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Modeling and Assessment of GPS/BDS Combined Precise Point Positioning Junping Chen 1,2, *, Jungang Wang 1,3 , Yize Zhang 1,3 , Sainan Yang 1 , Qian Chen 1 and Xiuqiang Gong 1 1

2 3

*

Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China; [email protected] (J.W.); [email protected] (Y.Z.); [email protected] (S.Y.); [email protected] (Q.C.); [email protected] (X.G.) Shanghai Key Laboratory of Space Navigation and Positioning Techniques, Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China College of Surveying and Geo-informatics Engineering, Tongji University, Shanghai 200092, China Correspondence: [email protected]; Tel.: +86-21-3477-5223

Academic Editor: Peter Teunissen Received: 31 May 2016; Accepted: 15 July 2016; Published: 22 July 2016

Abstract: Precise Point Positioning (PPP) technique enables stand-alone receivers to obtain cm-level positioning accuracy. Observations from multi-GNSS systems can augment users with improved positioning accuracy, reliability and availability. In this paper, we present and evaluate the GPS/BDS combined PPP models, including the traditional model and a simplified model, where the inter-system bias (ISB) is treated in different way. To evaluate the performance of combined GPS/BDS PPP, kinematic and static PPP positions are compared to the IGS daily estimates, where 1 month GPS/BDS data of 11 IGS Multi-GNSS Experiment (MGEX) stations are used. The results indicate apparent improvement of GPS/BDS combined PPP solutions in both static and kinematic cases, where much smaller standard deviations are presented in the magnitude distribution of coordinates RMS statistics. Comparisons between the traditional and simplified combined PPP models show no difference in coordinate estimations, and the inter system biases between the GPS/BDS system are assimilated into receiver clock, ambiguities and pseudo-range residuals accordingly. Keywords: multi-GNSS; BDS; precise point positioning; MGEX; ISB

1. Introduction To prepare for the next phase of generating products for all GNSS available, the IGS [1] initiated the IGS Multi-GNSS Experiment (MGEX) campaign. The MGEX campaign focuses on tracking the newly available GNSS signals including the Chinese BeiDou Navigation Satellite System (BDS). BDS had been providing regional service since December 2012 with five Geostationary Earth Orbit (GEO) satellites, five Inclined Geosynchronous Satellite Orbit (IGSO) satellites and four Medium Earth Orbit (MEO) satellites. Many studies are being carried out and much progress has been achieved in BDS-only and GPS/BDS-combined precise data analysis. Precise BDS orbits/clocks are currently available at many institutes and several IGS analysis centers [2–5]. Many publications demonstrate the daily static Precise Point Positioning (PPP, [6] using BDS observations, and precise orbits/clocks could reach a precision of a few cm [5,7,8]. With the additional tracking of BDS constellations and therefore, a significant increase of observed satellites, GPS/BDS combined PPP could improve solution availability and accuracy by improving tracking geometry with a reduction of the position dilution of precision (PDOP) [9], especially in environments like urban canyons and ravines where there are limits in sky view. Chen et al. [10] demonstrated that near real time GPS/BDS combined PPP solution show a higher degree of precision and better robustness, which

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is very important for tsunami early warning. Also, for GNSS-based precipitable water vapor retrieval, GPS/BDS combined PPP could obtain more accurate and reliable water vapor estimates than GPS-only or BDS-only PPP [11]. Guo et al. [12] evaluated the orbit and clock of different GNSS systems from different analysis centers comprehensively and has shown the accuracy improvement of Mulit-GNSS PPP rather than single-system PPP. Compared to the GPS-only and BDS-only PPP, the key issue in GPS/BDS PPP is the handling of inter system bias (ISB). Station-wise GPS/BDS inter-system bias includes two parts, one is the difference between system time GPST and BDT, and the other comes from the station-wise instrument hardware delay difference when tracking satellites of different systems. For most geodetic receivers and antennas, the instrument hardware delay is not calibrated and it will be assimilated into the receiver clock. Thus, the GPS/BDS inter-system bias shows up as the receiver clock differences between GPS-only and BDS-only PPP. For GPS/BDS combined PPP, two kinds of parameterization strategies are widely implement: (1) two receiver clock parameters at each epoch; (2) epoch-wise receiver clock and ISB parameter. Since for one station, the difference of hardware delay between different satellite systems is over one day interval [13–15], ISB is usually treated as a daily constant [9,16–18]. Chen et al. [19] developed a new simplified multi-GNSS PPP model, which does not include ISB parameter and observations of different GNSS systems could be treated in a unique way. Jiang et al. [20] demonstrated that a priori constraint of predicted ISB in GPS/BDS combined PPP could shorten convergence time and also, improve positioning accuracy slightly. Furthermore, many studies have been carried out over the handling of ISB in multi-GNSS data processing, including GPS/BDS inter-ambiguity resolution [21], GPS/BDS single-frequency short baseline RTK [22,23] and long baseline relative positioning [24], GPS/BDS/GLONASS/Galieo/QZSS ISBs analysis with different types of receivers [25], BDS/Galileo/QZSS and GPS single-frequency RTK [26]. In this paper, we present and evaluate GPS/BDS PPP models, where single system and combined PPP models are discussed. In the following, Section 2 presents the traditional and simplified GPS/BDS PPP model; Section 3 presents data analysis and compares daily positioning qualities; Section 4 compares the solved parameters between the traditional and simplified combined PPP models; finally, Section 5 summarizes the main points of this paper. 2. GPS/BDS Combined PPP Models 2.1. PPP Model for Single System Taking a GPS system as an example, pseudo-range and phase observation functions of the ionosphere-free combination between a receiver and a satellite G can be written as: PG “ ρG ` c ¨ dtG ` bG ` ZPDG ` ςG LG “ ρG ` c ¨ dtG ` BG ` N G ` ZPDG ` εG

(1)

where PG , LG are the pseudo-range and carrier phase observation; ρG is geometrical distance; c is light speed and c ¨ dtG is receiver clock correction; bG and BG are receivers pseudo-range and carrier phase hardware delay bias; N G is ambiguity, and ZPDG is the slant tropospheric delay, ςG and εG are noise terms. The pseudo-range hardware delay biases bG is assimilated into clock correction c ¨ dtG . The carrier phase hardware delay biases BG is satellite dependent and stable over time, and it will be grouped into ambiguity [27,28]. Thus, the ionosphere-free combination PPP observation can be rewritten as: PG “ ρG ` c ¨ dtG ` ZPDG ` ςG G

LG “ ρG ` c ¨ dtG ` N ` ZPDG ` εG

(2)

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G

where dtG and N are reformed station clock and ambiguity with: c ¨ dtG “ c ¨ dtG ` bG

(3)

G

N “ N G ` BG ´ bG For BDS phase and pseudo-range observations the same Equations (2) and (3) also apply. 2.2. Traditional GPS/BDS Combined PPP Model

The ionosphere-free pseudo-range and phase observations for the GPS/BDS combined PPP can be written as: PG “ ρG ` c ¨ dtG ` ZPDG ` ςG G

LG “ ρG ` c ¨ dtG ` N ` ZPDG ` εG PC “ ρC ` c ¨ dtC ` ZPDC ` ςC

(4)

C

LC “ ρC ` c ¨ dtC ` N ` ZPDC ` εC In Equation (4), the superscript and subscript G, C refers to GPS and BDS satellites. The slant tropospheric delay term ZPD could be modelled as the hydrostatic and wet parts with following: ZPD “ m f h ¨ ZHD ` m f w ¨ ZWD

(5)

In Equation (5), ZHD, ZWD are the hydrostatic and wet zenith path delays, m f h , m f w are the mapping functions. The hydrostatic part of ZPD is normally corrected using empirical models and the term ZTDw is estimated. The traditional GPS/BDS combined PPP model requires the estimation of an additional inter-system bias parameter. Defining GPS as the reference system, the inter-system bias could be written as, ISB “ c ¨ dtC ´ c ¨ dtG (6) “ c ¨ dtC ´ c ¨ dtG ` bC ´ bG “ c ¨ ∆dt ` ∆b Considering Equations (5) and (6), the traditional GPS/BDS combined PPP equations are written as, G ¨ ZWD ` ςG PG “ ρG ` c ¨ dtG ` m f w G

G ¨ ZWD ` εG LG “ ρG ` c ¨ dtG ` N ` m f w C C C ¨ ZWD ` ςC P “ ρ ` c ¨ dtG ` ISB ` m f w

(7)

C

C ¨ ZWD ` εC LC “ ρC ` c ¨ dtG ` ISB ` N ` m f w

2.3. Simplified GPS/BDS Combined PPP Model Based on the scaled sensitivity matrix (SSM, [19,29]) method, Chen et al. [19] proves that there is no correlation between ISB and coordinate parameters in multi-GNSS combined PPP, and ISB parameter can be removed in multi-GNSS PPP. Based on this conclusion, we remove the ISB term in Equation (7) and rewrite observation equations as: 1

1

G ¨ ZWD ` ςG PG “ ρG ` c ¨ dt ` m f w G1

1

G ¨ ZWD ` εG LG “ ρG ` c ¨ dt ` N ` m f w 1 1 C C C P “ ρ ` c ¨ dt ` m f w ¨ ZWD ` ςC

1

LC “ ρC ` c ¨ dt ` N 1

G1

C1

(8)

C ¨ ZWD ` εC ` m fw C1

where dt is defined as new receiver clock, N and N are new ambiguity terms. Compared to Equation (7), the ISB term is absorbed in the new terms and pseudo-range residuals with:

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1

dt “ dtG ` λ ¨ ISB G1

G

N “ N ´ λ ¨ ISB 1 ςG “ ςG ´ λ ¨ ISB 1 ςC “ ςC ` p1 ´ λq ¨ ISB N

C1

(9)

C

“ N ` p1 ´ λq ¨ ISB

where λ is called the element of the SSM, which quantitatively defines the ratio of ISB parameter assimilated into the receiver clock. As described in Chen et al. [19], the benefits of the new model are: (1) under some extreme circumstances, the traditional Multi-GNSS PPP model may final when there are only four satellites observed, while the new approach will still give coordinate solution; (2) the correlation between the clock and ambiguity parameters is reduced, thus making solution more stable using the new model; (3) Multi-GNSS PPP realization is simplified and unified. In this new model, observations of different GNSS systems are treated in a unified way as they were of the same satellite system. In this simplified model, the ISB is assimilated into a receiver clock and ambiguity parameter. Due to the much smaller weight assigned on pseudo-range observations, all pseudo-range observations will show big residuals. The GPS pseudo-range residuals are close to the amount of ISB value that assimilates into station clock parameter. The BDS pseudo-range residuals are close to the amount of ISB value that assimilates into BDS ambiguity. 3. Data Processing To assess GPS/BDS PPP performance under different processing models, and to validate the simplified GPS/BDS combined PPP model, daily static and kinematic PPP data analysis is performed. Data of 11 IGS MGEX stations in January 2014 is used. Figure 1 shows the geographic distribution of these stations. Sensorstracking 2016, 16, 1151 5 of 13

Figure Thegeographic geographicdistribution distributionofofGPS/BDS GPS/BDS stations stations used used in in the the data data analysis. analysis. Figure 1.1.The 3.1. Position Differences between Static PPP and IGS Daily Solutions For results evaluation and comparison, data processing is performed in the following four scenarios: GPS-onlyGPS/BDS PPP, BDS-only PPP, andestimates GPS/BDSwith combined with traditional and simplified We compared daily position the IGSPPP daily solutions covering the same models, where station-wise ISB is estimated as transformation a daily constant [28,36]. in a traditional model. period through a seven-parameter Helmert The RMS of each station is Precise satellite2 orbits andmean satellite clocks from SHA [17,30] used forcombined data analysis. apply the showed in Figure and the value is shown in Table 1. are GPS/BDS PPP We achieves IGS antenna phase center model for PPP GPS and observations phase-wind up modeling [32] bestabsolute performance compared with GPS-only BDS-only[31], PPP.the The improvement of combined PPPthe over GPS-only PPP is not obvious. BDS-only has the worst 2010 accuracy, which may be and station displacement arevery modeled according to thePPP IERS conventions [33]. A cut-off angle due to the less accurate BDS orbit and clock products and a regional satellite constellation with less satellites compared with GPS. We notice that stations JFNG (Wuhan, China), CUT0 (Perth, Australia), GMSD (Guts Masda, Japan), and REUN (Le Tampon, La Reunion, France) have the best accuracy in BDS PPP, which is due to the fact that more BDS satellites are tracked for these sites under the current BDS constellation. Also, the GPS/BDS combined PPP using a simplified and traditional model is nearly

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of 7˝ is set for usable measurements and data sampling is set to 30 s. An elevation-dependent weighting strategy is applied to measurements at low elevations and the noise ratio between pseudo-range and phase observation is 500:1. Moreover, we estimate wet zenith path delay every hour by applying the most recently developed GPT2 empirical slant delay model [34]. An improved version of the LTW_BS Figure 1. The geographic distribution of GPS/BDS stations used in the data analysis. software is used [35].

3.1. Position Position Differences Differences between between Static Static PPP PPP and and IGS IGS Daily Daily Solutions Solutions 3.1. We compared comparedGPS/BDS GPS/BDS daily estimates with with the the IGS IGS daily daily solutions solutions covering covering the the same same We daily position position estimates period through througha seven-parameter a seven-parameter Helmert transformation [28,36]. The of RMS each isstation is period Helmert transformation [28,36]. The RMS eachofstation showed showed in Figure 2 and the mean value is shown in Table 1. GPS/BDS combined PPP achieves the in Figure 2 and the mean value is shown in Table 1. GPS/BDS combined PPP achieves the best best performance compared GPS-only PPP BDS-only and BDS-only The improvement of combined performance compared with with GPS-only PPP and PPP. PPP. The improvement of combined PPP PPP over GPS-only PPP is not very obvious. BDS-only PPP has the worst accuracy, which may over GPS-only PPP is not very obvious. BDS-only PPP has the worst accuracy, which may be due be to dueless to the less accurate BDS orbit and clock products and a regional satellite constellation less the accurate BDS orbit and clock products and a regional satellite constellation with less with satellites satellites compared notice that JFNG stations JFNG (Wuhan, China), CUT0Australia), (Perth, Australia), compared with GPS.with We GPS. noticeWe that stations (Wuhan, China), CUT0 (Perth, GMSD GMSD (Guts Masda, Japan), and REUN (Le Tampon, La Reunion, France) have the best accuracy in (Guts Masda, Japan), and REUN (Le Tampon, La Reunion, France) have the best accuracy in BDS BDS PPP, which is due to the fact that more BDS satellites are tracked for these sites under the current PPP, which is due to the fact that more BDS satellites are tracked for these sites under the current BDS BDS constellation. the GPS/BDS combined using a simplified traditional model is nearly constellation. Also,Also, the GPS/BDS combined PPPPPP using a simplified and and traditional model is nearly the the same and their values differ at the level of µm, which will be discussed in detail in Sections 3.3 same and their values differ at the level of µm, which will be discussed in detail in Sections 3.3 and and 4. 4.

Figure 2. 2. RMS RMS of of coordinate coordinate differences differences between between daily daily static static PPP PPP and and IGS IGS daily daily solutions solutions for for each each Figure station in the North, East, and Up component, where PPP in different scenarios is shown in different station in the North, East, and Up component, where PPP in different scenarios is shown in different color. The The last last column column of of each each subplot subplot is is the the mean mean value valueof ofthe the11 11stations. stations. color. Table 1. Mean RMS of station coordinate differences of different scenarios.

GPS/BDS GPS-only BDS-only

N (cm)

E (cm)

U (cm)

0.34 0.36 2.09

0.58 0.59 2.48

1.46 1.54 2.97

We make 3D RMS statistics of the 11 stations over the whole month and plot the histogram of the three scenarios in Figure 3. The GPS/BDS combined PPP achieves the best accuracy, with a mean 3D

GPS/BDS GPS-only BDS-only

N (cm) 0.34 0.36 2.09

E (cm) 0.58 0.59 2.48

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U (cm) 1.46 1.54 2.97 6 of 13

We make 3D RMS statistics of the 11 stations over the whole month and plot the histogram of the three scenarios in Figure 3. The GPS/BDS combined PPP achieves the best accuracy, with a mean 3D RMS 1.5while cm, while those of GPS-only and BDS-only 1.63.7 cm and 3.7 cm, respectively. The RMS of 1.5ofcm, those of GPS-only and BDS-only is 1.6 cm is and cm, respectively. The percentage percentage of 3D RMS within the1.5] range [0, 1.5] cmand is 26%, and 59% GPS-only for BDS-only, GPS-only of 3D RMS within the range of [0, cm of is 26%, 40%, 59%40%, for BDS-only, and combined and combined which demonstrates a more accurate solution is obtained by adding solutions, whichsolutions, demonstrates a more accurate solution is obtained by adding observations from observations from different systems. different systems.

Figure 3. 3. Magnitude Magnitude distribution distribution of of 3D 3D RMS RMS of of coordinate coordinate differences differences between between daily daily static static PPP PPP and and Figure IGS IGS daily daily solutions. solutions. Subplot Subplot presents presentsthe theGPS/BDS GPS/BDS combined, GPS-only and BDS-only BDS-only solutions solutions from from left left to to right, right, respectively. respectively.The Thetext textof ofeach eachsubplot subplotshows showsthe themedian medianand andmean meanvalue valueof of3D 3DRMS. RMS.

3.2. Position Position Differences Differences between between Kinematic Kinematic PPP PPP and and IGS IGS Daily Daily Solutions Solutions 3.2. Toassess assesskinematic kinematic position position estimates estimates differences differences among among PPP PPP scenarios, scenarios, epoch-wise epoch-wise kinematic kinematic To coordinate estimates estimatesare arecompared comparedwith withthe theIGS IGSdaily dailyestimates. estimates. In In each each daily daily kinematic kinematic solution, solution, the the coordinate epoch-wise coordinates after the first 1 h (120 epochs) are used for statistics analysis. Figure 4 shows epoch-wise coordinates after the first 1 h (120 epochs) are used for statistics analysis. Figure 4 shows for all all11 11stations stationsthe the magnitude distribution of RMS 3D RMS of kinematic coordinate differences. The for magnitude distribution of 3D of kinematic coordinate differences. The mean mean 3Dare RMS are of 28.1, 9.2 cm and median are8.2, of 17.0, 8.2, for the BDS-only, 3D RMS of 28.1, 10.9, 9.210.9, cm and median 3D RMS3D areRMS of 17.0, 6.5 cm for6.5 thecm BDS-only, GPS-only GPS-only and combined GPS/BDS solutions. combined The solutions. The distribution magnitude also distribution shows a higher and GPS/BDS magnitude shows a also higher percentage of percentage of more accurate in GPS/BDS combined solutions. more accurate estimations in estimations GPS/BDS combined solutions. Kinematic PPP PPP of of station station JFNG JFNG on on DOY DOY (Day (Day of of Year) Year) 028, 028, 2014 2014 is is shown shown in in Figure Figure 5. 5. ItIt shows shows Kinematic that combined GPS/BDS kinematic PPP has much better stability and shorter convergence, while both that combined GPS/BDS convergence, while both GPS-only and and BDS-only BDS-only kinematic kinematic PPP PPP show show big big scatters scatters as as large large as as 0.5 0.5 m. m. The The number number of of tracked tracked GPS-only satellites is shown in the bottom-right subfigure of Figure 5, where number of available satellites of satellites is shown in the bottom-right subfigure of Figure 5, where number of available satellites of GPS/BDS combination GPS-only and and BDS-only BDS-only cases. cases. Statistics Statistics show show that that for for GPS/BDS combination almost almost doubles doubles that that of of GPS-only JFNG, GPS-only GPS-only kinematic kinematic PPP PPP has has aa better better accuracy accuracy than than that that of of BDS, BDS, even even ifif BDS-only BDS-only kinematic kinematic JFNG, PPP occasionally occasionallyperforms performsbetter betterthat thatGPS GPSwhen whenBDS BDShas has better satellite coverage with observations PPP better satellite coverage with observations of of good quality, such as JFNG at DOY 2014. good quality, such as JFNG at DOY 028,028, 2014.

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Figure4. 4. Magnitude Magnitude distribution distribution of of 3D 3D RMS RMS of of coordinate coordinate differences differences between between epoch-wise epoch-wisekinematic kinematic Figure Figure 4. Magnitude distribution of 3D RMS of coordinate differences between epoch-wise PPP and and IGSdaily dailysolutions. solutions. Subplot Subplot presents theGPS/BDS GPS/BDS combined, GPS-only andkinematic BDS-only PPP IGS presents the combined, GPS-only and BDS-only PPP and IGS daily solutions. Subplot presents the GPS/BDS combined, GPS-only and BDS-only solutions from left to right, respectively. The texts of each subplot shows the median and mean 3D solutions from right, respectively. The texts of each subplot shows the median and mean solutions from left to right, respectively. The texts of each subplot shows the median and mean 3D RMS. 3D RMS. RMS.

Figure 5. Epoch-wise kinematic PPP coordinate bias of GPS/BDS combined, GPS-only and BDS-only Figure 5. Epoch-wise Epoch-wise kinematic coordinate bias of GPS/BDS combined, GPS-only and BDS-only BDS-only Figure 5. kinematic GPS/BDS combined, GPS-only and solutions in the North, East andPPP Up component. Station: JFNG, DOY 028, 2014. Bottom-right subplot solutions innumber the North, North, East and andtracked Up component. component. Station: JFNG, JFNG, DOY DOY 028, 028, 2014. 2014. Bottom-right Bottom-right subplot subplot solutions in the East Up Station: shows the of satellites at each epoch. shows the the number number of of satellites satellites tracked tracked at at each each epoch. epoch. shows

3.3. Position Differences between Traditional and New Model 3.3. Position Position Differences Differences between between Traditional Traditional and New Model 3.3. Assessing the differences between the position estimates can directly illustrate to what extent Assessing the the can directly to what the Assessing differences betweenagree theposition position estimates can directly illustrate to what extent the traditional anddifferences simplified between models in theirestimates positioning results. Inillustrate the following weextent compare traditional andcombined simplified models in between their positioning results. In the following we we compare the the and simplified models agree in their positioning results. In the following compare thetraditional GPS/BDS static PPPagree results the traditional and simplified models. GPS/BDS combined PPP results between thecomputed traditional andand simplified models. the GPS/BDS combined PPP results between the traditional simplified models. For each of the static 11static IGS MGEX stations, we GPS/BDS combined static PPP position For each each we GPS/BDS combined static PPP position For of the 11 IGS MGEX stations, we computed computed GPS/BDS combined static PPP position differences between theIGS twoMGEX modelsstations, over 1 month. The biggest position difference is less than 1 mm differences the models over The biggest difference is less thanthan 1 mm and differences between thetwo two models over1 1month. month. The biggest position difference is less 1 mm and most between points have position differences of less than 0.1position mm. Figure 6 shows the magnitude most points have position differences of less than 0.1 mm. Figure 6 shows the magnitude distribution and most points have position differences of less than 0.1 mm. Figure 6 shows the magnitude distribution of these coordinate differences in the North, East, and Up component. The mean biases of differences indifferences the East, Up component. mean biases ´0.6, ´2.5 distribution these coordinate in theand North, East, andcoordinate UpThe component. Theare mean biases arethese −0.6,coordinate −2.5ofand 0.4 µm and RMS areNorth, 1.1, 3.9, and 1.9 µm for each component. In addition, and 0.491.5% µm areand 1.1,96.9% 3.9, and 1.9 each coordinate In addition, about 91.5% are −0.6, −2.5and and 0.4 North, µm RMS are 1.1,µm 3.9,for and 1.998.7% µm for coordinate component. In addition, about in RMS the in the East, and in each the component. Up component of all deviations are in the North, 96.9% in the East, and 98.7% in the Up component of all deviations are within twice the about 91.5% in the North, 96.9% in the East, and 98.7% in the Up component of all deviations are within twice the standard deviations. These results verify same position estimates of these two standard deviations. Thesedeviations. results verify the results same position estimates of these estimates two models. within twice the standard These verify the same position of these two models. models.

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6. Magnitude distribution of all positiondifferences differences of of GPS/BDS GPS/BDS combined static PPPPPP between FigureFigure 6. Magnitude distribution of all position combined static between the traditional and simplified models in the North, East, and Up components. All subplots exhibit the traditional and simplified models in the North, East, and Up components. All subplots exhibit normal6.distributions. The top-left of each subplot shows the bias and the static standard Figure Magnitude distribution ofcorner all position differences of GPS/BDS PPP deviation between normal distributions. The top-left corner of each subplot shows the biascombined and the standard deviation (σ), (σ),traditional as well as the of deviations are East, withinand 2σ Up andcomponents. 3σ. the andpercentages simplified models in thethat North, All subplots exhibit as well as the percentages of deviations that are within 2σ and 3σ. normal distributions. The top-left corner of each subplot shows the bias and the standard deviation

4. Parameter between Combined Models (σ), as wellDifferences as the percentages of deviations thatPPP are within 2σ and 3σ.

4. Parameter Differences between Combined PPP Models

In the following, we compare the station clocks, ambiguities and pseudo-range residuals 4. Parameter Differences between Combined PPP Models

the traditional and simplified GPS/BDS PPP and models. Data used isresiduals the kinematic Inbetween the following, we compare the station clocks,combined ambiguities pseudo-range between In the following, we compare the station clocks, ambiguities and pseudo-range residuals solution of station JFNG on DOY 028, 2014. the traditional and simplified GPS/BDS combined PPP models. Data used is the kinematic solution of between traditional and simplified GPS/BDS combined PPP models. Data used is the kinematic station JFNG the on DOY 028, 2014. 4.1. Kinematic Coordinate Difference solution of station JFNG on DOY 028, 2014.

4.1. Kinematic Coordinateabove, Difference As discussed in the simplified GPS/BDS PPP model where ISB is not considered, the

4.1. Kinematic Coordinate Difference coordinate solution is indentical with that of traditional model. Here the coordinate difference

As discussed above, in the simplified GPS/BDS PPP model where ISB is not considered, the As discussed above, in the simplified GPS/BDS PPP solution model where ISB is not considered, the8 between traditional and new model of JFNG’s kinematic is presented in Figure 7. After coordinate solution is indentical with that of traditional model. Here the coordinate difference between coordinate solution is indentical with that of traditional model. Here the coordinate difference epochs the cooridnate differences is less than 1 cm, and after 30 epochs the difference reduces to less traditional new of JFNG’s kinematic solution issolution presented in stable Figure 7.Figure After epochs between traditional and new model of JFNG’s kinematic is presented in 7.8After 8 the than 1and mm. Aftermodel 500 epochs the difference is less than 5 µm and remains until the last epoch. cooridnate is less than 1 is cm, and after 30and epochs difference reduces reduces to less than 1 mm. epochsdifferences the cooridnate differences less than 1 cm, after the 30 epochs the difference to less Afterthan 500 epochs the difference is less than 5 µm and remains stable until the last epoch. 1 mm. After 500 epochs the difference is less than 5 µm and remains stable until the last epoch.

Figure 7. Coordinate Difference between traditional and new model of JFNG at DOY 028, 2014. (a) Coordinate difference of the first 100 epochs; (b) Coordinate difference after 100 epochs. Figure 7. Coordinate Difference between traditional and new model of JFNG at DOY 028, 2014.

Figure 7. Coordinate Difference between traditional and new model of JFNG at DOY 028, 2014. 4.2. Station Clock and Ambiguity (a) Coordinate difference of the first 100 epochs; (b) Coordinate difference after 100 epochs. (a) Coordinate difference of the first 100 epochs; (b) Coordinate difference after 100 epochs. In the simplified GPS/BDS PPP model, station clock, and ambiguity estimates absorb the ISB 4.2. Station Clock and Ambiguity parameters together, and the percentage which goes into each parameter depends on the normal 4.2. Station Clock and Ambiguity In the simplified GPS/BDS PPP model, station clock, and ambiguity estimates absorb the ISB parameters together, and the percentage which goesclock, into each on the normal In the simplified GPS/BDS PPP model, station and parameter ambiguitydepends estimates absorb the ISB

parameters together, and the percentage which goes into each parameter depends on the normal equation. In Equation (9), the ISB assimilation shows the opposite sign for station clock and GPS

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equation. In Equation (9), the ISB assimilation shows the opposite sign for station clock and GPS ambiguity parameters, parameters, thus thus the the sum sum quantity quantity of of station station clock clockand andGPS GPSambiguity ambiguity for forthe thetraditional traditional ambiguity and simplified simplified model model isis theoretically theoretically the the same same at ateach eachepoch. epoch. The The upper upper subplot subplot of of Figure Figure 88 shows shows and the magnitude distribution of this sum quantity differences between the two models using all GPS the magnitude distribution of this sum quantity differences between the two models using all GPS satellite/station pairs satellite/station pairs at at all allepochs. epochs.The Themean meandifference differenceisis2.0 2.0µm µmand andall allthe thedifferences differencesare arebelow below1 1mm mmwith witharound around0.35% 0.35%bigger biggerthan than0.1 0.1mm. mm. According to to Equation Equation (9), (9), the the epoch-wise epoch-wise sum sum of of station station clock clock and and BDS BDS ambiguity ambiguity in in the the According simplifiedmodel modelis is theoretically thequantity sum quantity ofclock, station ISB,ambiguity and BDS simplified theoretically the the samesame as theassum of station ISB,clock, and BDS ambiguity in the traditional model. The bottomofsubplot shows the magnitude distribution in the traditional model. The bottom subplot Figureof 8 Figure shows 8the magnitude distribution of this of this sum quantity differences two using models all BDS satellite/station pairs at all sum quantity differences betweenbetween the two the models allusing BDS satellite/station pairs at all epochs. epochs. The mean difference 4.6 µm anddifferences all the differences are 1below 1 mm at around The mean difference is 4.6 µmis and all the are below mm at around 0.72%0.72% biggerbigger than than 0.1Both mm.plots Bothshow plotsthat show the differences are which so small, which further the same 0.1 mm. the that differences are so small, further proves the proves same coordinate coordinate these two models. estimates inestimates these twoinmodels.

Figure Figure 8.8. Magnitude Magnitude distribution distribution of of differences differences between between the the traditional traditional and and simplified simplified models models of of (a) epoch-wise sum of GPS ambiguity and station clock parameters; (b) epoch-wise sum of BDS (a) epoch-wise sum of GPS ambiguity and station clock parameters; (b) epoch-wise sum of BDS ambiguity, ambiguity, ISB ISBand andstation stationclock clockparameters. parameters. Both Bothplots plotsexhibit exhibitnormal normaldistributions. distributions. The The top-left top-left corner corner of of each each subplot subplot shows shows the the bias bias and andthe thestandard standarddeviation deviation (σ), (σ),whereas whereas the thetop-right top-right corner corner shows showsthe thepercentages percentagesof ofdeviations deviationsthat thatare arewithin within2σ 2σand and3σ. 3σ.

4.3. Station Station Clock Clock and and Pseudo-Range Pseudo-RangeResidual Residual 4.3. In Equation (9), assimilation shows opposite signals for station clock and GPS pseudoIn (9),the theISB ISB assimilation shows opposite signals for station clock and GPS range observation residuals, thus thethus epoch-wise sum quantity of station and GPSand pseudopseudo-range observation residuals, the epoch-wise sum quantity of clock station clock GPS range residualresidual is theoretically the same forsame the traditional and simplified model. The upper pseudo-range is theoretically the for the traditional and simplified model. Thesubplot upper of Figure shows9the station differences betweenbetween the traditional and simplified PPP models, subplot of 9Figure shows the clock station clock differences the traditional and simplified PPP where the differences are between the range −31 of and −28and m. ´28 The m. bottom subplotsubplot of Figure 9 shows models, where the differences are between the of range ´31 The bottom of Figure 9 the magnitude distribution of the epoch-wise sum quantity differences between the two models for shows the magnitude distribution of the epoch-wise sum quantity differences between the two models all all BDS satellite/station pairs at all TheThe mean difference is −0.7 µm µm andand all the differences are for BDS satellite/station pairs at epochs. all epochs. mean difference is ´0.7 all the differences below 0.1 0.1 mm. are below mm.

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Figure (a)(a)Station Station clock differences between simplified models; (b) Magnitude Figure 9.9.9.(a) traditional and simplified simplifiedmodels; models;(b) (b)Magnitude Magnitude Figure Stationclock clockdifferences differences between between traditional traditional and and distribution ofof differences ofofthe the epoch-wise and GPS pseudo-range residuals distribution of differences sum station clock and andGPS GPSpseudo-range pseudo-range residuals distribution differencesof theepoch-wise epoch-wise sum sum of of station station clock clock residuals between the traditional and simplified of bottom subplot shows the bias between the traditional The top-left top-left cornerof ofbottom bottomsubplot subplotshows shows the bias between the traditionaland andsimplified simplifiedmodels. models. The top-left corner the bias and the standard deviation (σ), the percentages of deviations that and the standard deviation corner shows showsthe thepercentages percentagesof ofdeviations deviations that and the standard deviation(σ), (σ),whereas whereasthe thetop-right top-right corner corner that are within 2σ2σ and 3σ. are within 2σ and 3σ. are within and 3σ.

4.4. Ambiguity and Pseudo-RangeResidual Residual 4.4. Ambiguity and Pseudo-Range 4.4. Ambiguity and Pseudo-Range Equation (9),the theISB ISBassimilation assimilationisis the the same same for for ambiguity ambiguity and InIn Equation (9), the ISB assimilation ambiguity and pseudo-range observation In Equation (9), andpseudo-range pseudo-rangeobservation observation residual in both the GPS and BDS systems. Thus, ambiguity differences and pseudo-range residual in both the GPS and BDS systems. Thus, the ambiguity differences and pseudo-range residual in both the GPS and BDS systems. Thus, the ambiguity differences and pseudo-range residual residualdifferences differences between thetwo twomodels models should theoretically the at epoch residual theoretically besame the same same atthe thesame same epoch for differences between between the two the models shouldshould theoretically be thebe at the same epoch forfor the the traditional and simplified model. In the following, we define the epoch-wise “double difference” the traditional and simplified model. In the following, we define the epoch-wise “double difference” traditional and simplified model. In the following, we define the epoch-wise “double difference” term termbybymaking differencebetween theambiguity ambiguity difference difference and residual difference term difference the and pseudo-range pseudo-range residual difference by making making difference betweenbetween the ambiguity difference and pseudo-range residual difference between between the two models, which by definition should theoretically be zero. between the twowhich models, by definition should theoretically the two models, by which definition should theoretically be zero. be zero.

Figure 10.10.(a,c) andBDS BDSPseudo-range Pseudo-range residuals between traditional and simplified Figure (a,c)GPS GPS and residuals between traditional and simplified models;models; (b,d) Figure 10. (a,c) GPS and BDS Pseudo-range residuals traditional and simplified models; (b,d) (b,d) magnitude distribution of the double differences for GPS and observations. The left corner magnitude distribution of the double differences forbetween GPS and BDSBDS observations. The left corner of magnitude distribution double differences GPS and BDS deviation observations. The left the corner of subplots (subplot b) and d)) the and the (σ), whereas right of right subplots (subplotof b)the and d))shows shows thebias biasfor and thestandard standard deviation (σ), whereas the right right subplots (subplot b) and d)) shows the bias and the standard deviation (σ), whereas the right corner right subplots(subplot (subplotb) b)and and d)) d)) shows shows the percentages within 2σ2σ corner of of right subplots percentagesof ofdeviations deviationsthat thatare are within corner of right subplots (subplot b) and d)) shows the percentages of deviations that are within 2σ and and 3σ.3σ. and 3σ.

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The left subplots of Figure 10 show the GPS and BDS pseudo-range residual differences between the two models, where we see that they present a similar shape but with different signs as presented in Equation (9). The right subplot of Figure 10 shows the magnitude distribution of the double difference quantity using all satellite/station pairs at all epochs. The mean double differences are ´2.5 and ´5.0 µm for GPS and BDS. All the double differences are below 0.1 mm for GPS, while all the BDS double differences are below 1 mm with around 0.5% bigger than 0.1 mm. 5. Conclusions and Suggestions In this study, we discuss the different parameterization of GPS/BDS PPP, where combined PPP between the traditional and a simplified model are compared. GPS/BDS PPP were performed with different models and scenarios using 1 month of GPS/BDS data from 11 IGS MGEX stations. Comparing daily static coordinate estimates of BDS-only, GPS-only and GPS/BDS combined solutions with the IGS daily estimates, the mean 3D RMSes were 3.7, 1.6 and 1.5 cm, respectively. In the kinematic case, the 3D RMSes were 17.0, 8.2, 6.5 cm. Magnitude distribution of the 3D RMS shows higher percentage of more accurate solutions in GPS/BDS combined solutions, which is due to the inclusion of data from different systems. Comparing GPS/BDS combined PPP coordinate estimates between the traditional and simplified models, mean biases of the differences are ´0.6, ´2.5, and 0.4 µm and RMSes are 1.1, 3.9, and 1.9 µm in the North, East, and Up components, respectively. The analysis of parameter and pseudo-range residual differences between the two models show the same quantity with deviation at the µm level for: (1) epoch-wise sum of station clock and GPS ambiguity; (2) epoch-wise sum of station clock, ISB, and BDS ambiguity; (3) epoch-wise sum of station clock and GPS pseudo-range residuals; (4) epoch-wise sum of station clock, ISB, and BDS pseudo-range residuals; (5) epoch-wise double difference of ambiguity and pseudo-range residuals. The overall comparisons verify the equivalence of the position estimates derived from the traditional and simplified GPS/BDS combined PPP models. Acknowledgments: The IGS community is acknowledged for providing RINEX data and orbit and clock products. This research is supported by 100 Talents Program of the Chinese Academy of Sciences, the National Natural Science Foundation of China (NSFC) (No. 11273046), the National High Technology Research and Development Program of China (Grant No. 2013AA122402 and 2014AA123102). Author Contributions: Junping Chen proposed the research topic, developed the algorithms and designed the experiment; Jungang Wang, Yize Zhang, Sainan Yang, Qian Chen and Xiuqiang Gong collected the data and performed the experiment; all the authors contributed to writing and improving the paper. Conflicts of Interest: The authors declare no conflict of interest.

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