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Absolute GPS to better than one meter -- [page 1 of 4]

Page 1      Page 2      Page 3      Page 4

C.O. Alley - Physics Department, Rm 2201 / University of Maryland / College Park, MD 20742

T. Van Flandern - Meta Research / PO Box 3504 / Sequim WA 98382-5040 / <tomvf@metaresearch.org>

We have analyzed raw SA-free two-frequency pseudorange and accumulated delta range data collected at the five Air Force monitor stations for 19 GPS satellites at 1.5-second intervals over a four-day time span in August, 1993. After the standard corrections for ionosphere and troposphere, and solving for a single bias correction and one rate correction per day for each clock over the four-day span, the rms of the difference between these observations and accurate JPL-determined orbits is 2.3 meters. With one clock rate correction every 16 hours and improvements in the two-frequency ionosphere modeling, that rms drops to 1.2 meters. We have now identified additional corrections that are expected to reduce this rms to 0.8 meters or less. The limit set by the present noise level is apparently about 0.2 meters. If the generality of the corrections we derived can be confirmed for another time span, it should become possible to utilize these corrections to improve the broadcast ephemeris information so that suitably programmed GPS receivers will provide station coordinates with an accuracy of better than one meter.

1. Input Parameters

The Global Positioning System (GPS) is a network of 24 satellites, each orbiting the Earth every 12 hours, with atomic clocks on board continually transmitting the time and location of the satellites. This enables receivers on the ground to measure the time of reception of the signals from several satellites, where each signal travels at the speed of light (one foot per billionth of a second). Then the signal arrival delays tell the receiver how far away each satellite is (called the "pseudo-range"). Finally, from the known positions of the satellites and the pseudo-ranges, the receiver can triangulate and determine its own precise three-dimensional location.

In this paper, we already know the locations of the receivers very well, but are interested in studying the intrinsic accuracy of the observations and existing analysis methods. The observed pseudo-range values generally differ from the predicted ones by a few meters, yet the intrinsic accuracy of the data should be an order of magnitude better than that. We analyze these differences and find several possible modeling improvements that will greatly improve the agreement between observations and predictions.

The data analyzed for this report consists of about 6,000,000 raw pseudo-range measurements at 1.5-second intervals recorded for 19 satellites in two frequencies at the five Air Force monitor stations during the slightly more than 4-day period from 15h GPS time on 1993 August 23 to 16h GPS time on 1993 August 27. (GPS time differs by a few seconds from Universal Time or Greenwich Mean Time.) This time span is from Julian date 2,449,223.125 to 2,449,227.167. All data used is unaffected by the Selective Availability (SA) or Anti-Spoofing (AS) security measures, and therefore represents the full precision that the system is capable of producing.

The five Air Force monitor stations are identified in Table 1, along with their coordinates used as inputs to this analysis. The data analysis can then be used to determine small corrections to these nominal positions. The site numbers are as used by the Air Force to identify its own stations in the data records. The stations are listed in order of longitude eastward around the globe from the Colorado Springs command center. Here and throughout this paper, brackets "[ ]" around symbols indicate quantities measured in the Earth-centered Earth-fixed (ECF) coordinate system. The same symbols without brackets are measured in the Earth-centered inertial (ECI) coordinate system, defined below. The ECF system has its [z] axis pointing toward the Earth’s north pole, its [x] axis in the equator toward the meridian of Greenwich (the zero meridian of longitude), and its [y] axis in the equator 90 ahead of the [x] axis in the direction of rotation.

# ID Descr. [x] [y] [z] E. Long. Lat. Ht.
7 CSOC Colo.Springs -1248597.172 -4819433.227 3976500.179 -104.525 +38.803 1909
6 ASCE Ascension 6118524.215 -1572350.831 -876464.163 -014.412 -07.951 +104
4 DIEG DiegoGarcia 1917032.272 6029782.265 -801376.134 +072.363 -07.267 -067
3 KWAJ Kwajelain -6160884.566 1339851.534 960842.885 +167.731 +08.722 +037
5 HAWA Hawaii -5511982.238 -2200248.261 2329481.530 -158.239 +21.561 +425

Table 1. Air Force monitor station rectangular coordinates & height above sea level in meters.

The GPS constellation when data was acquired in 1993 is described in Table 2, including some approximate orbital elements near the epoch of the observations. The first column is a simple sequential numbering of the 24 operating satellites in orbit at that time. An asterisk after this sequence number indicates the 19 satellites for which data was provided by the Air Force and used in this analysis. The Air Force’s criteria for selecting these satellites are unknown to us. Several other satellite identifiers in common usage are also listed:

  • PRN – Pseudo Random Noise number, the identity of the satellite as determined by the receiver. Since all GPS satellites transmit on the same frequency, they are distinguished by their pseudo random noise code. Some receivers list the satellites as "SV #", but really mean PRN #, NOT SVN #.

  • SVN – Satellite Vehicle NAVSTAR, also known as NAVSTAR number. This is the satellite identification used by the Air Force, and that will be used by default throughout this paper.

  • NORAD – now US SpaceCom, number is a sequential number of objects in orbit.

  • COSPAR – Committee On Space & Atmospheric Research (international), number is year of launch + sequence of launch in year + letter designating object launched.

The approximate orbital elements listed are for the epoch t0 1993 August 25d 15h 30m GPS time = Julian date 244 9225.145833, which is the mid-point of the span of observations. a = semi-major axis, e = eccentricity, I = inclination to equator, w = argument of perigee, W = right ascension of ascending node, l0 = mean anomaly at epoch t0, n = daily mean motion. The mean anomaly l at other epochs t is given by l = l0 +n (t - t0). 

#

PRN SVN NORAD COSPAR a e i w W l0 n
01* 01 32 22231 1992 079A 26558115 0.003924 54.76 294.92 210.81 163.16 722.12
02* 02 13 20061 1989 044A 26562171 0.012247 54.81 201.58 328.51 323.71 721.95
03* 03 11 16129 1985 093A 26558858 0.013420 64.53 140.64 52.27 159.17 722.09
04* 07 37 22657 1993 032A 26557684 0.005812 55.00 197.14 29.35 265.59 722.14
05* 09 39 22700 1993 042A 26559442 0.002251 54.70 298.35 270.26 186.61 722.07
06* 12 10 15271 1984 097A 26557655 0.013630 62.53 345.45 288.00 119.30 722.14
07* 13 09 15039 1984 059A 26559235 0.005084 63.77 214.21 50.31 205.02 722.07
08 14 14 19802 1989 013A 26560031 0.003647 55.09 171.04 151.44 241.15 722.04
09* 15 15 20830 1990 088A 26560610 0.007131 55.29 105.84 92.13 22.00 722.02
10* 16 16 20185 1989 064A 26559290 0.000592 54.91 210.79 152.10 76.22 722.07
11* 17 17 20361 1989 097A 26558498 0.007369 55.36 98.53 94.07 157.52 722.10
12* 18 18 20452 1990 008A 26561111 0.005194 54.09 69.60 208.55 253.97 722.00
13* 19 19 20302 1989 085A 26559055 0.000601 53.75 328.59 269.07 292.72 722.08
14 20 20 20533 1990 025A 26561498 0.004396 55.14 87.51 328.94 303.70 721.98
15* 21 21 20724 1990 068A 26561801 0.010736 54.73 152.10 149.83 10.05 721.97
16 22 22 22446 1993 007A 26560340 0.006483 54.81 336.83 329.38 328.93 722.03
17* 23 23 20959 1990 103A 26562754 0.007361 54.94 220.26 151.66 328.70 721.93
18* 24 24 21552 1991 047A 26563234 0.004792 55.59 229.35 89.61 124.88 721.91
19 25 25 21890 1992 009A 26561183 0.005883 54.31 155.49 268.95 230.62 722.00
20* 26 26 22014 1992 039A 26560403 0.008255 54.93 290.28 209.52 273.36 722.03
21* 27 27 22108 1992 058A 26559437 0.010533 54.52 129.80 269.59 103.84 722.07
22 28 28 21930 1992 019A 26559945 0.006324 55.45 161.60 29.67 72.73 722.05
23* 29 29 22275 1992 089A 26562918 0.005331 54.69 257.90 208.35 99.98 721.92
24* 31 31 22581 1993 017A 26562207 0.004231 54.96 38.70 29.55 161.05 721.95

Table 2. Satellite identifications and orbital elements. Asterisks indicate satellites used in this analysis.
a is in meters, i, w, W, l0 are in degrees, n is in /day.

All GPS satellites have orbital periods of about 11h 58m so they repeat the same ground tracks after two revolutions, since the Earth’s spin period in space (as opposed to its period with respect to the Sun) is 23h 56m. The orbital heights correspond to roughly four Earth radii from the center. The orbits are nearly circular, with the largest eccentricities being well under 2%. The first three of these satellites have inclinations of around 64, where the right ascension of ascending node changes at a rate of -0.032/day; while the later satellites have inclinations of about 55, where the nodal rate is -0.041/day.

Each satellite is launched with four atomic clocks on board. During the data period for this analysis, SV 10, 11 & 25 had rubidium clocks in use, while the others were using cesium clocks. Preliminary corrections to all the active clocks, as needed to synchronize them with the U.S. Naval Observatory (USNO) Master Clock, were provided by the Naval Research Laboratory, and are shown in Table 3. These are taken as first approximations. The data analysis itself determines the most appropriate clock corrections.

#

MJD

PHASE

FREQ.

REMARK

03

49224.500

-514110

+000611

KWAJ

04

49224.500

-648460

-001510

DIEG

05

49224.500

-438170

-000458

HAWA

06

49224.500

+004860

-000270

ASCE

07

49224.500

-388300

+000462

CSOC

08

49225.000

+000000

+000000

USNO

09

49224.500

+649260

+001384

10

49224.500

-719130

-085185

rubidium

11

49224.500

-607638

-008194

rubidium

11

49224.592

+006278

+000000

13

49224.500

-041680

-001887

14

49224.500

+002230

+000204

15

49224.500

+024880

+001898

16

49224.500

-059790

-001999

17

49224.500

-031110

-000753

18

49224.500

-001630

-000249

19

49224.500

+110470

+002480

20

49224.500

+025910

+001609

21

49224.500

-003300

-001439

22

49224.500

+046860

+003439

23

49224.500

+000430

+000223

24

49224.500

-097120

-002432

25

49222.300

-240160

+055050

rubidium

25

49224.500

+003740

+002966

26

49224.500

-005380

-001806

27

49224.500

+036920

+001333

28

49224.500

+028390

+002501

29

49224.500

+013290

+000668

31

49224.500

+033440

+002688

32

49224.500

-052160

-002295

37

49224.500

+020830

+002369

39

49224.500

-001840

-000410

Table 3. Clock corrections for ground stations (#03-08) and satellites (SV #09-39). MJD = modified Julian date, phase in ns, frequency in pp1015.
Note reversed signs for satellites (see text).

The first five entries in Table 3 are for the Air Force monitor station clocks. Unlike ordinary GPS receivers, these monitor stations have their own atomic clocks. The next entry, #08, is just a reminder that the USNO Master Clock is the standard for absolute time. The remaining entries are for the GPS satellites. A double entry indicates a change in the satellite clock settings (phase or frequency or both) at the specified epoch by the specified amount, which must be taken into account at all times after that epoch. Modified Julian date (MJD) conforms to the official International Astronomical Union definition, Julian date - 2,400,000.5, where the leading digits are dropped to shorten it, and the one-half day difference is to allow MJD to start at Greenwich midnight instead of noon. A variety of other unofficial and ad hoc usages of the term "modified Julian date" in the GPS community are ignored here. MJD 49,224.5 is 1993 August 25d 12h GPS time.

The column "phase" in Table 3 is the offset of that clock in nanoseconds (ns) when the USNO Master Clock ticks an integral second. For ground clocks, a positive phase means that clock ticks after the Master Clock, and therefore needs a positive time correction. For satellite clocks, the sign is reversed: a positive phase means the clock ticks before the Master Clock. The frequency column shows the rate of the clock relative to the Master Clock rate in parts per 1015, and follows the same sign conventions as phase. No aging corrections (acceleration terms) are needed over this four-day span.

[X]

- 6051549.36

[Y]

+14797744.41

[Z]

+21334249.68

[X]

-2677.29493

[Y]

- 182.03584

[Z]

- 628.77318

Table 4. State vector (m, m/s) for SV 32 at 1993 Aug. 23d 15h 45m 00s GPS time.

Table 4 shows a typical satellite state vector (rectangular coordinates and velocity components, with velocities indicated by   )  as provided by JPL based on their best orbit determinations using all available data. These orbits are described as accurate to about 20 cm or so. Units are meters and meters/second. This sample is for SV 32 at epoch 1993 Aug. 23d 15h 45m 00s GPS time. As for the monitor station coordinates, these are measured in the ECF (rotating) system. JPL provides a new state vector for each satellite every 15 minutes of GPS time. A 10th order Lagrangian interpolation is used to derive values for times between the 15-minute points.

This satellite and epoch will be used in the sample calculation illustrating our data reduction procedure. The effects of polar motion and variations in Earth’s rotation rate are already included in these state vectors in order to make them truly Earth-fixed. This saves us from having to compute those corrections again in this analysis.

The two frequencies used by the GPS satellites are labeled L1 and L2. L1 is the primary transmission frequency at 1575.42 MHz, while L2 at 1227.60 MHz is usually unavailable except to authorized users. The fundamental frequency of the GPS system is 10.23 MHz, and the L1 and L2 frequencies are 154 and 120 multiples of the fundamental frequency, respectively. This frequency information is of importance primarily for calculating corrections for ionospheric propagation delays.

column

at L1

at L2

day of year 235 235
hour 15 15
minute 45 45
second (T) 0 0
SV # 32 32
monitor station # 3 3
demodulator # 7 8
pseudo-range at T-4.5s 24589841.72 24589845.38
pseudo-range at T-3.0s 24588885.86 24588889.98
pseudo-range at T-1.5s 24587930.47 24587934.13
pseudo-range at T 24586975.07 24586978.73
signal strength 4078 4082
delta-range at T-4.5s -219999.3551 -219985.4465
delta-range at T-3.0s -220954.8531 -220940.9428

delta-range at T-1.5s

-221910.3354 -221896.4257
delta-range at T -222865.7888 -222851.8791

Table 5. Sample of raw data from Air Force. Shows all information for one satellite-monitor station pair over a 6-second interval for both frequencies (L1 and L2).

A sample observation from the raw data as provided by the Air Force is shown in Table 5. This also refers to SV 32 at the same epoch as in Table 4, as seen from the monitor station at Kwajelain. Data as measured at both frequencies, L1 and L2, is listed. Day of year 235 in 1993 is August 23. The column labeled "second (T)" specifies the exact epoch of GPS time for transmission of the signal from the satellite. It is always a multiple of six seconds for this data. Monitor station numbers are defined in Table 1. The demodulator number identifies a particular receiver with certain specific cable lengths. So each demodulator will have its own unique signal delay of a few meters, which must be determined from the data. The signal strength is on a scale of 0-4096, and can be a warning of spurious signals or bad data when it is low. The pseudo-ranges in meters are provided at the epoch T and at each of three 1.5-second intervals prior to epoch T. The accumulated delta-ranges, also in meters, are provided at the same four epochs. See the precise definitions of these measures in section 2.

GPS time is closely related to International Atomic Time (IAT). When dealing with the rotating Earth, it will also be convenient to consider Universal Time (UT), which is loosely the equivalent of Greenwich Mean Time (GMT). UT comes in several varieties necessitated by variations in the rotation rate of the Earth. Of interest here are UT1, which is closely related to the Earth’s orientation with respect to the stars; and UTC, which is broadcast by the various time services, and commonly used to record observational data. Roughly once per year as needed, a leap second is added to UTC to keep it close to UT1. GPS time and UTC always differ by an exact integral number of whole seconds, plus a small difference usually less than 100 ns arising from the independent ways these time scales are determined by atomic clocks. The integer part in 1993 was GPS-UTC = 9 seconds. Our adopted formula for GPS-UT1 is in section 2.

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