Some remrks on GNSS integer mbiguity vlidtion methods T. Li nd J. Wng* Ambiguity resolution is n indispensble step in fst nd high precision Globl Nvigtion Stellite System (GNSS) bsed positioning. In generl, mbiguity resolution consists of three steps. The first step is to estimte the mbiguities using lest-squres estimtion process, from which the so clled flot solution or rel vlued solution is obtined. Then in the second step, the flot solution is used to serch for the integer mbiguities. Once integer mbiguities re resolved, the lst step is to pply the integer mbiguities into the models so tht firly ccurte fixed solution cn be generted. Owing to the importnce of the integer mbiguities, one indispensble procedure tht needs to be implemented in the second step is integer mbiguity vlidtion. Over the pst decdes, considerble work hs been concentrted on this procedure nd vrious pproches hve been proposed, such s R-rtio test, F-rtio test, W-rtio test, integer perture estimtor, etc. However, their performnces re controversil. Therefore, in this contribution, n overview of the existing mbiguity vlidtion methods is firstly presented, nd then some numericl nlysis is crried out to evlute their performnces. Keywords: GNSS, Integer lest-squres, Ambiguity vlidtion, Integer perture, Rtio test Introduction In Globl Nvigtion Stellite System (GNSS) positioning, crrier phse mesurements re more precise thn the pseudornges. However, one troublesome problem with the crrier phse mesurements is tht ech crrier phse contins n mbiguity in the number of wvelengths, which hs n intrinsic nture of being integer. Consequently, the most importnt issue in precise GNSS positioning is tht integer mbiguities hve to be resolved correctly, nd once integer mbiguities re obtined, the precision of crrier phses cn be fully exploited in positioning. The whole process of mbiguity resolution includes three steps. Initilly, with n pproprite mthemticl model, the flot solution (or rel vlued solution) cn be estimted by lest-squres or Klmn filter. Then, serching strtegy will be crried out round the flot solution so s to find the integer cndidtes [3]. As there could be severl sets of integer mbiguity cndidtes, vlidtion procedure is required, nd the most likely cndidte should be vlidted nd seprted from the others bsed on sttisticl theory or other methods. To fix the rel vlued mbiguities to the integer cndidtes, in this pper, LAMBDA [12], [13], [14], ws dopted. The lst step is to estimte the unknown prmeters with the correct integer mbiguities. School of Surveying nd Sptil Informtion Systems, University of New South Wles, NSW 2052, Sydney, Austrli *Corresponding uthor, emil jinling.wng@unsw.edu.u For the mbiguity vlidtion step, severl procedures hve been developed, with the qudrtic form of residuls ssocited with the most likely set of integer mbiguities, nd the second qudrtic form of residuls ssocited with the second most likely set of integer mbiguities, such s F-rtio [5], R-rtio [4], W-rtio [24], difference test [23], projector test [7]. With given criticl vlue, the best cndidte will be identified. In such mbiguity vlidtion procedures, the fixed mbiguities re treted s constnt vlues. In nother pproch to mbiguity vlidtion, integer perture (IA) estimtor [18] hs been developed. In [28], the IA estimtor ws considered s frmework for ll other clssicl theory of hypothesis testing methods, nd the geometries of different vlidtion methods re then reflected in the perture pull-in region. Following the definition, mbiguity vlidtion is crried out by determining the criticl vlues bsed on the pre-defined filrte, which sounds more resonble. However, s commented in [9], the theory of the IA estimtor ssumes the most likely set of cndidtes re correct (or true integer vlues) in the designing stge. Through numericl tests, it is found tht determining the criticl vlues ccording to the IA estimtor is not tht strightforwrd, especilly in rel pplictions, which shows tht mbiguity vlidtion by the IA estimtor hs some limittions. In this contribution, the performnces of different mbiguity vlidtion methods hve been studied nd nlysed. This pper firstly strts with introducing the mthemticl models, nd then in the following section, n overview of these vlidtion methods s well s some discussion is presented. After tht, ccording to numericl nlysis, the performnce of different mbiguity ß 2012 Survey Review Ltd Received 5 December 2011; ccepted 5 December 2011 230 Survey Review 2012 VOL 44 NO 326 DOI 10.1179/1752270611Y.0000000027
vlidtion methods nd the limittion of IA theory re extensively nlysed nd discussed. Some conclusions nd suggestions re given in the finl section. Mthemticl models The initil GNSS observtion mthemticl models re constructed with unknown prmeters such s coordintes, integer mbiguities, ionosphere dely, troposphere dely, etc. Ionosphere dely nd troposphere dely certinly hve impcts on the ccurcy, nd in order to eliminte those unnecessry fctors, double differenced functionl models re employed s follows D+w~ 1 l D+rzD+Nze w (1) D+P~D+rze P (2) where D= is the double differencing opertor between stellites nd receivers, w nd P re crrier phse mesurements nd code mesurements respectively, l is the crrier phse wvelength, r is the geometric distnce between stellites nd receivers, N is the integer mbiguities in cycles nd e represents the mesurement errors. With n pproximte rover position given, the bove models re linerised s l~axzv (3) where ~ D+w{ 1 T l D+r,D+P{D+r T,, v~ e w,e P x~ ðx r,d+nþ T, A is the design mtrix of both coordintes A xr nd mbiguities A, r* is the pproximte distnce, nd x r is the coordintes. Thus, the stochstic model for the double differenced mesurements cn be expressed s Dy ð Þ~s 2 0 Q~s2 0 P{1 (4) where D is the covrince mtrix, s 2 0 is priori vrince fctor, nd Q nd P re the cofctor mtrix nd weight mtrix of the mesurements, respectively. By pplying the clssicl lest-squres pproch, whose criterion is v T Pv~min, the unknown prmeters nd their covrince re uniquely estimted s [25] T~ x ~ x r A T {1A PA T Pl Q~ x A T PA " # {1~ Q x Qr x r Q x r Q where represents the rel vlued solution of integer mbiguities. Then the posteriori vrince is 2 TPv s0 ~v ~ V 0 f f with f is the degree of freedom. An overview of existing mbiguity vlidtion methods The unknown prmeters estimted from norml lest-squres procedure re the so clled the flot solution. Considering the integer nture of the mbiguities, n integer lest-squres (ILS) problem needs to be tken into ccount, e.g. [8], [12], [16]. In this pper, (5) (6) LAMBDA is used to serch the integer mbiguities within the constructed hyper-ellipsoid. The remining problem is tht the best integer cndidte should be selected from ll the combintions of integer cndidtes, nd trditionlly, it is ccomplished by investigting the qudrtic forms of residuls. Another proposed theory is the integer perture estimtion, which hs hybrid nture of yielding integer outcomes s well s noninteger outcomes. This pproch llows users to control the pre-defined fil-rte by themselves, yet its ppliction in prctice requires more numericl tests nd nlysis. Bsed on the estimtion from lest-squres, mbiguity vlidtion will strt from, 2 s 0 nd Q, nd we ssume tht the flot solution of the mbiguity is normlly distributed s N,Q ð Þ. The qudrtic form of residuls for flot solution is define s V 0, while the qudrtic form of the fixed solution residuls is V, while TQ R~ {1 {^ {^ mens the geometric distnce between V 0 nd V [25]. We ssume V 1 nd V 2 re the minimum nd second minimum qudrtic forms of the residuls in the mbiguity fixed solutions, which correspond to ă 1, ă 2 nd R 1, R 2, then mbiguity vlidtion methods cn be generlly divided into two min groups [9]. Sttisticl tests bsed on the best nd second best mbiguity cndidtes Trditionlly, mbiguity vlidtion methods consider both the best nd second best mbiguity cndidtes nd the reltionship between them re nlysed from sttistic point of view. Most commonly used methods re the F- rtio, R-rtio, W-rtio, difference test nd projector test, nd their detiled definitions re expressed s follows: Rtio tests The first pproch proposed for mbiguity vlidtion is the F-rtio test [5], [27] (here inverse version is used for nlysis), which is the rtio between V 1 nd V 2 F~ V 1 ~ V 0zR 1 (7) V 2 V 0 zr 2 Normlly, the test sttistic for F-rtio is ssumed to hve n F-distribution, with degrees of freedom for V 1 nd V 2 respectively. Then, criticl vlue c could be obtined by specifying significnce level. However, ccording to, e.g. [24] nd [15], the numertor nd denomintor re not independent, nd F-rtio does not follow n F-distribution. Sometimes, there re lrge discrepncies between the obtined results nd their true vlues, which clerly show tht F-rtio is tht not relible, even though, certin empiricl vlues, e.g. 1?5 (inversed 0?67), 2 (inversed 0?5), 3 (inversed 0?3), perform stisfctorily [2], [10]. An lterntive populr test similr to the F-rtio test is the rtio between R 1 nd R 2, known s the R-rtio [4], [27]. The inverse of R-rtio is constructed s R~ R 1 ƒc (8) R 2 As described in the bove formul, the distribution of R- rtio is lso unknown. Hence, it would be incorrect to determine the performnce of the rtio test on the bsis of the distributionl results s provided by the clssicl theory of hypothesis testing [22], [28]. Theoreticlly, the Survey Review 2012 VOL 44 NO 326 231
chosen of criticl vlue is groundless by n empiricl vlue. In the originl work of [4], the criticl vlues rnges from 5 to 10 were given. However, in prctice, mny reserchers vlidte the mbiguities bsed on the R-rtio, with criticl vlues like 2 (inversed 0?5), 2?5 (inversed 0?4), see e.g. [6], [11], [22]. In [24], the discrimintion procedure is constructed either by compring the likelihood of two integer cndidtes or by rtificilly nesting the two compred models by nesting prmeter, nd then the W-rtio is defined s follows d W~ c (9) 1=2 ½vrðdÞŠ where d~v 2 {V 1, vrðdþ~r 2 Q d (10) where r 2 could be decided by users either from the priori vrince s 2 0 or from the posteriori vrince 2 s 0. By pplying the vrince covrince propgtion lw, the vrince for TQ {1 d could be Q d ~4 ^ 2{^1 ^ 2{^1. Assuming tht W nd W s re the two rtios corresponding to priori vrince s 2 0 nd posteriori vrince 2 s 0,they re supposed to hve truncted stndrd norml distribution nd truncted student t-distribution respectively, from which the criticl vlues could be esily obtined. Under the ssumption tht the fixed mbiguities re deterministic quntities, the W-rtios cn provide rigorous confidence level for the vlidtion test. Difference test In [23], the vlidtion pproch is to nlyse the difference between R 1 nd R 2. First, globl model test is crried out. When it pssed, the mbiguity could be vlidted by compring the difference test with its criticl vlue s follows R 2 {R 1 c (11) where c is non-negtive sclr, which is user defined tolernce level. This test ccepts the best solution if the flot solution is much closer to the best solution thn to the second best solution. Criticl vlues of 15 nd 12 were suggested in [23] nd [7]. Still, the determintion of the criticl vlue, which is lso empiricl, is controversil, s the distribution of the difference test is unknown. A looser setting of criticl vlues increses the performnce, but lso increses the number of wrongly ccepted cndidtes (type I errors). On the other hnd, the difference sttistic is dependent on the priori vrince fctor given in eqution (4). Projector test The projector test is bstrcted from [1], nd proposed in [7], lso deducted in [24], the null hypothesis is tht there is no outlier so ^1 is ccepted, while the lterntive hypothesis is tht there is n outlier in the direction of (^ 2{^1); consequently, we hve y~a ^ 1zA ^ 2{^1 cza xr x rzv With quntity c nd its vrince s TQ ^2{^ 1 {1 ^{^1 c^~ TQ, s^2c ~ 4V 1{c 2 Qd (12) {1 ðn{4þq d ^ 2{^1 ^ 2{^1 Then the definition of the test sttistic is c =s ~t(n24) c (in this contribution, only bseline components re considered) if ^ 2{^ 1 is selected. If ^2 is selected, the test sttistic is then non-centrl t-distribution. The quntity hs been proved to be lwys smller thn or equl to 0?5, so tht the definition nd the distribution re ctully not strict s well. Besides, s discussed in [24], the derivtion is unfortuntely not rigorous becuse the non-centrl prmeter is formulted with the estimted vrince insted of its known vlue. So fr, there hs been no criterion identifying which of the mbiguity vlidtion methods proposed is the best, since the criticl vlues re chosen differently for different distributions. Their performnces should be evluted nd tested through vrious mens, including theoreticl nlysis nd evlution with the ground truth for the integer mbiguities. Integer perture theory On the bsis of ILS estimtor, IA theory ws firstly introduced by [18], nd the IA estimtor - is developed s -~ X ( zv z z 1{ X h i ) v z (13) z [ Z n z [ Z n With the indictor function v z (x) defined s v z ðxþ~ 1 if x[v z (14) 0 otherwise where the V z re the perture pull-in regions, nd their union V5R n is the perture spce, which lso hs property of trnsltionl invrint. With the bove definition, three outcomes cn be distinguished s [V Success: correct integer estimtion [V=V Filure: incorrect integer estimtion [/ V Undecided: mbiguity not fixed to n integer The corresponding probbilities of success (P s ), filure (P f ) nd undecided (P u ) re given by ð P s ~Pð-~ Þ~ ðxþdx ð P f ~ V=V f P u ~1{P s {P f V f ðxþdx (15) In the cse of GNSS model, f ðxþ represents the probbility density function of the flot mbiguities, nd is usully ssumed to be normlly distributed. The IA estimtor llows the user to choose predefined fil-rte, nd then determine the criticl vlue ccordingly. With the determintion of the criticl vlue, the user is cpble of compring different vlidtion methods by clculting the corresponding success rtes. Ellipsoidl integer perture A rther strightforwrd wy of determining the IA pull-in region is the ellipsoidl integer perture (EIA) [19], which cn be constructed s E z ~E 0 zz, E 0 ~S 0 \C e,0, Vz[Z n (16) With n S 0 being the lest-squres o pull-in region nd C e,0 ~ x[r n 2 jjjxjj Q ƒc 2, n origin centred ellipsoidl 232 Survey Review 2012 VOL 44 NO 326
region of which the size is controlled by the perture prmeter (or criticl vlue, in the sequel, criticl vlue is used insted) e with kxk 2 Q ~x T Q {1 x The structure of the EIA pull-in region is considered to hve Chi-squre distribution, nd thus the EIA probbilities of success, filure, nd undecided re given s P f ~ X P x 2 ðn,l z Þƒc 2 z [ Z n = f0g P s ~P x 2 ðn,0þƒc 2 P u ~1{P f {P s (17) where x 2 ðn,l z Þ represents rndom vrible hving non-centrl Chi-squre distribution, nd n nd l z re the degree of freedom nd the non-centrl prmeter l z ~z T Q {1 z respectively. Optiml integer perture Optiml integer perture (OIA), which ims t the mximistion of the success rte, is proposed by [20], [28] s mx P s subject to : P f ~b with the pull-in region V 0 5S 0, nd b is the pre-defined fil-rte. The relistion of OIA could be considered s the rtio between the PDF of the mbiguity residul nd the PDF of the mbiguities s P f exp { ð1=2þe^zz 2 e^ e^ z [ Z ~ n Q ƒc (18) f e^ exp { ð1=2þe^ 2 Q with e^ ~ {^. According to the definition of OIA, it hs the dvntge of obtining the highest success rte. Wheres its ppliction in relity is quite time consuming, this cn be known from the PDF of the mbiguity residul. In [17], n exct formul of f e^ ðxþ is derived, nd in order to clculte f e^ ðxþ, summtion over infinite integers should be crried out. This is, however, impossible to operte in prctice. As consequence, severl finite integers re serched within n ellipsoidl spce, nd pplied [29]. In the next section, we will discuss the following two questions through numericl nlysis: (1) how to compre different vlidtion methods; nd (2) how to determine criticl vlues from given fil-rte for different vlidtion methods. Numericl nlysis According to the bove mentioned sttements, different mbiguity vlidtion pproches re hrd to compre directly, s the different chosen criticl vlues. However, under the frmework of the IA estimtion, this is possible if pre-defined fil-rte is provided. In this section, the comprison of different vlidtion methods is firstly studied by simultion, nd then the limittions of IA theory re discussed. In ddition, the performnce 1 Determintion of criticl vlues with given fil-rte of both IA nd non-ia bsed methods re nlysed in rel pplictions. Comprison of different vlidtion methods With the sme pre-defined fil-rte given, one is ble to select the criticl vlue for different vlidtion methods, nd then determine their corresponding success rtes. By compring their success-rtes, we re ble to figure out which method my hve the best performnce. Detiled procedures re shown s Fig. 1. The geometry informtion of n existing GNSS model is entirely reflected in Q, so Monte Crlo simultions were used to investigte the performnce of different vlidtion methods with the informtion of Q given. By following Fig. 1, rndom genertor ws used to simulte number of smples (e.g., 50 000) of the estimted flot mbiguities. Here, bivrite norml distribution with the following vrince covrince mtrix is considered first [28] Q~ ½0 : 1021, {0 : 0364; {0 : 0364, 0 : 1100Š; Survey Review 2012 VOL 44 NO 326 233
The reltionships between the fil-rte nd the criticl vlue for ech of the mbiguity vlidtion methods re lwys on one to one bsis, which implies tht the filrte chnges with different chosen criticl vlues for ech vlidtion method. The corresponding success rtes obtined with given fil-rte re shown in Tble 1. It is noted tht the OIA, which hs the highest success rte, performs the best, no mtter how big the fil-rte is. In this cse, the R-rtio, W -rtio nd difference test, which re ll treted s seprted IA method, hve performnce quite close to tht of the optiml IA estimtor. When the fil-rte increses from 0?005 to 0?02, these three tests hve quite similr success rte, but it is impossible to judge which one is the best mong these three. The reson is tht the smple size of simultion will hve some impct on the results. Such impcts re lrge enough to cuse slight differences in the success rtes. As for the projector test s n IA method, its success rte is smller thn the bove three. For the EIA, it is esy to mnipulte. However, its performnce is not tht good. In [9], combintion of R-rtio nd overlpped EIA ws proposed. To void the conservtiveness of the EIA, the EIA is llowed to be overlpped to increse the success rte, nd for the purpose of ensuring tht the best mbiguity cndidte is sttisticlly better thn the others, the R-rtio is combined with the overlpped EIA. Theoreticlly, however, the definition of the EIA is not overlpped, nd the overlpped EIA increses the fil-rte s well. Aprt from this, the combintion of the EIA nd the R-rtio mkes the success rte decrese, which is not the purpose of the overlpped EIA. F-rtio s n IA method performs slightly worse thn the other vlidtion methods. The reson could be tht, s shown in eqution (7), nd lso commented in [28], both flot solution residuls nd mbiguity residuls re considered, so tht the dimension of perture spce is different. Tble 2 presents four-dimensionl rel dt exmple, with the vrince-covrince mtrix s follows: Q~ ½1:1834, {1:3348, 0:2814, {0:6537; {1:3348, 1:6088, {0:2619, 0:8123; 0:2814, {0:2619, 0:0975, {0:1139; {0:6537, 0:8123, {0:1139, 0:4177Š Tble 1 Two-dimensionl cse for success rtes nd criticl vlues with the pre-defined-fil rte P f FTIA RTIA WTIA DTIA PTIA EIA OIA 0. 005 c 0. 237 0. 043 1. 148 7. 447 0. 494 0. 568 1. 047 P s 0. 146 0. 150 0. 150 0. 151 0. 148 0. 149 0. 162 0. 01 c 0. 288 0. 084 0. 981 6. 424 0. 594 0. 748 1. 066 P s 0. 244 0. 252 0. 250 0. 247 0. 246 0. 244 0. 253 0. 02 c 0. 355 0. 146 0. 801 5. 230 0. 835 0. 940 1. 112 P s 0. 353 0. 361 0. 358 0. 359 0. 355 0. 356 0. 363 For the fil-rte of 0?001, the OIA still yields the highest success rte, nd the difference test, in this cse, hs much better performnce thn the others. When the filrte is 0?01, the W-rtio test is more preferble thn ny other non-ia vlidtion methods. Menwhile, in both cses, the EIA hs the worst performnce. It should be mentioned tht for four-dimensionl simultion, it consumes hours of computtion for ech method when the user choose the Mtlb build-in function fzero ( good tool to find zero of function ner roughly given criticl vlue) to run the simultion. The numericl results revel three issues: (1) the OIA in these two cses performs the best in the simultions, but its efficiency is the worst mong ll the vlidtion methods, which indictes tht this method is hrdly pplicble in rel time GNSS positioning; (2) The higher the dimension, the more time the simultions will consume; (3) using the function fzero to serch for the criticl vlue, the pre-defined fil-rte should be resonbly determined by the user, since for certin stellite geometry, there is n upper bound for the fil-rte. Limittions of the IA methods: pre-defined filrte versus the criticl vlue The integer perture theory llows the user to determine the criticl vlues bsed on the user preferred fil-rte. Theoreticlly, once the stellite geometry is obtined, the reltionship between the pre-defined fil-rte nd the criticl vlue hs lredy been determined, which mens tht when fil-rte is pre-defined, the criticl vlue will be uniquely fixed under the current stellite geometry. However, the problem is tht their reltionship cnnot be mthemticlly represented; only simultions re fesible. Thus the wy of clculting criticl vlue with pre-defined fil-rte will be entirely dependent on simultions. In order to generte relible results, the number of smples should be s lrge s possible. As suggested in [19], [28], desirble smple size N should be lrger thn 5000. In order to show the influence of the smple size N, two simultions were crried out to compre with the vlues given in [28], note tht the sme Q 5[0?0865, 20?0364; 20?0364, 0?0847] ws utilised. First, the smple size ws incresed. As listed in Tble 3, we utilised the sme criticl vlue, nd the Tble 3 Comprison of R-rtio s n IA estimtor N c P f P s Ref. 28 0. 035 0. 001 0. 168 5000 0. 035 0. 001400 0. 165600 10 000 0. 035 0. 000700 0. 172000 50 000 0. 035 0. 000480 0. 170780 100 000 0. 035 0. 001260 0. 166340 500 000 0. 035 0. 001342 0. 167228 5 000 000 0. 035 0. 000733 0. 171139 Tble 2 Four-dimensionl cse for success-rtes nd criticl vlues with the pre-defined fil-rtes P f FTIA RTIA WTIA DTIA PTIA EIA OIA 0. 001 c 0. 9815 0. 3884 0. 8316 7. 9241 1. 6938 2. 1351 1. 0193 P s 0. 8978 0. 8803 0. 9015 0. 9071 0. 9012 0. 6644 0. 9072 0. 01 c 0. 9981 0. 9020 0. 0867 1. 0887 2. 3774 2. 7963 1. 5860 P s 0. 9831 0. 9834 0. 9835 0. 9825 0. 9832 0. 9016 0. 9836 234 Survey Review 2012 VOL 44 NO 326
2 Fil-rte chnges with both smple size nd number of simultion for R-rtio simulted results fluctuted ner the given vlue. Even if the smple size is constnt, s shown in Tble 4, the filrte nd success rte re not stble. The smple size N undoubtedly influences the fil-rte with the criticl vlue known (or criticl vlue with the fil-rte given). Hence, the smple size N, to some extent, determines the relibility of Integer Aperture theory. As shown in Figs. 2 4, when the smple size increses from 5000 to 50 000, the fil-rte nd the success rte re becoming smoother. This tells us tht the lrger the smple size, the more relible results we cn hve. However, s routine, the simulted results my be problemtic if the ccurcy of the criticl vlue is importnt. According to numericl nlysis, smple size lrger thn 50 000 will hve more relible solution. However, with powerful PC, such simultions (given the criticl vlue to clculte the fil-rte) for one solution will tke over 1?5 s nd thus hrdly be usble for rel time opertions. Limittions of the IA methods in rel pplictions However, the problem is tht incresing the smple size ggrvtes the computtion burden, which is troublesome problem in kinemtic positioning. The wy of determining the criticl vlue more rpidly nd relibly bsed on the pre-defined fil-rte is perhps nother mjor issue. A fesible wy could be using look-up tble (here only for R-rtio), which ws creted in [21], 3 Fil-rte chnges with both smple size nd number of simultion for W -rtio [22]. The look-up tble is generted bsed on the lower bound of ILS fil-rte (expressed by the fil-rte of integer bootstrpping), nd then pply liner interpoltion to find the corresponding criticl vlue. In order to nlyse the performnce of the look-up tble, rel dt were collected t smpling rte of 1 s on 12 September 2009, in Sydney, Austrli, nd then processed by different session lengths. With single frequency crrier phse mesurements nd n elevtion ngle of 15u, five stellites were used to execute the double difference so tht four mbiguities need to be resolved. As shown in Tble 5, it is obvious tht, by determining the criticl vlue c from the look-up tble, when the session length is longer thn 3 min, the resolved mbiguities were ccepted s undoubtedly correct. However, the truth is tht correct mbiguities (given by processing the whole dtset) could only be obtined when the session length is longer thn 6 min. This tells us tht using look-up tble to determine the criticl vlue is not verstile, nd fixed Tble 4 Ten simultions of R-rtio with the smple size N c P f P s Ref. 28 0. 035 0. 001 0. 168 5 000 000 0. 035 0. 001099 0. 167604 5 000 000 0. 035 0. 001221 0. 170165 5 000 000 0. 035 0. 000977 0. 168333 5 000 000 0. 035 0. 000833 0. 169142 5 000 000 0. 035 0. 001020 0. 166257 5 000 000 0. 035 0. 000976 0. 168333 5 000 000 0. 035 0. 001026 0. 166258 5 000 000 0. 035 0. 001120 0. 166259 5 000 000 0. 035 0. 001143 0. 167235 5 000 000 0. 035 0. 000932 0. 170143 4 Fil-rte chnges with both smple size nd number of simultion for difference test Survey Review 2012 VOL 44 NO 326 235
5 W-rtio (W nd W s ) vlue chnges with the epoch number 6 R-rtio nd F-rtio vlues chnge with epoch number look-up tble sometimes does not properly reflect the chnging of stellite geometry, even though it is more efficient to operte the simultions. In Tble 6, the differences between the look-up tble nd simultion re given. With the criticl vlues from the look-up tble, the simulted fil-rtes re not quite close to 0?001 nd 0?01 respectively, with notble error of 10 20%, nd in the cse of the sme fil-rtes given, the simulted criticl vlues lso hve some discrepncies with the look-up tble results. Note tht the Mtlb build-in function fzero ws utilised here to find the corresponding criticl vlues more efficiently nd rigorously (compred with the repets of the simultion). However, n even more noticeble phenomenon is neither using look-up tble nor simultion gives the correct decision in this cse (type I error). Performnces of both the IA nd non-ia bsed methods in rel ppliction Another rel dtset ws collected on 9 June 2010, in Sydney, Austrli. A session length of 10 min (smpling rte: 1 s) ws used here to briefly show the performnce of both trditionlly used methods to determine the criticl vlue nd using simultions bsed on the IA theory. Only single frequency code nd crrier phse were utilised, nd fter double differencing, there re Tble 5 Ambiguity vlidtion by using the look-up tbles to determine the criticl vlues 7 Difference nd projector tests vlues chnge with epoch number Tble 6 An insight into the determintion of criticl vlues from simultions nd look-up tble 0. 75 min c50. 068 c50. 233 P f 50. 001 P f 50. 01 P f 50. 0009 P f 50. 0087 c50. 076 c50. 248 1 min c50. 236 c50. 611 P f 50. 001 P f 50. 01 P f 50. 0008 P f 50. 0095 c50. 283 c50. 652 Integer mbiguity P f 50. 001 P f 50. 01 Session length/min sv17 sv3 sv18 sv21 RTIA c Criticl vlue c 01 P s Criticl vlue c 02 P s Vlidtion results* 0. 75 217 41 18 225 0. 899 0. 068 0. 034 0. 233 0. 247 CR 1 235 222 223 216 0. 947 0. 236 0. 569 0. 611 0. 914 CR 3 231 227 222 218 0. 267 1 0. 969 1 1. 0 WA 5 231 227 222 218 0. 439 1 1. 0 1 1. 0 WA 6 231 227 222 218 0. 980 1 1. 0 1 1. 0 WA 7 236 221 223 215 0. 525 1 1. 0 1 1. 0 CA 8 236 221 223 215 0. 391 1 1. 0 1 1. 0 CA 15 236 221 223 215 0. 009 1 1. 0 1 1. 0 CA *CR5correctly rejected, WA5wrongly ccepted, CA5correctly ccepted. 236 Survey Review 2012 VOL 44 NO 326
seven mbiguities to be estimted with the degree of freedom s 4, nd the dt were processed on n epoch by epoch bsis. Figures 5 7 show the rtio vlues between the best solution nd the second best solution. As shown in Tble 7, F-rtio dopts commonly used criticl vlue 0?67, which yields 560 ccepted epochs out of the totl 600 epochs. The R-rtio ccepts 506 epochs with n empiricl criticl vlue of 0?5. For the W -rtio nd W s -rtio here, with the truncted norml distribution nd the truncted Student s t-distribution respectively, by significnce level, the corresponding criticl vlue cn be generted. Five hundred nd sixty-eight epochs were ccepted for W with confidence level of 0?99. For the W s, confidence level of 0?99 with degree of freedom s 4 llows 438 epochs ccepted. For the difference test, if n empiricl vlue of 15 ws chosen, it is too conservtive to operte nd the ccepted number of epoch is zero. The empiricl vlue suggested in [7] is lso too conservtive in this cse. When specifying criticl vlue of 0?87 for the projector test [28], 412 epochs were ccepted. However, mong the 600 best mbiguity combintions, the totl numbers of correct mbiguities re 597, nd for the W-rtio test, nd the projector test, the three sets of wrong mbiguities were correctly rejected, wheres for the F-rtio nd R-rtio tests, those three epochs hd been wrongly ccepted. In the cse of running simultions to determine the criticl vlue, for ech epoch, with given fil-rte, the geometry of flot solution ws used to obtin the criticl vlues. Owing to the short observtion time spn, the stellite geometry chnges slightly, nd so does the criticl vlues determined from the simultions. Menwhile, s mentioned before, there re hevy computtion burdens for higher dimension cses (more thn 2 h per simultion); consequently, the pproximte criticl vlues were used, nd the performnce of the IA theory in rel pplictions ws shown in Tble 8. The results re, however, not preferble compring the results from the non-ia s listed in Tble 7 (except the WTIA, which performed the best in both pproches). When fil-rte of 0?01 is given, the corresponding criticl vlues cn be simulted, nd then the IA bsed criticl vlues were pplied. For the EIA, once the criticl vlue obtined, we could compre its rel fil-rte with the predefined fil-rte. For the OIA, eqution (18) ws implemented. Besides the hevy computtion burden, it cn be seen tht in this cse, the OIA does not mke the optiml decision, nd the non-ia methods outperform the IA bsed method for most of the time in this rel ppliction. Another issue is tht the success rtes re extremely low, much smller thn the ILS success rte (59?3%), nd the probbility of undecided solution is quite high. This phenomenon is not consistent with the rel performnce, which cn be judged from the number of the ccepted solutions. Moreover, the IA estimtor committed type I error s did some conventionl methods. Obviously, in this rel ppliction, the success rte does not reflect the true correct rte properly, nd the IA estimtor does not perform resonbly. Concluding remrks To sum up, this contribution presents n overview of current mbiguity vlidtion pproches which hve been numericlly compred nd nlysed. Trditionlly, the comprison mong different mbiguity vlidtion methods is difficult s the determintion of criticl vlues vry with different methods. Within the frmework of the IA theory, the users re ble to control the fil-rte of mbiguity vlidtion, nd different vlidtion methods could be compred. The two-dimensionl simultion indictes tht OIA performs the best s the mximistion of success rte, yet the computtion burden is extremely hevy. While for the other vlidtion methods, R-rtio, W-rtio nd difference test re preferred. However, higher dimensionl cse gives slightly different results, nd this shows tht in prcticl pplictions, such simultions cnnot show which mbiguity vlidtion method is the best in terms of mbiguity success rte, becuse the mbiguity success rte is not the correct rte of mbiguities, which hs lso been shown in the prcticl exmple discussed in Tble 8. Therefore, for the IA bsed methods, there re some issues tht need to be emphsised: (1) even if we cn determine the criticl vlue from given fil-rte by using look-up tble or simultion, the relibility of both the look-up tble nd the simultion needs to be pid more ttention, s they re lwys ffected by the smple size. In order to obtin relible results, the smple size should be s lrge s possible, which is, however, contrdictory to the computtionl efficiency. Numericl results hve shown tht simultions re Tble 7 The performnce of the non-ia bsed methods Non-IA bsed methods F-rtio R-rtio W -rtio W s -rtio D-test P-test Criticl vlues 0. 67 0. 50 2. 32 3. 75 15. 0 0. 87 No. of ccepted solutions 560 506 568 438 0 421 Performnce in three specil epochs* WA WA CR CR CR CR *The best solutions from those three epochs re not correct. Tble 8 The performnce of the IA bsed methods IA bsed methods FTIA RTIA WTIA DTIA PTIA EIA OIA Criticl vlues (P f 50. 01) 0. 57 0. 36 0. 97 5. 33 0. 34 1. 71 1. 03 No. of ccepted solutions 507 388 591 226 173 221 443 P s 0. 057 0. 058 0. 057 0. 054 0. 051 0. 051 0. 059 P u 0. 933 0. 932 0. 933 0. 936 0. 939 0. 939 0. 931 Performnce in three specil epochs* WA WA CR CR CR CR WA *The best solutions from those three epochs re not correct. Survey Review 2012 VOL 44 NO 326 237
extremely time consuming nd not stble, nd re ccordingly not pplicble in prctice; (2) it should be noted tht the success rte defined under the frmework of IA theory is not equl to correct rte, which is unknown during mbiguity resolution. The rel GPS dtset hs been used to evlute both the conventionl nd IA bsed methods with the ground truth. In terms of mbiguity correct rte, the W -rtio hs outperformed ll the other vlidtion methods. Moreover, it should be stressed tht the current mbiguity vlidtion methods endevour to discriminte the best integer mbiguity from the others, wheres the correctness of the resolved best integer mbiguity is only vlidted indirectly. Therefore, their performnces in vlidting the correctness of the mbiguities should be evluted by using the ground truth in vrious ppliction scenrios. Acknowledgements The first uthor is PhD student sponsored by Chinese Scholrship Council (CSC) for his studies t the University of New South Wles. References 1. Brd, W., 1968. 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