Enhanced baseline determination for formation flying LEOs by relative corrections of phase center and code residual variations

Bin YI ,Defeng GU ,Bing JU ,Ki SHAO ,Houzhe ZHANG

a College of Liberal Arts and Sciences,National University of Defense Technology,Changsha 410073,China

b TianQin Research Center for Gravitational Physics and School of Physics and Astronomy,Sun Yat-sen University(Zhuhai Campus),Zhuhai 519082,China

c National Key Laboratory of Science and Technology on Aerospace Flight Dynamics,Beijing Aerospace Control Center,Beijing 100094,China

KEYWORDS Ambiguity resolution;Formation flying;Precise baseline determination (PBD);Relative phase centre variations (RPCVs) estimation;SD MW combination residuals variations (SD MWVs)estimation

Abstract Formation flying Low Earth Orbiters (LEOs) are important for implementing new and advanced concepts in Earth observation missions.Precise Baseline Determination (PBD) is a prerequisite for LEOs to complete specified mission targets.PBD is usually performed based on space-borne GNSS data,the relative corrections of phase center and code residual variations play crucial roles in achieving the best relative orbit accuracy.Herein,the influences of antenna Relative Phase Centre Variations(RPCVs)and Single-Difference(SD)Melbourne-Wbbena(MW)Combination Residuals Variations(SD MWVs)on PBD are studied.The methods were tested using flight data from Gravity Recovery And Climate Experiment (GRACE) and GRACE Follow-On(GRACE-FO).Results showed that the maximum values for RPCVs and SD MWVs were 14 mm and 0.32 cycles,respectively.Then,the RPCVs correction significantly enhanced the baseline accuracy;the K-Band Ranging (KBR) measurement consistency improved by 30.1% and 37.5%for GRACE and GRACE-FO,respectively.The application of SD MWVs further improved the accuracy and reliability of PBD results.For GRACE,the ambiguities fixing success rate increased from 85.1% to 97.9% and a baseline consistency of 0.57 mm was achieved for the KBR measurements.It was found that the correction of both RPCVs and SD MWVs reduced the carrier phase observation minus computation residuals from double-difference ionosphere-free combination.In addition,in-flight data processing demonstrated that RPCVs and SD MWVs estimations for the current period could be used for the previous and subsequent periods.

1.Introduction

With the advantages of flexibility,redundancy,high efficiency,and low cost,formation flying LEOs are currently considered as a key technology for space missions.Prominent examples of operating formation flying LEOs include the TerraSAR-X add-on for Digital Elevation Measurement (TanDEM-X),the Prototype Research Instruments and Space Mission Technology Advancement (PRISMA),Swarm,Gravity Recovery And Climate Experiment(GRACE)and GRACE Follow-On(GRACE-FO).To fulfil their scientific mission requirements,a precise inter-satellite baseline determination is required.A well-known example is the TanDEM-X mission,which set a Precise Baseline Determination (PBD) precision requirement of 1 mm (1σ) for the generation of high quality global Digital Elevation Model (DEM) products.

With the development of the Global Navigation Satellite System (GNSS),it has been widely used in precises orbit determination for LEOs.Formation flying LEOs usually make use of multi-frequency GNSS receivers for precise orbit determination.By using Carrier-phase Differential GNSS(CDGNSS)observations,some common errors are eliminated and(sub-)millimetre level PBD precision can be achieved.The GRACE baseline solutions of 1.0 mm level precision were first obtained by Kroes et al.,combining reduced-dynamic filtering with Double-Difference(DD)ambiguities resolution.Jäggi et al.used batch least-squares estimation for PBD and the baseline precision for GRACE was 0.88 mm.Ju et al.proposed a thrust modelling and numerical integration strategy,which can largely eliminate the effects caused by GRACE orbital maneuvers,and a Root-Mean-Square (RMS) of the KBand Ranging (KBR) residuals of better than 0.7 mm can be achieved.In 2012,TanDEM-X’s baseline precision was proved by inter-agency comparison;the bias in each direction was better than 1 mm.In 2019,Swarm baseline comparisons with orbits from the German Space Operations Center showed a baseline consistency of 1-3 mm for Swarm-A/C.

The noise level of carrier phase measurements is about 1 mm for L1 and L2,and 2-3 mm for the IF L1/L2 combination.In order to fully exploit this accuracy level for PBD,careful modelling of antenna Phase Centre Variations (PCVs)is necessary,as the PCVs reach an order of magnitude of 3-4 mm.Usually,PCVs can be derived by analysing the in-flight data and using the residual approach,which is based on an averaging of the phase Observation Minus Computation (OC) residuals produced by the precise orbit determination process.Guo et al.used post-fit residuals of GPS IF phase measurements to estimate the PCVs of Haiyang 2A;the RMS of post-fit residuals decreased from 8.5 mm to 7.5 mm when PCV corrections were added.Shao et al.generated the PCVs map of Tiangong-2;the RMS of SLR validation residuals decreased by approximately 6% and 22% for reduceddynamic and kinematic orbit solutions after applying the PCVs correction,respectively.Mao et al.analysed each GPS frequency for separate PCVs of Swarm,and the reduceddynamic baselines from different software packages show an agreement of 1 mm in three directions.

Most of the related studies have focused on the absolute corrections of GNSS data for POD of a single LEOs.While for PBD,the influence of the GNSS date from the reference LEOs can be weakened by difference,and the influence is mainly derived from the inter-satellite relative errors of GNSS data,which represent differential error between two LEOs.Gu et al.proposed Relative PCVs (RPCVs) estimation with a relative IF phase pattern estimation;the GRACE baseline precision saw a significantly improvement by 33.3%.The effects of the inter-satellite relative corrections of GNSS data on PBD should be studied further.The first objective of this paper is to overcome the shortcomings existing in MW combinations by proposing a Single-Difference (SD) MW Combination Residuals Variations (SD MWVs) estimation based on the DD WL ambiguities constraint.The second objective of this paper is to investigate the influence of RPCVs correction on PBD.

To assess the influence of RPCVs and SD MWVs on baseline determination,the data collected onboard GRACE and GRACE-FO satellites were used.The GRACE mission consists of two identical formation flying LEOs,GRACE A and B,in an almost circular,near-polar orbit;the distance between the two satellites is approximately 220 km.Due to goodbattery health,GRACE finished its task after operating for over 15 years.On May 22,2018,the GRACE-FO mission launched.GRACE-FO also consists of two identical satellites,GRACE C and D.At present,only GRACE and GRACE-FO offer the possibility for continuous validation of the line-ofsight component of the inter-satellite vector by the very precise observations of the KBR system.There are no direct measures for precision assessment of the resulting baseline with the other missions,it makes GRACE and GRACE-FO suitable for this study.In Section 2,the reduced-dynamic PBD strategy is introduced and the method of generating the SD MWVs estimation is provided.In Sections 3.1 and 3.2,the results of the RPCVs and SD MWVs,respectively,are described and discussed.The influences on the PBD results of the RPCVs and SD MWVs are analyzed in Sections 4.1 and 4.2,respectively.Section 5 presents the conclusions.

2.PBD strategy and SD MWVs estimation

The code and phase observations of dual-frequency GNSS can be expressed as,

where the subscript(=1,2) indicates the two different signals;the superscriptindicates the GNSS satellite;the subscriptdenotes the LEOs;andrepresent the code and phase observations,respectively;ρ denotes the geometric distance;is the speed of light;δis the clock biases;is the frequency;andare the hardware delays of code and phase observations,respectively;andare the multipath errors of code and phase observations,respectively;λ is the wavelength;is the phase ambiguity;δis the receiver antenna PCVs;ε and εrepresent the observation noise and other unmodeled errors,respectively;is the first order ionospheric delay,which comprises the dominant error induced by the ionosphere,and the high-order ionospheric terms are neglected here due to their very limited effect.It should be noted that the relativity correction,GNSS transmitter antenna PCVs corrections,and phase wind-up correction are already applied.

Traditionally,DD GNSS observations are used for PBD as the carrier phase ambiguities become integers.This study assumes that the two receivers are of the same type.Then,the DD IF combinations (2) are adopted,in which clock biases,hardware delays,and ionospheric delays are removed.

Multipath errors are caused by the superposition of the direct signal with interfering reflected signals.Their severity is determined by the signal path difference,signal strength,polarization of the reflected signal,and receivers’ internal properties.Code multipath errors may attain meter level.Carrier-phase multipath errors are much smaller than code errors.Therefore,they are negligible compared with code multipath errors.In the PBD process,the code and carrier phase observations are usually weighted (see Table 1),which implies that the code has only a marginal effect on the quality of the orbit products,since they rely predominantly on the phase observations.However,as both code and carrier observations are used with equal weight when using MW combination(3)to resolve the integer carrier phase ambiguities,the success rate of the DD WL ambiguities resolution will strongly depend on code observations.

Initialization:=0, k=0,=0.

Form the SD observation with.

Determine the DD WL ambiguities,the success rate of the DD WL ambiguities is k.

Estimate the SD WL ambiguities with the DD WL ambiguities constraint.

Obtain the SD MW combination residuals,update.

Justify whether k-k＜θ,if so,stop the iteration;otherwise,=+1,go to Step 2.

The SD MW combination residuals are not affected by GNSS satellite ephemeris and PCVs errors.The reduceddynamic orbit determination method is applied to the intersatellite PBD of LEOs.This research was accomplished by the National University of Defense Technology Orbit Determination Toolkit (NUDTTK) software.A complete flowchart of the reduced-dynamic PBD with the RPCVs and SD MWVs estimation is shown in Fig.1 and the detailed models used in the PBD are listed in Table 1.The Least-squares Ambiguity Decorrelation Adjustment (LAMBDA) methodhas been implemented in integer ambiguities resolution.It should be noted that integer ambiguity validation is adopted.Theis taken as 0.3,and theis taken as 3.0.

Fig.1 Flow chart of PBD with RPCVs and SD MWVs.

Table 1 Summary of dynamic and measurement models used for PBD.

3.Estimation of RPCVs and SD MWVs

The space-borne dual-frequency (L1 and L2) GPS data from 30 days of operation were used to estimate the RPCVs and SD MWVs in this study,specifically,day of year (DOY) 60-89 in 2007 (March 1-30) for GRACE,and DOY 1-30 in 2019 (January 1-30) for GRACE-FO.

3.1.Results of RPCVs estimation

The O-C residuals can be used to measure the consistency of the applied models with the GPS observation data,while the PCV errors will make O-C residuals larger.In this paper,the carrier phase DD IF O-C residuals are analyzed,as the DD IF combinations are used for PBD.The carrier phase DD IF O-C residuals of GRACE and GRACE-FO obtained by PBD without PCVs correction are shown in Fig.2.Each point shown in the figure represents the RMS of the carrier phase DD IF O-C residuals based on the statistics of the data falling into an interval with a width of 2°.The smaller the elevation,the higher are the carrier phase DD IF O-C residuals(see Fig.2(a)).The carrier phase DD IF O-C residuals of GRACE-FO are larger than that of GRACE.The variation of carrier phase DD IF O-C residuals with respect to azimuth is more complicated(see Fig.2(b)).The residuals can reach up to 7 mm between 140° and 160° for the GRACE-FO mission.

The RPCVs and SD MWVs maps are defined in the Antenna-Fixed Reference System (AFS),for which the origin is the antenna reference point,the X-axis coincides with the Body Fixed System (BFS) X-axis,the Z-axis points toward the up direction,and the Y-axis completes the right-handed system.Case et al.and Wen et al.introduced the definitions of the BFSs of GRACE and GRACE-FO,respectively.As both RPCVs and SD MWVs depend on the direction of the GPS reception signal,the pattern can be estimated by azimuth/elevation bins.In this study,the residuals are sorted in the azimuth/elevation bins of 5°×5°;the pattern is obtained from the mean value of all the residuals that fall into this region.The iteration is performed until convergence is reached and no further improvement can be observed.

The RPCVs based on the carrier phase SD O-C residuals,which represent differential antenna PCVs between two LEOs.The results of the RPCVs estimation for GRACE and GRACE-FO are shown in Fig.3.For lower elevations,the RPCVs are typically much larger than for higher elevations.For GRACE (see Fig.3(a)),the larger values of the RPCVs occur at the upper part of the map.For GRACEFO (see Fig.3(b)),the maximum value of the RPCVs is up to 14 mm and the larger values of the RPCVs occur at azimuths 150°-180° and 330°-360°,which are consistent with the results in Fig.2.

Fig.2 RMS of carrier phase DD IF O-C residuals as a function of elevation and azimuth for GRACE(analyzed period:DOY 60-89,2007) and GRACE-FO (analyzed period:DOY 1-30,2019).

Fig.3 Results of RPCVs estimation for GRACE and GRACE-FO given in the AFS of GRACE B and GRACE D,respectively.

3.2.Results of SD MWVs estimation

The success rate of the DD WL ambiguities resolution is strongly depending on code observations.The Multipath Combinations (MPCs) can be used to eliminate the geometric distance,clock correction,and ionospheric delay,which are often used to assess multipath errors of the code observations.The MPCs have been used to evaluate the code observations quality in several previous studies,such as,CHAMP,GRACE,Fengyunand Tiangong-2.Fig.4 shows the magnitudes of the multipath errors.The averages of the Multipath Errors for P1 (MP1) for GRACE A,GRACE B,GRACE C,and GRACE D are 0.191 m,0.147 m,0.199 m,and 0.206 m,respectively;the Multipath Errors for P2 (MP2) are 0.363 m,0.154 m,0.119 m,and 0.127 m,respectively.The multipath errors of GRACE A are significantly higher than that of GRACE B,which has a significant influence on the WL ambiguities resolution.

Fig.4 Daily RMS of multipath errors.

The SD MW combination between two LEOs can be treated as a constant value in a continuous tracking arc,without the influence of observation noise,multipath,and hardware delays.Fig.5 shows the SD MW observations of GRACE on 3 March 2007.There are obvious fluctuations at the beginning and end of the arc in Fig.5.This phenomenon may be due to the smaller elevation at the beginning and end of the arc,where the multipath errors are worse.Fig.6 plots the histograms of the fractional parts of the SD WL ambiguities;the mean and standard deviations are displayed in the top right corner.Approximately 66% of the fractional parts are confined to 0.1 cycles for GRACE (see Fig.6(a)),and 80% for GRACE-FO (see Fig.6(b)).There are systematic biases that strongly depend on the code observations in the MW combinations.The success rate of the ambiguities resolution will be significantly affected without an estimation of such systematic biases.

Fig.5 SD MW combinations of GRACE.

Fig.6 Histogram of the fractional parts of SD WL ambiguities.

Fig.7 shows the results of the SD MWVs estimation.Similar to the results of the RPCVs,the SD MWVs also clearly demonstrate systematic deviation;much larger values occur at low elevations.For GRACE (see Fig.7(a)),the maximum value of the SD MWVs is up to 0.32 cycles.These large values are most prominent at the lower part of the map,which is due to the active occultation antenna of GRACE A.When the terminator-pointing occultation antennas are activated,there is a considerably elevated level of interference,which could significant affect the DD WL ambiguities resolution.In contrast,for GRACE-FO (see Fig.7(b)),the maximum value is only approximately 0.1 cycles,which is considerably smaller than that of GRACE.

Fig.7 Results of SD MWVs estimation for GRACE and GRACE-FO given in the AFS of GRACE B and GRACE D.

4.Results and discussion

The space-borne dual-frequency (L1 and L2) GPS data from 30 days of operation were used to estimate the RPCVs and SD MWVs in this study,specifically,Day of Year (DOY)60-89 in 2007 (March 1-30) for GRACE,and DOY 1-30 in 2019 (January 1-30) for GRACE-FO.The impacts of RPCVs and SD MWVs on PBD are analyzed by internal and external validations.The internal validation is provided by the carrier phase DD IF O-C residuals.Orbit comparison with JPL precise science orbit and KBR data are selected as external validation.JPL precise science orbit is distributed along with the GRACE GPS data and is a part of GRACE Level 1B product,which is based on un-differenced IF linear combination and un-differenced ambiguities resolution.The main objective of KBR is the ultra-precise satellite-to-satellite tracking with a precision of 10 μm.This allows to use KBR data for independent validation and precision assessment of baseline solutions obtained from other instruments.

4.1.Influence of RPCVs

As show in Table 2,the consistency with KBR observations is improved from 1.03 mm to 0.72 mm after the RPCVs correction for GRACE.Jäggi et al.estimated the PCVs based on 362 days of un-differenced IF combination carrier phase residuals in the year 2007,and standard deviation values of the mean KBR for that solution is 0.81 mm.Many other errors such as the GPS satellite ephemeris and PCV errors might be absorbed by the results of the absolute PCVs estimation.Here,the RPCVs estimation is based on the differential carrier phase measurements with fixed DD ambiguities,which better reflect the magnitude of the PCVs errors between two LEOs.The means of the orbit differences,in three-dimensions (3D),between the -RPCVs-SD MWVs and+RPCVs-SD MWVs orbit solutions,and the JPL orbit solutions,are 4.83 mm and 4.64 mm,respectively.The carrier phase DD IF O-C residuals are reduced from 3.46 mm to 3.03 mm when using the RPCVs correction.

The inclusion of the RPCVs correction improves the consistency with the KBR observations of GRACE-FO;the mean of the daily RMS decreases from 1.04 mm to 0.65 mm.The GRACE-FO orbit solutions computed with the RPCVs correction have better consistency with the JPL orbit solutions;the means of the orbit differences,in 3D,decreased from 4.44 mm to 3.99 mm.The carrier phase DD IF O-C residuals decreased from 5.16 mm to 4.15 mm after the RPCVs correction (see Table 2).

The results of the RPCVs estimation,based on March 2007(Fig.3(a)),are applied to the PBD of February 2007 (DOY 32-59) and April 2007 (DOY 91-120).Fig.8 shows the daily RMS of GRACE’s relative position for the different types of solution as compared to the KBR measurements.The mean KBR RMS decreased from 1.07 mm to 0.73 mm after the RPCVs correction.The details for February and April are shown in Table 3;the RMS of the differences of the KBR observations for February and April are improved by 27.6%and 33.0% after the RPCVs correction,respectively.

4.2.Influence of SD MWVs

For GRACE,the correction of the SD MWVs leads to an improved KBR consistency of 0.57 mm.The average daily RMS of the JPL comparison with the+RPCVs+SD MWVs solution is 4.56 mm.The use of the SD MWVs correction further reduces the carrier phase DD IF O-C residuals to 2.99 mm.Notably,the fixing rate of the DD WL and DD NL ambiguities are increased significantly after using the SD MWVs correction (see Table 2).The success rate of the ambiguities resolution depends on the quality of the float ambiguities estimation.The more precise the float ambiguities are,the higher is the probability of estimating the correct integer ambiguities.Fig.9 shows the daily accuracy of the float DD WL and Fig.10 shows the DD WL ambiguities’ fixing rate.After the SD MWVs correction,the accuracy of the float DD WL ambiguities increased from 0.145 cycles to 0.053 cycles.The fixing rate of the DD WL ambiguities increased from 85.4% to 98.4%.Fig.11(a) shows the number of ambiguities and the KBR residuals without the SD MWVs correction on 3 March 2007(DOY 62);the fixed rate of the DD ambiguities is 86.4%.It should be noted that the majority of the KBR residuals are within 3 mm.However,the KBR residuals are as high as 7.4 mm between 07:00 and 08:00.From 16:00 to 18:00,the KBR residuals as high as 11.4 mm.The main reason for the presence of these outliers is the failure to fix the ambiguities at that time.After the SD MWVs correction (see Fig.11(b)),the fixed rate of ambiguities is 96.0%.As shown in Fig.8,the solutions obtained from the +RPCVs-SD MWVs exhibit an agreement with KBR of 0.5-1.0 mm,with few significant outliers.The RMS of the KBR residuals of DOY 62,DOY 63,and DOY 113 decreased from greater than 2 mm to millimetre level after the SD MWVs correction.The SD MWVs correction can effectively improve the success rate of the DD ambiguities;the accuracy and reliability of the PBD results are further enhanced.

Fig.9 Daily accuracy of float DD WL ambiguities of GRACE.

Fig.10 Daily DD WL ambiguities fixing rate of GRACE.

Fig.11 Number of ambiguities and KBR residuals on 3 March 2007 of GRACE.

Table 2 Mean of daily RMS of differences KBR observations,orbit differences in 3D between the orbit solutions in this research and JPL orbit solutions,the mean of DD WL ambiguities and DD NL ambiguities fixing rate,and mean of daily RMS of carrier phase DD IF (DD LIF) O-C residuals.

Fig.8 Daily RMS of GRACE relative orbit determination from different type of solution when compared to the KBR measurements.

For GRACE-FO,after the SD MWVs correction the average daily RMS of the KBR is 0.65 mm.The accuracy of the PBD of GRACE-FO can also reach sub-millimetre level based on the DD IF combination and integer ambiguities resolution.The orbit difference,in 3D,with JPL is 3.97 mm when using the SD MWVs correction,and the carrier phase DD IF O-C residuals are 4.12 mm (see Table 2).The results of the+RPCVs-SD MWVs solution are essentially the same as the results of the +RPCVs+SD MWVs solution.For GRACE-FO,the DD WL ambiguities fixing rate as high as 98.9% without the SD MWVs correction (see Table 2).The inclusion of the SD MWVs correction makes further improvement difficult.

Table 3 Mean of daily RMS of differences KBR observations,and the mean of DD ambiguities fixing rate of GRACE.

The results of the SD MWVs estimation of GRACE based on March 2007(Fig.7(a))are applied to the PBD of February and April 2007.The RMS of KBR is 0.59 mm after the SD MWVs correction.The RMS of the differences of the KBR observations of February and April are improved by 20.6%and 18.1% after the SD MWVs correction,respectively (see Table 3).Both the RPCVs and SD MWVs estimation of the current period could be used for the previous and subsequent periods.

5.Conclusions

The inter-satellite relative corrections of phase center and code residual variations play important roles in precise relative orbit for formation flying LEOs.The influence of the RPCVs and SD MWVs on the PBD were assessed with GPS data from the GRACE and GRACE-FO missions in this paper.The conclusions are as follows.

(1) After the RPCVs correction,the consistency with the KBR measurements increased from 1.03 mm to 0.72 mm,and from 1.04 mm to 0.65 mm for GRACE and GRACEFO,respectively.The results confirm the importance of the RPCVs estimation based on in-fight data.

(2) For GRACE,the fixing rate of the DD ambiguities increased by 12.8% after the SD MWVs correction.When using the SD MWVs correction,an improved consistency of 0.57 mm is achieved for the KBR measurements.The SD MWVs correction can effectively solve the problem that of the success rate of the ambiguities resolution strongly depending on the code observations.

(3) More importantly,the results of the RPCVs and SD MWVs estimations both have good stability.In-flight data processing results shows that the results of the RPCVs and SD MWVs estimations for the current period could be applied to the previous and subsequent periods.

The RPCVs estimation and SD MWVs estimation are both applicable for PBD that based on CDGNSS techniques.In general,the satellite formations with relatively stable baselines between hundreds of meters up to 220 km(GRACE)are more suitable for application of CDGNSS techniques.When formation with baselines lengths up to thousands of kilometers,the unfavorable tracking conditions that are less suitable for application of CDGNSS techniques.The improvement of the satellite formation PROD with high-dynamic baselines will be discussed in our next study.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

This study was supported by the National Natural Science Foundation of China (Nos.41874028,61803018).