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GPS precise point positioning with ambiguity resolution
Available online at www.sciencedirect.com
ScienceDirect Advances in Space Research xxx (2019) xxx–xxx www.elsevier.com/locate/asr
Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/GPS precise point positioning with ambiguity resolution Xuexi Liu a, Weiping Jiang a,b, Zhao Li c,⇑, Hua Chen a, Wen Zhao a a
School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, People’s Republic of China b GNSS Research Center, Wuhan University, 129 Luoyu Road, Wuhan 430079, People’s Republic of China c Department of Land Surveying and Geo-Informatics, The Hong Kong Polytechnic University, 181 Chatham Road South, Hung Hom, Kowloon 999077, Hong Kong Received 1 November 2018; received in revised form 18 February 2019; accepted 19 February 2019
Abstract Precise point positioning (PPP) usually takes about 30 min to obtain centimetre-level accuracy, which greatly limits its application. To address the drawbacks of convergence speed and positioning accuracy, we develop a PPP model with integrated GPS and BDS observations. Based on the method, stations with global coverage are selected to estimate the fractional cycle bias (FCB) of GPS and BDS. The short-term and long-term time series of wide-lane (WL) FCB, and the single day change of narrow-lane (NL) FCB are analysed. It is found that the range of GPS and BDS non-GEO (IGSO and MEO) WL FCB is stable at up to a 30-day-time frame. At times frame of up to 60 days, the stability is reduced a lot. Whether for short-term or long-term, the changes in the BDS GEO WL FCB are large. Moreover, BDS FCB sometimes undergoes a sudden jump. Besides, 17 and 10 stations were used respectively to investigate the convergence speed and positioning errors with six strategies: BDS ambiguity-float PPP (Bfloat), GPS ambiguity-float PPP (Gfloat), BDS/GPS ambiguity-float PPP (BGfloat), BDS ambiguity-fixed PPP (Bfix), GPS ambiguity-fixed (Gfix), and BDS/GPS ambiguity-fixed (BGfix). The average convergence speed of the ambiguity-fixed solution is greatly improved compared with the ambiguity-float solution. In terms of the average convergence time, the Bfloat is the longest and the BGfix is the shortest among these six strategies. Whether for ambiguityfloat PPP or ambiguity-fixed PPP, the convergence reduction time in three directions for the combined system is the largest compared with the single BDS. The average RMS value of the Bfix in three directions (easting (E), northing (N), and up (U)) are 2.0 cm, 1.5 cm, and 5.9 cm respectively, while those of the Gfix are 0.8 cm, 0.5 cm, and 1.7 cm. Compared with single system, the BDS/GPS combined ambiguity-fixed system (BGfix) has the fastest convergence speed and the highest accuracy, with average RMS as 0.7 cm, 0.5 cm, and 1.9 cm for the E, N, U components, respectively. Ó 2019 Published by Elsevier Ltd on behalf of COSPAR.
Keywords: BDS (BeiDou); GPS; Combined system; Precise point positioning; Fractional cycle bias; Convergence time
1. Introduction
⇑ Corresponding author.
E-mail addresses: [email protected] (X. Liu), [email protected] (W. Jiang), [email protected] (Z. Li), [email protected]. cn (W. Zhao).
Precise point positioning (PPP) (Zumberge et al., 1997) has been widely used in precise satellite orbit determination, high-precision coordinate frame maintenance, global or regional scientific investigation, aeronautical dynamic measurement, hydrographic surveying and charting, etc,
https://doi.org/10.1016/j.asr.2019.02.026 0273-1177/Ó 2019 Published by Elsevier Ltd on behalf of COSPAR.
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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X. Liu et al. / Advances in Space Research xxx (2019) xxx–xxx
in the past twenty years. It can provide centimetre-level positioning with only one receiver on a global scale. However, to achieve accurate and stable positioning results, the convergence time for PPP is long, since the difficulty of ambiguity resolution (AR) can influence the convergence of PPP (Li and Zhang, 2014; Liu et al., 2016). The emergence of new PPP AR technology provides an opportunity to solve this problem. To achieve the desired precision in a short time, it is necessary to study an effective method for PPP AR, among which the fractional cycle bias (FCB) of ambiguities should be corrected to realize rapid PPP AR. The carrier phase observations contain FCB from receiver and satellite, which destroys the integer properties of ambiguity in undifferenced (UD) observation equations (Blewitt, 1989). These uncorrected FCB will reduce the success rate of AR and it is hard to achieve effective AR with increasing bias (Teunissen, 1999). To overcome the influence of FCB, Gabor et al. first proposed a method for estimating FCB using a global reference station network and conducted correlation analysis. This estimation method was based on the separation of FCB and integer ambiguity by a specific method (Gabor and Nerem, 1999). Later, Ge et al. (2006) and Leandro and Santos (2006) further expanded Gabor’s research on FCB. Some PPP integer AR methods have been developed and implemented with real data sets in recent years. Ge et al. (2008) proposed a single-differenced (SD) PPP ambiguity resolution method which shows that the singlestation ambiguity corrected by FCB can achieve a fixing rate of over 80%. Collins et al. (2008) also studied phase biases with a different processing method which is known as the decoupled clock model. A decoupled clock model was proposed and verified that the correction of FCB could improve the convergence speed of ambiguity (Collins et al. 2010). Laurichesse et al. (2009) also developed an integer phase clock model to recover the integer properties of ambiguity, which used different clock terms for code and phase observations. This model utilized the wide-lane (WL) FCBs to resolve the integer wide-lane ambiguity, whereas the integer L1 ambiguity was directly estimated. This method was later adopted by the Centre National d’E´tudes Spatiales (CNES) to generate the integerrecovered precise clock bias (Loyer et al. 2012). Geng et al. (2010a,b) analyzed the three methods mentioned above and showed that the three methods were equivalent in theory and had comparable performance in practice. Most recently, Shi and Gao (2014) conducted the comparison among the three PPP AR methods with respect to integer property recovery method, system redundancy, and required corrections. The above traditional single system PPP cannot cope with some complex environments, for example, urban canyons, which have poor satellite structure and few satellites available. Since the accuracy, availability, and reliability of satellite navigation are directly related to the number of satellites tracked, the positioning accuracy in urban canyons and other complex environments would be affected
(Cai and Gao, 2013; Cai et al., 2015). The BeiDou Navigation Satellite System (BDS) began to provide navigation, positioning, and time services for the Asia Pacific region since 2012. The BDS-3 which is a global system will be completed by 2020 (http://en.beidou.gov.cn/). As expected, the combination of the BDS and GPS could increase the number of visible satellites, which helps improve the geometric structure of the satellite and reduce the PDOP value, so that the PPP can be used in more complex environments. However, most of the research has been conducted on single GPS PPP AR, and only a few teams focus on the ambiguity resolution of BDS PPP (Geng et al., 2010a; Zhang and Li, 2013; Li et al., 2016b; Wang et al., 2017; Li et al., 2018a,b). Li et al. (2016a) performed a comprehensive analysis for the PPP AR of the BDS and GPS combined systems, but the long-term FCB of GPS and BDS have not been investigated. Liu et al. (2016) used BDS to accelerate the convergence of GPS PPP, yet did not study BDS AR in detail. Li et al. (2018a) studied on the subject of multi-GNSS AR, however the effect of single BDS AR was neglected. Recently, Geng et al. (2018) developed an approach of forming inter-system ambiguities using GPS and BDS data modulated on multi-frequencies so as to investigate the potential of inter-system PPP-AR in further speeding up initializations, but did not show the details of BDS PPP AR. Therefore, we think that there is still of necessity to carry out detailed researches on these topics, so as to better understand the long-time change of WL and NL FCB, the convergence time and the precise of BDS/GPS PPP AR. Based on this, we will focus on long-term FCB estimation and analysis based on a BDS/GPS combined system observation model to compare single system AR with combined system AR. The remainder of this paper is organised as follows: Section 2 describes the BDS/GPS PPP model and the derived formulae, Section 3 introduces the results of FCB estimation and focuses on the time series characteristics of BDS FCB, Section 4 presents experiments to analyse convergence rate of the six schemes, Section 5 analyses the performance of PPP AR for different systems including BDS PPP AR, GPS PPP AR, and BDS/GPS PPP AR in terms of positioning accuracy and convergence time, conclusions are presented in Section 6.
2. Methods In PPP, ionospheric-free (IF) linear combination is normally used to eliminate the first-order ionospheric delays in the pseudo-range and carrier-phase measurements. The corresponding equations are as follows: K s;K K s;K s;K s;K P s;K þ T s;K ð1Þ r;IF ¼ qr þ c dt r dt r þ br;IF bIF þ eP ;IF K s;K s;K þ T s;K Ls;K r;IF ¼ qr þ c dt r dt r s;K s;K K K þ kIF N r;IF þ Br;IF BIF þ es;K L;IF
ð2Þ
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
X. Liu et al. / Advances in Space Research xxx (2019) xxx–xxx
where superscript K represents the satellite system; P IF and LIF are the IF code and carrier-phase observations respectively; q is the geometric distance between the satellite and the receiver; c represents the speed of light in vacuum; dtr and dts represent the receiver and satellite clocks, respectively; T is the tropospheric delay; N sr;IF is the IF ambiguity for the corresponding wavelength kIF ; br;IF , and Br;IF are the IF receiver code and phase hardware biases respectively, and bsIF and BsIF are the IF satellite code and phase hardware biases;eP ;IF and eL;IF are the code and phase observation errors including multipath and noises. Both the receiver and satellite phase centre offsets and variations, earth tides, phase wind-up, and the relativistic delays must be corrected in model. The re-parameterized ambiguities, receiver and satellite code and phase hardware biases are as: br;IF ¼ f 21 br;1 f 22 br;2 = f 21 f 22 ð3Þ 2 s 2 s 2 s 2 ð4Þ bIF ¼ f 1 b1 f 2 b2 = f 1 f 2 N sr;IF ¼ c f 1 N sr;1 f 2 N sr;2 =kIF f 21 f 22 ð5Þ Br;IF ¼ cðf 1 Br;1 f 2 Br;2 Þ=kIF f 21 f 22 ð6Þ 2 2 s s s ð7Þ BIF ¼ c f 1 B1 f 2 B2 =kIF f 1 f 2 where br;1 , br;2 and bs1 , bs2 are the receiver and satellite code hardware biases on frequencies L1 and L2 respectively.Br;1 , Br;2 and Bs1 , Bs2 are the receiver and satellite phase hardware biases on frequencies L1 and L2 respectively. N sr;1 , N sr;2 are the integer ambiguities on frequencies L1 and L2 . To reduce satellite orbit and clock error, precise ephemeris and clock from the International GNSS Service (IGS) analysis centre are usually used in PPP data processing. According to the processing strategy proposed by the IGS Analysis Centre, it is known that cdts provided by IGS absorbs the satellite code hardware biases bsIF . From formula (1), we can see that the code observation provides an absolute reference for the receiver clock, so the IF receiver hardware biases br;IF are absorbed by the actual estimated receiver clock parameters. At the same time, ambiguity parameters are strongly correlated with phase hardware biases, which are difficult to separate from each other. In addition, it is generally assumed that the phase hardware bias has extremely high stability over a certain period of time, so the ambiguity parameter is considered to absorb phase hardware bias when we calculate the floating PPP solution (Chen et al., 2015). For convenience, by removing part of the upper and lower superscript from formulae (1) and (2), the IF combination of GPS and BDS PPP observation equation can be rewritten as:
P GIF ¼ qG þ c d tGr þ T G þ eGP;IF ð8Þ LGIF ¼ qG þ c d tGr þ T G þ kGIF N Gr;IF þ d Gr;IF d s;G þ eGL;IF IF ð9Þ
3
ð10Þ P CIF ¼ qC þ c d tCr þ T C þ eCP;IF LCIF ¼ qC þ c d tCr þ T C þ kCIF N Cr;IF þ d Cr;IF d s;C þ eCL;IF IF ð11Þ with:
c d tr ¼ cdtr þ br;IF
ð12Þ
d r;IF ¼ Br;IF br;IF =kIF
ð13Þ
d sIF
¼ BsIF
bsIF =kIF
ð14Þ
From the above formulas, we can see that the satellite code hardware bias is introduced into the phase observation equation due to the introduction of the satellite clock based on code observation. In addition, since the code observation equation is a reference equation, the code receiver clock is also introduced into the phase observation equation. At the same time, because of the satellite and the receiver phase hardware biases, together with the initial phase biases, which are difficult to separate from the integer ambiguity, the integer properties of ambiguity are destroyed. The BDS/GPS PPP solution is carried out by Panda software (Shi et al., 2008). Turbo-Edit method is used for cycle slip detection in pre-processing (Blewitt, 1990). All the observations are weighted according to their elevation and the elevation cut-off angle for data is set to 7 degrees. The BDS GEO carrier observation noise at zenith direction is set to 0.009 m while the other satellites are set to 0.003 m. GPS, BDS non-GEO and BDS GEO code observation noise at zenith direction are set to 0.3 m, 0.6 m and 3 m, respectively. The critical criterion of the ratio value is 2.0. More specific solutions are shown in Table 1. 3. FCB estimation The IF ambiguity can usually be decomposed into float WL ambiguity and float narrow-lane (NL) ambiguity (Ge et al., 2008). !, s;K s;K s;K cf c 2 b b b N N N ð15Þ kIF r;IF ¼ r;WL þ f 1 þ f 2 r;NL f 21 f 22 with s;K b s;K ¼ N s;K N r;WL þ d r;WL d WL r;WL
ð16Þ
s;K s;K b N r;NL ¼ N r;NL þ d r;NL d NL
ð17Þ
s;K
b s;K is the float IF ambiguity, N b s;K and N s;K where N r;WL reprer;IF r;WL b s;K and sent the float and integer WL ambiguity, while N r;NL N s;K denote the float and integer NL ambiguity r;NL respectively. The method of FCB estimation proposed by Ge et al. (2008) is applied here which are shown in Fig. 1. To fix the ambiguity in PPP, the WL and NL FCB must be derived at the server and provided to the user along with
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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Table 1 Data processing strategies and models of PPP. Options
Processing strategies
Observations interval Elevation Satellite orbit/clock Phase center correction
30 s 7° GFZ precise ephemeris GPS:IGS14.ATX; BDS:ESA recommended value Only GPS Model correction IERS Conventions 2010 IERS Conventions 2010 IERS Conventions 2010 IERS Conventions 2010 Ionosphere-free combination Saastamoinen + random walk Static: as a constant estimation; kinematic: random walk Random walk
PCV Phase Wind-up Solid earth tide Ocean load Polar motion Relativistic effect Ionospheric delay Tropospheric delay Receiver coordinates Receiver clock
the corresponding satellite orbit and clock. The client uses WL FCB and NL FCB to fix the WL ambiguity and NL ambiguity respectively. Then IF ambiguity are reconstructed which are used as a constraint to float PPP. Finally, a fixed PPP solution is obtained. The short-term and long-term FCB of GPS and BDS will be calculated and analysed in this part. 3.1. GPS FCB estimation The experimental data relating to GPS FCB estimation cover 60 days from day of year (DOY) 1–60 in 2016. The data are obtained from the global IGS network, with a total of 130 stations. The distribution of stations is shown by red circles in Fig. 2. All stations provide 24 h observations with a sampling interval of 30 s. It should be noted that in order to do the experiment, these stations should be evenly distributed around the globe (as far as possible). 3.1.1. GPS WL FCB Here, the GPS WL FCB is estimated by using the aforementioned experimental data. The WL FCB values
for 1–30 days, 31–60 days, and 1–60 days are shown in the left panels of Fig. 3, respectively. To investigate the influence of time length on WL FCB, the WL FCB timevarying characteristic map is drawn into these left panels of Fig. 3. These WL FCB data are referenced to G01. Due to poor data quality, G04, G10, and G32 satellites are excluded. The left panels of Fig. 3 show that the change of WL FCB within 30 days is relatively stable. Most of the changes are within 0.05 cycles while a few satellites change more with time, such as G03 and G30, but the range of variation within 30 days is no more than 0.1 cycles. The stability of the WL FCB may due to the different hardware conditions or observation errors of each satellite. From the left bottom panel of Fig. 3, we can see that if the time span is increased, the value of WL FCB fluctuates significantly, which shows that, over longer times, the stability of WL FCB decreases. The fluctuation range of most satellites within 60 days is about 0.1 cycles, while a small part of the satellite fluctuation range is about 0.15 cycles, such as at G25 and G30. The phase bias of SD WL formed by any two satellites should be consistent over a wide area, which is the basis of phase bias estimation methods. The regularity of the phase bias with time is an important condition for the service of PPP. It is also of a certain significance to make medium- and long-term predictions of WL FCB. 3.1.2. GPS NL FCB The rapid change of NL FCB makes it unwise to estimate daily values of NL FCB, thus, it is also unnecessary to study the stability of long-term NL FCB. Here we randomly select DOY 16 in 2016 to estimate the NL FCB. The estimated time interval of NL FCB is 5 min, so there are 288 epochs in a day. The NL FCB time series is shown in Fig. 4. It can be seen from Fig. 4 that the change in NL FCB is much greater than that in WL FCB. Some satellites could change by 0.2 cycles within one day, such as G02, G05, G29, and G27, while most other satellites vary less than 0.1 cycles in a day.
Server end
User end MW combination
WL Fixed WL FCB
WL FCB calculation
Float PPP
NL Fixed
Float PPP
NL FCB NL FCB calculation
Fixed PPP
Fig. 1. Flow chart for PPP processing with ambiguity resolution.
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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Fig. 2. Distribution of the reference network for FCB estimation. Their supported constellations are shown in different colours, GPS in red, BDS in blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3.2. BDS FCB estimation Due to the fact that the BDS has not yet completed its global network, the BDS observations could only be received in some areas. Here, 83 stations which can receive BDS observations in the Multi-GNSS Experiment (MGEX) are selected to estimate the BDS WL FCB (Montenbruck et al., 2017). Its spatial distribution is shown by blue circles in Fig. 2. The estimation strategy and method of the BDS FCB are basically the same as that of GPS. However, because the satellite-induced code biases of BDS GEO cannot be corrected, the precision of GEO precise products is also lower than that of IGSO and MEO products, BDS GEO FCB and non-GEO (IGSO and MEO) FCB are estimated separately, in order to obtain a higher precision FCB and avoid GEO observation error pollution of IGSO and MEO observations (Wanninger and Beer, 2015). It can improve the accuracy of BDS FCB. 3.2.1. BDS WL FCB The right panels of Fig. 3 show the BDS WL FCB value from 1–30 days, 31–60 days, and 1–60 days, respectively. BDS GEO data are referenced to C04 while BDS nonGEO are referenced to C06. As can be seen from the right panels of Fig. 3, within one month, the fluctuation of BDS GEO FCB is within 0.4 cycles (indicates a significant change). The WL FCB of C02 and C05 exhibit strong fluctuation, with a maximum range of 0.7 cycles, indicating that these two satellites are unstable. C01 and C03 are more stable than C02 and C05. Their WL FCB variations within one day are less than
0.2 cycles, and the variations within the adjacent 5–10 days are stable. Besides, there is a sudden jump seen in the GEO WL FCB data. The WL FCB of BDS MEO and the IGSO are very stable compared with the GEO satellite data. The range of changes within a month is very small, basically changing within 0.1 cycles. For period of five to 10 days, the range of variation is within 0.05 cycles or less. BDS MEO and IGSO also have a sudden jump in the WL FCB, but the WL FCB data are stable before and after the jump. We think that the jump may be caused by satellite clock error since we found that the precise clock corrections of corresponding satellites provided by GFZ experienced a datum offset on that day, and we think there may be some relationship between them (Li et al., 2016a). Another possibility is that this phenomenon may also due to the re-initialization of the satellite signals. Further studies still needed to investigate the actual reasons. The right bottom panel of Fig. 3 shows that, whether for GEO or non-GEO, the variation of BDS WL FCB within two months is larger than that in one month. The BDS GEO WL FCB data are within 0.7 cycles in a given twomonths period while the non-GEO data are within 0.15 cycles. In the long-term, the frequency of jumps in WL FCB increases. Therefore, we conclude that the BDS GEO WL FCB cannot be predicted in the long-term, while the WL FCB of MEO and IGSO can be predicted, but the phenomenon of sudden jumps should be heeded. 3.2.2. BDS NL FCB Because the NL FCB changes rapidly, we select DOY 8 in 2016 to do BDS NL FCB estimation, and the result
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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X. Liu et al. / Advances in Space Research xxx (2019) xxx–xxx
BDS
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C01 C02 C03 C05 C07 C08 C09 C10 C11 C12 C14
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Fig. 3. Estimated GPS (left) and BDS (right) WL FCB. The time span is different in each subgraph. G01 is taken as the reference for GPS. C04 and C06 are taken as the reference for BDS GEO and non-GEO respectively. Different colours represent different satellites.
time series are shown in Fig. 5. The C01 was eliminated due to the poor data quality. From Fig. 5, we can see that the BDS GEO NL FCB fluctuates within 0.8 cycles in a day, but changes less than 0.1 cycles within
100 min. Compared with GEO satellites, BDS MEO and IGSO have a much smaller change in the NL FCB. Except for C14, all non-GEO changed within 0.2 cycles. There is also a sudden jump in the C14 on
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
X. Liu et al. / Advances in Space Research xxx (2019) xxx–xxx
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G02 G03 G05 G06 G07 G08 G09 G11 G12 G13 G14 G15 G16 G17 G18 G19 G20 G21 G22 G23 G24 G25 G26 G27 G28 G29 G30 G31
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Hours Fig. 4. Estimated GPS NL FCB on DOY 16 of 2016 referenced to G01. Different colours represent different satellites.
C02
C03
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Fig. 5. Estimated BDS NL FCB on DOY 8 of 2016 referenced to C04 and C06 for GEO and non-GEO respectively. Different colours represent different satellites.
this day, but its value is stable before and after the jump, and the amplitude of the change is within 0.2 cycles for a certain time. Compared Fig. 5 with
Fig. 4, we found that the changes in GPS NL FCB are more gentle than in BDS NL FCB while there are more sudden jumps in BDS NL FCB.
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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4. Comparison of convergence rates for single and combined systems After getting WL and NL FCB, we can carry out PPP AR on the user side. In this section, the convergence rate of floating and fixed solutions for BDS, GPS and BDS/ GPS will be compared. Six strategies are implemented here: BDS ambiguityfloat PPP (Bfloat), GPS ambiguity-float PPP (Gfloat), BDS/GPS ambiguity-float PPP (BGfloat), BDS ambiguity-fixed PPP (Bfix), GPS ambiguity-fixed (Gfix), and BDS/GPS ambiguity-fixed (BGfix). 17 stations from the Asia Pacific region were used (as shown in Fig. 6). The time span is 30 days, DOY 1 to 30 in 2016. Floating and fixed solutions with sampling rate as 30 s for the three systems are calculated respectively by the above methods. For each station, there’s one solution per day, and each solution includes station’s position difference at each epoch (30 s sampling) with respect to (w.r.t) the IGS provided coordinate (we treat it as the true value) in the easting (E), northing (N), and up (U) directions. Therefore, we totally obtain 510 (equals to 17 * 30) solutions in each direction. The convergence rates in the E, N, and U directions are then assessed in detail. The convergence time is defined as follows: from the first epoch, suppose at epoch t, together with its following 10 epochs, the position difference in the E/N/U directions are less than 5 cm, then the time span from the first epoch to the epoch t is defined as the convergence time in this direction. Some days will not
return results because of data loss. Figs. 7 and 8 show the percentage of float and fixed PPP solutions for the three systems with different convergence time in the E, N, and U directions respectively. Different colour in the pie charts represents the proportion of the solutions with convergence time inside different time span. For example, red represents the percentage of solutions that converge to 5 cm within in 20 min. In terms of float PPP solution, it can be seen from Fig. 7 that Bfloat shows the slowest convergence, with convergence time less than 20 min only account for 4.0%, 15.1%, and 3.8% in the E, N, and U directions respectively. Gfloat exhibits much faster convergence, with 44.3%, 88.0%, and 36.4% of the solutions converge within 20 min for the E, N, U components, while for BGfloat is 46.3%, 90.2%, and 40.6%. Besides, most GPS and GPS/ BDS combined solutions (>70%) converge within 40 min, especially for the N component. However, for Bfloat, there are still 27.1%, 20.4%, and 43.9% of the solutions show a convergence time of longer than 120 min in the E, N, and U directions respectively. Therefore, we conclude that in general the convergence rate of the combined system is significantly higher than that of the single system, especially when compared with BDS. This is one of the advantages of the combined system, as the combined system has more satellites, better spatial structure, and more observation redundancy, which can accelerate convergence. There’s also another interesting finding that whether it is a single or combined system, the convergence speed in the N direc-
Fig. 6. Distribution of the reference network in Asia Pacific region. The violet circles denote the 17 stations used for convergence speed. The black triangles denote the 10 stations used for analysis of accurate of PPP AR. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
X. Liu et al. / Advances in Space Research xxx (2019) xxx–xxx
[0,20)
[20,40)
[60,80)
8.4%
9.8%
BDS(float)
[40,60)
[80,100) 8%
10.9%
15.3%
27.1%
9
[100,120)
more than 120
5.3%
6%
15.5%
43.9%
8.2% 20.4% 11.3%
15.3%
4%
3.8%
24.2%
20.2%
15.1%
15.3%
12.2%
34.8%
GPS(float)
38.6% 13.5%
11.3% 5.1% 1.8% 0.4% 2.2%
44.3%
88% 9.3% 1.1% 1.1% 0.4%
5.8% 1.3% 1.1% 3.3%
BDS/GPS(float)
36.4% 36.8%
38.4% 90.2%
13.3%
7.8% 4.2% 1.6% 0.4% 1.3%
46.3%
3.5% 0.7% 1.1% 4%
8% 0.7% 1.1%
E
40.6%
N
U
Fig. 7. Statistics of convergence time for static BDS, GPS and BDS/GPS float PPP in E, N, and U directions.
BDS(fix)
[0,20)
[20,40) 9.9%
13.3%
[40,60)
[60,80)
[80,100)
11.1%
5.6% 23.9%
[100,120)
9.3%
5.6%
5.6%
16.3%
more than 120 42.2%
10.8% 17.4%
15.8%
12% 3.8%
27.5%
2.9% 25.1%
15.3%
14%
GPS(fix)
49.4%
12.4%
52.4% 89.6% 9.5%
4.5% 0.9%
0.7% 0.2%
45.1%
35.9%
BDS/GPS(fix)
48.3%
7% 2.3% 1.4% 0.2% 0.9%
46.7% 91.9% 7.7%
3.4% 0.7% 0.5% 0.2%
47%
E
2.5% 0.9% 1.8%
8.1% 40.4%
N
U
Fig. 8. Statistics of convergence time for static BDS, GPS and BDS/GPS fixed PPP in E, N, and U directions.
tion is significantly faster than that of the other two directions. As for the fixed solution, Fig. 8 shows that Bfix exhibiting the slowest convergence, with only 3.8%, 15.3%, and
2.9% of the solutions converge within 20 min for the E, N, and U components respectively, while for those converge longer than 120 min still account for 23.9%, 17.4%, and 42.2%. Gfix ranks the second, and BGfix shows the
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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X. Liu et al. / Advances in Space Research xxx (2019) xxx–xxx
fastest convergence. Solutions with convergence time less than 20 min for Gfix are 45.1%, 89.6%, and 35.9% in the E, N, and U directions, while for BGfix are 47.0%, 91.9%, and 40.4%. Besides, nearly all of the Gfix and BGfix solutions converge within 120 min, while the majority (>87%) even converge within 40 min, especially for the N component (100%). Compare Fig. 8 and Fig. 7, we can conclude that in general, the convergence time of the fixed solution for the combined system is shorter than that of the single system, especially compared with the BDS. In the case where the convergence time is within 20 min for these three systems, the fixed solution is equivalent to the floating solution. However, the proportion of BDS fixed solution with convergence time exceeding 120 min is less than that of the floating solution, while nearly all the fixed solution of the GPS and the combined system converge within 120 min. In addition, for both fixed and floating solution, all the three systems show the fastest convergence in the N direction, followed by the E and U components. To investigate the convergence time difference between floating and fixed solution, Table 2 shows the average convergence time and its reduction rate at 17 stations for the 30 days in E, N, and U directions for the three systems. It can be seen from Table 2 that in general the average convergence speed of the fixed solution is greatly improved compared with the floating solution. This is consistent with Figs. 7 and 8. Bfloat shows the longest average convergence time, with 109.22 min, 87.77 min, and 241.01 min respectively in E, N, and U directions, while BGfix is the fastest, with average convergence time only at 21.66 min, 11.67 min, and 25.89 min. The reduction rate from Bfloat to Bfix is 15.4%, 12.5%, and 5.2% for the E, N, and U components respectively, while both GPS and the BDS/GPS combined system exhibit even higher reduction rate of more than 20% on average. Dec represents the reduction rate of the average convergence time from floating to fixed solution. Table 3 illustrates the reduction rate of the average convergence time between different systems for both the fixed and floating solutions. We can see that both GPS and the combined systems show very large reduction (>72%) in the average convergence time relative to single BDS, no matter for fixed or floating solution. Compared with GPS, however, the reduction for the combined system is quite small, less than 10% and 4% for the floating and fixed solution respectively. These results further reflect the supe-
Table 3 Statistical results of the reduction rate of the average convergence time within 30 days for 17 stations (unit: %). Reduction rate
E(float)
N(float)
U(float)
E(fix)
N(fix)
U(fix)
GtoB BGtoB BGtoG
72.7 75.3 9.7
83.8 84.4 3.7
84.9 85.7 4.7
76.3 76.6 1.0
84.2 84.8 3.7
88.5 88.7 1.3
GtoB: reduction rate of the average convergence time for GPS compared to BDS. BGtoB: reduction rate for BDS/GPS compared to BDS. BGtoG: reduction rate for BDS/GPS compared to GPS.
riority of the fixed solution relative to the floating solution, and also the combined system relative to single system. 5. Result analysis: PPP AR With WL FCB and NL FCB, the integer properties of ambiguity can be restored and result of the PPP AR can be obtained. Here, 10 stations in the Asia Pacific region are selected to carry out the related experiments on the PPP AR of BDS, GPS, and BDS/GPS. The time span of the experimental data is 30 days, DOY 1 to 30, in 2016. The stations used are shown by the black triangles in Fig. 6. The coordinate differences between our results and corresponding values from IGS are calculated (Figs. 9– 12). The floating solution is also plotted on the same figure for comparison. Statistical information of all obtain results is shown in Table 4. 5.1. GPS PPP AR We randomly select station KARR in our analysis of the GPS PPP AR on DOY 17, 2016 (Fig. 9). As seen from Fig. 9, the fixed solution in the horizontal direction converge to its true value in about 15 min. Although the floating solution can quickly converge to 5 cm accuracy, it remains much slower than the fixed solution. The accuracy of the GPS floating solution and fixed solution are similar at about 1 cm. The observation time for GPS floating solution (24 h), was so long that it has enough time to converge to high accuracy. In the up direction, the convergence of the fixed solution is slightly faster than that of the floating solution. After convergence, the accuracies of both fixed and floating solution are similar at about 1.5 cm. Fig. 9 also shows that the convergence of the fixed solution in the horizontal direction is faster than that in the up direction. Once the ambiguity is fixed to the correct value,
Table 2 Statistical results of average convergence time within 30 days for 17 stations. System
BDS GPS BDS/GPS
Average convergence time (unit: min)
Reduction rate from floating to fixed
E(float)
N(float)
U(float)
E(fix)
N(fix)
U(fix)
E(dec/%)
N(dec/%)
U(dec/%)
109.22 29.87 26.97
87.77 14.25 13.72
241.01 36.50 34.79
92.44 21.88 21.66
76.79 12.12 11.67
228.52 26.23 25.89
15.4 26.8 19.7
12.5 15.0 15.0
5.2 28.1 25.6
Dec represents the reduction rate of the average convergence time from floating to fixed solution.
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
X. Liu et al. / Advances in Space Research xxx (2019) xxx–xxx
11
0.2 fix
float
E/m
0.1 0.0 -0.1 -0.2 0
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0.2 fix
12
15
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float
N/m
0.1 0.0 -0.1 -0.2 0
3
0.2 fix
Hours
float
U/m
0.1 0.0 -0.1 -0.2 0
3
Hours
Fig. 9. Position differences for GPS floating and fixed PPP in static mode, at station KARR, on DOY 17, 2016.
high-precision distance will be obtained, which makes the positioning converge to its true value instantaneously. The advantage of the static fixed solution relative to the floating solution only lies in the accelerated convergence and improved accuracy in the east direction. 5.2. BDS PPP AR We randomly select station JFNG to analyse the BDS PPP AR on DOY 8, 2016 (Fig. 10). From Fig. 10 we can see that, in the horizontal direction, once the ambiguity is fixed correctly, the accuracy of the fixed solution is higher than that of the floating solution, which can quickly converge to a high precision value. In the up direction, the accuracy of BDS fixed solution and floating solution is so poor that it takes a long time to converge. There is also an obvious systematic deviation in the up direction. Besides, there are some zig-zag lines in the figure because the ambiguity parameters need to be re-initialised when a cycle slip occurred. Here, we set BDS ratio to 2.0. 5.3. BDS/GPS PPP AR Fig. 11 shows the position differences for BDS/GPS floating and fixed PPP in static mode at station GMSD on DOY 8, 2016. In the horizontal direction, the precision of the fixed solution of BDS/GPS PPP is higher than that of the floating solution, and the convergence of the fixed solution is much faster than that of the floating solution,
especially in the E direction. After convergence, the precision of both fixed and floating solutions of BDS/GPS PPP are all within 1 cm in the horizontal direction. As for the up direction, the initial precision of the fixed solution is much higher than that of the floating solution. As time increases, however, the precision of the floating solution can also reach a high level. After convergence, the fixed and floating solutions of the combined system are all within 2 cm. In addition, we can see that the combined system has a small systematic deviation in the up direction, which may probably be due to the introduction of the systematic deviation of BDS. Fig. 12 gives the position differences of BDS PPP, GPS PPP, and BDS/GPS PPP fixed solutions at station JFNG on DOY 8, 2016. It can be see that the precision and convergence rate of the combined BDS/GPS and the single GPS system are similar in the horizontal direction, which are higher than the fixed solutions of the single BDS. In the up direction, the precision of the combined system is similar to the single GPS, but the convergence speed is faster. We think that this is due to the introduction of the BDS observations, increase of observation number, and improvement of the spatial structure thereof. Another interesting phenomenon is that for both the horizontal and up components, the convergence speed and accuracy of BDS fixed solutions are lower than those of the other two systems, especially for the up direction. This indicates that the single BDS is still very unstable and the correction models for BDS are not accurate enough.
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
12
X. Liu et al. / Advances in Space Research xxx (2019) xxx–xxx 0.2 fix
float
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0.1 0.0 -0.1 -0.2 0
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0.1 0.0 -0.1 -0.2 0
3
0.2 fix
Hours
float
U/m
0.1 0.0 -0.1 -0.2 0
3
Hours
Fig. 10. Position differences for BDS floating and fixed PPP in static mode, at station JFNG, on DOY 8, 2016.
0.2 fix
float
E/m
0.1 0.0 -0.1 -0.2 0
3
6
9
12
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0.1 0.0 -0.1 -0.2 0
3
6
9
6
9
0.2 fix
Hours
float
U/m
0.1 0.0 -0.1 -0.2 0
3
Hours
Fig. 11. Position differences for BDS/GPS floating and fixed PPP in static mode, at station GMSD, on DOY 8, 2016.
The fixed and floating solutions of three systems for 10 stations are shown in Figs. 13 and 14, respectively. The statistical results of average RMS in the E, N, and U directions are listed in Table 4. As can be seen from Fig. 13, the accuracy of Bfloat for almost all stations is the lowest, followed by Gfloat, while
BGfloat has the highest precision in the east direction. In the north direction, except for station XMIS, all stations have Bfloat with the worst accuracy, while the accuracy of Gfloat and BGfloat are similar. In the up direction, the accuracy of Bfloat is the lowest in all stations. Due to the poor accuracy of BDS in the up direction, the error
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
X. Liu et al. / Advances in Space Research xxx (2019) xxx–xxx
13
0.2 B(fix)
G(fix)
BG(fix)
E/m
0.1 0.0
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-0.1 -0.2 0
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0.2 B(fix)
G(fix)
Hours
BG(fix)
U/m
0.1 0.0
-0.1 -0.2 0
3
6
9
Hours
Fig. 12. Position differences for BDS PPP AR, GPS PPP AR, BDS/GPS PPP AR in static mode, at station JFNG, on DOY 8, 2016.
Table 4 Average RMS for fixed and floating solution of BDS, GPS, and BDS/GPS (cm). System
E(float)
N(float)
U(float)
E(fix)
N(fix)
U(fix)
BDS GPS BDS/GPS
2.8 1.5 1.1
2.0 1.0 0.9
6.8 2.1 2.2
2.0 0.8 0.7
1.5 0.5 0.5
5.9 1.7 1.9
of BDS will be introduced into the combined system. If the up direction of BDS is too poor, then the error in the up direction of the combined system will be worse than that of the single GPS. Similar as Fig. 13, Fig. 14 gives that almost all stations, BGfix exhibits the best accuracy, Gfix the second, while Bfix is the worst in the east direction. In the north direction, the accuracy of Bfix is the worst, while the accuracy of BGfix is slightly higher than that of Gfix. This is because the precision of Gfix in the north direction is already very high (i.e., at millimetre level), so that the fixed solution of the combined system will not improve much compared with single GPS. In the up direction, the accuracy of Bfix is the worst at all stations, while BGfix and Gfix has the similar accuracy. The reason why the fixed solution of the combined system is worse than the single GPS is also due to the systematic error introduced in the up direction of BDS. From Table 4, we can see that the accuracy of the fixed solutions of the three systems are higher than that of floating solution. Whether adopting a floating or fixed solution, the accuracy in the north direction is the highest, followed by the east component, while up component is the worst.
Compared with floating solution, the accuracy of BDS, GPS, and BDS/GPS fixed solution increase by 28.6%, 46.7%, and 36.4% respectively for the east component. In the north direction, the accuracy increase by 25.0%, 50.0%, and 44.4% for the above three systems. The up component shows the smallest accuracy increase, at 13.2%, 19.0%, and 13.6%, respectively. In addition, the accuracy of both BDS fixed and floating solutions is the worst in three directions, especially in the up direction (at about 5 cm). BGfix shows the highest accuracy in the horizontal direction, but its precision in the up direction is slightly lower than Gfix. The reason that the RMS of the upcomponent solution of the BGfix is worse than that of the Gfix is because BDS has no precise antenna phase center values. On the other hand, the accuracy of BDS observations and the precision of products are lower than GPS. Compared with Figs. 13 and 14, we find that in general the accuracy of the stations at middle and high latitudes are better than that at low latitudes. For example, the accuracy of stations such as PNGM, PONH, PTVL, SOLO, and XMIS are better than that of other stations. This is mainly because the ionosphere above the equator is more active than in middle and high latitudes, and the ionospheric combination cannot eliminate higher-order ionospheric delays in the pseudo-range and carrier-phase measurements, which would affect the final coordinates. 6. Conclusion The purpose of this work is to study the technology and method of BDS/GPS combined PPP and its ambiguity resolution. The method and mathematical model of BDS/
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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Bfloat Gfloat BGfloat
E/cm
4
2
0 gmsd
karr
nnor
pngm
pohn
ptvl
sin1
solo
xmis
karr
nnor
pngm
pohn
ptvl
sin1
solo
xmis
karr
nnor
pngm
pohn
ptvl
sin1
solo
xmis
Station
Bfloat Gfloat BGfloat
4
N/cm
jfng
2
0 gmsd
U/cm
8
jfng
Station
Bfloat Gfloat BGfloat
6 4 2 0 gmsd
jfng
Station Fig. 13. The average RMS for float solution of BDS, GPS and BDS/GPS in E, N, and U directions.
E/cm
4
Bfix Gfix BGfix
2
0 gmsd
jfng
karr
nnor
pngm
pohn
ptvl
sin1
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ptvl
sin1
solo
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ptvl
sin1
solo
xmis
Station
N/cm
4
Bfix Gfix BGfix
2
0 gmsd
jfng
karr
nnor
pngm
pohn
Station
U/cm
8
Bfix Gfix BGfix
6 4 2 0 gmsd
jfng
karr
nnor
pngm
pohn
Station Fig. 14. The average RMS for fixed solution of BDS, GPS, and BDS/GPS in E, N, and U directions.
GPS integrated PPP AR are described. The long-term GPS and BDS WL FCB are calculated using two month’s observation data. Results show that the variation of the GPS FCB is less than 0.05 cycles within 30 days. Within 60 days,
it exhibits a small increase, but still it is no more than 0.1 cycles. The ranges of BDS GEO and non-GEO WL FCB within 30 days are less than 0.4 and 0.1 cycles respectively. If the time span increase to 60 days, the value increased to
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
X. Liu et al. / Advances in Space Research xxx (2019) xxx–xxx
0.7 and 0.15 cycles for GEO and non-GEO. The variation in GPS NL FCB in one day is mostly less than 0.1 cycles. However, the BDS GEO NL FCB would change by up to 0.7 cycles in a given day, while the non-GEO data were all within 0.2 cycles. In addition, there is a sudden jump in WL and NL FCB of BDS GEO and non-GEO schemes. The convergence speed of six schemes (Bfloat, Gfloat, BGfloat, Bfix, Gfix, and BGfix) has been deeply investigated. The convergence time of Bfloat within 40 min is 24.2%, 39.3%, and 16.0% in the E, N, and U directions respectively, which is the slowest among the six solutions examined here, while the BGfix shows the fastest convergence, with convergence rate as 95.3%, 100%, and 87.1%. In general, the average convergence speed of the fixed solution is greatly improved compared to that of the floating solution. For both the floating and fixed solutions, the reduction in convergence time in three directions for the combined system is the largest compared with that of single BDS. In terms of floating solution, the combined system convergence time decreases by 9.7%, 3.7%, and 4.7% respectively in the E, N, U directions compared with single GPS, while for the fixed solution, the reduction in convergence time of the combined system only improve a little relative to single GPS. The results of BDS PPP AR, GPS PPP AR, and BDS/ GPS PPP AR are studied. We found that the accuracy of a fixed solution for the three systems is greatly improved relative to the floating solution. As for the different fixed solution, the accuracy of single BDS PPP fixed solution shows the worst result, with the average RMS value in the E, N, and U directions as 2.0 cm, 1.5 cm, and 5.9 cm. The BDS/GPS PPP fixed solution has the highest precision in the horizontal direction, but its precision in the vertical direction is similar to that of GPS, with the average RMS value as 0.7 cm, 0.5 cm, and 1.9 cm in the E, N, and U directions respectively. We conclude that the BDS/GPS combined ambiguityfixed system can improve convergence speed and positioning accuracy compared to that achieved by using single system. Because BDS has not yet been completed, the stability and accuracy of BDS FCB are slightly worse than those of GPS. With the continuous development of the BDS network how to use BDS GEO and real-time PPP with multi-GNSS will be one of our focus in future research. Acknowledgments The contribution of data from MGEX is appreciated. This research is supported by the National Science Foundation for Distinguished Young Scholars of China (Grant No. 41525014), the Natural Science Innovation Group Foundation of China (No. 41721003), the Major Technology Innovation Project of Hubei Province of China (2018AAA066), the Program for Changjiang Scholars of the Ministry of Education of China, and the Major Project of Beijing Future Urban Design Innovation Center, Beijing University of Civil Engineering and Architecture (UDC
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Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
ScienceDirect Advances in Space Research xxx (2019) xxx–xxx www.elsevier.com/locate/asr
Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/GPS precise point positioning with ambiguity resolution Xuexi Liu a, Weiping Jiang a,b, Zhao Li c,⇑, Hua Chen a, Wen Zhao a a
School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, People’s Republic of China b GNSS Research Center, Wuhan University, 129 Luoyu Road, Wuhan 430079, People’s Republic of China c Department of Land Surveying and Geo-Informatics, The Hong Kong Polytechnic University, 181 Chatham Road South, Hung Hom, Kowloon 999077, Hong Kong Received 1 November 2018; received in revised form 18 February 2019; accepted 19 February 2019
Abstract Precise point positioning (PPP) usually takes about 30 min to obtain centimetre-level accuracy, which greatly limits its application. To address the drawbacks of convergence speed and positioning accuracy, we develop a PPP model with integrated GPS and BDS observations. Based on the method, stations with global coverage are selected to estimate the fractional cycle bias (FCB) of GPS and BDS. The short-term and long-term time series of wide-lane (WL) FCB, and the single day change of narrow-lane (NL) FCB are analysed. It is found that the range of GPS and BDS non-GEO (IGSO and MEO) WL FCB is stable at up to a 30-day-time frame. At times frame of up to 60 days, the stability is reduced a lot. Whether for short-term or long-term, the changes in the BDS GEO WL FCB are large. Moreover, BDS FCB sometimes undergoes a sudden jump. Besides, 17 and 10 stations were used respectively to investigate the convergence speed and positioning errors with six strategies: BDS ambiguity-float PPP (Bfloat), GPS ambiguity-float PPP (Gfloat), BDS/GPS ambiguity-float PPP (BGfloat), BDS ambiguity-fixed PPP (Bfix), GPS ambiguity-fixed (Gfix), and BDS/GPS ambiguity-fixed (BGfix). The average convergence speed of the ambiguity-fixed solution is greatly improved compared with the ambiguity-float solution. In terms of the average convergence time, the Bfloat is the longest and the BGfix is the shortest among these six strategies. Whether for ambiguityfloat PPP or ambiguity-fixed PPP, the convergence reduction time in three directions for the combined system is the largest compared with the single BDS. The average RMS value of the Bfix in three directions (easting (E), northing (N), and up (U)) are 2.0 cm, 1.5 cm, and 5.9 cm respectively, while those of the Gfix are 0.8 cm, 0.5 cm, and 1.7 cm. Compared with single system, the BDS/GPS combined ambiguity-fixed system (BGfix) has the fastest convergence speed and the highest accuracy, with average RMS as 0.7 cm, 0.5 cm, and 1.9 cm for the E, N, U components, respectively. Ó 2019 Published by Elsevier Ltd on behalf of COSPAR.
Keywords: BDS (BeiDou); GPS; Combined system; Precise point positioning; Fractional cycle bias; Convergence time
1. Introduction
⇑ Corresponding author.
E-mail addresses: [email protected] (X. Liu), [email protected] (W. Jiang), [email protected] (Z. Li), [email protected]. cn (W. Zhao).
Precise point positioning (PPP) (Zumberge et al., 1997) has been widely used in precise satellite orbit determination, high-precision coordinate frame maintenance, global or regional scientific investigation, aeronautical dynamic measurement, hydrographic surveying and charting, etc,
https://doi.org/10.1016/j.asr.2019.02.026 0273-1177/Ó 2019 Published by Elsevier Ltd on behalf of COSPAR.
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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in the past twenty years. It can provide centimetre-level positioning with only one receiver on a global scale. However, to achieve accurate and stable positioning results, the convergence time for PPP is long, since the difficulty of ambiguity resolution (AR) can influence the convergence of PPP (Li and Zhang, 2014; Liu et al., 2016). The emergence of new PPP AR technology provides an opportunity to solve this problem. To achieve the desired precision in a short time, it is necessary to study an effective method for PPP AR, among which the fractional cycle bias (FCB) of ambiguities should be corrected to realize rapid PPP AR. The carrier phase observations contain FCB from receiver and satellite, which destroys the integer properties of ambiguity in undifferenced (UD) observation equations (Blewitt, 1989). These uncorrected FCB will reduce the success rate of AR and it is hard to achieve effective AR with increasing bias (Teunissen, 1999). To overcome the influence of FCB, Gabor et al. first proposed a method for estimating FCB using a global reference station network and conducted correlation analysis. This estimation method was based on the separation of FCB and integer ambiguity by a specific method (Gabor and Nerem, 1999). Later, Ge et al. (2006) and Leandro and Santos (2006) further expanded Gabor’s research on FCB. Some PPP integer AR methods have been developed and implemented with real data sets in recent years. Ge et al. (2008) proposed a single-differenced (SD) PPP ambiguity resolution method which shows that the singlestation ambiguity corrected by FCB can achieve a fixing rate of over 80%. Collins et al. (2008) also studied phase biases with a different processing method which is known as the decoupled clock model. A decoupled clock model was proposed and verified that the correction of FCB could improve the convergence speed of ambiguity (Collins et al. 2010). Laurichesse et al. (2009) also developed an integer phase clock model to recover the integer properties of ambiguity, which used different clock terms for code and phase observations. This model utilized the wide-lane (WL) FCBs to resolve the integer wide-lane ambiguity, whereas the integer L1 ambiguity was directly estimated. This method was later adopted by the Centre National d’E´tudes Spatiales (CNES) to generate the integerrecovered precise clock bias (Loyer et al. 2012). Geng et al. (2010a,b) analyzed the three methods mentioned above and showed that the three methods were equivalent in theory and had comparable performance in practice. Most recently, Shi and Gao (2014) conducted the comparison among the three PPP AR methods with respect to integer property recovery method, system redundancy, and required corrections. The above traditional single system PPP cannot cope with some complex environments, for example, urban canyons, which have poor satellite structure and few satellites available. Since the accuracy, availability, and reliability of satellite navigation are directly related to the number of satellites tracked, the positioning accuracy in urban canyons and other complex environments would be affected
(Cai and Gao, 2013; Cai et al., 2015). The BeiDou Navigation Satellite System (BDS) began to provide navigation, positioning, and time services for the Asia Pacific region since 2012. The BDS-3 which is a global system will be completed by 2020 (http://en.beidou.gov.cn/). As expected, the combination of the BDS and GPS could increase the number of visible satellites, which helps improve the geometric structure of the satellite and reduce the PDOP value, so that the PPP can be used in more complex environments. However, most of the research has been conducted on single GPS PPP AR, and only a few teams focus on the ambiguity resolution of BDS PPP (Geng et al., 2010a; Zhang and Li, 2013; Li et al., 2016b; Wang et al., 2017; Li et al., 2018a,b). Li et al. (2016a) performed a comprehensive analysis for the PPP AR of the BDS and GPS combined systems, but the long-term FCB of GPS and BDS have not been investigated. Liu et al. (2016) used BDS to accelerate the convergence of GPS PPP, yet did not study BDS AR in detail. Li et al. (2018a) studied on the subject of multi-GNSS AR, however the effect of single BDS AR was neglected. Recently, Geng et al. (2018) developed an approach of forming inter-system ambiguities using GPS and BDS data modulated on multi-frequencies so as to investigate the potential of inter-system PPP-AR in further speeding up initializations, but did not show the details of BDS PPP AR. Therefore, we think that there is still of necessity to carry out detailed researches on these topics, so as to better understand the long-time change of WL and NL FCB, the convergence time and the precise of BDS/GPS PPP AR. Based on this, we will focus on long-term FCB estimation and analysis based on a BDS/GPS combined system observation model to compare single system AR with combined system AR. The remainder of this paper is organised as follows: Section 2 describes the BDS/GPS PPP model and the derived formulae, Section 3 introduces the results of FCB estimation and focuses on the time series characteristics of BDS FCB, Section 4 presents experiments to analyse convergence rate of the six schemes, Section 5 analyses the performance of PPP AR for different systems including BDS PPP AR, GPS PPP AR, and BDS/GPS PPP AR in terms of positioning accuracy and convergence time, conclusions are presented in Section 6.
2. Methods In PPP, ionospheric-free (IF) linear combination is normally used to eliminate the first-order ionospheric delays in the pseudo-range and carrier-phase measurements. The corresponding equations are as follows: K s;K K s;K s;K s;K P s;K þ T s;K ð1Þ r;IF ¼ qr þ c dt r dt r þ br;IF bIF þ eP ;IF K s;K s;K þ T s;K Ls;K r;IF ¼ qr þ c dt r dt r s;K s;K K K þ kIF N r;IF þ Br;IF BIF þ es;K L;IF
ð2Þ
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
X. Liu et al. / Advances in Space Research xxx (2019) xxx–xxx
where superscript K represents the satellite system; P IF and LIF are the IF code and carrier-phase observations respectively; q is the geometric distance between the satellite and the receiver; c represents the speed of light in vacuum; dtr and dts represent the receiver and satellite clocks, respectively; T is the tropospheric delay; N sr;IF is the IF ambiguity for the corresponding wavelength kIF ; br;IF , and Br;IF are the IF receiver code and phase hardware biases respectively, and bsIF and BsIF are the IF satellite code and phase hardware biases;eP ;IF and eL;IF are the code and phase observation errors including multipath and noises. Both the receiver and satellite phase centre offsets and variations, earth tides, phase wind-up, and the relativistic delays must be corrected in model. The re-parameterized ambiguities, receiver and satellite code and phase hardware biases are as: br;IF ¼ f 21 br;1 f 22 br;2 = f 21 f 22 ð3Þ 2 s 2 s 2 s 2 ð4Þ bIF ¼ f 1 b1 f 2 b2 = f 1 f 2 N sr;IF ¼ c f 1 N sr;1 f 2 N sr;2 =kIF f 21 f 22 ð5Þ Br;IF ¼ cðf 1 Br;1 f 2 Br;2 Þ=kIF f 21 f 22 ð6Þ 2 2 s s s ð7Þ BIF ¼ c f 1 B1 f 2 B2 =kIF f 1 f 2 where br;1 , br;2 and bs1 , bs2 are the receiver and satellite code hardware biases on frequencies L1 and L2 respectively.Br;1 , Br;2 and Bs1 , Bs2 are the receiver and satellite phase hardware biases on frequencies L1 and L2 respectively. N sr;1 , N sr;2 are the integer ambiguities on frequencies L1 and L2 . To reduce satellite orbit and clock error, precise ephemeris and clock from the International GNSS Service (IGS) analysis centre are usually used in PPP data processing. According to the processing strategy proposed by the IGS Analysis Centre, it is known that cdts provided by IGS absorbs the satellite code hardware biases bsIF . From formula (1), we can see that the code observation provides an absolute reference for the receiver clock, so the IF receiver hardware biases br;IF are absorbed by the actual estimated receiver clock parameters. At the same time, ambiguity parameters are strongly correlated with phase hardware biases, which are difficult to separate from each other. In addition, it is generally assumed that the phase hardware bias has extremely high stability over a certain period of time, so the ambiguity parameter is considered to absorb phase hardware bias when we calculate the floating PPP solution (Chen et al., 2015). For convenience, by removing part of the upper and lower superscript from formulae (1) and (2), the IF combination of GPS and BDS PPP observation equation can be rewritten as:
P GIF ¼ qG þ c d tGr þ T G þ eGP;IF ð8Þ LGIF ¼ qG þ c d tGr þ T G þ kGIF N Gr;IF þ d Gr;IF d s;G þ eGL;IF IF ð9Þ
3
ð10Þ P CIF ¼ qC þ c d tCr þ T C þ eCP;IF LCIF ¼ qC þ c d tCr þ T C þ kCIF N Cr;IF þ d Cr;IF d s;C þ eCL;IF IF ð11Þ with:
c d tr ¼ cdtr þ br;IF
ð12Þ
d r;IF ¼ Br;IF br;IF =kIF
ð13Þ
d sIF
¼ BsIF
bsIF =kIF
ð14Þ
From the above formulas, we can see that the satellite code hardware bias is introduced into the phase observation equation due to the introduction of the satellite clock based on code observation. In addition, since the code observation equation is a reference equation, the code receiver clock is also introduced into the phase observation equation. At the same time, because of the satellite and the receiver phase hardware biases, together with the initial phase biases, which are difficult to separate from the integer ambiguity, the integer properties of ambiguity are destroyed. The BDS/GPS PPP solution is carried out by Panda software (Shi et al., 2008). Turbo-Edit method is used for cycle slip detection in pre-processing (Blewitt, 1990). All the observations are weighted according to their elevation and the elevation cut-off angle for data is set to 7 degrees. The BDS GEO carrier observation noise at zenith direction is set to 0.009 m while the other satellites are set to 0.003 m. GPS, BDS non-GEO and BDS GEO code observation noise at zenith direction are set to 0.3 m, 0.6 m and 3 m, respectively. The critical criterion of the ratio value is 2.0. More specific solutions are shown in Table 1. 3. FCB estimation The IF ambiguity can usually be decomposed into float WL ambiguity and float narrow-lane (NL) ambiguity (Ge et al., 2008). !, s;K s;K s;K cf c 2 b b b N N N ð15Þ kIF r;IF ¼ r;WL þ f 1 þ f 2 r;NL f 21 f 22 with s;K b s;K ¼ N s;K N r;WL þ d r;WL d WL r;WL
ð16Þ
s;K s;K b N r;NL ¼ N r;NL þ d r;NL d NL
ð17Þ
s;K
b s;K is the float IF ambiguity, N b s;K and N s;K where N r;WL reprer;IF r;WL b s;K and sent the float and integer WL ambiguity, while N r;NL N s;K denote the float and integer NL ambiguity r;NL respectively. The method of FCB estimation proposed by Ge et al. (2008) is applied here which are shown in Fig. 1. To fix the ambiguity in PPP, the WL and NL FCB must be derived at the server and provided to the user along with
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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Table 1 Data processing strategies and models of PPP. Options
Processing strategies
Observations interval Elevation Satellite orbit/clock Phase center correction
30 s 7° GFZ precise ephemeris GPS:IGS14.ATX; BDS:ESA recommended value Only GPS Model correction IERS Conventions 2010 IERS Conventions 2010 IERS Conventions 2010 IERS Conventions 2010 Ionosphere-free combination Saastamoinen + random walk Static: as a constant estimation; kinematic: random walk Random walk
PCV Phase Wind-up Solid earth tide Ocean load Polar motion Relativistic effect Ionospheric delay Tropospheric delay Receiver coordinates Receiver clock
the corresponding satellite orbit and clock. The client uses WL FCB and NL FCB to fix the WL ambiguity and NL ambiguity respectively. Then IF ambiguity are reconstructed which are used as a constraint to float PPP. Finally, a fixed PPP solution is obtained. The short-term and long-term FCB of GPS and BDS will be calculated and analysed in this part. 3.1. GPS FCB estimation The experimental data relating to GPS FCB estimation cover 60 days from day of year (DOY) 1–60 in 2016. The data are obtained from the global IGS network, with a total of 130 stations. The distribution of stations is shown by red circles in Fig. 2. All stations provide 24 h observations with a sampling interval of 30 s. It should be noted that in order to do the experiment, these stations should be evenly distributed around the globe (as far as possible). 3.1.1. GPS WL FCB Here, the GPS WL FCB is estimated by using the aforementioned experimental data. The WL FCB values
for 1–30 days, 31–60 days, and 1–60 days are shown in the left panels of Fig. 3, respectively. To investigate the influence of time length on WL FCB, the WL FCB timevarying characteristic map is drawn into these left panels of Fig. 3. These WL FCB data are referenced to G01. Due to poor data quality, G04, G10, and G32 satellites are excluded. The left panels of Fig. 3 show that the change of WL FCB within 30 days is relatively stable. Most of the changes are within 0.05 cycles while a few satellites change more with time, such as G03 and G30, but the range of variation within 30 days is no more than 0.1 cycles. The stability of the WL FCB may due to the different hardware conditions or observation errors of each satellite. From the left bottom panel of Fig. 3, we can see that if the time span is increased, the value of WL FCB fluctuates significantly, which shows that, over longer times, the stability of WL FCB decreases. The fluctuation range of most satellites within 60 days is about 0.1 cycles, while a small part of the satellite fluctuation range is about 0.15 cycles, such as at G25 and G30. The phase bias of SD WL formed by any two satellites should be consistent over a wide area, which is the basis of phase bias estimation methods. The regularity of the phase bias with time is an important condition for the service of PPP. It is also of a certain significance to make medium- and long-term predictions of WL FCB. 3.1.2. GPS NL FCB The rapid change of NL FCB makes it unwise to estimate daily values of NL FCB, thus, it is also unnecessary to study the stability of long-term NL FCB. Here we randomly select DOY 16 in 2016 to estimate the NL FCB. The estimated time interval of NL FCB is 5 min, so there are 288 epochs in a day. The NL FCB time series is shown in Fig. 4. It can be seen from Fig. 4 that the change in NL FCB is much greater than that in WL FCB. Some satellites could change by 0.2 cycles within one day, such as G02, G05, G29, and G27, while most other satellites vary less than 0.1 cycles in a day.
Server end
User end MW combination
WL Fixed WL FCB
WL FCB calculation
Float PPP
NL Fixed
Float PPP
NL FCB NL FCB calculation
Fixed PPP
Fig. 1. Flow chart for PPP processing with ambiguity resolution.
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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Fig. 2. Distribution of the reference network for FCB estimation. Their supported constellations are shown in different colours, GPS in red, BDS in blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3.2. BDS FCB estimation Due to the fact that the BDS has not yet completed its global network, the BDS observations could only be received in some areas. Here, 83 stations which can receive BDS observations in the Multi-GNSS Experiment (MGEX) are selected to estimate the BDS WL FCB (Montenbruck et al., 2017). Its spatial distribution is shown by blue circles in Fig. 2. The estimation strategy and method of the BDS FCB are basically the same as that of GPS. However, because the satellite-induced code biases of BDS GEO cannot be corrected, the precision of GEO precise products is also lower than that of IGSO and MEO products, BDS GEO FCB and non-GEO (IGSO and MEO) FCB are estimated separately, in order to obtain a higher precision FCB and avoid GEO observation error pollution of IGSO and MEO observations (Wanninger and Beer, 2015). It can improve the accuracy of BDS FCB. 3.2.1. BDS WL FCB The right panels of Fig. 3 show the BDS WL FCB value from 1–30 days, 31–60 days, and 1–60 days, respectively. BDS GEO data are referenced to C04 while BDS nonGEO are referenced to C06. As can be seen from the right panels of Fig. 3, within one month, the fluctuation of BDS GEO FCB is within 0.4 cycles (indicates a significant change). The WL FCB of C02 and C05 exhibit strong fluctuation, with a maximum range of 0.7 cycles, indicating that these two satellites are unstable. C01 and C03 are more stable than C02 and C05. Their WL FCB variations within one day are less than
0.2 cycles, and the variations within the adjacent 5–10 days are stable. Besides, there is a sudden jump seen in the GEO WL FCB data. The WL FCB of BDS MEO and the IGSO are very stable compared with the GEO satellite data. The range of changes within a month is very small, basically changing within 0.1 cycles. For period of five to 10 days, the range of variation is within 0.05 cycles or less. BDS MEO and IGSO also have a sudden jump in the WL FCB, but the WL FCB data are stable before and after the jump. We think that the jump may be caused by satellite clock error since we found that the precise clock corrections of corresponding satellites provided by GFZ experienced a datum offset on that day, and we think there may be some relationship between them (Li et al., 2016a). Another possibility is that this phenomenon may also due to the re-initialization of the satellite signals. Further studies still needed to investigate the actual reasons. The right bottom panel of Fig. 3 shows that, whether for GEO or non-GEO, the variation of BDS WL FCB within two months is larger than that in one month. The BDS GEO WL FCB data are within 0.7 cycles in a given twomonths period while the non-GEO data are within 0.15 cycles. In the long-term, the frequency of jumps in WL FCB increases. Therefore, we conclude that the BDS GEO WL FCB cannot be predicted in the long-term, while the WL FCB of MEO and IGSO can be predicted, but the phenomenon of sudden jumps should be heeded. 3.2.2. BDS NL FCB Because the NL FCB changes rapidly, we select DOY 8 in 2016 to do BDS NL FCB estimation, and the result
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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Fig. 3. Estimated GPS (left) and BDS (right) WL FCB. The time span is different in each subgraph. G01 is taken as the reference for GPS. C04 and C06 are taken as the reference for BDS GEO and non-GEO respectively. Different colours represent different satellites.
time series are shown in Fig. 5. The C01 was eliminated due to the poor data quality. From Fig. 5, we can see that the BDS GEO NL FCB fluctuates within 0.8 cycles in a day, but changes less than 0.1 cycles within
100 min. Compared with GEO satellites, BDS MEO and IGSO have a much smaller change in the NL FCB. Except for C14, all non-GEO changed within 0.2 cycles. There is also a sudden jump in the C14 on
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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G02 G03 G05 G06 G07 G08 G09 G11 G12 G13 G14 G15 G16 G17 G18 G19 G20 G21 G22 G23 G24 G25 G26 G27 G28 G29 G30 G31
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Hours Fig. 4. Estimated GPS NL FCB on DOY 16 of 2016 referenced to G01. Different colours represent different satellites.
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Fig. 5. Estimated BDS NL FCB on DOY 8 of 2016 referenced to C04 and C06 for GEO and non-GEO respectively. Different colours represent different satellites.
this day, but its value is stable before and after the jump, and the amplitude of the change is within 0.2 cycles for a certain time. Compared Fig. 5 with
Fig. 4, we found that the changes in GPS NL FCB are more gentle than in BDS NL FCB while there are more sudden jumps in BDS NL FCB.
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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4. Comparison of convergence rates for single and combined systems After getting WL and NL FCB, we can carry out PPP AR on the user side. In this section, the convergence rate of floating and fixed solutions for BDS, GPS and BDS/ GPS will be compared. Six strategies are implemented here: BDS ambiguityfloat PPP (Bfloat), GPS ambiguity-float PPP (Gfloat), BDS/GPS ambiguity-float PPP (BGfloat), BDS ambiguity-fixed PPP (Bfix), GPS ambiguity-fixed (Gfix), and BDS/GPS ambiguity-fixed (BGfix). 17 stations from the Asia Pacific region were used (as shown in Fig. 6). The time span is 30 days, DOY 1 to 30 in 2016. Floating and fixed solutions with sampling rate as 30 s for the three systems are calculated respectively by the above methods. For each station, there’s one solution per day, and each solution includes station’s position difference at each epoch (30 s sampling) with respect to (w.r.t) the IGS provided coordinate (we treat it as the true value) in the easting (E), northing (N), and up (U) directions. Therefore, we totally obtain 510 (equals to 17 * 30) solutions in each direction. The convergence rates in the E, N, and U directions are then assessed in detail. The convergence time is defined as follows: from the first epoch, suppose at epoch t, together with its following 10 epochs, the position difference in the E/N/U directions are less than 5 cm, then the time span from the first epoch to the epoch t is defined as the convergence time in this direction. Some days will not
return results because of data loss. Figs. 7 and 8 show the percentage of float and fixed PPP solutions for the three systems with different convergence time in the E, N, and U directions respectively. Different colour in the pie charts represents the proportion of the solutions with convergence time inside different time span. For example, red represents the percentage of solutions that converge to 5 cm within in 20 min. In terms of float PPP solution, it can be seen from Fig. 7 that Bfloat shows the slowest convergence, with convergence time less than 20 min only account for 4.0%, 15.1%, and 3.8% in the E, N, and U directions respectively. Gfloat exhibits much faster convergence, with 44.3%, 88.0%, and 36.4% of the solutions converge within 20 min for the E, N, U components, while for BGfloat is 46.3%, 90.2%, and 40.6%. Besides, most GPS and GPS/ BDS combined solutions (>70%) converge within 40 min, especially for the N component. However, for Bfloat, there are still 27.1%, 20.4%, and 43.9% of the solutions show a convergence time of longer than 120 min in the E, N, and U directions respectively. Therefore, we conclude that in general the convergence rate of the combined system is significantly higher than that of the single system, especially when compared with BDS. This is one of the advantages of the combined system, as the combined system has more satellites, better spatial structure, and more observation redundancy, which can accelerate convergence. There’s also another interesting finding that whether it is a single or combined system, the convergence speed in the N direc-
Fig. 6. Distribution of the reference network in Asia Pacific region. The violet circles denote the 17 stations used for convergence speed. The black triangles denote the 10 stations used for analysis of accurate of PPP AR. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
X. Liu et al. / Advances in Space Research xxx (2019) xxx–xxx
[0,20)
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9
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6%
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8.2% 20.4% 11.3%
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4%
3.8%
24.2%
20.2%
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15.3%
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GPS(float)
38.6% 13.5%
11.3% 5.1% 1.8% 0.4% 2.2%
44.3%
88% 9.3% 1.1% 1.1% 0.4%
5.8% 1.3% 1.1% 3.3%
BDS/GPS(float)
36.4% 36.8%
38.4% 90.2%
13.3%
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46.3%
3.5% 0.7% 1.1% 4%
8% 0.7% 1.1%
E
40.6%
N
U
Fig. 7. Statistics of convergence time for static BDS, GPS and BDS/GPS float PPP in E, N, and U directions.
BDS(fix)
[0,20)
[20,40) 9.9%
13.3%
[40,60)
[60,80)
[80,100)
11.1%
5.6% 23.9%
[100,120)
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more than 120 42.2%
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12% 3.8%
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2.9% 25.1%
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14%
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49.4%
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52.4% 89.6% 9.5%
4.5% 0.9%
0.7% 0.2%
45.1%
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BDS/GPS(fix)
48.3%
7% 2.3% 1.4% 0.2% 0.9%
46.7% 91.9% 7.7%
3.4% 0.7% 0.5% 0.2%
47%
E
2.5% 0.9% 1.8%
8.1% 40.4%
N
U
Fig. 8. Statistics of convergence time for static BDS, GPS and BDS/GPS fixed PPP in E, N, and U directions.
tion is significantly faster than that of the other two directions. As for the fixed solution, Fig. 8 shows that Bfix exhibiting the slowest convergence, with only 3.8%, 15.3%, and
2.9% of the solutions converge within 20 min for the E, N, and U components respectively, while for those converge longer than 120 min still account for 23.9%, 17.4%, and 42.2%. Gfix ranks the second, and BGfix shows the
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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fastest convergence. Solutions with convergence time less than 20 min for Gfix are 45.1%, 89.6%, and 35.9% in the E, N, and U directions, while for BGfix are 47.0%, 91.9%, and 40.4%. Besides, nearly all of the Gfix and BGfix solutions converge within 120 min, while the majority (>87%) even converge within 40 min, especially for the N component (100%). Compare Fig. 8 and Fig. 7, we can conclude that in general, the convergence time of the fixed solution for the combined system is shorter than that of the single system, especially compared with the BDS. In the case where the convergence time is within 20 min for these three systems, the fixed solution is equivalent to the floating solution. However, the proportion of BDS fixed solution with convergence time exceeding 120 min is less than that of the floating solution, while nearly all the fixed solution of the GPS and the combined system converge within 120 min. In addition, for both fixed and floating solution, all the three systems show the fastest convergence in the N direction, followed by the E and U components. To investigate the convergence time difference between floating and fixed solution, Table 2 shows the average convergence time and its reduction rate at 17 stations for the 30 days in E, N, and U directions for the three systems. It can be seen from Table 2 that in general the average convergence speed of the fixed solution is greatly improved compared with the floating solution. This is consistent with Figs. 7 and 8. Bfloat shows the longest average convergence time, with 109.22 min, 87.77 min, and 241.01 min respectively in E, N, and U directions, while BGfix is the fastest, with average convergence time only at 21.66 min, 11.67 min, and 25.89 min. The reduction rate from Bfloat to Bfix is 15.4%, 12.5%, and 5.2% for the E, N, and U components respectively, while both GPS and the BDS/GPS combined system exhibit even higher reduction rate of more than 20% on average. Dec represents the reduction rate of the average convergence time from floating to fixed solution. Table 3 illustrates the reduction rate of the average convergence time between different systems for both the fixed and floating solutions. We can see that both GPS and the combined systems show very large reduction (>72%) in the average convergence time relative to single BDS, no matter for fixed or floating solution. Compared with GPS, however, the reduction for the combined system is quite small, less than 10% and 4% for the floating and fixed solution respectively. These results further reflect the supe-
Table 3 Statistical results of the reduction rate of the average convergence time within 30 days for 17 stations (unit: %). Reduction rate
E(float)
N(float)
U(float)
E(fix)
N(fix)
U(fix)
GtoB BGtoB BGtoG
72.7 75.3 9.7
83.8 84.4 3.7
84.9 85.7 4.7
76.3 76.6 1.0
84.2 84.8 3.7
88.5 88.7 1.3
GtoB: reduction rate of the average convergence time for GPS compared to BDS. BGtoB: reduction rate for BDS/GPS compared to BDS. BGtoG: reduction rate for BDS/GPS compared to GPS.
riority of the fixed solution relative to the floating solution, and also the combined system relative to single system. 5. Result analysis: PPP AR With WL FCB and NL FCB, the integer properties of ambiguity can be restored and result of the PPP AR can be obtained. Here, 10 stations in the Asia Pacific region are selected to carry out the related experiments on the PPP AR of BDS, GPS, and BDS/GPS. The time span of the experimental data is 30 days, DOY 1 to 30, in 2016. The stations used are shown by the black triangles in Fig. 6. The coordinate differences between our results and corresponding values from IGS are calculated (Figs. 9– 12). The floating solution is also plotted on the same figure for comparison. Statistical information of all obtain results is shown in Table 4. 5.1. GPS PPP AR We randomly select station KARR in our analysis of the GPS PPP AR on DOY 17, 2016 (Fig. 9). As seen from Fig. 9, the fixed solution in the horizontal direction converge to its true value in about 15 min. Although the floating solution can quickly converge to 5 cm accuracy, it remains much slower than the fixed solution. The accuracy of the GPS floating solution and fixed solution are similar at about 1 cm. The observation time for GPS floating solution (24 h), was so long that it has enough time to converge to high accuracy. In the up direction, the convergence of the fixed solution is slightly faster than that of the floating solution. After convergence, the accuracies of both fixed and floating solution are similar at about 1.5 cm. Fig. 9 also shows that the convergence of the fixed solution in the horizontal direction is faster than that in the up direction. Once the ambiguity is fixed to the correct value,
Table 2 Statistical results of average convergence time within 30 days for 17 stations. System
BDS GPS BDS/GPS
Average convergence time (unit: min)
Reduction rate from floating to fixed
E(float)
N(float)
U(float)
E(fix)
N(fix)
U(fix)
E(dec/%)
N(dec/%)
U(dec/%)
109.22 29.87 26.97
87.77 14.25 13.72
241.01 36.50 34.79
92.44 21.88 21.66
76.79 12.12 11.67
228.52 26.23 25.89
15.4 26.8 19.7
12.5 15.0 15.0
5.2 28.1 25.6
Dec represents the reduction rate of the average convergence time from floating to fixed solution.
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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Fig. 9. Position differences for GPS floating and fixed PPP in static mode, at station KARR, on DOY 17, 2016.
high-precision distance will be obtained, which makes the positioning converge to its true value instantaneously. The advantage of the static fixed solution relative to the floating solution only lies in the accelerated convergence and improved accuracy in the east direction. 5.2. BDS PPP AR We randomly select station JFNG to analyse the BDS PPP AR on DOY 8, 2016 (Fig. 10). From Fig. 10 we can see that, in the horizontal direction, once the ambiguity is fixed correctly, the accuracy of the fixed solution is higher than that of the floating solution, which can quickly converge to a high precision value. In the up direction, the accuracy of BDS fixed solution and floating solution is so poor that it takes a long time to converge. There is also an obvious systematic deviation in the up direction. Besides, there are some zig-zag lines in the figure because the ambiguity parameters need to be re-initialised when a cycle slip occurred. Here, we set BDS ratio to 2.0. 5.3. BDS/GPS PPP AR Fig. 11 shows the position differences for BDS/GPS floating and fixed PPP in static mode at station GMSD on DOY 8, 2016. In the horizontal direction, the precision of the fixed solution of BDS/GPS PPP is higher than that of the floating solution, and the convergence of the fixed solution is much faster than that of the floating solution,
especially in the E direction. After convergence, the precision of both fixed and floating solutions of BDS/GPS PPP are all within 1 cm in the horizontal direction. As for the up direction, the initial precision of the fixed solution is much higher than that of the floating solution. As time increases, however, the precision of the floating solution can also reach a high level. After convergence, the fixed and floating solutions of the combined system are all within 2 cm. In addition, we can see that the combined system has a small systematic deviation in the up direction, which may probably be due to the introduction of the systematic deviation of BDS. Fig. 12 gives the position differences of BDS PPP, GPS PPP, and BDS/GPS PPP fixed solutions at station JFNG on DOY 8, 2016. It can be see that the precision and convergence rate of the combined BDS/GPS and the single GPS system are similar in the horizontal direction, which are higher than the fixed solutions of the single BDS. In the up direction, the precision of the combined system is similar to the single GPS, but the convergence speed is faster. We think that this is due to the introduction of the BDS observations, increase of observation number, and improvement of the spatial structure thereof. Another interesting phenomenon is that for both the horizontal and up components, the convergence speed and accuracy of BDS fixed solutions are lower than those of the other two systems, especially for the up direction. This indicates that the single BDS is still very unstable and the correction models for BDS are not accurate enough.
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
12
X. Liu et al. / Advances in Space Research xxx (2019) xxx–xxx 0.2 fix
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Fig. 10. Position differences for BDS floating and fixed PPP in static mode, at station JFNG, on DOY 8, 2016.
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Fig. 11. Position differences for BDS/GPS floating and fixed PPP in static mode, at station GMSD, on DOY 8, 2016.
The fixed and floating solutions of three systems for 10 stations are shown in Figs. 13 and 14, respectively. The statistical results of average RMS in the E, N, and U directions are listed in Table 4. As can be seen from Fig. 13, the accuracy of Bfloat for almost all stations is the lowest, followed by Gfloat, while
BGfloat has the highest precision in the east direction. In the north direction, except for station XMIS, all stations have Bfloat with the worst accuracy, while the accuracy of Gfloat and BGfloat are similar. In the up direction, the accuracy of Bfloat is the lowest in all stations. Due to the poor accuracy of BDS in the up direction, the error
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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13
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0.1 0.0
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9
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Fig. 12. Position differences for BDS PPP AR, GPS PPP AR, BDS/GPS PPP AR in static mode, at station JFNG, on DOY 8, 2016.
Table 4 Average RMS for fixed and floating solution of BDS, GPS, and BDS/GPS (cm). System
E(float)
N(float)
U(float)
E(fix)
N(fix)
U(fix)
BDS GPS BDS/GPS
2.8 1.5 1.1
2.0 1.0 0.9
6.8 2.1 2.2
2.0 0.8 0.7
1.5 0.5 0.5
5.9 1.7 1.9
of BDS will be introduced into the combined system. If the up direction of BDS is too poor, then the error in the up direction of the combined system will be worse than that of the single GPS. Similar as Fig. 13, Fig. 14 gives that almost all stations, BGfix exhibits the best accuracy, Gfix the second, while Bfix is the worst in the east direction. In the north direction, the accuracy of Bfix is the worst, while the accuracy of BGfix is slightly higher than that of Gfix. This is because the precision of Gfix in the north direction is already very high (i.e., at millimetre level), so that the fixed solution of the combined system will not improve much compared with single GPS. In the up direction, the accuracy of Bfix is the worst at all stations, while BGfix and Gfix has the similar accuracy. The reason why the fixed solution of the combined system is worse than the single GPS is also due to the systematic error introduced in the up direction of BDS. From Table 4, we can see that the accuracy of the fixed solutions of the three systems are higher than that of floating solution. Whether adopting a floating or fixed solution, the accuracy in the north direction is the highest, followed by the east component, while up component is the worst.
Compared with floating solution, the accuracy of BDS, GPS, and BDS/GPS fixed solution increase by 28.6%, 46.7%, and 36.4% respectively for the east component. In the north direction, the accuracy increase by 25.0%, 50.0%, and 44.4% for the above three systems. The up component shows the smallest accuracy increase, at 13.2%, 19.0%, and 13.6%, respectively. In addition, the accuracy of both BDS fixed and floating solutions is the worst in three directions, especially in the up direction (at about 5 cm). BGfix shows the highest accuracy in the horizontal direction, but its precision in the up direction is slightly lower than Gfix. The reason that the RMS of the upcomponent solution of the BGfix is worse than that of the Gfix is because BDS has no precise antenna phase center values. On the other hand, the accuracy of BDS observations and the precision of products are lower than GPS. Compared with Figs. 13 and 14, we find that in general the accuracy of the stations at middle and high latitudes are better than that at low latitudes. For example, the accuracy of stations such as PNGM, PONH, PTVL, SOLO, and XMIS are better than that of other stations. This is mainly because the ionosphere above the equator is more active than in middle and high latitudes, and the ionospheric combination cannot eliminate higher-order ionospheric delays in the pseudo-range and carrier-phase measurements, which would affect the final coordinates. 6. Conclusion The purpose of this work is to study the technology and method of BDS/GPS combined PPP and its ambiguity resolution. The method and mathematical model of BDS/
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
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Bfloat Gfloat BGfloat
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GPS integrated PPP AR are described. The long-term GPS and BDS WL FCB are calculated using two month’s observation data. Results show that the variation of the GPS FCB is less than 0.05 cycles within 30 days. Within 60 days,
it exhibits a small increase, but still it is no more than 0.1 cycles. The ranges of BDS GEO and non-GEO WL FCB within 30 days are less than 0.4 and 0.1 cycles respectively. If the time span increase to 60 days, the value increased to
Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026
X. Liu et al. / Advances in Space Research xxx (2019) xxx–xxx
0.7 and 0.15 cycles for GEO and non-GEO. The variation in GPS NL FCB in one day is mostly less than 0.1 cycles. However, the BDS GEO NL FCB would change by up to 0.7 cycles in a given day, while the non-GEO data were all within 0.2 cycles. In addition, there is a sudden jump in WL and NL FCB of BDS GEO and non-GEO schemes. The convergence speed of six schemes (Bfloat, Gfloat, BGfloat, Bfix, Gfix, and BGfix) has been deeply investigated. The convergence time of Bfloat within 40 min is 24.2%, 39.3%, and 16.0% in the E, N, and U directions respectively, which is the slowest among the six solutions examined here, while the BGfix shows the fastest convergence, with convergence rate as 95.3%, 100%, and 87.1%. In general, the average convergence speed of the fixed solution is greatly improved compared to that of the floating solution. For both the floating and fixed solutions, the reduction in convergence time in three directions for the combined system is the largest compared with that of single BDS. In terms of floating solution, the combined system convergence time decreases by 9.7%, 3.7%, and 4.7% respectively in the E, N, U directions compared with single GPS, while for the fixed solution, the reduction in convergence time of the combined system only improve a little relative to single GPS. The results of BDS PPP AR, GPS PPP AR, and BDS/ GPS PPP AR are studied. We found that the accuracy of a fixed solution for the three systems is greatly improved relative to the floating solution. As for the different fixed solution, the accuracy of single BDS PPP fixed solution shows the worst result, with the average RMS value in the E, N, and U directions as 2.0 cm, 1.5 cm, and 5.9 cm. The BDS/GPS PPP fixed solution has the highest precision in the horizontal direction, but its precision in the vertical direction is similar to that of GPS, with the average RMS value as 0.7 cm, 0.5 cm, and 1.9 cm in the E, N, and U directions respectively. We conclude that the BDS/GPS combined ambiguityfixed system can improve convergence speed and positioning accuracy compared to that achieved by using single system. Because BDS has not yet been completed, the stability and accuracy of BDS FCB are slightly worse than those of GPS. With the continuous development of the BDS network how to use BDS GEO and real-time PPP with multi-GNSS will be one of our focus in future research. Acknowledgments The contribution of data from MGEX is appreciated. This research is supported by the National Science Foundation for Distinguished Young Scholars of China (Grant No. 41525014), the Natural Science Innovation Group Foundation of China (No. 41721003), the Major Technology Innovation Project of Hubei Province of China (2018AAA066), the Program for Changjiang Scholars of the Ministry of Education of China, and the Major Project of Beijing Future Urban Design Innovation Center, Beijing University of Civil Engineering and Architecture (UDC
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Please cite this article as: X. Liu, W. Jiang, Z. Li et al., Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/ GPS precise point positioning with ambiguity resolution, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.02.026