A Search Space Optimization Technique for Improving Ambiguity Resolution and Computational Efficiency

Transcription

1 A Search Sace Otimization echnique for Imroving Ambiguity Reolution and Comutational Efficiency Donghyun Kim and Richard B. Langley Geodetic Reearch Laboratory, Deartment of Geodey and Geomatic Engineering Univerity of Ne Brunick, Fredericton, N.B., Canada ABSRAC An Otimal Method for Etimating GPS Ambiguitie (OMEGA) that enable very high erformance and comutational efficiency ha been develoed and demontrated. hi method emloy to earch ace reduction rocee a caling and a creening roce that are related to the earch ace tranformation and the ambiguity candidate filtering in multi-earch level. o obtain the highet efficiency, an otimization rocedure, hich determine the arameter to minimize the number of candidate under given condition, i imlemented in cloed-form before the earch-verification. he method i eentially baed on the leat-quare-aroach originally rooed by Hatch but ue a modified and more efficient roce. o imroved algorithm are introduced in thi aer. Firt, an alternative algorithm for the ectral decomoition, hich reduce the dimenion of the reidual vector to it degree of freedom, i given in cloed form. hi algorithm i imlemented in the comutational te of the quadratic form of the reidual in order to increae comutational efficiency. Second, an efficient error model for the threhold of the filter equation that i ued to derive the earch ace caling roce i given. hi error model ho to advantage: 1) it bound noie ignal of the filter equation; 2) it give efficient threhold o that the caling effect for the earch ace can be increaed. INRODUCION In navigation and urveying ytem uing GPS carrier hae data, the erformance of ambiguity reolution and comutational efficiency are of great concern. hee caabilitie are often traded off in deigning the ytem. One oible ay to overcome the trade-off i to reduce the number of ambiguity candidate before or at the earch-verification te. he earch ace tranformation [Abidin, 1993; eunien, 1994; Martin-Neira et al., 1995] and ambiguity candidate filtering in multi-earch level [Chen and Lachaelle, 1995; eunien, 1997] are effective technique for that uroe. When e ue a roce imilar to the leat-quarearoach of Hatch [199] at the earch-verification te, it i oible to imlement otimization rocedure reducing the number of candidate before imlementing the te. We hoed that thi could be achieved uing the deign matrix of the linearized double-difference obervable in Kim and Langley [1999]. hee otimization rocedure include to earch ace reduction rocee (i.e., a caling and a creening rocee) and to-te otimization rocee a global otimization to find a matrix S minimizing the total earch ace volume and a local otimization to find a reordered matrix S minimizing the total earch ace volume out of all oible combination of S, the matrix S i comuted uing the deign matrix. o increae comutational efficiency, further attention ha been given to the quadratic form of the reidual and to error model for the threhold of the filter equation. he quadratic form of the reidual i generally ued for the ambiguity accetance tet and i the ractical comutational art in the ambiguity earch-verification te. herefore, e need to decreae comutational burden uing a more efficient comutational algorithm. A ignificant reaon for needing a more efficient error model for the filter threhold i that the filter threhold are related to a magnification factor of the caling roce. he more efficient the error model, the more the caling effect. he GPS Obervable o imlify dicuion, e ill aume that the float etimate of ambiguitie and their error model are given. For the double-difference obervable recorded on hort baeline, the atellite and the receiver clock biae are removed, and the reidual atmoheric effect are negligible. Ignoring multiath, e have

2 l = Ax+ N+ e [ ] [ ] [ ] E e = E l =, Cov e = l i the nx1 initial micloure vector of the difference beteen the double-difference obervation and their etimate; n i the number of the doubledifference obervation; x i the 3x1 vector of the unknon remote (rover) tation oition comonent; A i the deign matrix for the unknon oition; N i the nx1 vector of ambiguity arameter; e i the nx1 vector of Cov i the double-difference obervation noie; E[ i] and [ ] rereent the mathematical exectation and the variancecovariance oerator, reectively. he Modified Leat-Square Aroach Uing the ame terminology a Hatch [199], e outline the modified roce for the leat-quare aroach in able 1. In the comutational equation, the ubcrit and rereent the rimary and the round i i the econdary grou of atellite; and [ ] rounding-to-the-nearet-integer oerator. able 1. Modified leat-quare-aroach. Proceing Comutational equation te Original aroach Modified aroach Potential 3 3 x =A ( l -N ), N Z x =-AN, N Z olution Secondary n 3 N = round, Z l -AA (l -N ) N ambiguitie Innovation l = l =l vector l =l +AA ( l -N)-N l =l +AA N -N Reidual v=(i-aa )l, A =(A A) A, l= l l When comared ith the original leat-quare aroach, the modified aroach give exactly the ame reidual. o rove the equivalence of both aroache, e ill define: δx = Al l = A δx = AA l hen, e have folloing equality: l I = A ( A A) A l l AA A = = A. ( A A) A Al Al. (1) (2) (3) Subtracting the reidual for both aroache and alying equation (3) give ( ) ( o m o m) l l A ( ) v = v v = I AA l l = I AA = A l = l l A ubcrit o and m rereent the original and modified aroache. he Filter Equation for the Secondary Innovation Vector When e comute the reidual uing the modified aroach in able 1, the only variable arameter i the econdary innovation vector. In accord ith the leatquare rincile, the otimal etimate for the econdary innovation vector i given a: ( v v) ˆ l =, l =AA l. We can recognize that the otimal etimate i indeendent of the earch-verification te, ince it i derived from the deign matrix and the initial micloure vector for the rimary grou. hee arameter are contant in a nahot (i.e., ingle eoch) aroach, uch a the leat-quare aroach. One natural idea to utilize the otimal etimate i to define a filter equation a: = l -l ˆ = l round -AA (l -N ) l -AA (l -N ). Uing the filter equation and a certain threhold vector τ, e can define a filter a: ab [ ] (4) (5) (6) τ (7) [ ] [ ] ab = τ = τ τ τ 1 2 n n 3. he dimenion of n-3 for the filter equation and threhold vector come from the dimenion difference beteen the (n-dimenional) double-difference obervation and the (3-dimenional) unknon remote tation oition comonent. UADRAIC FORM OF HE RESIDUALS For the ambiguity accetance tet, e have to comute the quadratic form of the reidual for all ambiguity candidate in the ambiguity earch-verification te. o (8)

3 aroache can be conidered for decreaing comutational burden earch ace (or ambiguity candidate) reduction and comutational algorithm imrovement. We ill focu on the comutational algorithm for the quadratic form of the reidual in thi aer. Uing the comutational equation of the reidual in able 1, the quadratic form of the reidual i given a: v v = l Σl (9) v v = v Pv = v P v + v P v + v P v + vpv P P P = =. P P Subtituting equation (15) and (16) into (18) give (18) (19) Σ= I AA I AA (1) ( ) ( ). Sectral decomoition for a ingular ymmetric matrix Σ i exreed a [Bailevky,1983]: Σ= EΛE (11) the nx(n-3) matrix E contain the nx1 latent vector E,E, 1 2,E n-3 and the (n-3)x(n-3) diagonal matrix Λoe n-3 nonzero latent root λ1, λ2,, λn 3. Subtituting equation (11) into (9) give n 3 v v = ϖ Λ ϖ = λϖ i i (12) i= 1 ϖ = El. (13) We can find an examle of thi algorithm in Martin- Neira et al. [1995]. An alternative algorithm for the ectral decomoition aroach can be derived uing the folloing relational equation. From equation (4), reriting the reidual give v v = = ( I-AA ) l m v (14) = ( I-AA ) l o = ( I-AA ). We have, therefore, to relational equation from equation (14) a: G v (15) = v = G G v (16) G G G = I AA =. G G (17) Uing the artitioned matrice and vector, the quadratic form of the reidual can be exreed a: v v = vrv = Ω (2) - - R = GGPGG + GGP+ PGG + P (21) Ω= G RG. Furthermore, the folloing equality can be roved: R (22) = v v = A A A A = v v v v ( ). (23) o comare the comutational efficiency of different algorithm, e outline three algorithm for the comutation of the quadratic form of the reidual in able 2. he comutational algorithm I and II rereent a normal and a ectral decomoition aroach. he comutational algorithm III i an alternative aroach for the comutational algorithm II. he comutational te are earated into three art for clarity common, external, and internal. he external art i comuted before the ambiguity earch-verification te. On the other hand, the internal art i comuted in the earch loo. herefore, the comutational efficiency of the internal art i uually of great concern. able 2. Comarion of the comutational algorithm. I II III Common G = I-AA, A = MA, M = ( A A ) External Σ= G G Internal Σ= GG Σ= EΛE v = -AMA Ω = G G l =, l ( or ) = l +AA ( l -N )-N V= l Σl i i i= 1 v ϖ = El n 3 V λϖ V = Ω = ERROR MODEL FOR HE FILER HRESHOLD In general, the threhold vector τ in equation (7) can be derived from the quadratic form of the filter equation a:

4 c (24) i the variance-covariance matrix of and c i a oitive contant hich can be elected baed on the robability ditribution of. herefore, e can determine the threhold ith a certain confidence by chooing c at a certain confidence level. hen, one imle ay to et the threhold i τi = σi c, i = 1,2,, n 3 (25) σ i i the quare root of the i th diagonal element in. Hoever, e can recognize in equation (6) that it i not eay to etablih correctly the variance-covariance matrix and the robability ditribution of. For thi reaon, e have ued an alternative aroach for the quadratic form of the filter equation. In general, the n 3 ellioidal region in R centered on the etimator ˆx of a certain vector x can be exreed a [Giri, 1977]: (x x) ˆ (x x) ˆ c (26) xˆ ˆx i the variance-covariance matrix of ˆx. he family of ellioid obtained by varying c (c>) ha the ame center ˆx, their hae and orientation are determined by ˆx, and their ize are determined by c. It hould be noted that the oitive contant c doe not have robabilitic ene unle the robability ditribution of x i knon. From equation (26), the threhold are given a: τi = σxi ˆ c, i = 1,2,, n 3 (27) σ ˆxi i the quare root of the i th diagonal element in. One condition required in thi aroach i that the ˆx etimator ˆx hould be the unbiaed etimator of x; then, e can get different threhold according to the eighting cheme of ˆx. hree cae from (a) to (c) in able 3 ho the unbiaed etimator of x and their error model. Becaue both x and ˆx are robabilitic variable, the condition of unbiaedne become [ ] = E[ ] E x x ˆ. (28) For each cae, the condition of unbiaedne can be roved if the folloing equality hold (more detail in eunien [1998]): E round[ y] = AA N (29) y = l AA (l N ). (3) hree error model for the filter threhold (i.e., the error model from (a) to (c)) ere teted and dicued in Kim and Langley [1999]. A rigorou error model, hich i related to deciding the value of the oitive contant c in a robabilitic ene, can be derived from equation (2) and (24). From thee to equation, e can exre a correct error model of the filter equation a: ( ) 1 =Ω = v G G. (31) If the double-difference obervation noie e in equation (1) ha a normal ditribution, e have ( n α) 2 v v = ~ χ 3, (32) α i the level of ignificance. herefore, e can determine the oitive contant c ith a certain confidence level in accord ith α. able 3 ho the ummary of the error model for the filter threhold. In each error model, the variance-covariance matrice ith ubcrit are the ub-matrice artitioned from the variance-covariance matrix of the double-difference obervation in equation (1). able 3. Error model for the filter threhold. (a) round [ ] + ( ) (b) round[ y] (c) x ˆx or ˆx or y AA l N l l (d) - RESULS y ˆ +AA A A - -A A - A A - - l AA A A G v G ( G ) v o tet the efficiency of the comutational algorithm for the quadratic form of the reidual and the erformance of the error model for the filter threhold, e roceed ome of the tet data recorded at one reference tation in the Canadian Coat Guard (CCG) DGPS and OF netork (Figure 1 and 2) and that recorded imultaneouly on board a hydrograhic ounding hi (Figure 3) at roi-rivière, on the St. Larence River, 13km utream (outhet) of uébec City, on 3 Setember Figure 4 ho the track of the veel. he data et contain both L1 and L2 obervation recorded at a one econd amling interval for to hour in kinematic mode. Baeline length beteen the reference tation and hydrograhic ounding hi a about 4 km.

5 46.22 Latitude (degree) Figure 1. CCG DGPS and OF netork: coverage of the St. Larence River Longitude (degree) Figure 4. Veel track. Comutational Algorithm Efficiency Uing each comutational algorithm for the quadratic form of the reidual in able 2, the FLOPS tet a conducted to comare the efficiency of the algorithm. able 4 ho the tet reult. et condition ere given a: 1) three-level earch loo ere built; 2) the earch range for each earch level a given a 2 m, therefore, tenty-one candidate ere given for each earch level; 3) even atellite ere ued. able 4. FLOPS tet reult Figure 2. roi-rivière DGPS reference tation. I II Common Figure 3. GPS-equied hydrograhic ounding hi. he reult of our analye are reented in to art: the floating oint oeration count (FLOPS) tet of the comutational algorithm for the quadratic form of the reidual; and the erformance of the error model for the filter threhold. III 1,588 External 868 5,278 1,4 Internal 1,917,27 722, ,575 otal 1,919, , ,23 he normal comutational algorithm I gave the ort comutational efficiency and the alternative algorithm III for the ectral decomoition aroach gave the bet reult. Comared ith the normal algorithm I, the efficiency of the ectral decomoition algorithm II a imroved by about 62%. In the cae of the alternative algorithm III, the efficiency a imroved by about 64%. In both the external and internal comutation art, the alternative algorithm III a uerior to the ectral decomoition algorithm II. We can ay therefore, that the alternative algorithm III i the mot efficient algorithm of the three algorithm teted for comuting the quadratic form of the reidual no matter hich condition are given.

6 SV9-SV8 SV3-SV29 SV23-SV21 SV25-SV3 Figure 5. Performance of the filter threhold: the x-axi in each anel rereent GPS time in econd (from to 32982); the y-axi rereent the filter threhold in cycle (from to ); olid line are the filter threhold for each error model; and dot are the value of the filter equation. Error Model Performance o comare the erformance of the error model, the filter threhold ere comuted uing equation (25) and (27) for each error model in able 3. he oitive contant value c a determined from the chi-quared ditribution at the 95% confidence level ith n-3 degree of freedom (i.e., e aumed that the double-difference obervation noie follo the normal ditribution.). A a mentioned reviouly, hoever, thi choice of c cannot guarantee the ame confidence level to the error model from (a) to (c) in able 3, becaue thee error model are not baed on a rigorouly-defined robability ditribution. Even though thi limitation exit, e conidered, a a matter of convenience, that the oitive contant c ha the ame confidence level for all error model in our invetigation. Figure 5 ho ho ell the filter threhold (olid line) bound the value of the filter equation (dot). All error model excet for the firt error model erformed ell according to the noie level of the filter equation. Furthermore, the filter threhold for the firt error model ere almot contant in thee examle. o invetigate the erformance of the filter threhold in detail, e lotted the filter threhold for each error model and for the double-difference obervation of PRN 8 and 9 earately in Figure 6. We need in general an error model that rotect all the value of the filter equation for true ambiguitie in the given confidence level, hich at the ame time i alo efficient. he third error model atifie thee criteria a hon by the examle in Figure 6. On the other hand, it i evident that the firt error model doe not atify thee criteria. Furthermore, the firt error model did not react ell to the noie level of the filter equation. For the econd and fourth error model, our invetigation have hon that both of thee error model have imilar characteritic (ometime, almot identical). In fact, the equation for the fourth error model can be develoed in a imilar form to that of the econd error model in able 3. According to the artitioned deign matrice, both error model ould be almot identical. he erformance of thee error model i not much different from that of the third error model a hon by the examle in Figure 6. CONCLUSIONS he ork reorted in thi aer ha been a conecutive tudy of our reviou ork reorted in Kim and Langley [1999]. o aect error model for the filter threhold and comutational algorithm in the earchverification te have been invetigated in detail.

7 (a) (b) (c) (d) Figure 6. Performance of the filter threhold: the x-axi rereent GPS time in econd (from to 32982); the y-axi rereent the filter threhold in cycle (from to ). For the comutational algorithm of the quadratic form of the reidual in able 2, e found that the alternative algorithm III for the ectral decomoition i the mot efficient. Regarding the error model for the filter threhold in able 3, e found that three of the four error model erformed ell, i.e., the filter threhold bounded almot all the noie ignal of the filter equation. Only the firt of the four erformed oorly. We alo found that the econd and fourth error model have imilar behavior. Becaue of the meaning of the oitive contant c, e have conidered the fourth error model a the mot rigorou aroach in our invetigation. ACKNOWLEDGEMEN hi ork ha been conducted under the GEOIDE Netork of Centre of Excellence (roject ENV#14). he uort of the Canadian Coat Guard, the Canadian Hydrograhic Service, VIASA, Geomatic Canada, and the Centre de Recherche en Géomatique, Univerité Laval i gratefully acknoledged. REFERENCES Abidin, H.Z. (1993). On the contruction of the ambiguity earching ace for on-the-fly ambiguity reolution. Navigation: Journal of he Intitute of Navigation, Vol. 4, No. 3, Bailevky, A. (1983). Alied Matrix Algebra in the Statitical Science. Elevier Science Publihing Co., Inc., North-Holland, Ne York. Chen, D. and G. Lachaelle (1995). A comarion of the FASF and leat-quare earch algorithm for onthe-fly ambiguity reolution. Navigation: Journal of he Intitute of Navigation, Vol. 42, No. 2, Giri, N. C. (1977). Multivariate Statitical Inference. Academic Pre, Ne York. Hatch, R. (199). Intantaneou ambiguity reolution. Proceeding of KIS 9, Banff, Canada, Setember 13, Kim, D. and R. B. Langley (1999). An otimized leatquare technique for imroving ambiguity reolution erformance and comutational efficiency. Proceeding of ION GPS 99, Nahville, enneee, Setember 147 (in re). Martin-Neira, M., M. oledo, and A. Pelaez (1995). he null ace method for GPS integer ambiguity reolution. Proceeding of DSNS 95, Bergen, Noray, Aril 24-28, Paer No. 31, 8. eunien, P.J.G. (1994). A ne method for fat carrier hae ambiguity etimation. Proceeding of IEEE PLANS 94, La Vega, NV, Aril 115, eunien, P.J.G. (1997). A canonical theory for hort GPS baeline. Part III: the geometry of the ambiguity earch ace. Journal of Geodey, Vol. 71, No. 8, eunien, P.J.G. (1998). A cla of unbiaed integer GPS ambiguity etimator, Artificial Satellite, Vol. 33, No. 1,. 3.