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 11^h 58^m 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 23^h 56^m. 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.

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 25^d 12^h 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.

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. 23^d 15^h 45^m 00^s 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 10^th 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.

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.