VLBI for better gravimetry in SELENE

The Japanese lunar explorer SELENE (SElenological and Engineering Explorer), to be launched in 2007, will for the first time utilize VLBI observations in lunar gravimetry investigations. This will particularly improve the accuracy to which the low degree gravitational harmonics and the gravity field near the limb can be measured, and when combined with Doppler measurements will enable three-dimensional information to be extracted. Differential VLBI Radio sources called VRAD experiment involves two on-board sub-satellites, Rstar and Vstar. These will be observed using differential VLBI to measure the trajectories of the satellites with the Japanese network named VERA (VLBI Exploration of Radio Astrometry) and an international VLBI network.     

The simulation of lunar gravity field recovery from D-VLBI of Chang’E-1 and SELENE lunar orbiters

The lunar gravity field is a foundation to study the lunar interior structure, and to recover the evolution history of the Moon. It is still an open and key topic for lunar science. For above mentioned reasons, it becomes one of the important scientific objectives of recent lunar missions, such as KAGUYA (SELENE) the Japanese lunar mission and Chang’E-1, the Chinese lunar mission. The Chang’E-1 and the SELENE were successfully launched in 2007. It is estimated that these two missions can fly around the Moon longer than 6 months simultaneously. In these two missions, the Chinese new VLBI (Very Long Baseline Interferometry) network will be applied for precise orbit determination (POD) by using a differential VLBI (D-VLBI) method during the mission period. The same-beam D-VLBI technique will contribute to recover the lunar gravity field together with other conventional observables, i.e. R&RR (Range and Range Rate) and multi-way Doppler. Taking VLBI tracking conditions into consideration and using the GEODYNII/SOVLE software of GSFC/NASA/USA [8 and 10], we simulated the lunar gravity field recovering ability with and without D-VLBI between the Chang’E-1 and SELENE main satellite. The cases of overlapped flying and tracking period of 30 days, 60 days and 90 days have been analyzed, respectively. The results show that D-VLBI tracking between two lunar satellites can improve the gravity field recovery remarkably. The results and methods introduced in this paper will benefit the actual missions.     

Long-wavelength lunar gravity field recovery from simulated orbit and inter-satellite tracking data

As has been demonstrated recently, inter-satellite Ka-band tracking data collected by the GRAIL (Gravity Recovery And Interior Laboratory) spacecraft have the potential to improve the resolution and accuracy of the lunar gravity field by several orders of magnitude compared to previous models. By means of a series of simulation studies, here we investigate the contribution of inter-satellite ranging for the recovery of the Moon’s gravitational features; the evaluation of results is made against findings from ground-based Doppler tracking. For this purpose we make use of classical dynamic orbit determination, supported by the analysis of satellite-to-satellite tracking observations. This study sheds particularly light on the influence of the angular distance between the two satellites, solar radiation modeling and the co-estimation of the lunar Love number k₂. The quality of the obtained results is assessed by gravity field power spectra, gravity anomalies and precision orbit determination. We expect our simulation results to be supportive for the processing of real GRAIL data.     

Proposal of application of same beam VLBI measurements in precision orbit determination of lunar orbiter and return capsule and lunar gravity field simulation in Chang’E-3 mission

High accuracy differenced phase delay can be obtained by observing multiple point frequencies of two spacecraft using the same beam Very Long Baseline Interferometry (VLBI) technology. Its contribution in lunar spacecraft precision orbit determination has been performed during the Japanese lunar exploration mission SELENE. In consideration that there will be an orbiter and a return capsule flying around the moon during the Chinese lunar exploration future mission Chang’E-3, the contributions of the same beam VLBI in spacecraft precision orbit determination and lunar gravity field solution have been investigated. Our results show that the accuracy of precision orbit determination can be improved more than one order of magnitude after including the same beam VLBI measurements. There are significant improvements in accuracy of low and medium degree coefficients of lunar gravity field model obtained from combination of two way range and Doppler and the same beam VLBI measurements than the one that only uses two way range and Doppler data, and the accuracy of precision orbit determination can reach meter level.     

Simulation study for the determination of the lunar gravity field from PRARE-L tracking onboard the German LEO mission

A simulation study has been performed at GFZ Potsdam, which shows the anticipated improvement of the lunar gravity field model with respect to current (LP150Q model) or near-future (SELENE) knowledge in the framework of the planned German Lunar Explorations Orbiter (LEO) mission, based on PRARE-L (Precise Range And Range-rate Equipment – Lunar version) Satellite-to-Satellite (SST) and Satellite-Earth-Satellite (SEST) tracking observations. It is shown that the global mean error of the lunar gravity field can be reduced to less than 0.1 mGal at a spatial resolution of 50 km. In the spectral domain, this means a factor of 10 (long wavelengths) and some 100 (mid to short wavelengths) improvement as compared to predictions for SELENE or a factor of 1000 with respect to LP150Q. Furthermore, a higher spatial resolution of up to 28 km seems feasible and would correspond to a factor of 2–3 improvement of SELENE results. Moreover, PRARE-L is expected to derive the low-degree coefficients of the lunar gravity field with unprecedented accuracy. Considering long mission duration (at least 1 year is planned) this would allow for the first time a precise direct determination of the low-degree tidal Love numbers of the Moon and, in combination with high precision SEST, would provide an experimental basis to study relativistic effects such as the periselenium advance in the Earth–Moon system.     

A simulation of Martian gravity field recovery by using a near equatorial orbiter

An improvement to the Martian gravity field may be achieved by means of future orbiting spacecraft with small eccentricity and low altitude exemplified through a newly proposed mission design that may be tested in upcoming reconnaissance of Mars. Here, the near equatorial orbital character (with an inclination approximating 10°, eccentricity as 0.01 and semi-major axis as 4000 km) is considered, and its contribution to Martian gravity field solution is analyzed by comparing it with a hypothetical polar circular orbiter. The solution models are evaluated in terms of the following viewpoints: power spectra of gravity field coefficients, correlations of low degree zonal coefficients, precise orbit determination, and error distribution of both Mars free air gravity anomaly and areoid. At the same time, the contributions of the near equatorial orbiters in low degree zonal coefficients time variations are also considered. The present results show that the near equatorial orbiter allows us to improve the accuracy of the Martian gravity field solution, decrease correlation of low degree zonal coefficients, retrieve much better time variable information of low degree zonal coefficients, improve precise orbit determination, and provide more accurate Mars free air gravity anomaly and areoid around the equatorial region.     

Comparison analyses on the 150 × 150 lunar gravity field models by gravity/topography admittance,correlation and precision orbit determination

We analyzed the 150 × 150 lunar gravity field models, LP150Q, GLGM-3 and SGM150, using the power spectrum on the lunar nearside and farside, the lunar global and localized gravity/topography admittance and correlation, and Chang’E-2 precision orbit determination to investigate which model is a more effective tool to estimate geophysical parameters and determine the lunar satellite precision orbit. Results indicate that all gravity field models can be used to estimate the lunar geophysical parameters of the nearside of the Moon. However, SGM150 is better in such computation of the farside. Additionally, SGM150 is shown to be the most useful model for determining the lunar satellite orbit.     

Simulation-based evaluation of a cold atom interferometry gradiometer concept for gravity field recovery

The prospects of future satellite gravimetry missions to sustain a continuous and improved observation of the gravitational field have stimulated studies of new concepts of space inertial sensors with potentially improved precision and stability. This is in particular the case for cold-atom interferometry (CAI) gradiometry which is the object of this paper. The performance of a specific CAI gradiometer design is studied here in terms of quality of the recovered gravity field through a closed-loop numerical simulation of the measurement and processing workflow. First we show that mapping the time-variable field on a monthly basis would require a noise level below $5\text{mE}/\sqrt{\text{Hz}}$. The mission scenarios are therefore focused on the static field, like GOCE. Second, the stringent requirement on the angular velocity of a one-arm gradiometer, which must not exceed ${10}^{-6}$?rad/s, leads to two possible modes of operation of the CAI gradiometer: the nadir and the quasi-inertial mode. In the nadir mode, which corresponds to the usual Earth-pointing satellite attitude, only the gradient $V_{\mathit{yy}}$, along the cross-track direction, is measured. In the quasi-inertial mode, the satellite attitude is approximately constant in the inertial reference frame and the 3 diagonal gradients $V_{\mathit{xx}}\text{,}{V}_{\mathit{yy}}$ and $V_{\mathit{zz}}$ are measured. Both modes are successively simulated for a 239?km altitude orbit and the error on the recovered gravity models eventually compared to GOCE solutions. We conclude that for the specific CAI gradiometer design assumed in this paper, only the quasi-inertial mode scenario would be able to significantly outperform GOCE results at the cost of technically challenging requirements on the orbit and attitude control.     

Regional gravity field model in Pakistan area from the combination of CHAMP,GRACE and ground data using least squares collocation: A case study

This study describes a methodology of recovery of the Earth’s gravity field from CHAMP and GRACE satellites data in Pakistan using least squares collocation (LSC) based downward continuation technique. The CHAMP height anomalies and GRACE gravity disturbances derived from the observed satellite data have been used in combination solution using LSC with observed gravity values at the Earth surface. The combined covariance functions of height anomalies and/or gravity disturbances at satellite altitudes and observed gravity anomalies at Earth surface have been used as the basis for combination and downward continuation solution. The variance of predicted gravity anomalies from GRACE gravity disturbances is relatively lower than the corresponding results of gravity anomalies from CHAMP height anomalies. This fact may be attributed partly to the amplification of noise and partly to the unstable inverse transformation process of height anomalies to gravity anomalies. The impact of data error variance has been studied in the context of smoothing and noise reduction in the final solution of downward continuation using least squares collocation. The raising of data error suppresses the noise and as a result a smooth final solution is obtained. The prediction results appear to be dependent on the quality of data and goodness of combined covariance function, which are fairly comparable for the CHAMP and GRACE data. The recovered gravity field from satellite data appears to contribute mainly to medium and long wavelength parts of total gravity field spectrum. Due to flexibility of data handling in least squares collocation, this procedure is applicable to any observable of gravity field being at different altitudes and with different data spacing.     

On the selection of optimal global geopotential model for geoid modeling: A case study in Pakistan

A statistical comparison has been made between gravity field parameters derived from different global geopotential models (GGMs) and observed gravity anomalies, gravimetric geoid and GPS-Leveling data. The motivation behind this study is the selection of best possible global geopotential model that best matches statistically with the local observed data in Pakistan. This will facilitate in decreasing the load on observed data for the development of regional gravimetric geoid in remove-compute-restore technique when used in the Stokes’s integral for computation of the residual part. It is observed that combined geopotential models such as EGM96 and PGM200A, EIGEN-GL04C and EIGEN-CG03C reflect the better match in the total spectral range of gravity and GPS-Leveling data. Results of the precise local geoid model also indicate similar characteristics. A very-high-degree model “EGM2008” (degree/order 2160) exhibits relatively superior statistical fit with observed ground data in Pakistan region. For satellite-only models an increasing trend in the standard deviation can be seen with maximum of about ∼4 m in difference between GPS-Leveling and corresponding GGM’s geoid with increase in the order from 50 to 120 and then it decreases afterwards. However, for the EIGEN-CHAMP03SP, standard deviation saturates to a value of 3.4 m. This is an indication of contamination in the long to medium wavelength part, i.e. 50–100° for the satellite-only models. Moreover, the models DEOS-CHAMP-01C, GGM02C and then ITG-GRACE03 appear to have better fit for medium to long wavelength and can possibly be recommended for use as long wavelength part with the local observed data. While a hybrid geopotential model selection can be achieved through the selection from either of DEOS-CHAMP-01C, GGM02C, GGM02S, EIGEN-GRACE02S or ITG-GRACE03 in the long wavelength (to degree and order 40) and EGM96, PGM200A, EIGEN-GL04C, EIGEN-CG03C or even EGM2008 in medium to short wavelength, i.e. from degree 41 to maximum degree and order.     

Inversion of satellite gradiometry data using statistically modified integral formulas for local gravity field recovery

The satellite gravity gradiometric data can be used directly to recover the gravity anomaly at sea level using inversion of integral formulas. This approach suffers by the spatial truncation errors of the integrals, but these errors can be reduced by modifying the formulas. It allows us to consider smaller coverage of the satellite data over the region of recovery. In this study, we consider the second-order radial derivative (SORD) of disturbing potential (T_rr) and determine the gravity anomaly with a resolution of 1° × 1° at sea level by inverting the statistically modified version of SORD of extended Stokes’ formula. Also we investigate the effect of the spatial truncation error on the quality of inversion considering noise of T_rr. The numerical investigations show satisfactory results when the area of T_rr coverage is the same with that of the gravity anomaly and the integral formula is modified by the biased least-squares modification. The error of recovery will be about 6 mGal after removing the regularization bias in the presence of 1 mE noise in T_rr measured on the orbit.