Single receiver phase ambiguity resolution with GPS data

J Geod(2010)84:327–337

DOI10.1007/s00190-010-0371-9

ORIGINAL ARTICLE

Single receiver phase ambiguity resolution with GPS data Willy Bertiger·Shailen D.Desai·Bruce Haines·

Nate Harvey·Angelyn W.Moore·Susan Owen·

Jan P.Weiss

Received:10December2009/Accepted:11February2010/Published online:21March2010

?US Government2010

Abstract Global positioning system(GPS)data processing algorithms typically improve positioning solution accuracy by?xing double-differenced phase bias ambiguities to inte-ger values.These“double-difference ambiguity resolution”methods usually invoke linear combinations of GPS carrier phase bias estimates from pairs of transmitters and pairs of receivers,and traditionally require simultaneous measure-ments from at least two receivers.However,many GPS users point position a single local receiver,based on publicly avail-able solutions for GPS orbits and clocks.These users cannot form double differences.We present an ambiguity resolution algorithm that improves solution accuracy for single receiver point-positioning users.The algorithm processes dual-frequency GPS data from a single receiver together with wide-lane and phase bias estimates from the global network of GPS receivers that were used to generate the orbit and clock solutions for the GPS satellites.We constrain(rather than?x)linear combinations of local phase biases to improve compatibility with global phase bias estimates.For this pre-cise point positioning,no other receiver data are required. When tested,our algorithm significantly improved repeat-ability of daily estimates of ground receiver positions,most notably in the east component by approximately30%with respect to the nominal case wherein the carrier biases are estimated as real values.In this“static”test for terrestrial receiver positions,we achieved daily repeatability of1.9, 2.1and6.0mm in the east,north and vertical(ENV)com-ponents,respectively.For kinematic solutions,ENV repeat-ability is7.7,8.4,and11.7mm,respectively,representing improvements of22,8,and14%with respect to the nominal. W.Bertiger(B)·S.D.Desai·B.Haines·N.Harvey·A.W.Moore·S.Owen·J.P.Weiss

Jet Propulsion Laboratory,California Institute of Technology,

Pasadena,CA,USA

e-mail:willy.bertiger@https://www.wendangku.net/doc/115289930.html, Results from precise orbit determination of the twin GRACE satellites demonstrated that the inter-satellite baseline accu-racy improved by a factor of three,from6to2mm up to a long-term bias.Jason-2/Ocean Surface Topography Mission precise orbit determination tests results implied radial orbit accuracy significantly below the10mm level.Stability of time transfer,in low-Earth orbit,improved from40to7ps. We produced these results by applying this algorithm within the Jet Propulsion Laboratory’s(JPL’s)GIPSY/OASIS soft-ware package and using JPL’s orbit and clock products for the GPS constellation.These products now include a record of the wide-lane and phase bias estimates from the under-lying global network of GPS stations.This implies that all GIPSY–OASIS positioning users can now bene?t from this capability to perform single-receiver ambiguity resolution. Keywords Ambiguity?xing·GPS·POD·Precise point positioning·WLPBLIST

1Introduction

Since the1980s,global positioning system(GPS)data pro-cessing algorithms,which estimate positions and other parameters,have frequently“resolved ambiguities”—?xed linear combinations of phase bias estimates—to improve solution accuracy(Blewitt1989).Until recently,ambiguity resolution algorithms explicitly differenced phase bias esti-mates,or phase bias data,from a pair of receivers and a pair of transmitters,in order to cancel receiver and transmitter hardware delays.Dual-frequency ambiguity resolution algo-rithms typically require two steps:resolution of the wide-lane,followed by resolution of the narrow-lane(Melbourne 1985;Wubbena1985).

328W.Bertiger et al.

During the past few years,several authors have suggested hardware delay calibration,allowing ambiguity resolution at a single dual-frequency receiver(Laurichesse et al.2008). Laurichesse et al.(2008)calibrated GPS transmitter wide-lane delays to allow single-receiver wide-lane resolution,and produced a special“integer”GPS clock solution to allow single-receiver narrow-lane resolution.Satellite Laser Rang-ing(SLR)data,compared to pre-and post-resolution results, strongly indicated improved Jason-1orbit solutions.A single GRACEA/GRACEB test day registered2-mm relative accu-racy,accepting as truth the micron-level accurate GRACE K-Band ranging system,up to an overall bias in each connected K-band arc on each day(Dunn et al.2003).

We present a different approach to single-receiver ambi-guity resolution,implemented in a complete operational system that computes bias-resolved solutions for low earth orbiting(LEO)or ground receivers,running the Jet Pro-pulsion Laboratory’s(JPL’s)GIPSY–OASIS software.Our system requires orbit,clock,wide-lane and phase bias infor-mation,computed by GIPSY–OASIS operational processes. Orbit,clock,wide-lane and phase bias information from the JPL International GNSS Service(IGS)(Dow et al.2009) Analysis Center’s contribution to the IGS products, https://www.wendangku.net/doc/115289930.html,/components/prods.html,are avail-able via anonymous ftp at ftp://https://www.wendangku.net/doc/115289930.html,/pub/ JPL_GPS_Products.

Like all IGS analysis centers,JPL solves for GPS orbits and clocks by processing data from a globally distributed set of static ground receivers.In addition,we save wide-lane (WL)and dual-frequency phase bias(PB)estimate informa-tion for each phase-connected GPS data arc processed dur-ing our global computation,in a small(<200kB)wide-lane phase bias information,or WLPBLIST,?le.A WLPBLIST contains a line for each continuously tracked phase arc in our global solution.Each line records GPS transmitting satellite name,GPS ground receiver name,phase arc start time,phase arc stop time,estimated wide-lane value,and the estimated dual-frequency phase bias for this arc.

To resolve ambiguities while point positioning a single local receiver,we form double difference combinations of local phase bias estimates,from our current run,and global phase bias estimates,drawn from the WLPBLIST?le.The values of the phase bias in the WLPBLIST?le are relative to the assumed antenna offsets.For the time period tested here, this has been the IGS recommended standard.The values have changed only for the addition of new GPS satellites and receiver antenna types.The exact antenna calibration used is recorded and made published with the WLPBLIST.We then add constraint equations in our Kalman?lter/smoother to nudge double differences towards integer values.The next section of this note presents our algorithm in more detail.

JPL began operational production of the WLPBLIST?les in2009,and they are being generated as part of our efforts to reprocess historical GPS data from1994to2009.The WLPB-LIST?les are now included in JPL’s delivery of GPS orbit and clock products for GIPSY/OASIS users.A GIPSY/OASIS user,using version5.1and higher,can therefore produce an ambiguity-resolved point-positioning solution for a single receiver with a single command(gd2p.pl)that requires a few command line arguments.

2Algorithm description

We designed a variant of the standard(ionosphere-free wide-lane)double difference ambiguity resolution algorithm described by Blewitt(1989)to resolve ambiguities at a single receiver based on a global network solution.

Roughly described,the algorithm in Blewitt(1989)does the following.

1)For every receiver A and every transmitter I,de?ne

n1(I,A)≡integer ambiguity in L1carrier phase

cycle count

n2(I,A)≡integer ambiguity in L2carrier phase

cycle count

n W(I,A)≡n1(I,A)?n2(I,A)

2)At every epoch,for every pair of receivers{A,B}and

every pair of transmitters{I,II}in common view,double difference ionosphere-free wide-lanes(WL)(Melbourne 1985;Wubbena1985)to resolve n W.

WL=

f1·L1?f2·L2

f1?f2

?

f1·P1+f2·P2

f1+f2

n W({I,II},{A,B})≡n W(I,A)?n W(II,A)

?n W(I,B)+n W(II,B)

≈WL(I,A)?W L(II,A)

?WL(I,B)+W L(II,B)

3)Having identi?ed n W,double difference dual fre-

quency phase bias(PB)estimates from the Kalman?l-ter/smoother,form the narrow-lane double-difference combination,and resolve n1.

n1({I,II},{A,B})

=

P B({I,II},{A,B})?λD· n W({I,II},{A,B})

λN

λD≈1.98·λ1

λN≈10.7cm

Single receiver phase ambiguity resolution with GPS data329

4)Having identi?ed n W and n1,apply a hard con-

straint to the phase bias double difference.Since we con-strain double differences tightly,we only need to resolve

a maximal linearly independent set of ambiguities.

PB({I,II},{A,B})

≡PB(I,A)?PB(II,A)?PB(I,B)+PB(II,B)

=λD· n W+λN· n1

Our single receiver ambiguity resolution algorithm differs from the(ionosphere-free wide-lane)algorithm described in Blewitt(1989)in the following ways.

1)Process data from a global network of GPS stations(with

network ambiguity resolution)to position GPS stations and satellites and solve for their respective clocks.Save information about phase biases and wide-lanes from arcs in the global solution to a WLPBLIST?le that is subse-quently made available to point-positioning users.

2)Point position the single receiver of interest based on the

global network solution of GPS orbits and clocks.

3)Processing information from our single-receiver point

positioning run,and global arc information from the WLPBLIST,form a list of possible double differences involving the local receiver being point-positioned(L),

a station from the global network solution(G),and a

pair of transmitters{I,II}in common view.No observa-tional data is needed for this list from the global stations or transmitters.Only the time intervals from the WLPB-LIST?le are required and the local receiver information.

4)Double difference wide-lanes to resolve n W using

wide-lane estimates for station G from the WLPBLIST ?le,

5)Having identi?ed n W,read phase bias estimates for

station G from the WLPBLIST?le and double difference phase bias estimates to resolve n1.

6)Having identi?ed n W and n1,apply a soft con-

straint to the phase bias double difference.We apply a soft constraint rather than a hard constraint to allow for inaccuracies in the global solution,and because the prob-ability of mis-resolution is typically fairly high.Since we constrain ambiguities loosely,we resolve every ambigu-ity we can,rather than restrict ourselves to a linearly independent set of ambiguities.

7)If necessary,iterate steps3)–6)to converge towards a

better solution.

Since we apply single receiver ambiguity resolution in a vari-ety of situations,ranging from static GPS ground receivers to low-Earth orbiting(LEO)satellites,our algorithm and soft-ware allows the user to specify a number of variables and options,some of which are worth mentioning here.1)Double-differencing wide-lanes cancels receiver and

transmitter hardware wide-lane biases.If these hardware biases remain stable from epoch to epoch over the length of an arc,we can use every available point in all four arcs of the double difference to estimate n W.On the other hand,if hardware delays vary from epoch to epoch,biases only cancel if we double-difference wide-lane observations at a common epoch.GPS transmitter wide-lane biases are fairly stable.Wide-lane biases at most(but not all)IGS stations are also stable,but some other networks contain a high proportion of stations with unstable wide-lane receiver biases.Since LEO receivers process much shorter arcs,wide-lane biases drift less over the length of an arc,and the relative advantage of using every available point is more https://www.wendangku.net/doc/115289930.html,cking information to the contrary,we typically assume stable wide-lane biases for LEO receivers,and unstable biases for ground stations.

2)At well-behaved receiver pairs,mis-resolved wide-lanes

are usually off by+/?1cycle,which results in a nar-row-lane bias of+/?0.53cycles(modulo an integer) conveniently far away from any https://www.wendangku.net/doc/115289930.html,ually,we take advantage of this behavior when computing ambigu-ity resolution con?dence.Unfortunately,for the Jason-2 GPS receiver,half-cycle issues in phase data compli-cate resolution(Bertiger et al.2010),so we apply a cruder con?dence calculation that does not assume inte-ger-based behavior.

3)With shorter arcs,both the wide-lane and the narrow-lane

are harder to resolve at a LEO receiver than at a static ground station,so we accept a lower con?dence level.To compensate for lower con?dence,we apply looser con-straints to resolved ambiguities.These looser constraints do not affect the?nal result as effectively,so we iterate ambiguity resolution when positioning a LEO,usually 10times(a number chosen by trial and error).We typ-ically do not iterate when positioning a static ground station.

3Low earth orbiter results

Since the TOPEX/Poseidon satellite launched in1992,a number of LEO satellites whose missions require precisely determined orbits have carried GPS receivers(Bertiger et al. 1994).For this investigation,we studied GPS data processing results from two LEO missions:

1)Jason-2/Ocean Surface Topography Mission(OSTM)

satellite(Neeck and Vaze2008),is a follow-on to TOPEX/Poseidon,in a1,300km altitude orbit.It carries

a radar altimeter to measure sea surface height.Space-

craft radial position estimates directly affect sea surface

330W.Bertiger et al.

height estimates.Data are processed at JPL by an

automated operational system that has used single

receiver ambiguity resolution since June2009.

2)Gravity Recovery and Climate Experiment(GRACE)

mission,twin satellites(GRACEA and GRACEB)in a

common orbit at500km altitude,separated by200km.A

K-band biased ranging system between the two GRACE

satellites measures separation,up to a bias,with micron-level accuracy(Dunn et al.2003).Twin spacecraft serve as test masses for recovery of Earth’s mass distribution (Tapley et al.2004).K-band ranging requires time syn-chronization between GRACEA and GRACEB with bet-ter than150ps accuracy to meet mission requirements.

The synchronization is accomplished by processing GPS data for GRACEA/B orbital positions and clocks.Data are processed daily at JPL by an automated operational system which has used single receiver ambiguity resolu-tion since1May2009.Necessary but not suf?cient tests of the150ps requirement are presented below.

3.1GRACE results

The operational processing of the GRACE data to synchro-nize time between the GRACE spacecraft and form the micron-level dual one-way K-band measurement was changed on1May2009to use the bias resolution method for a single receiver.Double-differenced biases between the two GRACE satellites are not formed,and POD for each of the twins is performed independently.In the operational system, the GPS phase data are decimated to5-min points and the pseudorange data are carrier smoothed to5-min points.We routinely monitor several system performance statistics. 1)Standard deviation of(K-Band biased range—GPS-

based positioning range).The GRACEA/B range is com-puted from independent GPS data processing orbital solutions for each satellite and compared to the K-band measurements.The standard deviation of the difference is computed on each continuous K-band arc on each day (since K-band rarely loses lock,typically only one arc per day,very seldom more than three).By the nature of K-band tracking,each continuous K-band arc has one undetermined bias,which we remove by least-squares estimation.Even after removal of a bias,GRACEA/B baseline length varies complexly with time,exhibiting features ranging from micron-level to meter-level.

2)RMS orbit overlaps.On each day,we compute GRACE

orbit solutions by processing30h of GPS data centered on that day,so each day’s solution overlaps with the previous day’s solution for6h.The RMS of the differ-ence is computed over the center5h of the6-h overlap, and separated into radial,cross-track,and along-track

Ambiguity Resolution On

Fig.1Kband Range–GPS Range and along track overlaps.KBR–GPS improved dramatically on1May,when ambiguity resolution began, overlap improvement for GRACEA and GRACEB lags a day since they are computed by comparison with the previous day

components.This5h time span starts at21:30on the ?rst day and ends at02:30on the second day.

3)(GRACEA–GRACEB)relative clock overlaps.The

(GRACEA–GRACEB)relative clock solution is com-puted on each day and the mean and standard deviation of relative clock solution differences from day-to-day are computed using the center5h of the6-h overlap. Figures1and2display results for these three metrics from GRACE operational processing before(7April2009to30 April2009)and after(1–22May2009)we added single receiver ambiguity resolution to our automatic process.The average(K-band biased range-GPS range)daily standard deviation improved from14.3to4.1mm,a factor of more than3.Along-track mean RMS overlaps for GRACEA improved from8.0to2.5mm,and for GRACEB from6.8to 2.1mm.Relative clock solution average standard deviation improved from41.0to7.2ps,while the mean relative clock differences did not improve by a comparable factor,prob-ably because pseudorange data dominates determination of mean relative clocks.Improved P-code antenna calibrations for GPS transmitters might improve the mean relative clock consistency.Our automated process currently uses IGS-rec-ommended antenna calibrations,which do not distinguish between code and phase data types.

The GPS receivers on-board GRACE sample pseudorange at10-s intervals and phase every second.In the past,our sys-tem estimating GPS orbits and clocks routinely produced GPS clock values at5-min intervals,but did not routinely produce them at higher rates(Jefferson et al.1999).Our cur-rent system routinely produces30-s GPS clock solutions, which enables improved processing of higher rate GRACE

Single receiver phase ambiguity resolution with GPS data

331

Fig.2Operational clock synchronization statistics before and after ambiguity resolution

data.While our results with 5-min data meet mission requirements,processing 30-s data gives better results (J?ggi et al.2009).J?ggi’s team examined the effects of antenna cal-ibration techniques on GRACE precision orbit determination (POD).The best solution over the best 60-day time period (in 2007)cited in their paper gave a KBR-GPS range mean daily standard deviation (SD)of 5.9mm,processing 30-s data.When they double-differenced data between GRACEA and GRACEB,and resolved ambiguities,mean SD for all of 2007(excluding a few days)improved to 0.81mm,improving on a previous result (Kroes 2006)without antenna calibra-tion.

One would expect a further improvement in baseline deter-mination when ?xing double differences directly between GRACEA and GRACEB,from direct cancellation of com-mon mode errors (e.g.,Kroes 2006).In this note,however,

Table 1Stochastic acceleration parameters;operational and re-tuned strategy Parameter Process noise Update (s)

Time

(nm/s 2)

correlation (s)

Operations

Constant along track 3003001,800Constant radial 503001,800Constant cross-track 100

300

1,800

Re-tuned

Constant along track 303007,200Constant radial 53007,200Constant cross track 103007,2001/rev along track 56,75021,6001/rev cross track

5

6,750

21,600

2

345678Fig.3Bene?ts of single receiver ambiguity resolution for GRACE baseline determination using 30-s data.Accuracy measured by the K-band instrument (KBR)

we focus only on single receiver ambiguity resolution,appli-cable to single-satellite LEO missions.

We re-processed GRACE data from 1May 2009to 20June 2009,using 30-s rather than 5-min samples.Simply process-ing the data at the 30-s rate did not significantly improve the key KBR-GPS metric,so we also adjusted our empirical sto-chastic acceleration parameters.Table 1contrasts the nom-inal parameterization for processing the 5-min.data in the operational process with the tuned solution strategy adopted for the 30-s data.The most significant difference is the addi-tion of an empirical once-per-revolution acceleration.With a 5-min data rate bias resolved orbits improved the KBR-GPS range standard deviation from 4.1to 2.9mm when the oper-ational parameterization in Table 1was replaced with the re-tuned parameters for the test period.Processing 30-s data with the tuned parameter set yielded KBR-GPS agreement at the 6.1and 2.1mm level (mean of the standard deviation)before and after single receiver ambiguity resolution (Fig.3).Figure 4shows GRACEA orbit overlaps for 30-s data and re-tuned parameters with Table 2summarizing the daily sta-tistics.

3.2Jason-2/Ocean Surface Topography Mission

Launched 20June 2008,Jason-2/Ocean Surface Topogra-phy Mission (OSTM)carries a radar altimeter that mea-sures satellite-ocean separation with roughly 3-cm accuracy at 1Hz sampling (Neeck and Vaze 2008).Jason-2also car-ries three measurement systems for orbit determination from GPS,Satellite Laser Ranging (SLR),and Doppler Orbitog-raphy and Radiopositioning Integrated by Satellite (DORIS)data.The French space agency,Centre National d’études

332

W.Bertiger et al.

2

4

6

8

Fig.4GRACEA 5-h RMS overlaps,30-s data rate,re-tuned param-eters,with and without ambiguity resolution.Closed circles indicate ambiguity resolved RMS overlaps

Spatiales (CNES),produces of?cial orbits for geophysical data records (GDR)based on GPS,SLR,and DORIS data (Cerri et al.2010).

Jason-2/OSTM orbits at a substantially higher altitude than do the GRACE satellites,1,300km rather than 500km,which affects positioning in two ways;one helpful,one harmful.

1)Less atmosphere,so lower drag,at 1,300km,reduces

errors in our force model,allowing a more dynamic model,less affected by bad or missing data.

2)The radiation environment is significantly harsher at

1,300km,forcing frequent GPS receiver resets,which concentrate near the South Atlantic Anomaly.Jason-2/OSTM model details may be found in Bertiger et al.(2010).We applied three metrics to assess the effect of sin-gle receiver ambiguity resolution on Jason-2orbit estimate precision,with results summarized in the next two tables and a ?gure.

1)RMS orbit overlaps from July 2008to June 2009

(Table 3):On each day,we solve for Jason-2orbits pro-cessing 30h of data centered on noon,so each day’s solution overlaps with the previous day’s solution for 6h.The RMS difference over the center 4h of the 6-h overlap is computed and separated into radial,cross-track,and along-track components,before and after sin-gle receiver ambiguity resolution.The radial component,key for Jason-2/OSTM,improved by a factor of 1.7.

2)Scatter of SLR residuals from July 2008to May 2009

(Table 4):We produced orbit solutions by processing only GPS data,so SLR data provide an independent val-idation of our pre-and post-ambiguity resolution orbits.Since Jason-2requirements address only radial accuracy,we restricted our consideration to SLR data above 60?elevation as viewed from the ground.To reduce SLR measurement error,we focused on four of the high-est quality SLR ground stations:Monument,Yaragadee,Graz and McDonald.Ambiguity resolution improved scatter substantially at all four stations.Our ambiguity resolved results compare favorably with CNES’s of?cial results (GDR-C)(Cerri et al.2010),which did ?t to the SLR data.Typical scatter,pre and post-ambiguity resolu-tion,lies below a centimeter,suggesting sub-centimeter radial accuracy,up to an overall bias.

3)Differences of sea-surface height (SSH)estimates at

crossover locations (Fig.5):The data from the radar altimeter are used in neither the CNES-determined orbits nor the JPL GPS-determined orbits discussed here.At the locations where the ground track paths from ascend-ing and descending passes cross each other on the ocean surface,we can use the difference in the radar altim-eter’s measurement of the SSH to infer relative radial orbit error.At the crossover points,the radial orbit error is fully expressed in the SSH error.In addition to the orbit error,however,there are measurement errors in the radar and true changes in SSH due to different sampling times at the crossover point.In order to minimize these other error sources,we examine only those crossover points for which the ascending and descending passes are separated by fewer than three days with moderately calm oceans (significant wave heights of 1–4m and sur-face wind speeds of 4–10m/s)and with the absolute value of atmospheric pressure loading correction less than 15cm.We refer to this selection process as super-editing.Since Jason-2/OSTM repeats its ground track on the Earth every 10days,we computed the variance of the sea height crossover difference over each of these 10-day cycles.Figure 5,shows the reduction in cross-over variance between the orbits determined with and without ambiguity resolution.A positive value indicates that ambiguity resolution reduced crossover variance.The variance is reduced on all 2610-day cycles with an average reduction of 45mm 2.

4Static ground receiver and baseline results

To test the new single receiver method on static ground receiver positioning,we examined station coordinate repeat-ability before and after ambiguity resolution.Table 5shows

Single receiver phase ambiguity resolution with GPS data

333

Table 2Mean RMS overlaps with and without ambiguity resolution,1May 2009through 20June 2009,30-s data,re-tuned parameters

Radial (mm)

Radial Amb.Cross Trk.Cross Trk.

Along Trk.Along Trk.

Res.(mm)(mm)Amb.Res.(mm)(mm)Amb.Res.(mm)GRACEA 2.81.44.42.26.32.3GRACEB

2.9

1.4

4.2

2.1

6.3

2.3

Table 3Average RMS overlaps from consecutive 30-h processing arcs,11July 2008through 5June 2009

Radial (mm)

Radial Amb.Cross Trk.Cross Trk.Amb.Along Trk.(mm)Along Trk.

Res.(mm)(mm)Res.(mm)(mm)Amb.Res.(mm)Jason-2/OSTM

3.1

1.8

3.6

3.0

7.6

4.6

Table 4For each SLR tracking pass the mean of the one-way SLR range residual is computed.The standard deviation (sigma)and mean of these biases are computed for three different Jason-2/OSTM orbits 12July 2008through 31May 2009

CNES,

JPL reduced

JPL ambiguity

#Arcs GDR-C sigma/mean (mm)

dynamic sigma/mean (mm)resolved sigma/mean (mm)Monument 8.6/10.28.1/10.1 6.4/10.820Yaragadee 6.5/2.17.9/7.8 6.2/9.0190Graz 6.5/?6.810.3/?9.68.0/?8.975McDonald 8.8/?6.89.9/9.68.3/10.519All (weighted)

6.8/1.1

8.7/3.8

6.8/4.8

304

Fig.5Difference in sea height variance at super-edited cross-over points (variance without ambiguity resolution)–(variance with ambi-guity resolution),mean variance reduction is 45mm 2

key parameter assumptions for static positioning.The static point positioning procedure is similar to the one in Zumberge et al.(1997),but with updated models and more highly auto-mated software.Troposphere and clock parameters are updated every 5min and the GPS data are sampled every 5min.We processed 6months (1June 2008to 30November 2008)of data from 209stations in 24-h intervals.We ignore stations that have do not have quality data for at least 80%of the days.Figure 6shows a map of the locations of the stations

considered in this study.Table 6summarizes the scatter about the IGS de?ned coordinates for the 106stations that de?ne the IGS05frame.Results for each component (east,north,ver-tical)for two tropospheric mapping functions (Niell/VMF1)both with and without bias resolution are given.Bias resolu-tion makes a significant improvement in the east component in both cases.There are only modest improvements in the other components due to bias resolution.The newer VMF1troposphere model makes a significant improvement in the vertical component as seen by others (Boehm et al.2006).Bias ?xing has traditionally been used on baselines where the data may be explicitly double differenced to remove hard-ware delays.This method has long been implemented in the GIPSY–OASIS software package and is part of many other GPS software systems as well.Here we compare baseline results using our new single station bias resolution method and the method implemented in GIPSY–OASIS based on Blewitt (1989).The software wrapping Blewitt’s algorithm in GIPSY–OASIS is referred to as the “Network Processor”(Liu et al.2009;Owen et al.2006).We processed 301base-lines from 1June 2008to 30November 2008,using IGS station data.Only baselines between 10and 10,000km are considered.Figure 7shows the daily scatter about the mean as a function of baseline length (log scale)for each of the 301baselines considered and a linear ?t to the scatter.We see that the daily scatter of the single station bias resolution method is approximately equivalent to that of the traditional double

334W.Bertiger et al.

Table5Parameters estimated and key models applied in24-h static positioning tests

Parameter Relevant model Solution properties

Station coordinates IERS03Earth models(McCarthy and Petit2004)Constant over24h

Zenith troposphere delay Niell(Niell2006)or VMF1mapping functions(Boehm et al.2006)Random walk50μm/sqrt(h) Gradient troposphere delay Bar-Sever et al.(1998)Random walk5μm/sqrt(h) Station clock White-noise unconstrained

Fig.6Positions of the209

stations considered in the

baseline/station position

calculations

Table6Standard deviation of the station east,north,and vertical com-

ponents for24-h static positioning from1June2008to30November

2008using106IGS frame definition stations

East(mm)North(mm)Vertical(mm)

Unresolved/Niell2.92.17.0 Resolved/Niell1.92.06.8 Unresolved/VMF12.92.16.0 Resolved/VMF11.92.16.0

difference method.The ef?cacy of both methods decay sim-ilarly as the length of the baseline increases.At the longest baselines,improvements with explicit double differences or bias resolution over precise point positioning are small(note that the number of samples at the longest baselines is also small).

Figure8shows another view of the baseline scatter for the same set of data as Fig.7.The fractional improvement in baseline length repeatability as a function of baseline length is again indistinguishable.

Bias Resolved (single station)

Fig.7The daily baseline scatter about the mean as a function of base-line is plotted for each of the301considered baselines along with a linear?t.The scatter in the single station ambiguity resolution(red dot) and traditional double difference ambiguity resolution(blue triangle) are shown along with the pre-resolution scatter(green dot),precise point positioning,phase biases adjusted as real numbers only,along with a linear?t in each case

Single receiver phase ambiguity resolution with GPS data335 DD res DD res

Fig.8Baseline scatter using double differenced(σDD)?xed phase

ambiguities versus single receiver bias resolution(σres)

(σ

ppp ? σ

res

)/(σ

ppp

+ σ

res

)/2

Fig.9Baseline determination over24-h with precise point position-ing(ppp)versus bias resolution(res),improvements in baseline length scatter

Finally,Fig.9compares the new bias resolution method to precise point positioning.For baselines under1,000km,bias resolution improves baseline repeatability by about10%.The results for traditional double differencing,compared to pre-cise point positioning without ambiguity resolution,would, of course,be the similar.

The IGS station distribution did not include many base-lines in the range of1–10km.In order to study the methods in this regime we used309baselines from the Los Angeles California area in daily solutions during the month of June 2008.Figure10shows the scatter in baseline calculations. The average scatter for both explicit double differencing and bias resolution is1.4mm.The scatter in the unresolved pre-cise point positioning is2.2mm;thus either method of bias

Explicit Double Diff.

Precise Point Positioning

Fig.10The scatter of daily short baselines within the Los Angeles Basin for June2008is plotted as a function of baseline length.The average scatter for the single station ambiguity resolution and tradi-tional ambiguity resolution is1.4mm,while the scatter of the unresolved baselines is2.2mm

adjustment improves short baselines by about36%compared to precise point positioning.Although there is a slight slope to the linear?ts shown in the plots,this may be due to the small number of samples at the shortest baseline lengths.

5Kinematic ground receiver positioning

Kinematic point positioning of15static terrestrial GPS sta-tions is performed with and without single receiver ambiguity resolution for the6-month period from12April to11Octo-ber2009.The stations were selected from the IGS network and provide examples from different regions of the Earth, including island sites and sites on different continents.Kine-matic solutions are generated for each day using30h of data centered at noon of each day,and with positions estimated at5-min intervals.We?rst consider the scatter of the5-min kinematic position solutions from the middle24h of the 30-h solution window with respect to the static point position solution for that day.Each of the15stations demonstrated a reduction in the median of its daily RMS of position dif-ferences in all three components.As shown in Table7,the median of the daily RMS of these differences over all2,559 station days improves in all three components,east,north and vertical,by22,9,and14%,respectively.Days with incom-plete or missing data are excluded from these statistics.

Second,there is a6-h period in each day,where kinematic position solutions from neighboring days overlap.The scatter of the differences between the middle4h of these overlapping solutions is a measure of the impact of ambiguity resolution on the day-to-day consistency of kinematic point positioning

336W.Bertiger et al. Table7Impact of ambiguity resolution on kinematic point positioning of15global GPS sites for6-month period from12April to11October 2009

Median of daily RMS of differences Between5-min kinematic solutions and daily static solution Median of daily RMS of differences Between overlapping kinematic solutions

No ambiguity resolution With ambiguity resolution No ambiguity resolution With ambiguity resolution

East(mm)9.97.77.13.0

North(mm)9.28.44.32.7

Vertical(mm)13.611.77.75.5

(last two columns of Table7).Of the15stations considered, only the vertical component of the overlaps of one station, Tahiti,degraded by10%when using ambiguity resolution, while all components of the other stations improved signifi-cantly.The median of the daily RMS of kinematic position overlap differences over all2475station days with overlaps improved by58,36,and29%in the east,north and vertical components,respectively.

6Summary

The new ambiguity resolution method integrating operational GPS orbit and clock products(ftp://https://www.wendangku.net/doc/115289930.html,/ pub/JPL_GPS_Products)and JPL’s GPS receiver processing software(GIPSY-OASIS II,https://www.wendangku.net/doc/115289930.html,/ gipsy/software.html)can be used to process GPS data under a variety of circumstances.For static positioning of receivers on the surface of the Earth,the new method is competitive with traditional bias?xing methods used to determine base-lines between two stations.The new method improves static precise point positioning of a single receiver,yielding repeat-ability in east,north,and vertical components of1.9,2.1,and 6.0mm.Kinematic positioning with stochastic updates every 5min improved with bias resolution in east,north,and ver-tical by about22,9,and14%for a receiver on the Earth.

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