Scale factors of the thermospheric neutral density – a comparison of SLR and accelerometer solutions

2021 ◽  
Author(s):  
Lea Zeitler ◽  
Armin Corbin ◽  
Kristin Vielberg ◽  
Sergei Rudenko ◽  
Anno Löcher ◽  
...  
Keyword(s):  
Time Resolution ◽  
Aerodynamic Drag ◽  
Precise Orbit Determination ◽  
Neutral Density ◽  
Observation Data ◽  
Accelerometer Data ◽  
Scale Factors ◽  
Weather Events ◽  
Leo Satellites ◽  
Gravity Recovery

<p>The aerodynamic drag depending on the neutral density of the thermosphere is the largest non-gravitational force that decelerates Low Earth Orbiting (LEO) satellites with altitudes lower than 1000 km.  Consequently, the knowledge of the thermospheric neutral density is of crucial importance for many applications in geo-scientific investigations, such as precise orbit determination (POD), re-entry prediction, manoeuvre planning or satellite lifetime predictions. The accuracy of existing thermosphere models depends on observation data of the thermosphere, which are quite sparse. Evaluations of different thermosphere models indicate considerable differences, especially for time epochs of severe space weather events. Hence, an improvement of thermosphere models is absolutely necessary.</p><p>In this study, discrepancies between the empirical thermosphere model NRLMSISE-00 and the results of two geodetic observation techniques are discussed. For this purpose, two approaches are applied to calculate scale factors between the modelled density from the NRLMSISE-00 model and those from geodetic techniques. The first approach applies the POD of LEO satellites to estimate scale factors with a time resolution of 12 hours derived from Satellite Laser Ranging (SLR) tracking measurements. The SLR missions used here include the spherical satellites Starlette, Westpac, Blits, Stella and Larets. As our second approach, scale factors are computed by evaluating the aerodynamic acceleration using the on-board accelerometer data of the Challenging Mini-satellite Payload (CHAMP) mission and the Gravity Recovery and Climate Experiment (GRACE) mission. Here, the time resolution of scale factors is fixed to be 12 hours to be comparable with the first approach. Finally, we investigate the resulting scale factors from the above mentioned satellites at various altitudes, e.g. 960 km for Starlette and 400 km for GRACE. Especially, the temporal variation as well as the altitude dependency of the scale factors will be discussed.</p>

scholarly journals Comparison of accelerometer data calibration methods used in thermospheric neutral density estimation

2018 ◽  
Vol 36 (3) ◽  
pp. 761-779 ◽  
Author(s):  
Kristin Vielberg ◽  
Ehsan Forootan ◽  
Christina Lück ◽  
Anno Löcher ◽  
Jürgen Kusche ◽  
...  
Keyword(s):  
Solar Activity ◽  
Radiation Pressure ◽  
Precise Orbit Determination ◽  
High Solar Activity ◽  
Neutral Density ◽  
Accelerometer Data ◽  
The Earth ◽  
Calibration Methods ◽  
Leo Satellites ◽  
Gravity Recovery

Abstract. Ultra-sensitive space-borne accelerometers on board of low Earth orbit (LEO) satellites are used to measure non-gravitational forces acting on the surface of these satellites. These forces consist of the Earth radiation pressure, the solar radiation pressure and the atmospheric drag, where the first two are caused by the radiation emitted from the Earth and the Sun, respectively, and the latter is related to the thermospheric density. On-board accelerometer measurements contain systematic errors, which need to be mitigated by applying a calibration before their use in gravity recovery or thermospheric neutral density estimations. Therefore, we improve, apply and compare three calibration procedures: (1) a multi-step numerical estimation approach, which is based on the numerical differentiation of the kinematic orbits of LEO satellites; (2) a calibration of accelerometer observations within the dynamic precise orbit determination procedure and (3) a comparison of observed to modeled forces acting on the surface of LEO satellites. Here, accelerometer measurements obtained by the Gravity Recovery And Climate Experiment (GRACE) are used. Time series of bias and scale factor derived from the three calibration procedures are found to be different in timescales of a few days to months. Results are more similar (statistically significant) when considering longer timescales, from which the results of approach (1) and (2) show better agreement to those of approach (3) during medium and high solar activity. Calibrated accelerometer observations are then applied to estimate thermospheric neutral densities. Differences between accelerometer-based density estimations and those from empirical neutral density models, e.g., NRLMSISE-00, are observed to be significant during quiet periods, on average 22 % of the simulated densities (during low solar activity), and up to 28 % during high solar activity. Therefore, daily corrections are estimated for neutral densities derived from NRLMSISE-00. Our results indicate that these corrections improve model-based density simulations in order to provide density estimates at locations outside the vicinity of the GRACE satellites, in particular during the period of high solar/magnetic activity, e.g., during the St. Patrick's Day storm on 17 March 2015.

scholarly journals Integrated Precise Orbit Determination of Multi-GNSS and Large LEO Constellations

2019 ◽  
Vol 11 (21) ◽  
pp. 2514 ◽  
Author(s):  
Xingxing Li ◽  
Keke Zhang ◽  
Fujian Ma ◽  
Wei Zhang ◽  
Qian Zhang ◽  
...  
Keyword(s):  
Orbit Determination ◽  
Precise Orbit Determination ◽  
Global Network ◽  
Computational Time ◽  
Observation Data ◽  
Accuracy Improvement ◽  
Precise Orbit ◽  
Leo Satellites ◽  
Geo Satellites ◽  
Gnss Observations

Global navigation satellite system (GNSS) orbits are traditionally determined by observation data of ground stations, which usually need even global distribution to ensure adequate observation geometry strength. However, good tracking geometry cannot be achieved for all GNSS satellites due to many factors, such as limited ground stations and special stationary characteristics for the geostationary Earth orbit (GEO) satellites in the BeiDou constellation. Fortunately, the onboard observations from low earth orbiters (LEO) can be an important supplement to overcome the weakness in tracking geometry. In this contribution, we perform large LEO constellation-augmented multi-GNSS precise orbit determination (POD) based on simulated GNSS observations. Six LEO constellations with different satellites numbers, orbit types, and altitudes, as well as global and regional ground networks, are designed to assess the influence of different tracking configurations on the integrated POD. Then, onboard and ground-based GNSS observations are simulated, without regard to the observations between LEO satellites and ground stations. The results show that compared with ground-based POD, a remarkable accuracy improvement of over 70% can be observed for all GNSS satellites when the entire LEO constellation is introduced. Particularly, BDS GEO satellites can obtain centimeter-level orbits, with the largest accuracy improvement being 98%. Compared with the 60-LEO and 66-LEO schemes, the 96-LEO scheme yields an improvement in orbit accuracy of about 1 cm for GEO satellites and 1 mm for other satellites because of the increase of LEO satellites, but leading to a steep rise in the computational time. In terms of the orbital types, the sun-synchronous-orbiting constellation can yield a better tracking geometry for GNSS satellites and a stronger augmentation than the polar-orbiting constellation. As for the LEO altitude, there are almost no large-orbit accuracy differences among the 600, 1000, and 1400 km schemes except for BDS GEO satellites. Furthermore, the GNSS orbit is found to have less dependence on ground stations when incorporating a large number of LEO. The orbit accuracy of the integrated POD with 8 global stations is almost comparable to the result of integrated POD with a denser global network of 65 stations. In addition, we also present an analysis concerning the integrated POD with a partial LEO constellation. The result demonstrates that introducing part of a LEO constellation can be an effective way to balance the conflict between the orbit accuracy and computational efficiency.

How can Thermospheric Neutral Density (TND) estimates from CHAMP and GRACE accelerometer observations be used to improve empirical models?

2020 ◽  
Author(s):  
Ehsan Forootan ◽  
Saeed Farzaneh ◽  
Mona Kosary ◽  
Maike Schumacher
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Orbit Determination ◽  
Ensemble Kalman Filter ◽  
Precise Orbit Determination ◽  
Accurate Estimation ◽  
Neutral Density ◽  
Square Root ◽  
Precise Orbit ◽  
Leo Satellites

<p>An accurate estimation of the Thermospheric Neutral Density (TND) is important to compute drag forces acting on Low-Earth-Orbit (LEO) satellites and debris. Empirical thermospheric models are often used to compute TNDs (along-track of LEO satellites) for the Precise Orbit Determination (POD) experiments. However, recent studies indicate that the TNDs of available models do not perfectly reproduce TNDs derived from accelerometer observations. In this study, we use TND estimates from the Challenging Minisatellite Payload (CHAMP) and Gravity Recovery and Climate Experiment (GRACE) missions and merge them with the NRLMSISE00 from the Mass Spectrometer and Incoherent Scatter family. The integration is implemented by applying a simultaneous Calibration and Data Assimilation (C/DA) technique. The application of C/DA is advantageous since it uses model equation to interpolate and extrapolate TNDs that are not covered by CHAMP and GRACE. It also modifies the model's selected parameters to simulate TNDs that are closer to those of CHAMP and GRACE. The C/DA of this study is implemented daily using CHAMP- and/or GRACE-TNDs, while using the Ensemble Kalman Filter (EnKF) and Ensemble Square-Root Kalman Filter (EnSRF) as merger. Compared to the original model, on average, we found 27% (in the range of 2% to 56%) improvements in the estimation of TNDs. In addition, the results of the C/DA are compared with the TND outputs of the JB2008 model along the CHAMP and GRACE orbits, whose results indicate that the daily C/DA outputs are 60% closer to the observed TNDs (that are not used for the C/DA). Overall, our assessment indicates that EnSRF results in more realistic TND simulation and prediction compared to those derived from EnKF. We show that the improved TND estimates of this study will be beneficial for Precise Orbit Determination (POD) studies.  </p><p><strong>Keywords: </strong>Thermosphere, Calibration and Data Assimilation (C/DA), NRLMSISE00, Ensemble Kalman Filter (EnKF), Ensemble Square-Root Kalman Filter (EnSRF)</p>

A simultaneous calibration and data assimilation (C/DA) to improve NRLMSISE00 using thermospheric neutral density (TND) from space-borne accelerometer measurements

2020 ◽  
Vol 224 (2) ◽  
pp. 1096-1115
Author(s):  
E Forootan ◽  
S Farzaneh ◽  
M Kosary ◽  
M Schmidt ◽  
M Schumacher
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Precise Orbit Determination ◽  
Low Earth Orbit ◽  
Model Parameters ◽  
Neutral Density ◽  
Incoherent Scatter ◽  
Drag Forces ◽  
Gravity Recovery ◽  
Limited Accuracy

SUMMARY Improving thermospheric neutral density (TND) estimates is important for computing drag forces acting on low-Earth-orbit (LEO) satellites and debris. Empirical thermospheric models are often used to compute TNDs for the precise orbit determination experiments. However, it is known that simulating TNDs are of limited accuracy due to simplification of model structure, coarse sampling of model inputs and dependencies to the calibration period. Here, we apply TND estimates from accelerometer measurements of the Challenging Minisatellite Payload (CHAMP) and the Gravity Recovery and Climate Experiment (GRACE) missions as observations to improve the NRLMSISE-00 model, which belongs to the mass spectrometer and incoherent scatter family of models. For this, a novel simultaneous calibration and data assimilation (C/DA) technique is implemented that uses the ensemble Kalman filter and the ensemble square-root Kalman filter as merger. The application of C/DA is unique because it modifies both model-derived TNDs, as well as the selected model parameters. The calibrated parameters derived from C/DA are then used to predict TNDs in locations that are not covered by CHAMP and GRACE orbits, and forecasting TNDs of the next day. The C/DA is implemented using daily CHAMP- and/or GRACE-TNDs, for which compared to the original model, we find 27 per cent and 62 per cent reduction of misfit between model and observations in terms of root mean square error and Nash coefficient, respectively. These validations are performed using the observations along the orbital track of the other satellite that is not used in the C/DA during 2003 with various solar activity. Comparisons with another empirical model, that is, Jacchia-Bowman, indicate that the C/DA results improve these quality measurements on an average range of 50 per cent and 60 per cent, respectively.

Kinematic orbit positioning applying the raw observation approach

2020 ◽  
Author(s):  
Barbara Suesser- Rechberger ◽  
Torsten Mayer-Guerr ◽  
Sandro Krauss
Keyword(s):  
Three Dimensional ◽  
Precise Orbit Determination ◽  
Satellite Orbit ◽  
Satellite System ◽  
Low Earth Orbit ◽  
Observation Data ◽  
Kinematic Orbit ◽  
Leo Satellites ◽  
Least Squares Adjustment ◽  
Global Navigation Satellite

<p>The kinematic strategy for precise orbit determination (POD) of low earth orbit (LEO) satellites uses only geometric observations to estimate the satellite orbit and does not take any forces into account. This strategy requires a large amount of observation data for one epoch to determine the three-dimensional satellite position. One possibility to get these data is the usage of the spaceborne global navigation satellite system (GNSS) technology, which provides a high number of accurate observations. Following Zehentner (2016) the kinematic orbit positioning applying the raw observation approach by using a least-squares adjustment has shown promising results with a high accuracy.</p><p>By applying this approach the kinematic orbits for several LEO satellite missions are estimated and subsequently validated by a comparison with state of the art gravity field solutions. Furthermore due to the fact that solar events causes an orbit decay, these precise determined orbit data are used to analyze solar event impacts on LEO satellites.</p>

Optical GNSS Receiver for the ESA's NGGM-MAGIC Mission and for LEO Satellites with the Highest Orbit Accuracy

2021 ◽  
Author(s):  
Drazen Svehla
Keyword(s):  
Laser Beam ◽  
Gravity Field ◽  
Carrier Phase ◽  
Beam Steering ◽  
Precise Orbit Determination ◽  
Optical Carrier ◽  
Gnss Receiver ◽  
Cw Laser ◽  
Leo Satellites ◽  
Leo Satellite

<p>Precise orbit determination (POD) of LEO satellites is done with a geodetic grade GPS receiver measuring carrier-phase between a LEO and GPS satellites, and in some cases this is supported with a DORIS instrument measuring Doppler between LEO and ground DORIS stations. Over the last 20 years we have demonstrated 1-2 cm accurate LEO POD and about 1 mm for inter-satellite distance. In order to increase the accuracy of the single satellite POD or satellites in LEO formation we propose an “optical GNSS receiver”, a cw-laser on a LEO satellite to measure Doppler between a LEO and GNSS satellite(s) equipped with SLR arrays and to develop it for the next gravity field mission.      </p><p>The objective of the ESA mission NGGM-MAGIC (Next Generation Gravity Mission - Mass-change and Geosciences International Constellation) is the long-term monitoring of the temporal variations of Earth’s gravity field at high resolution in time (3 days) and space (100 km), complementing the GRACE-FO mission from NASA at 45° orbit inclination. Currently, the GRACE-type mission design is based on optical carrier-phase measurements between two LEO satellites flying in a formation and separated by 200 km.</p><p>We propose an extension of the GRACE-type LEO-LEO concept by the “optical GNSS receiver” to provide Doppler measurements between a LEO satellite and GNSS satellite(s) equipped with SLR corner cubes by means of a cw-laser onboard a LEO satellite. Such a “vertical” LEO-GNSS observable is missing in the classical GRACE-type LEO-LEO concept. If Doppler measurements are carried out from the two GRACE-type satellites in the LEO orbit to the same GNSS satellite and by forming single-differences to that GNSS satellite one can remove any GNSS-orbit related error in the measured LEO-GNSS Doppler. In this way, radial orbit difference can be obtained between the two GRACE-type satellites (free of all GNSS orbit errors) and complement “horizontal” LEO-LEO measurements between the two GRACE-type satellites in the LEO orbit.</p><p>The non-mechanical laser beam steering has been developed for an angle window of -40° to +40° and it does not require a rotating and a big telescope in LEO (no clouds and atmosphere turbulences in LEO). Therefore, in such a beam-steering window, one could always observe with a fiber cw-laser one GNSS satellite close to the zenith from both GRACE-type satellites. The non-mechanical beam steering concept in zenith direction can be supported by a small 10-cm like (fixed) Ritchey-Chrétien telescope (COTS), a Cassegrain reflector design widely used for LEO satellites, e.g., for James Webb Space Telescope or for an optical Earth imaging with Cubesats with the 50 cm resolution.</p><p>Considering that several GNSS satellites in the field of view could be observed from a LEO satellite with this approach (including LAGEOS-1/2 and Etalon satellites) and the non-mechanical laser beam steering could be extended towards the LEO horizon, an “optical” GNSS receiver is a new concept for POD of LEO satellites. Here, we provide simulations of this new concept for LEO POD with GNSS/SLR constellations equipped with SLR arrays and discuss all new applications this new concept could bring.</p>

scholarly journals LEO-Constellation-Augmented BDS Precise Orbit Determination Considering Spaceborne Observational Errors

2021 ◽  
Vol 13 (16) ◽  
pp. 3189
Author(s):  
Min Li ◽  
Tianhe Xu ◽  
Haibo Ge ◽  
Meiqian Guan ◽  
Honglei Yang ◽  
...  
Keyword(s):  
Orbit Determination ◽  
Precise Orbit Determination ◽  
Satellite System ◽  
Satellite Orbits ◽  
Navigation Satellite ◽  
Noise Levels ◽  
Precise Orbit ◽  
Leo Satellites ◽  
Solar Storm ◽  
Observational Errors

The precise orbit determination (POD) accuracy of the Chinese BeiDou Navigation Satellite System (BDS) is still not comparable to that of the Global Positioning System because of the unfavorable geometry of the BDS and the uneven distribution of BDS ground monitoring stations. Fortunately, low Earth orbit (LEO) satellites, serving as fast moving stations, can efficiently improve BDS geometry. Nearly all studies on Global Navigation Satellite System POD enhancement using large LEO constellations are based on simulations and their results are usually overly optimistic. The receivers mounted on a spacecraft or an LEO satellite are usually different from geodetic receivers and the observation conditions in space are more challenging than those on the ground. The noise level of spaceborne observations needs to be carefully calibrated. Moreover, spaceborne observational errors caused by space weather events, i.e., solar geomagnetic storms, are usually ignored. Accordingly, in this study, the actual spaceborne observation noises are first analyzed and then used in subsequent observation simulations. Then, the observation residuals from the actual-processed LEO POD during a solar storm on 8 September 2017 are extracted and added to the simulated spaceborne observations. The effect of the observational errors on the BDS POD augmented with different LEO constellation configurations is analyzed. The results indicate that the noise levels from the Swarm-A, GRACE-A, and Sentinel-3A satellites are different and that the carrier-phase measurement noise ranges from 2 mm to 6 mm. Such different noise levels for LEO spaceborne observations cause considerable differences in the BDS POD solutions. Experiments calculating the augmented BDS POD for different LEO constellations considering spaceborne observational errors extracted from the solar storm indicate that these errors have a significant influence on the accuracy of the BDS POD. The 3D root mean squares of the BDS GEO, IGSO, and MEO satellite orbits are 1.30 m, 1.16 m, and 1.02 m, respectively, with a Walker 2/1/0 LEO constellation, and increase to 1.57 m, 1.72 m, and 1.32 m, respectively, with a Walker 12/3/1 constellation. When the number of LEO satellites increases to 60, the precision of the BDS POD improves significantly to 0.89 m, 0.77 m, and 0.69 m for the GEO, IGSO, and MEO satellites, respectively. While 12 satellites are sufficient to enhance the BDS POD to the sub-decimeter level, up to 60 satellites can effectively reduce the influence of large spaceborne observational errors, i.e., from solar storms.

GRACE and GRACE-FO Level-1 V04 Data Processing Status

2020 ◽  
Author(s):  
Christopher Mccullough ◽  
Tamara Bandikova ◽  
William Bertiger ◽  
Carmen Boening ◽  
Sung Byun ◽  
...  
Keyword(s):  
Data Processing ◽  
Attitude Determination ◽  
Precise Orbit Determination ◽  
Mass Change ◽  
Sensor Data ◽  
The Earth ◽  
Analysis Center ◽  
Level 1 ◽  
Gravity Recovery

<p>The Gravity Recovery and Climate Experiment Follow-On (GRACE-FO), launched in May 2018, provides invaluable information about mass change in the Earth system, continuing the legacy of GRACE. Fundamental requirements for successful mass change recovery are precise orbit determination and inter-satellite ranging, determination of the relative clock alignment of the ultra-stable oscillators (USOs), precise attitude determination, and accelerometry. NASA/Caltech Jet Propulsion Laboratory is the official Level-1 data processing and analysis center, and is currently processing software version 04. Here we present analysis of the aforementioned GRACE-FO sensor data, as well a preview of an upcoming GRACE reprocessing, and a discussion of measurement performance.</p>

scholarly journals Error Correction of Meteorological Data Obtained with Mini-AWSs Based on Machine Learning

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Ji-Hun Ha ◽  
Yong-Hyuk Kim ◽  
Hyo-Hyuc Im ◽  
Na-Young Kim ◽  
Sangjin Sim ◽  
...  
Keyword(s):  
Machine Learning ◽  
Error Correction ◽  
Atmospheric Pressure ◽  
Observation Interval ◽  
Meteorological Data ◽  
Weather Prediction ◽  
Observation Data ◽  
Pressure Data ◽  
Weather Forecasts ◽  
Weather Events

Severe weather events occur more frequently due to climate change; therefore, accurate weather forecasts are necessary, in addition to the development of numerical weather prediction (NWP) of the past several decades. A method to improve the accuracy of weather forecasts based on NWP is the collection of more meteorological data by reducing the observation interval. However, in many areas, it is economically and locally difficult to collect observation data by installing automatic weather stations (AWSs). We developed a Mini-AWS, much smaller than AWSs, to complement the shortcomings of AWSs. The installation and maintenance costs of Mini-AWSs are lower than those of AWSs; Mini-AWSs have fewer spatial constraints with respect to the installation than AWSs. However, it is necessary to correct the data collected with Mini-AWSs because they might be affected by the external environment depending on the installation area. In this paper, we propose a novel error correction of atmospheric pressure data observed with a Mini-AWS based on machine learning. Using the proposed method, we obtained corrected atmospheric pressure data, reaching the standard of the World Meteorological Organization (WMO; ±0.1 hPa), and confirmed the potential of corrected atmospheric pressure data as an auxiliary resource for AWSs.

scholarly journals Real-Time Kinematic Precise Orbit Determination for LEO Satellites Using Zero-Differenced Ambiguity Resolution

2019 ◽  
Vol 11 (23) ◽  
pp. 2815 ◽  
Author(s):  
Xingxing Li ◽  
Jiaqi Wu ◽  
Keke Zhang ◽  
Xin Li ◽  
Yun Xiong ◽  
...  
Keyword(s):  
Real Time ◽  
Ambiguity Resolution ◽  
Orbit Determination ◽  
Precise Orbit Determination ◽  
The Real ◽  
Precise Orbit ◽  
Ambiguity Fixing ◽  
Real Time Kinematic ◽  
Leo Satellites ◽  
Fixed Solution

The rapid growing number of earth observation missions and commercial low-earth-orbit (LEO) constellation plans have provided a strong motivation to get accurate LEO satellite position and velocity information in real time. This paper is devoted to improve the real-time kinematic LEO orbits through fixing the zero-differenced (ZD) ambiguities of onboard Global Navigation Satellite System (GNSS) phase observations. In the proposed method, the real-time uncalibrated phase delays (UPDs) are estimated epoch-by-epoch via a global-distributed network to support the ZD ambiguity resolution (AR) for LEO satellites. By separating the UPDs, the ambiguities of onboard ZD GPS phase measurements recover their integer nature. Then, wide-lane (WL) and narrow-lane (NL) AR are performed epoch-by-epoch and the real-time ambiguity–fixed orbits are thus obtained. To validate the proposed method, a real-time kinematic precise orbit determination (POD), for both Sentinel-3A and Swarm-A satellites, was carried out with ambiguity–fixed and ambiguity–float solutions, respectively. The ambiguity fixing results indicate that, for both Sentinel-3A and Swarm-A, over 90% ZD ambiguities could be properly fixed with the time to first fix (TTFF) around 25–30 min. For the assessment of LEO orbits, the differences with post-processed reduced dynamic orbits and satellite laser ranging (SLR) residuals are investigated. Compared with the ambiguity–float solution, the 3D orbit difference root mean square (RMS) values reduce from 7.15 to 5.23 cm for Sentinel-3A, and from 5.29 to 4.01 cm for Swarm-A with the help of ZD AR. The SLR residuals also show notable improvements for an ambiguity–fixed solution; the standard deviation values of Sentinel-3A and Swarm-A are 4.01 and 2.78 cm, with improvements of over 20% compared with the ambiguity–float solution. In addition, the phase residuals of ambiguity–fixed solution are 0.5–1.0 mm larger than those of the ambiguity–float solution; the possible reason is that the ambiguity fixing separate integer ambiguities from unmodeled errors used to be absorbed in float ambiguities.

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