Rapid re-convergences to ambiguity-fixed solutions in precise point positioning

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1 J Geod (2010) 84: DOI /s ORIGINAL ARTICLE Rapd re-convergences to ambguty-fxed solutons n precse pont postonng Janghu Geng Xaoln Meng Alan H. Dodson Maorong Ge Felx N. Teferle Receved: 25 May 2010 / Accepted: 4 August 2010 / Publshed onlne: 14 August 2010 Sprnger-Verlag 2010 Abstract Integer ambguty resoluton at a sngle recever can be acheved f the fractonal-cycle bases are separated from the ambguty estmates n precse pont postonng (PPP). Despte the mproved postonng accuracy by such nteger resoluton, the convergence to an ambguty-fxed soluton normally requres a few tens of mnutes. Even worse, these convergences can repeatedly occur on the occason of loss of trackng locks for many satelltes f an open sky-vew s not constantly avalable, consequently totally destroyng the practcablty of real-tme PPP. In ths study, n case of such re-convergences, we develop a method n whch onospherc delays are precsely predcted to sgnfcantly accelerate the nteger ambguty resoluton. The effectveness of ths method conssts n two aspects: frst, wde-lane ambgutes can be rapdly resolved usng the onosphere-corrected wde-lane measurements, nstead of the nosy Melbourne Wübbena combnaton measurements; second, narrow-lane ambguty resoluton can be accelerated under the tght constrants derved from the onosphere-corrected unambguous wde-lane measurements. In the test at 90 statc statons sufferng from smulated total loss of trackng locks, 93.3 and 95.0% of re-convergences to wde-lane and narrow-lane ambguty resolutons can be acheved wthn fve epochs of 1-Hz measurements, respectvely, even though the tme J. Geng (B) X. Meng A. H. Dodson Insttute of Engneerng Surveyng and Space Geodesy, Unversty of Nottngham, Nottngham, NG7 2TU, UK e-mal: sxjg@nottngham.ac.uk M. Ge GeoForschungsZentrum Helmholtz Center, Potsdam, Germany F. N. Teferle Faculty of Scence, Technology and Communcaton, Unversty of Luxemburg, 1359 Luxemburg, Luxemburg latency for the predcted onospherc delays s up to 180 s. In the test at a moble van movng n a GPS-adverse envronment where satellte number sgnfcantly decreases and cycle slps frequently occur, only when the predcted onospherc delays are appled can the rate of ambguty-fxed epochs be dramatcally mproved from 7.7 to 93.6% of all epochs. Therefore, ths method can potentally releve the unrealstc requrement of a contnuous open sky-vew by most PPP applcatons and mprove the practcablty of realtme PPP. Keywords Precse pont postonng Ambguty resoluton Rapd re-convergence Predcted onospherc delays 1 Introducton Double-dfference carrer-phase ambgutes are normally fxed to ntegers n precse Global Postonng System (GPS) applcatons. Nonetheless, undfferenced ambgutes n precse pont postonng (PPP) (Zumberge et al. 1997) lose ther nteger propertes because they absorb the fractonal-cycle bases (FCBs) that orgnate from both recever and satellte hardware (Geng et al. 2010a). Fortunately, these nteger propertes can be retreved by correctng the float ambguty estmates wth the FCBs that are derved from a network of reference statons (Ge et al. 2008). Due to the onosphere-free observable used n PPP, nteger ambguty resoluton s performed through two sequental steps: wde-lane ambgutes are frst resolved usng the Melbourne Wübbena combnaton observable (Melbourne 1985; Wübbena 1985) and narrow-lane ambgutes are then resolved based on the resultng nteger wde-lane ambgutes (Geng et al. 2009; Laurchesse et al. 2009). Geng et al. (2010b) llustrated that ambguty

2 706 J. Geng et al. resoluton can sgnfcantly mprove the postonng accuracy from decmeter to centmeter level n real-tme PPP. Nevertheless, a long convergence perod to such an ambguty-fxed soluton wll consderably devalue the accuracy mprovement contrbuted by the nteger resoluton. Accordng to a substantal data processng, Geng et al. (2010b) concluded that at least 10 mn of measurements are requred to resolve about 90% of wde-lane ambgutes, whereas over 10 mn for a narrow-lane ambguty resoluton. A wde-lane ambguty s estmated usng the nosy Melbourne Wübbena combnaton measurements, and thus a long observaton perod s requred to suffcently smooth the nose. On the other hand, rapd narrow-lane ambguty resoluton s handcapped by the mprecse pseudorange measurements whch cannot suffcently shrnk the search space for the nteger ambgutes sufferng from a rather short wavelength of 10.7 cm (Teunssen 1996). Hence, a long observaton perod has to be used to average out suffcently precse pseudorange measurements n order to derve accurate ambguty estmates (Teunssen et al. 1997). To date, however, t s stll extremely dffcult to develop an effectve method to accelerate these convergences. Even worse, these long-perod convergences can repeatedly occur on the occason of loss of trackng locks for many satelltes. Frequent re-convergences are common problems n ndustral applcatons and can totally destroy the practcablty of real-tme PPP. As a result, a contnuous open sky-vew s constantly requred by most PPP applcatons to mantan a centmeter-level postonng accuracy (Bsnath and Gao 2007). To releve ths unrealstc requrement and to mprove the practcablty of PPP, Banvlle and Langley (2009) dentfed the nteger cycle slps caused by such lock losses through dfferencng between epochs, smply assumng that onospherc delays canceled out durng such dfferences. However, they dd not perform the nteger resoluton on undfferenced ambgutes. On the other hand, Geng (2009) developed a method where onospherc delays were precsely predcted to tghtly constran the narrow-lane resoluton to sgnfcantly accelerate re-convergences. Encouragngly, the re-convergence perods were shortened from about 1,000 to 20 s on average wth a success rate of around 90%. However, Melbourne Wübbena combnaton measurements were stll used to fx wde-lane ambgutes, thereby hardly guaranteeng a rapd re-convergence of only a few seconds. In ths study, we am at mprovng ths method to acheve rapd re-convergences to ambguty-fxed solutons wthn a few epochs of 1-Hz measurements. Note that convergence throughout ths study means achevng an ambguty-fxed soluton. In the followng, Sect. 2 detals the theory of ths method; Sect. 3 presents a case study at statc statons, nvestgates the mpacts of tme latency on the predcton error of onospherc delays and dscusses the effcency of rapd re-convergences; Sect. 4 presents a case study at a moble van, nvestgates the performance of rapd re-convergences n a GPS-adverse envronment where satellte number sgnfcantly decreases and cycle slps frequently occur, and compares the results wth the network real-tme knematc (NRTK) soluton; fnally, Sect. 5 summarzes the man ponts and addresses the perspectve of a wde-area PPP-RTK servce supported by rapd ambguty resoluton. 2 Method In terms of the prevous secton, the keys to acceleratng convergences consst n both avodng the Melbourne Wübbena combnaton measurements and shrnkng the search space for nteger ambgutes. The volume of ths search space s governed by the varance covarance matrx of ambguty estmates, and t approxmately observes the followng rule (Teunssen 1996, 1997): V σ P (1) nλ where V s the volume of search space; σ P s the precson of pseudorange or unambguous measurements; n s the number of epochs; λ s the wavelength of carrer-phase measurements; and denotes a proportonal relatonshp. To acheve rapd convergences, ncreasng n s unrealstc, and thus we can only dmnsh σ P or enlarge λ. In ths secton, we thereby derve how to precsely predct onospherc delays and apply these delays to accelerate both the wde-lane and narrowlane ambguty resolutons, then dscuss the mplementaton of ths method, and fnally address the ambguty search and valdaton n ths study. 2.1 Precsely predct onospherc delays In general, GPS carrer-phase measurements on frequency g(g = 1, 2) at a partcular epoch for a sngle recever can be wrtten as λ g j g = ρ j + λ g ϕ j + ct ct j + T j +λ g ( B g + B j g + N j g κ j fg 2 ) + ε j g (2) where the superscrpt j ( j = 1,...) denotes a satellte; j g s the carrer-phase measurement where antenna phase center correctons have been appled; λ g s the wavelength and f g s the frequency; c s the lght speed; ρ j s the geometrc dstance between the recever and the satellte j; ϕ j s the phase wnd-up correcton (Wu et al. 1993); t and t j are the recever and satellte clocks, respectvely; T j denotes a slant tropospherc delay; κ j denotes a slant onospherc delay; B fg 2 g and B j g are the recever and satellte FCBs, respectvely; N j g

3 Rapd re-convergences to ambguty-fxed solutons n precse pont postonng 707 denotes an nteger ambguty; fnally, ε j g denotes unmodeled errors ncludng multpath effects. From Eq. 2, a wde-lane measurement at ths epoch can be formed as λ w j w = κ j + ρ j + ct ct j + T j f 1 f 2 ) +λ w (B w + Bw j + N j w + ε j w (3) where λ w s the wde-lane wavelength of about 86 cm; j w = j 1 j 2, B w = B 1 B 2, Bw j = B j 1 B j 2, N j w = N j 1 N j 2 and ε j w = f 1 f 1 f 2 ε j 1 f 2 f 1 f 2 ε j 2 ; and note that the phase wnd-up effects on dual frequences cancel out. In ths study, Eq. 3 s hereafter used to estmate the onospherc κ delay, namely the term of j f 1 f 2 n theory, at all ambgutyfxed epochs. For a low-dynamc recever, the onospherc delays for each satellte normally manfest a strong temporal correlaton over a few mnutes (Da et al. 2003; Kashan et al. 2007), whch s the foremost foundaton of our method for rapd re-convergences. Despte ths favorable characterstc of onospherc delays, we have to further quantfy the resdual bases possbly assmlated nto the estmate of j κ f 1 f 2, and nvestgate ther temporal characterstcs. Frst, the recever clock seems an unstable quantty over tme. Hence, a sngle dfference between satelltes j and k at ths epoch can be formed, namely λ w jk w = κ jk ( + λ w f 1 f 2 +T jk B jk w ) + N jk w + ρ jk ct jk + ε jk w (4) where the recever clock t and the recever FCB B w are consequently elmnated. The satellte FCB Bw jk s normally deemed very stable over at least 24 h. Ge et al. (2008) verfed ths by dervng an agreement of better than 0.05 cycles between 14 days of wde-lane FCB estmates. Moreover, the nteger N jk w has been known n the ambguty-fxed soluton at ths epoch. Second, ρ jk depends on both the satellte and the recever postons. To date, predcted GPS satellte orbts by IGS over 24 h have reached an accuracy of around 5 cm ( jpl.nasa.gov/components/prods.html) (Dow et al. 2009). Resdual radal orbt errors can be mostly mtgated by satellte clocks (Senor et al. 2008), f a consstent yaw-atttude model s appled to the satelltes (Kouba 2009). Moreover, GPS satellte orbts are tghtly constraned by dynamc force models, and thus the resdual orbt errors should change smoothly n theory. Of partcular note, unannounced satellte maneuvers can sgnfcantly degrade the orbt accuracy. For the precse predcton of onospherc delays, ths can be a lmtng factor, but s hereafter gnored because these maneuvers are normally rare. On the other hand, the recever poston can be known to the centmeter-level accuracy n the ambguty-fxed soluton at ths epoch (Geng et al. 2010b). Thrd, due to the hgh correlaton between t jk and the satellte orbts (Senor et al. 2008), favorable user range errors of centmeter level can be acheved (Hauschld and Montenbruck 2009). Nonetheless, ths t jk s normally a predcted value due to the communcaton delays n real-tme applcatons. Fortunately, Senor et al. (2008) reported that the Rubdum and Cesum clocks on current GPS satelltes can be precsely predcted to a precson of better than 0.1 ns up to a perod of 40 s whch covers typcal communcaton delays. Hence, we hereafter assume that errors of satellte clocks are always smaller than 0.1 ns. Fourth, T jk can be mostly mtgated by estmatng a zenth tropospherc delay (ZTD) based on a mappng functon. Resdual tropospherc delays are only a few mllmeters at hgh elevatons, but can be up to several centmeters at an elevaton of smaller than 10 (Stoew et al. 2007). However, the troposphere condton around a low-dynamc recever slghtly changes over several mnutes f nether rapd weather fronts nor large heght varatons occur for ths recever (Gregorus and Blewtt 1999; Shan et al. 2007). Consderng ths smlar temporal characterstc to that of onospherc delays, we can alternatvely combne the two atmospherc delays and then predct ther sum over tme. Fnally, ε jk w s the most uncertan quantty n Eq. 4 and we beleve that the multpath effect domnates among all possble unmodeled errors. Dlßner et al. (2008) showed that the carrer-phase bases caused by multpath effects stemmng from solely radatng near felds can theoretcally reach a few centmeters for low-elevaton satelltes. Although Han and Rzos (2000) descrbed the strong temporal correlaton of the multpath sgnatures at statc antennas, such sgnatures can be qute random and thus unpredctable at moble antennas. Therefore, at an ambguty-fxed epoch, knowng the quanttes of N jk w,ρjk, t jk and T jk, we can deduct them from jk w and obtan the onospherc-delay estmate, namely Î jk w = λ w jk w λ w N jk jk w ˆρ + ct jk ˆT jk = κ jk + λ w Bw jk f 1 f + e jk w + ε jk w (5) 2 where ˆρ jk and ˆT jk are the estmates of ρ jk and T jk, respectvely, whch are derved from the ambguty-fxed soluton of ths epoch; e jk w contans the errors of the satellte product and the tropospherc delay. Accordng to the quanttatve assessments above, we can nfer that e jk w + ε jk w can easly amount to over 10 cm for low-elevaton satelltes. However, Bw jk changes neglgbly over a long tme and e jk w changes mnmally or predctably over a few mnutes, thereby mnmally affectng the temporal-correlaton characterstc of

4 708 J. Geng et al. κ jk f 1 f 2. Hence, Î jk w can be precsely predcted to the succeedng epochs over a few mnutes, especally at statc antennas where multpath effects are also temporally correlated. Note that the error caused by the recever poston estmate s gnored n Eq. 5 because ths error wll be absorbed by the poston estmate where the predcted onospherc delays are appled (refer to Sect. 2.2). In ths study, Î jk w s smply called onospherc delay for brevty although t contans more than a true onospherc delay. For the predctng strategy, we suggest the lnear fttng model where estmated onospherc delays wthn a sldng tme wndow are used to ft a lnear functon, and a predcted delay s then extrapolated usng ths functon (Da et al. 2003). In ths study, the tme gap between a predcted and the latest estmated onospherc delays s named as the latency of ths predcted delay. Ths tme latency can be caused by data gaps, for example. The predcton error at an epoch s quantfed by dfferencng the predcted and estmated onospherc delays at ths epoch. Geng (2009) llustrated that the predcton error s ncreased when the elevaton angle becomes smaller and the tme latency becomes longer, and ths ncreasng rate depends on the onosphere condton. Hence, besdes the elevaton-dependent weghtng on the predcted onospherc delays by Geng (2009), namely { p (E) = E 90 2snE 7 E < 30 (6) where E s the elevaton angle n degrees, a latency-dependent weghtng s also requred, such as { 1.0 τ<30 s p (τ) = tan ( π 4 30 ) (7) τ τ 30 s where τ s the tme latency n seconds and the 30 s wll be llustrated n Sect Rapdly retreve nteger ambgutes Integer ambgutes can be rapdly retreved by applyng the above precsely predcted onospherc delays after a reconvergence occurs. Ths s acheved through two steps. Frst, at a partcular epoch where the ambguty-fxed soluton has been lost, wde-lane measurements are formed usng Eq. 4 and are corrected by the known satellte products, the latest ZTD estmate, and the predcted onospherc delays. If the tme latency s shorter than a few mnutes, the wdelane-measurement errors caused by FCBs, satellte products and atmospherc delays can be canceled out or mostly mtgated by the counterparts n the predcted onospherc delays (Da et al. 2003; Kashan et al. 2007). Note that unmodeled errors are not easly predcted and thereby supposed to be suffcently small n ths study. Fnally, we obtan λ w jk w + ct jk T jk I jk w = ρ jk + λ w N jk w + ζ jk w (8) where T jk s the slant tropospherc delay computed usng the latest ZTD estmate that s derved from prevous solutons; I jk w s the predcted onospherc delay; ζ jk w contans the unmodeled errors and the predcton error of onospherc delays, and t actually quantfes the precson of Eq. 8; moreover, only the postons and ambgutes are unknown. Constraned by the onosphere-free pseudorange measurements, the wde-lane ambguty N jk w can be resolved usng Eq. 8 f ζ jk w s small enough (e.g. <1/4λ w). In terms of Eq. 1, thanks to the relatvely long wavelength for wde-lane ambgutes, onosphere-free pseudorange measurements can normally suffcently shrnk the search space to only a few nteger canddates, thereby mprovng the search effcency and acceleratng the dentfcaton of nteger ambgutes. Second, at ths epoch, once the wde-lane ambguty s successfully resolved, Eq. 8 becomes a precse unambguous measurement, namely λ w jk w + ct jk T jk I jk w λ w N jk w = ρ jk + ζ jk w (9) In ths study, t s Eq. 9, rather than the pseudorange measurements, that s supermposed to the normal equaton of PPP and thus constrans the nteger dentfcaton of narrowlane ambgutes durng rapd re-convergences. In terms of Eq. 1, f the magntude of ζ jk w s far smaller than a few centmeters, applyng Eq. 9 can consderably shrnk the search space for narrow-lane ambgutes to a few nteger canddates. As a result, narrow-lane ambguty resoluton can be accelerated. Note that the predcton error of onospherc delays domnates ζ jk w f the tme latency s long. An mprecsely predcted onospherc delay wll weaken the constrant mposed by Eq. 9, possbly falng a rapd re-convergence. Fnally, we stress that both rapd resolutons above mnmally rely on the satellte-geometry change, but on the suffcent shrnk of search space for nteger canddates. Such shrnks are acheved by applyng the onosphere-free pseudorange and the onosphere-corrected unambguous wde-lane measurements whch are precse enough relatve to the wde-lane and narrow-lane wavelengths, respectvely. As a result, float ambguty estmates are close to the correct ntegers and these ntegers can then be relatvely easly dentfed, even though only one epoch of measurements s used. 2.3 Remarks on the method mplementaton From the method descrbed above, the prerequste of ths method s how to precsely predct onospherc delays, whch depends on three aspects: Temporal propertes of all errors n the onospherc-delay estmate by Eq. 5. Although the true onospherc delay domnates ts temporal varaton, other errors ncludng

5 Rapd re-convergences to ambguty-fxed solutons n precse pont postonng 709 clock predctons and multpath effects should also be carefully regarded; Predctng strategy. Currently, t s not easy to precsely predct the onosphere condton over a long tme due to ts complcated relatonshps wth the geomagnetc feld, the solar actvty, etc. especally durng an onospherc storm. For example, sudden onospherc dsturbances and scntllatons may occur durng hgh onosphere actvtes n polar and equatoral regons (Basu et al. 2002). As a result, temporal onospherc rregulartes cannot be precsely depcted beforehand n real-tme applcatons. In ths case, only the lnear trend of the onosphere varaton can usually be quantfed wth a relatvely hgh confdence level (Da et al. 2003). However, we should keep n mnd that the resdual varatons can sometmes be very large and thus fal the precse predcton of onospherc delays; Model consstency between the onospherc-delay estmaton usng Eq. 5 and the wde-lane ambguty resoluton usng Eq.8. Note that both models are derved from Eq.4. In ths manner, ther common bases can naturally cancel out wthout mparng the nteger ambgutes, hence explanng why we do not care about the bases assmlated nto the estmated onospherc delays, but ther predcton errors. Heretofore, we need to acknowledge that ths method apples to only re-convergences because onospherc delays are estmated only at ambguty-fxed epochs as demonstrated n Sect Nonetheless, f a dense network of reference statons s avalable, onospherc delays can also be estmated at the reference statons and then nterpolated at sngle users to assst rapd convergences. In ths case, the frst ambguty-fxed soluton can also be rapdly acheved. 2.4 Ambguty search and valdaton For the ambguty search, we used the least-squares ambguty decorrelaton adjustment method (Teunssen 1995) to search for nteger canddates. All avalable satelltes over an elevaton angle of 7 were used for wde-lane ambguty resoluton, whereas those over 15 for narrow-lane ambguty resoluton (Geng et al. 2009). For the ambguty valdaton, the rato test, where the rato of the second mnmum to the mnmum quadratc form of resduals s assessed, was used wth a threshold of 2 whch was adopted by many authors (e.g. Teunssen and Verhagen 2009; Wang et al. 1998). Of partcular note, however, a fxed threshold here s not reasonable n theory because t should vary wth the strength of the underlyng GPS model and the degrees of freedom (Teunssen and Verhagen 2009). In ths case, an mproved rato test or GPS model may potentally mprove the effcency of ambguty resoluton, thus possbly further acceleratng the convergences (Verhagen and Teunssen 2006; Wang et al. 1998, 2002). However, these ssues are beyond the scope of ths study and thus gnored throughout. 3 A case study at statc statons In ths secton, we nvestgate the predcton error of onospherc delays, and assess ths method at statc statons by quantfyng the success rate of rapd re-convergences and the tme spent on achevng the ambguty-fxed soluton. 3.1 Data processng Seven days of 24-h 1-Hz GPS data from July 6 12 n 2008 at about 90 statons across Europe were collected from the European Reference Frame Internet Protocol project (Bruynnx 2004) and the Ordnance Survey of Great Brtan real-tme network (Fg. 1). The Kp ndex, whch quantfes the onosphere-actvty level, was below 4 on average, thus showng a moderate onosphere condton. At these statons, postons were estmated as whte nose at each epoch, and ZTDs were estmated every three hours. To test rapd reconvergences, we ntroduced a total sgnal nterrupton 20 mn after each successful convergence, and thus each staton suffered from about 70 re-convergences per day f there were no large data gaps. After each total sgnal nterrupton, a number of epochs of measurements were removed to smulate a tme latency for the predcted onospherc delays. Ths dataset s qute representatve for the md-lattude regons durng moderate onosphere condtons because t covers 24 h per day and contans recevers workng n dfferent envronments. In addton, the IGS predcted orbts and Earth rotaton parameters were used. The satellte clock and FCB determnaton refer to Geng (2009) and Geng et al. (2010b) (see Fg. 1 Dstrbuton of 1-Hz statons across Europe among whch black sold squares denote the statons used for real-tme satellte clock and FCB determnaton

6 710 J. Geng et al. Fg. 2 RMS of the predcton errors of onospherc delays under dfferent tme wndow wdths whch are labeled alongsde the sold crcles, dfferent tme latences and dfferent elevaton angles: (a) >30 and (b) 30 Fg. 1), and communcaton delays were gnored n ths smulated real-tme study. These also apply to the case study n Sect Predcton error of onospherc delays The predcton error of onospherc delays s subject to both the tme latency and the elevaton angle. In Fg. 2, the onospherc delays at a typcal staton ACOR are dvded nto two groups usng an elevaton angle of 30. The RMS of ther predcton errors for all satellte pars durng 24 h s plotted aganst both the tme latency and the tme wndow wdth. As expected, the RMS s ncreased when the tme latency s prolonged. However, enlargng the wndow wdth can slow down ths ncreasng rate. Because a large wndow wdth degrades the computaton effcency, we fnally choose 120 s as a trade-off for the onospherc delays above 30 and 240 s for those below 30 n ths study. In ths case, the error for a 300-s predcton s only 3.7 cm for the onospherc delays over 30 and 7.3 cm for those below 30. In addton, the predcton error remans steady when the tme latency s shorter than 30 s, thus explanng why dentcal weghts are posed wthn 30 s n Eq Performance of rapd re-convergences Applyng the precsely predcted onospherc delays can sgnfcantly mprove the success rate of rapd re-convergences and shorten the tme spent on these re-convergences. Table 1 shows the statstcs about the rapd re-convergences acheved wthn fve epochs of 1-Hz measurements under dfferent tme latences. When the tme latency s 10 s, 99.4 and 98.7% of re-convergences to wde-lane and narrow-lane ambguty resolutons can be rapdly acheved, respectvely. Even though the tme latency s up to 180 s, these two percentages can stll reach 93.3 and 95.0%. For comparson, f we do not apply the precsely predcted onospherc delays, only 78.9% of all re-convergences can be acheved wthn 20 mnutes wth a mean tme of 694 s. Hence, the mprovement on the re-convergence effcency by ths method s encouragngly sgnfcant. However, when the tme latency s up to 300 s, the above two percentages steadly fall to 85.8 and 89.4%. Ths deteroraton can be understood n terms of the amplfed predctonerror of onospherc delays. From Eq. 8, onospherc delays of large predcton errors wll enlarge the magntude of ζ jk w, hence possbly basng the correspondng wde-lane ambguty estmates and degradng the relablty of nteger resolutons. Furthermore, even though a rapd wde-lane ambguty resoluton can be acheved, a large ζ jk w wll lead to a poor precson of an unambguous measurement by Eq. 9, hence weakenng the resultng constrant on the subsequent narrowlane ambguty resoluton. Fnally, when the tme latency s only 10 s, the falure rates for the wde-lane and narrowlane resolutons account for 0.6 and 1.3%, respectvely, most of whch are caused by low satellte avalablty and poor satellte geometry. On the other hand, Table 1 also shows the mean tmes to resolve wde-lane and narrow-lane ambgutes. When the tme latency s 10 s, only one epoch of data s requred to resolve wde-lane ambgutes, whereas almost no more epochs are requred to resolve narrow-lane ambgutes. Overall, ncreasng the tme latency wll lead to a longer tme spent on achevng the ambguty-fxed soluton. However, these tmes are ncreased moderately or even mnmally, mplyng that most rapd re-convergences are acheved usng only one epoch of measurements, even though the tme latency s up to 300 s. In fact, ths fndng demonstrates that the satellte-geometry change mnmally contrbutes to the rapd re-convergences n ths study.

7 Rapd re-convergences to ambguty-fxed solutons n precse pont postonng 711 Table 1 Performance of rapd re-convergences wthn fve epochs of 1-Hz measurements under dfferent tme latences Tme latency (s) Wde-lane ambguty resoluton Narrow-lane ambguty resoluton Success rate (%) Tme to fx (s) Success rate (%) Tme to fx (s) /36665 (99.4) /36618 (98.7) /35822 (98.4) /35720 (98.6) /34632 (97.5) /34472 (98.2) /32333 (95.9) /32060 (96.7) /30070 (93.3) /29500 (95.0) /28024 (89.7) /27088 (92.5) /26137 (85.8) /24465 (89.4) 0.15 A success rate s the rato (wthn parentheses) between the number (before slashes) of rapd resolutons and the total number (behnd slashes) of resolutons for the wde-lane or narrow-lane ambgutes. A tme to fx s the real-valued mean of all tmes spent on resolvng the wde-lane or narrow-lane ambgutes. Note that one second here represents one epoch of measurements, and the tme to fx a narrow-lane ambguty begns to be measured when the correspondng wde-lane ambguty s successfully resolved 4 A case study at a moble van In ths secton, we assess ths method at a van movng n a GPS-adverse envronment, then nterpolate onospherc delays based on a dense reference network and fnally compare the results wth those from a commercal NRTK servce. 4.1 Data processng A van-borne 1-Hz GPS data set was collected n the cty of Nottngham on May 11, 2009, coverng about 2.5 h from 11:30 to 14:04 (Local tme). The Kp ndex was around 1 durng ths perod, showng a quet onosphere condton. As shown n Fg. 3, the van stopped or moved n four phases: (1) t stopped from 11:40 to 12:27 on a car park wth an open sky vew; (2) t moved from 12:27 to 13:12 along the red routes escorted by tall buldngs, large trucks and thck trees; (3) t stopped agan from 13:12 to 13:44 on the car park; (4) fnally, t moved agan from 13:44 to 14:04 along the blue routes on the car park wth a few thck trees around. A reference staton was located on top of the IESSG buldng whch was wthn 2 km from the van. Moreover, ths reference staton was surrounded by other sx reference statons wthn a dstance from about km (upper nset). They were used to nterpolate onospherc and tropospherc delays. In addton, the lower nset shows the satellte number whch kept steady durng the stoppng phases, but changed dramatcally durng the movng phases due to sgnal obstructons. At ths van, because the antenna s atttudes durng the movng phases were unknown, we appled the phase wndup correctons by assumng that the antenna s rotaton was always around ts boresght. In ths case, the phase wnd-up effects caused by ths rotaton can be fully assmlated nto the recever clock wthout affectng the correcton proposed by Wu et al. (1993)(Banvlle and Langley 2009). Moreover, no antenna phase center correctons were appled. We estmated Fg. 3 Van trajectory consstng of four phases: (1) t stopped on the car park wth an open sky vew; (2) t moved along the red routes escorted by tall buldngs, large trucks and thck trees; (3) t stopped agan on the car park; (4) t moved along the blue routes on the car park wth a few thck trees around. A reference staton was located at IESSG. The upper nset shows the dstances between sx reference statons and IESSG, whereas the lower nset shows the satellte number and the tme spans for the four phases. LT denotes local tme only one ZTD durng ths experment because the tme span was short and the area was small. A ground truth was computed usng the Leca Geo Offce software of verson 3.0 and the IESSG staton was used as reference. We beleve that the

8 712 J. Geng et al. Fg. 4 Poston dfferences from the ground truth that s derved from a post-processed short-baselne soluton for the East, North and Up components. From top to bottom show the ambguty-float soluton, the ambguty-fxed solutons wthout and wth rapd re-convergences (RRC), the ambguty-fxed soluton supported by a dense network (network PPP) and the network RTK soluton. The symbol denotes the tme when the ambguty-fxed soluton s acheved, whereas denotes the tme when the ambguty-fxed soluton s totally lost 3D accuracy of ths ground truth should be better than a few centmeters. In addton, an NRTK soluton provded by the Leca SmartNet servce was also collected for comparson. 4.2 Rapd convergences n a GPS-adverse envronment In a GPS-adverse envronment, frequent and severe sgnal obstructons can sgnfcantly jeopardze the postonng performance and the ambguty resoluton. In Fg. 4, for the ambguty-float solutons, a clear trend s present for the East component and a bas for the North component. Durng the frst movng phase when sgnals were frequently obstructed, all three components, especally vertcal, experence large poston errors of over a few decmeters. If the ambgutes are resolved, the postonng accuracy can be clearly mproved for all three components, but ths mprovement s present only durng the stoppng phases as shown n Fg. 4. Ambgutyfxed solutons were quckly and totally lost after the movng phases began. Ths s because re-convergences occurred due to the sgnal obstructons. Overall, the rate of ambgutyfxed epochs s only 7.7% of all epochs durng the movng phases. Nevertheless, by further applyng our method for rapd re-convergences, ambguty-fxed solutons can be well mantaned durng the movng phases. From Fg. 4, the accuraces of both horzontal components durng the movng phases are better than 5 cm, whereas that of vertcal s better than 10 cm. Large poston errors are present at only a few epochs where ambguty-fxed solutons cannot be acheved. More encouragngly, the rate of ambguty-fxed epochs s sgnfcantly ncreased to 93.6% of all epochs durng the movng phases, showng a sgnfcant mprovement over the ambguty-fxed solutons wthout rapd re-convergences. Nevertheless, the frst ambguty-fxed soluton s acheved after a few tens of mnutes. To solve ths problem, the sx surroundng reference statons were used to nterpolate the

9 Rapd re-convergences to ambguty-fxed solutons n precse pont postonng 713 onospherc delays at ths van. As a result, only fve epochs of 1-Hz measurements are used to acheve the frst ambguty-fxed soluton. Moreover, the epochs wth large poston errors become much fewer. Actually, the rate of ambgutyfxed epochs reaches up to 97.2% of all epochs durng the movng phases. Ths further mprovement s largely attrbuted to the nterpolated onospherc delays whch are more relable than the predcted ones. 4.3 Comparsons wth an NRTK soluton Normally, a NRTK servce can generate centmeter-level postonng accuracy usng double-dfference measurements based on an RTK network. At the bottom of Fg. 4,wepresent the poston dfferences of an NRTK soluton aganst the ground truth. Note that the gaps are caused by the unavalablty of NRTK correcton messages. Large poston errors occur from 13:40 to 13:50, and thus are excluded from the followng statstcs. It can be clearly seen that bases are present for all three components, especally for the Up one. We beleve that these bases are largely caused by the antenna phase center correctons appled to the NRTK soluton. Dsregardng these dscrepances, we fnd that the ambguty-fxed solutons enhanced by rapd convergences perform comparably wth the NRTK soluton. Specfcally, the standard devatons of the dfferences for the East, North and Up components durng the movng phases are 1.0, 1.3 and 2.7 cm for the ambguty-fxed solutons wth rapd re-convergences, respectvely, whereas 0.6, 1.0 and 1.4 cm for the NRTK soluton. Nonetheless, from Fg. 4, the poston dfferences durng the movng phases for the NRTK soluton scatter clearly more tghtly than those of the ambguty-fxed PPP solutons enhanced by rapd convergences. On the contrary, the poston dfferences durng the stoppng phases for all these solutons scatter comparably. Ths hence demonstrates that the nternal consstency between the ground truth and the NRTK soluton due to the software smlarty cannot adequately explan the dstnct scatters durng the movng phases. Therefore, we argue that ths ssue mght be because the phase wnd-up correctons cannot be precsely computed durng the movng phases, snce the antenna s true atttudes were unknown. 5 Conclusons and perspectve In ths study, we develop a method where the onospherc delays are predcted to the succeedng epochs to accelerate the ambguty resoluton n case of re-convergences. In theory, the effectveness of ths method conssts n two aspects: on the one hand, wde-lane ambgutes can be rapdly resolved usng the onosphere-corrected wde-lane measurements, nstead of the nosy Melbourne Wübbena combnaton measurements; on the other hand, narrow-lane ambgutes can be rapdly resolved under the tght constrants derved from the onosphere-corrected unambguous wde-lane measurements. Ths method s verfed usng two tests under moderate onosphere condtons. At statc statons sufferng from smulated total losses of trackng locks, even f the tme latency for the predcted onospherc delays s up to 180 s, 93.3 and 95.0% of re-convergences to wde-lane and narrow-lane ambguty resolutons can be acheved wthn fve epochs of 1-Hz measurements, respectvely. If ths tme latency s prolonged, the predcton error of onospherc delays s ncreased, consequently reducng both percentages above. On the other hand, for most rapd re-convergences, only one epoch of measurements s needed, mplyng that the contrbuton of satellte-geometry changes to these rapd re-convergences s mnmal n ths study. At a moble van movng n a GPS-adverse envronment, ambguty-fxed solutons can be well mantaned when rapd re-convergences are appled. The rate of ambguty-fxed epochs s sgnfcantly mproved from 7.7 to 93.6% of all epochs durng the movng phases. Addtonally, the RMS of poston dfferences from the ground truth that s derved from a post-processed shortbaselne soluton are 1.0, 1.3 and 2.7 cm for the East, North and Up components, respectvely, whch are comparable wth those of the NRTK soluton. Fnally, the onospherc delays can also be nterpolated from a dense network of reference statons. Although such networks are normally not avalable for PPP, we can envson that f precse onospherc delays can be derved from a sparse network at scales of several hundred klometers usng an onospherc tomography technque (e.g. Hernández- Pajares et al. 2000), a PPP-RTK servce (Geng et al. 2010b) that can rapdly provde ambguty-fxed solutons may potentally preval aganst current RTK postonng servces based on dense networks. Acknowledgments Ths study s based on an mproved Postonng and Navgaton Data Analyst (PANDA) software package whch was orgnally developed by Wuhan Unversty n Chna (Sh et al. 2008). 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