A study on GPS PDOP and its impact on position error

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1 IndianJurnalfRadi& SpacePhysics V1.26,April1997,pp A study n GPS and its impact n psitin errr P Banerjee,AnindyaBse& B SMathur TimeandFrequencySectin,NatinalPhysicalLabratry,NewDelhi Received19June 1996;revised23August1996;accepted15Nvember1996 It is well established that 3-dimensinal psitin errr in psitin fix thrugh GPS netwrk shuld increase with value prvided the range errr estimates remain cnstant. T study this aspect an elabrate experimental bservatin was carried ut fr ne cmplete year. These data have been systematically analysed. It has been seen that fr maximum percentage f time ptimum values lie between 2 and 4. Study n the variatins f 3-dimensinal psitin errrs n reveals that psitin errrs increase prprtinately up t the value f 6. Aftelwards the change f psitin errrs with is almst randm in nature. Pssibly beynd this value f, the psitin errrs are gverned by sme ther factrs. 1 Intrductin Glbal Psitining System (GPS) is a satellite based radi psitining system. It is nw being used wrldwide. T find ut three psitin crdinates and time (i.e. 4 parameters), it is necessary t receive signals frm fur satellites simultaneusly. GPS cnstellatin is nw full with 21 satellites with 3 active spare satellites in rbit. S needed 4 satellites fr psitin fix are always available wrldwide. The gemetrical cnfiguratin f navigating satellites als plays a vital rle in the accuracy f GPS psitining. The effect f gemetry f the satellites is mathematically expressed by Gemetrical Dilutin f Precisin (GDOP). Effect f the gemetry f the navigating satellites n varius parameters are nrmally quantified by the terms, TDOP, HDOP and VDOP respectively fr 3-dimensinal psitin, time, hrizntal (2-dimensinal) psitin and vertical psitin errrs. Mathematical treatments n this aspect have been reprted in literaturel-3 We have experimentally studied the effect f n psitin errr fr a lng perid. In this paper sme interesting bservatins n this aspect have been elabrated based n a detailed analysis f these data. 2 Backgrund The gemetrical cnfiguratin f navigating satellites is dependent n the psitin f the receiver and it changes with time due t the relative mtin f satellites. It has been well established 1, that the amunt f the ranging errr cmbined with the gemetry f the fur selected satellites decides the magnitude f the user psitin errr in GPS navigatin fix. Psitin Dilutin f Precisin () refers t the gemetric cnfiguratin f satellites cncerning 3-dimensinal psitin fix. This is related t 3-dimensinal psitin errr by 3D errr in psitin 1 a = () x (user t satellites range errr 1 a)... (1) It is dear frm Eq. (1) that minimum value f is desirable fr best psitin fixing. The GPS cnstellatin being full, any time mre than fur satellites are visible everywhere. S it is necessary t select the best set f fur satellites giving minimum. Mdern GPS receivers determine the relative psitins f all visible satellites and find ut fr all cmbinatins f 4 satellites. Fr navigatin fix, it selects the set f 4 satellites with minimum. T study the impact f n psitin errr, it is abslutely necessary t knw the precise crdinates f the lcatin where the study is t be carried ut. GPS can give psitin accuracy f the rder f 100 m. S t have a check n the psitin accuracy ne has t knw the lcatin t the rder f a few tens f metres. The psitin given by GPS receiver is in WGS-84 (height, latitude and lngitude). T find ut the shift f psitin given by GPS frm the crrect psitin ne may nrmally have t use elabrate frmulatin.

2 lor INDIAN J RADIO & SPACE PHYS, APRIL 1997 But in this applicatin the errr is nt suppsed t exceed a few hundreds f metres. It implies that errr in latitude and lngitude wuld be abut a fractin f secnd f angle. S t simplify the calculatin withut sacrificing the required accuracy, the psitin errr (PE) may be defined as PE= J~h2+(~Ln x Rx COSL,)2+(~L, x Rf where L, Nminal value f latitude f the place ~h Departure f height frm knwn height in metres... (2) ~LIl Departure flngitude (in minute f angle) frm the knwn lngitude f~l, Departure f latitude (in minute f angle) frm the knwn latitude R 1852 m/minute f angle The abve frmulatin assumes that 1 minute f earth curvature is equivalent t 1852 m. 3 Mrphlgy f data GPS signals have.been mnitred ver ne cmplete calendar year at NPL, New Delhi, thrugh an 8-channel gedetic GPS receiver. The data have been recrded at 10 s sampling rate fllwing a fixed schedule n every Wednesday. f ptimally selected 4 satellites as given by the receiver used and the crrespnding psitin crdinates as navigatin slutin has been recrded regularly during the campaign. The data recrded at NPL fr ne cmplete year have been analysed t find ut the behaviur f and its impact n psitin errr. T find the psitin errr f the antenna lcatin at NPL based n the relatin given by Eq. (2), the actual psitin f the NPL antenna has been fund ut. This has. been achieved by averaging the psitin crdinates (generated by GPS receiver itself) ver a huge number f data samples recrded ver a year. The crdinates thus btained may be fairly accurate as the averaging has smthed ut randm fluctuatins. Thus the Selective Availability (SA) effect (Ref. 4) has als been significantly diluted as the averaging is ver a large number (10 6 ) f samples and ver a large span f time (1 year). The crdinates thus btained have been used t calculate the psitin errr fr each sample f data using Eq. (2). A large fractin f ttal data has been pltted in Fig. 1 t shw the variatin f psitin errr 600,--,-----,-----,------,-----,-----,------,-----,-----,-----, S ~ 350 Z 300 I 0 /f ~ E-< >--< 250 if) 0 P Q I 150 (") Fig. 1- Variatin f psitin errr with at NPL, New Delhi

3 BANERJEE et al.: A STUDY ON GPS & ITS IMPACT ON POSITION ERROR 109 ij) 25 +' (j) (\j I:lD c.. i:: () >-. 30 ij) Fig. 2-Percentages f ccurrence f ptimum values. with. Because f verlapping f data, the figure des nt clearly depict the nature f dependence f psitin errr f. Fr the cnvenience f this study, the data have been gruped int sets f values. Fr example, ne grup has been frmed with data having values greater than 1 and less than r equal t 2. Similarly, the next grup cmprises data with values greater than 2 and less than r equal t 3, and s n. Nine such grups have been frmed fr values up t 10. The percentage f ccurrence f values in different grups has been calculated. T d s, the number f samples in each grup has been cunted against the ttal number f samples (58,746) t find ut the percentage f ccurrence. These data crrespnd t a large span f time fr a particular lcatin. The percentage f data may be cnsidered t crrespnd t the maximum value f the respective grup withut lss f generality, and it has been illustrated in Fig. 2. In the graph the higher limits f the respective grups f values have been used as the crrespnding data pints. It is seen that mst f the time (i.e. - 67%) lies between 2 and 3. Thugh the GPS cnstellatin is full during the bservatin perid, fr a significant time (mre than 20%) is fund t lie between 3 and 4. Prbability f lying belw 2 and abve 6 is very lw. The errr in psitin fixing has been calculated fr each sample. These errrs have als been gruped int similar categries f sets as explained in Fig. 2. Samples f each mnth have been separated ut. The average f the psitin errr fr each grup f values has been separately calculated ut. Jitter (standard deviatin f three-dimensinal psitin errr) f each set f psitin errr ver the average value has als been fund ut. The data sets thus generated fr each mollth separately have been illustrated in Figs 3 and 4. Apparently up t the' value f 4, there is a prprtinal increase f psitin errr with the increase f value. Within this limit, the abslute values f average and jitter remain mstly within sme tlerable range f 100 m. Keeping Eq. (1) in mind, this apparently shw s that" pssibly range errr:remains fairly;'cnstant ver this regin. But interestingly, the nature f variatin f psitin errr beynd the value f greater than 4 is different. It may als be likely that when value exceeds 4 r s, the psitin errrs are nt gverned by Eq. (1) but sme ther parameters that have nt been cnsidered in Eq. (1) but they becme imprtant t be cnsidered. Figures 3 and 4 have been reprduced in Fig. 5 by wrking ut the grand average f all data. This illustrates the similar nature f variatin except that the prprtinal increase f psitin errr extends t value f 6. Interestingly, the psitin errr smetimes decreases at the value f greater than 6.

4 -,"'IJITT ~'I.! "rwllillil!ll lilt' I' It I '~'I 110 INDIAN J RADIO & SPACE PHYS, APRIL ' E 200 e Q; g 150 '':; '0; c. ~ 50 ~..~".~ ~ 100 I '0; >,- 0E 60 Ol ';::; <ll m 0Q Q) Q; c 40 ~ "') ce Q; ~ Fig. 3- Yariatin f average psitin errr with POOP. ~------~.- Oct {J-- Sep.95 I +- Aug.95 --<)--- Jul Jun tr--- May ~.Apr.95 Fig. 4-Jitter an average psitin errrs fr different values, Jitter deterirates prprtinally up t the value f 4. Beynd this value, the nature is randm. Data fr each mnth has been pltted separately. --' --' <Il --' S>-..~ ~ QJ 0,D ~ 'UC tlij ;> <Il 0 40 >-. --' > QJ..c: QJ., 20 ~ ~ Fig, 5-Dependence f psitin errrs (bth average and jitter) n, Psitin errrs prprtinally increase with up t the value f 6, beynd this value f the psitin errrs d nt change cnsistently. This plt includes all data taken ver ne year I I' " P j I ~ii I I I

5 BANERJEE et ai.: A STUDY ON GPS & ITS IMPACT ON PsmON ERROR Cncluding remarks The year lng bservatin f GPS signal reveals sme interesting aspects which therwise have nt been anticipated. The values mstly remain between 2 and 4 with full GPS cnstellatin. The psitin errr des increase with the increase f POOP value as expected but up t a certain value f POOP (say 6). Beynd this POOP value, the psitin errr des nt increase prprtinately, rather the nature f change is randm. Acknwledgements The authrs are thankful t the Department f Electrnics, and Defence Research and Develpment Organisatin, Gvernment f India, fr supprting this wrk. They als express their thanks t Dr R K Arra, DE, and Dr K K Ramachand, CABS, fr their encuragement. References I Milliken R J & Zller C J, J Inst Navig (UK), 25(2) (Summer 1978) Phillips A H, J Inst Navig (UK), 31(4) (Winter ) Banerjee P & Bse A, Indian J Radi & Space Phys, 25 (1996) Hfmann-Wellenhf B, Lichtenegger H & Cllins J, GPS Thery and Practices (Springer-Verleg Wien, New Yrk), 1993,17-18., \