Range-rate Observations Research Articles

An Improved Average Acceleration Approach of Modelling Earth Gravity Field Based on K-Band Range-Rate Observations

The conventional average acceleration approach relies on K-band range observation, containing an unknown bias, which leads to possible degradation of the precision of Earth’s gravity field modelling. It also suffers from correlated errors caused by three-point numerical differentiation. In this study, an improved approach is proposed that makes use of K-band range-rate observations instead and overcoming the influence of correlated errors by introducing a whitening filter. GRACE-Follow On data spanning the period from January 2019 to December 2022 were processed by the proposed approach and a series of time-varying gravity field models was derived, referred to as SSM-AAA-GFO in this paper. This model series is compared comprehensively with three official model series. Results demonstrate that all model series are highly coincident below degree 30 and reflect similar time-varying gravity field signals in both large and small basins. After filtering, SSM-AAA-GFO shows uncertainty, in the form of equivalent water height below 2.5 cm, which is comparable with three official model series. The comparison results confirm the effectiveness of the proposed approach for precisely modelling a time-varying gravity field based on K-band range-rate observations.

An Improved Acceleration Approach by Utilizing K-Band Range Rate Observations

During gravity field modeling, the conventional acceleration approach rarely incorporates KBR inter-satellite range rate data from the GRACE mission. To propose an improved acceleration method, this study introduces initial orbital position and velocity vectors to be estimated along with a combination of Cowell, KSG, and Adams integrators. In addition to achieving a full-rank design matrix regarding orbit corrections when constructing observation equations, the proposed method is capable of utilizing range rate observations for gravity field estimation. To verify the reliability of this approach, GRACE data from April 2002 to December 2016 was used to calculate a time series of monthly gravity solutions up to a degree and order of 96, referred to as Tongji-Acc RL06 in this paper. The computed time series are compared with the official models (i.e., CSR RL06, GFZ RL06, and JPL RL06) in terms of geoid degree variances, signal contents over distinct areas, and noise levels in desert regions. The investigations lead to the following conclusions: (a) the geoid degree variances indicate that Tongji-Acc RL06 exhibits comparable signal levels (approximately below 20 degrees) to the other three models while demonstrating lower noise at higher degrees (above 40 degrees); (b) the analysis over the globe, typical river basins, and land–ice regions illustrates that the solutions derived using the proposed acceleration method agree well with the official models based on the dynamic approach; (c) especially over the two large-scale river basins (i.e., Amazon and Zambezi) and another two small-scale river basins (i.e., Tennessee and Irrawaddy), Tongji-Acc RL06 significantly improves the SNR values; and (d) in the cases of the Sahara and Karakum deserts, Tongji-Acc RL06 achieves noise reductions of over 55.8% and 61.5% relative to CSR RL06, respectively. In general, the signal and noise analyses demonstrate that the proposed acceleration-based approach can effectively extract gravity field signals from KBR inter-satellite range rate observations with improved SNR, while significantly reducing the high-frequency noise.

Combined monthly GRACE-FO gravity fields for a Global Gravity-based Groundwater Product

SUMMARY The Combination Service for Time-variable Gravity fields (COST-G) operationally provides combinations of monthly Earth gravity field models derived from observations of the microwave ranging instrument of the GRACE Follow-on (GRACE-FO) satellite mission, applying the quality control and combination methodology originally developed by the Horizon 2020 project European Gravity Service for Improved Emergency Management for the data of the GRACE satellites. In the frame of the follow-up Horizon 2020 project Global Gravity-based Groundwater Product (G3P), the GRACE-FO combination is used to derive global grids of groundwater storage anomalies. To meet the user requirements and achieve optimal signal-to-noise ratio, the combination has been further developed and extended to incorporate: • new time-series based on the alternative accelerometer transplant product generated in the frame of the project by the Institute of Geodesy at the Graz University of Technology, which specifically improves the estimation of the C30 coefficient and also reduces the noise at medium to short wavelengths, and • the new time-series AIUB–GRACE-FO–RL02 of monthly GRACE-FO gravity fields, which is derived at the Astronomical Institute of the University of Bern by applying empirical noise modelling techniques. The COST-G quality control confirms the consistency of the contributing GRACE-FO time-series concerning the signal amplitude of seasonal hydrology in large river basins and the secular mass change in polar regions, but it also indicates rather diverse noise characteristics. The difference in the noise levels is taken into account in the combination process by relative weights derived by variance component estimation on the solution level. The weights are expected to be inverse proportional to the noise levels of the individual gravity field solutions. However, this expectation is violated when applying the weighting scheme as developed for the GRACE combination. The reason is found in the high-order coefficients of the gravity field, which are poorly determined from the low–low range-rate observations due to the observation geometry and suffer from aliasing due to the malfunctioning accelerometer onboard one of the GRACE-FO satellites. Hence, for the final G3P-combination a revised weighting scheme is applied where the gravity field coefficients beyond order 60 are excluded from the determination of the weights. The quality of the combined gravity fields is assessed by comparison of the noise content and the signal-to-noise ratio with the individual time-series. Independent validation is provided by the COST-G validation centre at the GFZ German Research Centre for Geosciences, where orbit fits of the low-flying Gravity and steady-state Ocean Circulation Explorer satellite are performed that confirm the high quality of the combined GRACE-FO gravity fields. By the end of the G3P project, the new combination scheme is implemented by COST-G as the new COST-G–GRACE-FO–RL02 and continued to be used for the operational GRACE-FO combination.

Performance analysis of crosslink radiometric measurement based autonomous orbit determination for cislunar small satellite formations

Recent advances in space technology provide an opportunity for small satellites to be launched in cislunar space. However, tracking these small satellites still depends on ground-based operations. Autonomous navigation could be a possible solution considering the challenges presented by costly ground operations and limited onboard power available for small satellites. There have been various studies on autonomous navigation methods for cislunar missions. One of them, LiAISON, provides an autonomous orbit determination solution solely using inter-satellite measurements. This study aims at providing a detailed performance analysis of crosslink radiometric measurements based on autonomous orbit determination for cislunar small satellite formations considering the effects of measurement type, measurement accuracy, bias, formation geometry, and network topology. This study shows that range observations provide better state estimation performance than range-rate observations for the autonomous navigation system in cislunar space. Line-of-sight angle measurements derived from radiometric measurements do not improve the overall system performance. In addition, less precise crosslink measurement methods could be an option for formations in highly observable orbital configurations. It was found that measurement biases and measurements with high intervals reduce the overall system performance. In case there are more than two spacecraft in the formation, the navigation system in the mesh topology provides a better overall state estimation than the centralized topology.

GRACETOOLS—GRACE Gravity Field Recovery Tools

This paper introduces GRACETOOLS, the first open source gravity field recovery tool using GRACE type satellite observations. Our aim is to initiate an open source GRACE data analysis platform, where the existing algorithms and codes for working with GRACE data are shared and improved. We describe the first release of GRACETOOLS that includes solving variational equations for gravity field recovery using GRACE range rate observations. All mathematical models are presented in a matrix format, with emphasis on state transition matrix, followed by details of the batch least squares algorithm. At the end, we demonstrate how GRACETOOLS works with simulated GRACE type observations. The first release of GRACETOOLS consist of all MATLAB M-files and is publicly available at Supplementary Materials.

A Terrestrial Reference Frame realised on the observation level using a GPS-LEO satellite constellation

Applying a one-step integrated process, i.e. by simultaneously processing all data and determining all satellite orbits involved, a Terrestrial Reference Frame (TRF) consisting of a geometric as well as a dynamic part has been determined at the observation level using the EPOS-OC software of Deutsches GeoForschungsZentrum. The satellite systems involved comprise the Global Positioning System (GPS) as well as the twin GRACE spacecrafts. Applying a novel approach, the inherent datum defect has been overcome empirically. In order not to rely on theoretical assumptions this is done by carrying out the TRF estimation based on simulated observations and using the associated satellite orbits as background truth. The datum defect is identified here as the total of all three translations as well as the rotation about the z-axis of the ground station network leading to a rank-deficient estimation problem. To rectify this singularity, datum constraints comprising no-net translation (NNT) conditions in x, y, and z as well as a no-net rotation (NNR) condition about the z-axis are imposed. Thus minimally constrained, the TRF solution covers a time span of roughly a year with daily resolution. For the geometric part the focus is put on Helmert transformations between the a priori and the estimated sets of ground station positions, and the dynamic part is represented by gravity field coefficients of degree one and two. The results of a reference solution reveal the TRF parameters to be estimated reliably with high precision. Moreover, carrying out a comparable two-step approach using the same data and models leads to parameters and observational residuals of worse quality. A validation w.r.t. external sources shows the dynamic origin to coincide at a level of 5 mm or better in x and y, and mostly better than 15 mm in z. Comparing the derived GPS orbits to IGS final orbits as well as analysing the SLR residuals for the GRACE satellites reveals an orbit quality on the few cm level. Additional TRF test solutions demonstrate that K-Band Range-Rate observations between both GRACE spacecrafts are crucial for accurately estimating the dynamic frame’s orientation, and reveal the importance of the NNT- and NNR-conditions imposed for estimating the components of the dynamic geocenter.

Antarctica, Greenland and Gulf of Alaska land-ice evolution from an iterated GRACE global mascon solution

AbstractWe have determined the ice mass evolution of the Antarctic and Greenland ice sheets (AIS and GIS) and Gulf of Alaska (GOA) glaciers from a new GRACE global solution of equal-area surface mass concentration parcels (mascons) in equivalent height of water. The mascons were estimated directly from the reduction of the inter-satellite K-band range-rate (KBRR) observations, taking into account the full noise covariance, and formally iterating the solution. The new solution increases signal recovery while reducing the GRACE KBRR observation residuals. The mascons were estimated with 10 day and 1 arcdeg equal-area sampling, applying anisotropic constraints. An ensemble empirical mode decomposition adaptive filter was applied to the mascon time series to compute annual mass balances. The details and causes of the spatial and temporal variability of the land-ice regions studied are discussed. The estimated mass trend over the total GIS, AIS and GOA glaciers for the time period 1 December 2003 to 1 December 2010 is −380 ± 31 Gt a−1, equivalent to −1.05 ± 0.09 mm a−1 sea-level rise. Over the same time period we estimate the mass acceleration to be −41 ± 27 Gt a−2 , equivalent to a −0.11 ± 0.08 mm a−2 sea-level acceleration. The trends and accelerations are dependent on significant seasonal and annual balance anomalies.

Error calibration of geopotential harmonics in recent and past gravitational fields

The geopotential coefficients of a number of recent and past models have been examined with the aim of calibrating their formal (or published) errors, principally by direct comparison with the same coefficients of a precise and wholly independent satellite-only reference field determined (in 2010) predominantly from over 7 years of μ/s low–low GRACE inter-satellite range rate observations (ITG-GRACE2010S). In all of these comparisons the reference field used, over specified spectral ranges, has much smaller reported errors than the ones to be calibrated. In particular we find that a recently published field (in 2010) using gravity gradient and position data from ESA’s GOCE satellite, GO_CONS_GCF_2_TIM_R1, has formal errors which are significantly optimistic for the lowest degrees (n) but with increasing realism where the gradiometer gains influence over the position information (15 < n < 120). Other GOCE models do not afford an unbiased error calibration. Validating error calibration with the independent reference model, of two past comprehensive fields (containing both satellite and surface data), confirms that one of them (JGM3, published in 1994, complete to n = 70) reported generally realistic formal errors while another (EGM96, published in 1998, complete to n = 360, tested only for n < 150) had significantly optimistic ones for most n < 100 but with better realism when affected only by the surface information.

Low-low satellite-to-satellite tracking: a comparison between analytical linear orbit perturbation theory and numerical integration

Low-low satellite-to-satellite tracking (ll-SST) range-rate observations have been predicted by two methods: one based on a linear perturbation theory in combination with the Hill equations, and one based on solving the equations of motion of two low-flying satellites by numerical integration. The two methods produce almost equivalent Fourier spectra of the range-rate observations after properly taking into account a few resonant terms. For a typical GRACE-type configuration, where the two satellites trail each other at a distance of 300 km at an altitude of 460 km, and in the presence of the EGM96 gravity field model, complete to spherical harmonic degree and order 70, the agreement between the Fourier spectra is about 1 mm/s compared to a root-mean-square (RMS) value of more than 220 mm/s for the range-rate signal. The discrepancy of 1 mm/s can be reduced significantly when not taking into account perturbations caused by the J2 term. Excluding the J2 term, the agreement between the two methods improves to 0.4 mm/s compared to a RMS value of 6 mm/s for the range-rate signal. These values are 0.01 and 2.3 mm/s when ignoring the spectrum for frequencies below two cycles per orbital revolution, reducing the discrepancy even further to about 0.5% of the signal. The selected linear perturbation theory is thus capable of modeling gravity field induced range-rate observations with very high precision for a large part of the spectrum.

Global lunar gravity recovery from satellite-to-satellite tracking

A feasibility study is presented of high resolution and accuracy determination of the global grayity field of the Moon from a combination of low-low satellite-to-satellite range-rate observations and conventional tracking from stations on Earth. The European Moon ORbiting Observatory (MORO) mission, studied as a candidate for the third mediumsized science mission (M3) under the European Space Agency's Horizon 2000 scientific programme, is adopted for the simulation purposes. Global coverage and mapping of the fine details of the gravity field is achieved by satellite-to-satellite tracking (SST) of a small sub-satellite deployed by MORO in its 100 km polar orbit, whereas the long-wavelength features are obtained from Earth-based tracking. The combination of SST and Earth-based tracking therefore represents a powerful tool over a wide range of wavelengths. Moreover, tracking from Earth provides a clear reference for the satellite orbits, which are hard to determine from SST data only. The baseline mission proposal foresees a co-orbiting satellite pair in which the two satellites follow each other in essentially the same orbital plane with only a small spacing in time. The MORO gravimetry experiment requirements prescribe a surface level radial acceleration accuracy of a few mGal with a surface resolution of 50–100km. A number of satellite tracking configurations has been investigated and the influence of noise and systematic errors has been studied. Using perfect measurements and in the absence of systematic model errors the gravity field of the Moon, assumed to be pertectly represented by the 60 × 60 Lun60d model of Konopliv et al. (1993) with a surface resolution of 91 km, has been recovered with a radial acceleration accuracy better than 0.003 mGal using a 3° in-plane satellite spacing. Introducing uncorrelated random noise to the tracking links and a 10% model error in the direct solar radiation pressure model, still an accuracy better than 5 mGal can be achieved. Finally, it is shown that a small angular separation in right ascension of the ascending node of the orbital planes of MORO and its sub-satellite adds extra cross-track information to the SST range-rate signal and thus enables better determination of high sectorial and near-sectorial terms of the gravity field.

Filtering of satellite-to-satellite observations

This study addresses criticism of the low-low satellite-to-satellite methodology to obtain the small-scale gravitational structure of a primary gravitating body. Specifically, the aliasing of satellite-to-satellite range-rate observations from errors in the initial conditions of the orbits is studied numerically. It is shown that high-pass filtering effectively eliminates the aliasing of initial condition errors. In the case of the Lunar Polar Orbiter, ephemeris errors of at least 1 km can be tolerated. With two spacecraft in polar orbits, a data storage capability would provide global data coverage. Solution for the small-scale gravitational structure would probably be best achieved by an iterative approach using regional solutions. To this end, a simplified algorithm has been developed that can be used in feasibility studies. The performance of the algorithm is compared with simulated data.

Lunar orbit determination in the presence of unmodeled accelerations

A technique for estimating the state of an artificial satellite in the presence of unmodeled accelerations is presented. The unmodeled acceleration is approximated by a first-order Gauss-Markov sequence which can be separated into a timewise correlated component and a purely random component. Using this approximation, a sequential procedure for estimating the position, velocity, and the unmodeled acceleration is developed. The method is evaluated by reducing range-rate observations obtained by tracking the Apollo 10 and 11 spacecraft during the lunar orbit phase of the mission. Numerical results are presented which show that the observation residual pattern lies within the observation noise standard deviation. The values of the estimated components of the unmodeled acceleration are repeatable from orbit to orbit within a given mission and from mission to mission when the same ground track is covered. Finally, the variation in the radial component of the unmodeled acceleration shows a high correlation with the reported location of the lunar surface mascons.

Ranging and Data Transmission Using Digital Encoded FM-"Chirp" Surface Acoustic Wave Filters

A digital encoded multislope chirp modem/demodem unit has been implemented using two 3-port surface acoustic wave filters. Each filter operating at a 30-MHz center frequency provides either positive or negative slopes as digital 1's or 0's with a time-bandwidth product of 280, an unweighed bandwidth of 5.6 MHz, and a time dispersion of 50 /spl mu/S.The modems were used to calibrate and compare the operational performance of the conventions multitone CW ranging system with a multislope chirp ranging system. Range measurements and range-rate observations were made to a synchronous satellite with both systems using a ground communication satellite terminal. Both ranging techniques provided accuracies well within the predictable satellite range of 20 000 nmi; the multitone system provided a theoretical range resolution of 1 m, and the chirp system 0.4 m. Data transmission was also accomplished, using 12-bit binary code at a 1.25-Mbit rate. The significant advantage to be noted with the chirp system is the ability to obtain continuous range data from the satellite repeater simultaneously while other modes of information are being transmitted, and to combine the ranging and data-link transmission on a time-order basis using digital encoded chirp sequences. The chirp system was found the more desirable technique, since it provides a more direct range measurement, with minimal calibration requirements, and provided greater processing gain with relative ease and reliability. The data transmission at low data rates provided little deterioration in theoretical compressive gain. However, at the higher data rate a greater loss was encountered due to power sharing of the overlapping chirp coded carriers in the limiting satellite.

The improvement of the lunar ephemeris

At present the fundamental lunar ephemeris is based on Brown's theory of the motion of the Moon with improvements based on the bypassing of Brown's Tables, the removal of the great empirical term, the substitution of the relevant constants of the IAU system of astronomical constants and the retransformation of Brown's series in rectangular coordinates to spherical coordinates. Even so this ephemeris does not represent adequately the recent range and range-rate radio observations, and it will be inadequate for use in the analysis of laser observations of corner reflectors on the Moon. Numerical integrations for these purposes have already been made at the Jet Propulsion Laboratory, but improved theoretical developments are also required; new solutions of the main problem are in hand elsewhere. Work at H.M. Nautical Almanac Office is aimed at obtaining improved values of the constants of the lunar orbit by a rediscussion of occultation observations made since 1943 and at the redevelopment of the series for the planetary perturbations using more precise theories of the motion of the Sun and planets. The techniques and preliminary results of exploratory numerical integrations were briefly described.

Accuracy of determining the state of a lunar satellite and the lunargravitational field.

Tracking geometry and dynamics of lunar satellite, estimating orbital elements and lunar gravitational field parameters through earth based range and range rate observations