energy exchange of a satellite orbiting around the Earth are proposed as analysis tool to validate gravity field solutions together with orbits derived with precise reduced dynamic or kinematic orbit determination procedures. The theoretical foundation of the analysis tool is presented as well as simulation results and applications to real orbits of a GPS satellite and of CHAMP based on various gravity field solutions. It is shown that the validation procedure can be used to detect deficiencies in the orbit modelling and in gravity field recovery results.

Do We Need New Gravity Field Recovery Techniques for the New Gravity Field Satellites

The classical approach of satellite geodesy consists in deriving the spherical harmonic coefficients representing the gravitational potential from an analysis of accumulated orbit perturbations of artificial satellites with different altitudes and orbit inclinations. This so-called differential orbit improvement technique required the analysis of rather long arcs of days to weeks; it was the adequate technique for satellite arcs poorly covered with observations, mainly precise laser ranging to satellites. The situation changed dramatically with the new generation of dedicated gravity satellites such as CHAMP, GRACE and – in a couple of months – GOCE. These satellites are equipped with very precise sensors to measure the gravity field and the orbits. The sensors provide a very dense coverage with observations independent from Earth based observation stations. The measurement concepts can be characterized by an in-situ measurement principle of the gravitational field of the Earth. In the last years various recovery techniques have been developed which exploit these specific characteristics of the in-situ observation strategy. This paper gives an overview of the various gravity field recovery principles and tries to systemize these new techniques. Alternative in-situ modelling strategies are presented based on the translational and rotational integrals of motion. These alternative techniques are tailored to the in-situ measurement characteristics of the innovative type of satellite missions. They complement the scheme of in-situ gravity field analysis techniques.

Towards Improved Lunar Reference Frames: LRO Orbit Determination

Lunar reference systems are currently realized by sets of coordinates of the few laser reflectors deployed by Apollo astronauts and unmanned Soviet spacecrafts. Expanding this coordinate knowledge to other features identifiable in images of the lunar surface requires highly accurate orbits of the acquiring spacecraft. To support such activities using images and altimetry data from the Lunar Reconnaissance Orbiter (LRO), an independent processing facility for tracking observations to LRO has been established. We present orbits from 1 year radio Doppler, radio ranging and laser ranging data obtained by different combinations of data types. To obtain an external confirmation for the achieved orbit accuracy, coordinates of the Apollo 15 reflector were measured in LRO images by photogrammetric techniques and compared to reference values from Lunar Laser Ranging (LLR). Coordinate differences were found to be at the 10 m level.

A Validation Procedure for Satellite Orbits and Force Function Models Based on a New Balance Equation Approach

Since the availability of CHAMP and GRACE data, the energy approach has become an important tool for the recovery of the gravity field based on continuously observed satellite orbits. Up to now, only the total energy of the satellite’s three-dimensional motion has been considered which is known as the Jacobi integral if formulated in a constantly rotating Earth-fixed reference frame. Beside this, additional energy integrals can be found for the components of the satellite’s motion and various combinations hereof, starting from the three scalar components of Newton’s equation of motion. Furthermore, integrals of motion based on the linear momentum and the angular momentum can be formulated which show even better mathematical characteristics than the Jacobi integral for the determination of the gravity field. Therefore, this new approach seems to be appropriate to validate the consistency of gravity field models and precisely observed satellite orbits and to improve, subsequently, these gravity field models. The advantages and critical aspects of this approach are investigated in this paper. First results with real data were presented using kinematic CHAMP orbits.

Precise orbits of the Lunar Reconnaissance Orbiter from radiometric tracking data

Since 2009, the Lunar Reconnaissance Orbiter (LRO) acquires images and altimetric profiles of the lunar surface. Assembling these data to maps and terrain models requires the precise knowledge of the spacecraft trajectory. In this contribution, we present 5 years of LRO orbits from radiometric data processed with a software tailored to this mission. The presented orbits are the first independent validation of the LRO science orbits from NASA and are available for public use. A key feature of our processing is the elaborate treatment of model and observation errors by empirical parameters and an adaptive data weighting by variance component estimation. The quality of the resulting orbits is assessed by analyzing overlapping arcs. For our solution based on arcs of 2.5 days, such analysis yields a mean error of 2.81 m in total position and 0.11 m in radial direction. It is shown that this result greatly benefits from the adaptive data weighting, reducing the error by 2.54 and 0.13 m, respectively. Unfortunately, the precision achieved varies strongly, dependent on the view onto the orbital ellipse which changes with the lunar cycle. To mitigate this dependency, the arc length was extended in steps up to 10.5 days, leading in the best case to a further improvement of 0.80 m.

Consistency of Geoid Models, Radar Altimetry, and Hydrodynamic Modelling in the North Sea

ABSTRACT Radar altimetry, when corrected for tides, atmospheric forcing of the sea surface, and the effects of density variations and mean and time-variable currents, provides an along-track realization of the marine geoid. In this study we investigate whether and how such an ‘altimetric-hydrodynamic’ geoid over the North Sea can serve for validating satellite-gravimetric geoids. Our results indicate that, using ERS-2 and ENVISAT along-track altimetry and water levels from the high-resolution operational circulation model BSHcmod, we do find distinct differences in RMS fits for various state-of-the art satellite-only models (beyond degree 145 for GRACE-only, and beyond degree 185 for GOCE models) and for combined geoid models, very similar as seen in GPS-levelling validations over land areas. We find that, at spectral resolution of up to about 200, an RMS fit as low as about 7 cm can be obtained for the most recent GOCE-derived models such as GOCO05S. This is slightly above what we expect from budgeting individual errors. Key to the validation is a proper treatment of the spectral mismatch between satellite-gravimetric and altimetric-hydrodynamic geoids. Comparison of data fits and error budget suggests that geoid truncation errors residual to EGM2008 (i.e. EGM2008 commission and omission error) may amount up to few cm.

Assessment of the impact of one-way laser ranging on orbit determination of the Lunar Reconnaissance Orbiter

From June 2009 to September 2014, ten stations of the International Laser Ranging Service acquired more than 5000 h of one-way ranging data to NASA’s Lunar Reconnaissance Orbiter (LRO). The LRO campaign was intended to support the determination of precise science orbits for which S-band Doppler and range measurements are the primary source. The laser ranges were finally not used for the NASA orbits, since the quality of the radiometric data exceeded the expectations. In this contribution, we explore the potential of the LRO laser ranges using an independent software which is able to process all types of LRO tracking data. It is shown that the laser ranges agree with the radiometric data at the level of 5 cm, provided a proper modeling of the LRO clock error is done, including relativistic effects. This accuracy does not fully attain that of the Doppler data, hence the impact of the laser ranges on orbit determination is small, improving the orbit on average by no more than 7 cm. Additional tests without the Doppler data show, however, that the laser ranges would have been of great use if radiometric tracking to LRO were less successful.

Energy Balance Relations for Validation of Gravity Field Models and Orbit Determinations Applied to the CHAMP Mission

An extended Jacobi integral describes the energy balance of the motion of a satellite referred to a terrestrial reference frame along its orbit. In addition to its classical form inertia forces and non-conservative force function contributions are included. If force function models and observed satellite orbits are consistent the energy balance sums up to a constant. Deviations from it can be caused either by orbit errors or by insufficiencies in the force function models. Therefore, the energy integral offers itself as a validation tool for consistency checks of force functions and orbit determination results. A basic question is the separation of the various sources of inconsistency. In this paper the theoretical foundation of the validation procedure is presented. It is shown that the validation method can be used to detect deficiencies in the orbit modeling and in the gravity field recovery results. Examples are presented how to separate the various causes of inconsistency. Applications to the results of the CHAMP mission demonstrate the procedure.