To illustrate our calculation procedure with a numerical example, we begin with the ECF state vector of Table 4, which is for SV 32 at GPS time 1993 Aug. 23^d 15^h 45^m 00^s = JD 2,449,223.15625 exactly. If we need it for some non-tabular moment, we could obtain that by Lagrangian interpolation. Then we will compare the computed ranges and observed pseudo-ranges from Kwajelain at that epoch. The ECF state vector for the monitor station is from Table 1, where of course the velocity components in the ECF frame for a station fixed on the Earth are all zero. Additional quantities that will be needed for the calculations may be extracted from Table 2 through Table 8.

Our first step will be to rotate both of these state vectors into the ECI frame. For the satellite state vector, we use [2] to get UT1 from the GPS time, then [3] to get GMST_t, the Greenwich Sidereal Time at the time of satellite transmission. For the receiver state vector, the time of reception is found from [6]. Then we use [4] and [5] to rotate from ECF to ECI. The extracted values and results to this point are summarized in Table 9.

Next, we form the computed PR and ADR from [1], and take the observed PR and ADR at frequencies L1 and L2 from Table 5. It is somewhat arbitrary whether to apply corrections to the observed or computed values. But we will attempt to apply them so as to bring both values as close as possible to the actual geometric distance through space traveled by the signal. Therefore, clock corrections from Table 3 are applied to the observed PR and ADR, as are ionosphere and troposphere delay corrections from [7] and [8], the orbital relativity correction from [9], and the geometric offset correction from [11]. The solid Earth tide corrections from [10] will be used to correct the computed PR and ADR, since these change the true geometric range.

The final step is to form the (O-C) from the (Observed PR minus Computed PR). The results of computations for all these corrections and various intermediate quantities are summarized in Table 10. The final residual differs from this (O-C) in that additional corrections to the clocks, station coordinates, and the demodulator delays are derived from the analysis of all observations, as described in the next section.

5. Data Analysis

Reducing our data in this way, we can compute calculated ranges to compare with the observed PR values. The differences (O-C) in the sense of observed minus calculated can then be examined to determine how good our modeling has been. For each solution, we choose to analyze PR or ADR values, or some combination such as the average of the two. If we analyze ADR values, then each uninterrupted span of data during which the ADR value is accumulated requires solving for a single arbitrary additive constant to convert ADR into a pseudo-range-like quantity. Typically, this is one constant per passage of each satellite for each monitor station, or one constant for several hours of 1.5-second data. Because the ADR values need this additive constant, and because of occasional cycle slips corrupting the ADR counts, we prefer to use PR values alone. However, the higher precision of ADR measurements makes them very useful for reducing the scatter in PR measurements over any short arc.

Our second decision before performing an analysis run is what clock corrections to solve for. Typically, we must solve for one arbitrary offset correction and one rate correction for each clock in the system, both for satellite and ground clocks. Of course, one clock must be treated as the "master clock" providing the definitions of epoch and length of the second, or time would be indeterminate. For that purpose we adopt the Colorado Springs (CSOC) monitor station clock as the most closely tied to the U.S. Naval Observatory master clock. So we accept the clock corrections to the CSOC clock in Table 3 as exact, and solve for additional corrections needed to all other clocks.

Existing atomic clocks are not generally expected to remain well-behaved over a four-day time span because their rates are not stable for that long. Therefore we may choose to solve for additional clock corrections – for example, one additional rate correction for each clock each day. Because clock rate instability seemed to become significant in a bit less than a day, and to avoid confusion with 12-hour and 24-hour signals in the data, we adopted one clock rate correction every 16 hours for most of our analyses.

Our next decision is whether to solve for corrections to the adopted coordinates of the monitor stations, or whether to keep those coordinates fixed at their adopted values. For best results, corrections should be solved for, because the adopted coordinates are not as accurate as the precision our solutions can reach. However, sometimes other quantities of interest could become indeterminate if solved for simultaneously with station coordinates.

Each monitor station also has two delay corrections associated with it, mainly caused by cable lengths. The delay is different for the L1 and L2 band signals. These two delay corrections, one for each frequency, are called "demodulator" corrections. These corrections are not guaranteed to remain fixed over time because demodulators and cabling are sometimes changed. But over our four-day span we have assumed that the two corrections per demodulator do remain unchanged. There are potentially twelve odd-numbered demodulators associated with L1 and twelve even-numbered demodulators associated with L2 for each monitor station, totaling 60 pairs over the five monitor stations. However, ten of those pairs were inactive during this time span, leaving 50 pairs of demodulator corrections to be solved for.

We further note that the existence of two unknown demodulator corrections means that the two simultaneous measures of PR for L1 and L2, which we called PR₁ and PR₂ in Equations [7], may differ by a constant. But this will imitate a delay produced by the ionosphere. So to eliminate indeterminacy in our solutions, we solve instead for one constant bias in PR for each odd-numbered demodulator, except for one odd-numbered demodulator per monitor station for which no correction to PR is allowed. And we solve for one constant bias in (PR₂ - PR₁) for each even-numbered demodulator, except for one even-numbered demodulator overall for which no correction to (PR₂ - PR₁) is allowed. We chose the demodulators that seemed to need the smallest corrections in trial solutions as our reference demodulators.

Finally, we determine if there are any special parameters we might wish to solve for to test various theses about the data, and if we wish to use the entire four-day data span or some subset of it. We initially did not provide for corrections to the adopted satellite orbits, since those orbits were advertised to be good to of order 20 cm. However, we discovered that was apparently not always the case, as we will see in our discussion of the analysis.

So a typical analysis run might have almost 300 unknowns. However, this is only because we have so many receivers, satellites, and clocks in the GPS system. The partial derivatives of these unknowns are generally minimally correlated, and there are plenty of observations. So typical solutions are well-determined, and we seldom had an unexpected problem with matrices becoming nearly singular.