HOW BAD RECEIVER COORDINATES CAN AFFECT GPS TIMING

Transcription

1 HOW BAD RECEIVER COORDINATES CAN AFFECT GPS TIMING H. Chadsey U.S. Naval Observatory Washigto, D.C Abstract May sources of error are possible whe GPS is used for time comparisos. Some of these mo have bee listed by Louadowski[l~. Because of the complexity of the system, a error source could have more tha oe dect. This paper will preset theoretical ad observatioal results by o-g a receivot's coordiates. Re c(lleula&ios show how a error as d s 3 )I1PteTS i ay direckio ca suu i a timig error of more tha 1 aosecods. Tbe GPS receiver must be surveyed 1 Mer tha.2-meter accuracy for the timig error to be subaosecod. INTRODUCTION GPS is a receive-oly system. The user's equipmet does ot trasmit a sigal other tha the itermittet frequecies used iterally to the receiver. The system relies o kowig the positio of the trasmitter (the GPS satellite), the time of sigal trasmissio, ad the positio of the receiver so the receiver ca determie its time ad time offset from some referece (for time trasfer operatios). For mobile operatios, the iformatio from at least four satellites is eeded so the receiver ca fid its positio, time, ad time offset. If the satellite is at its stated locatio ad the correctios for propagatio are correct, the source of error i time trasfer mode of operatio must be the receiver coordiates. THEORETICAL CALCULATIONS A perso must first uderstad the differet coordiate systems used ad put all positios i a commo system. The GPS atea used was surveyed by The Defese Mappig Agecy ito the World Geodetic Survey 1984 (WGS-84) coordiatesiz1. The WGS-84 is based o the Earth's ceter of mass. The Z-axis is i the directio of the Covetioal Terrestrial Pole (CTP) for polar motio. The X-axis is the itersectio of the WGS-84 referece meridia plae ad the plae of the CTP's equator. The referece meridia is the zero meridia as defied by the BIH for epoch o the basis of the coordiates adopted for the BIH statios. The Y-axis completes a right-haded, Earth-fixed orthogoal coordiate system. Programs from the Defese Mappig Agecy ad Mihra Miraia (USNO) were used to covert the WGS-84 coordiates to Earth-Cetered, Earth-Fixed (ECEF), which is used by the GPS system. The coordiates for oe GPS atea at USNO are:

2 WGS-84 ECEF ECEF N 38 55'13397" X m Re m W 77 3'58.431" Y m A 555m Z S ' S where R, is the radius of the Earth (ECEF) at the receiver's locatio, ad P' the Z-axis rather tha from the X-Y plae. is measured from A satellite directly at zeith is meters from the receiver accordig to actual measured values. The height of the satellite above the receiver is meters. The ext step is to uderstad how chagig the positio of the satellite will chage the geometry of the satellite-receiver relatioship ad the path legth. Let: R = radius of Earth Re = height of satellite above ceter of Earth (assumed costat) h = height of satellite above receiver h+dh = height of satellite above receiver plus additioal distace due to chage of satellite-receiver geometry c = agle betwee zeith of receiver ad locatio of the satellite b = agle satellite is above the horizo We have the followig relatios: a = arcsi (R x si (b + 9)) Rt

3 x si (c) h+dh = si (b + 9) The agle b was varied from 9 to '. This resulted i c varyig from O to ' ad dh varyig from meters to meters. These variatios were the trasformed to those see by the idividual receiver coordiates. These values were the coverted to ECEF coordiates X, Y, 2. This assumes that the receiver is at its proper coordiates. I order to uderstad of how dh chages as the satellite chages positio whe the receiver is NOT i its proper locatio, oe must vary the surveyed latitude, logitude, ad altitude (i WGS-84 coordiates) ad determie the "ew" coordiates i the ECEF coordiate system. The altitude chages i direct proportio to the radius. However, latitude ad logitude do ot have such a simple trasform. The latitude of the satellite is give by: SLAT = S 21r x B where S is secods per 36 degrees (1296) ad B is the Earth's polar radius ( meters). For the USNO receiver, SLAT =.32448" per meter. The logitude is give by: SLON = S 27r x cos(lat) x A where A is the Earth's equatorial radius ( meters) , SLON = " per meter. At USNO's Latitude of The receiver offsets, symmetric about zero, were 15m, lor, 5m, 4m, 3m, 2m, lm,.9m,.8m,.7m,.6m,.5m,.4m, 3m, 2m,.lm, ad.5m. These offsets were trasformed to altitude, latitude, ad logitude offsets i the WGS-84 coordiate system. The ew positios were the trasformed ito the ECEF coordiate system. With the satellite ad receiver i ECEF coordiates ad kowig the o-offset h+dh values, a simple computer program ca solve the time error equatio. The time error equatio is: where s ad r represet satellite ad receiver respectively. The results are plotted as time offset vs. offset vs. agle of satellite above the horizo i Figure 1. A error of as small as 3 meters offset i ay of the three coordiates ca result i

4 a time error of more tha 1 aosewds. For the time error to be subaosecod, the GPS atea must be surveyed to better tha.2 meter accuracy. Theoretical calculatios for offsettig a receiver's coordiates, holdig the other variables fixed, show some iterestig results. First, a time error of 2 aosecods would require a atea's coordiates to be off by more tha five meters. Secod, the errors are three-dimesioally symmetric. OBSERVATIONAL RESULTS The theoretical results are iterestig, but mea othig without some proof of observatio. For this, two keyed dual-frequecy receivers were used. First, both receivers were set i the time trasfer mode of operatio with their correct coordiates i their databases (Figures 2 ad 3). After several days of observatio, receiver 1 cotiued to operate with the correct coordiates, while receiver 2 had its coordiates offset chaged daily. Receiver 2's offset were 15m, 1m, 5m, lm,.5m, ad back to Om (for two days) to verify each offset ru. The offsets were applied i altitude (Figures 4,5, ad 6), latitude (Figures 7 ad 8). ad logitude (Figures 9, 1, ad 11). The closure checks of zero offset showed that o parameters chaged durig the observatios. The bias of approximately 5.6 aosecods was betwee this pair of receivers. I a follow-o observatioal set betwee oe of these receivers ad aother, the bias was 3.5 aosecods. All receivers were calibrated by the maufacturer. CONCLUSION The theoretical ad observatioal results agree with commo sese that a approximate three aosewds per meter error would be preset because of receiver coordiates beig offset. However, more importat facts were foud from the observatioal data. First, although the receivers used to collect the observatioal data met specificatio, there was a offset betwee them. I a follow-up observatio series, usig oe of these two receivers ad a third, this offset was foud to still be preset but of a differet value. (The offset values differed by 2-3 aosecods.) Further ivestigatio is eeded to resolve these differeces for higher precisio time trasfers. Secod, although keyed dual-frequecy receivers were used, evidetly there are some differeces betwee satellites. Averagig does decrease this effect. Higher accuracy time trasfers will require more ivestigatio of this effect. Oe eeds to kow if averagig is the right thig to do or if some problem must be fcied. REFERENCES [I] W. Lewadowski 1994, "GPS commo-view time trasfer, " Proceedigs of the 25th Aual Precise Time ad Time Iterval (PTTI) Applicatios ad Plaig Meetig, 29 November-2 December 1993, Maria Del Rey, Califoria, pp [2] "Departmet of Defese World Geodetic System 1984, ' 1992, Defese Mappig Agecy Techical Report

5 Time Error vs. Receiver Offset Figure 1 TIME ERROR Figure 2 Theoretical mius Experimetal Results - each satellite observatio averaged idepedetly 2 3

6 igure Theoretical mius Experimetal Results all satellite observatios averaged daily I I I - L Figure 4 a 4 O 3 s e 2 c 1 ; -1 Receiver Altitude Coordiate Offset each satellite observatio averaged idepedetly S Figure 5 a " 4 s 3 2 C 1 d s Theoretical mius Experimetal Results each satellite observatio averaged idepedetly 14

7 Figure 6 heo ore tical mius Experimetal Results all satellite observatios averaged daily Figure 7 Theoretical mius Experimetal Results each satellite observatio averaged idepedetly a 4 s 3 e 2 C 1 " d - S I Figure 8 Theoretical mius Experimetal Results all satellite observatios averaged daily a 6.4 s 6. e c d c -

8 Figure 9 Receiver Logitude Coordiate Offset each satellite observatio averaged idepedetly a 3 2 O 1 S -1-2 O -3-4 " " 1 ' 1 " ~ ' " " 5 - s Figure 1 -* Theoretical mius Experimetal Results each satellite observatio averaged idepedetly a 33 2 e 1 c -1 d -2 s i Figure 11 Theoretical mius Experimetal Results all satellite observatios averaged daily

9 Questios ad Aswers WLODZIMIERZ LEWANDOWSKI (BIPM): The receivers you had compared, they had exactly the same software or were they differet? HAROLD A. CHADSEY (USNO): These were two idetical receivers ruig the same software ad firmware iterally. WLODZIMIERZ LEWANDOWSKI (BIPM): The differeces were ot comig, for example, aomolies from the software? HAROLD A CHADSEY (USNO): It defiitely was't a problem of oe was a TrueTime receiver ad oe was a S-TEL or somethig like that. There is a slight possibility that there may have bee a small fractioal differece i the software. But i talkig with the maufacturer, they said that those two receivers had the same software ad same firmware versios i them. Ad whe they left the factory, they were calibrated.