Single-Day GPS Position Estimates:
Solar or Sidereal Day?

GIPSY processing of the data sets revealed that the following days had poor pseudorange observables for one or more stations on the network: 01jan30, 01feb27, 01mar09, 01apr01, 01apr05, 01may11. For the sake of simplicity, these days were thrown out of the final data analysis.

The GIPSY software package outputs final position solutions in a binary format, and I used Dr. Kristine Larson's mythesis code to extract useful baseline statistics from these binary files. Using offsets and standard deviations reported by mythesis, I plotted east, north, and vertical offsets for the baseline measurements. For these offsets, the ENV system is defined at the first station of the baseline; therefore, offsets reported are for the second station relative to the first station of each baseline. Error bars and data points for baseline length offsets relative to the a priori baseline length estimation were also plotted. Using a simple weighted least squares MATLAB routine, a line was fit to the solar and sidereal data to clarify apparent trends.

MDO1-CAT1 baseline (long baseline)

Sidereal versus solar day processing did not show significant differences when viewing offsets in the north and vertical components of this almost 1400 km baseline. However, the east component and the apparent baseline length showed marked changes between sidereal and solar day processing, indicating that differences in the east component impacted the baseline calculation. Note that the solar and sidereal day data points are nearly identical for the first 20 days of processing; after this point the data trends begin to diverge significantly.

If one were to assume the same 'starting point' for the solar day and sidereal day baseline length estimations and propagate the trends outlined here forward for 180 days, the baseline length estimations would differ by only 2.9691 cm, or 2.13 x 10-6 % of the total baseline length. For both east and length least squares estimates, the difference between sidereal and solar day trends (i.e., slope [m] and intercept [b] of the best-fit lines) is outside the level of uncertainty inherent in the weighted least squares estimation (see "Standard deviations of fit" on each plot). Therefore, we can assume that the divergence of solar and sidereal best-fit lines is not a false correlation.

Below I have plotted the difference between each solar and sidereal day data point on each day for the MDO1-CAT1 baseline; these plots do not account for any formal errors, and exclude an extreme outlier at day 10 (not seen on the above plots as it lies outside the scale). The red lines are second-degree polynomial fits to the data (using MATLAB functions). As observed previously, there seems to be an increased separation between solar and sidereal day results for the east and baseline length components as the days progress. After approximately day 130, the differences begin to go back to zero according to the polynomial fit. The north component is reliably zero mean, whereas the vertical component shows a minor version of the same trend seen in the east and baseline length.

Basic statistics for the MDO1-CAT1 baseline indicate that sidereal and solar day processing have equivalent scatters about the best-fit line. The small differences in scatter about the mean (RMS, or root-mean-square) are easily accounted for by considering marked differences in slope for the best-fit line for the east and baseline length data.

CAT1-PVE3 baseline (short baseline)

Most of the trends observed above in the long baseline were also expressed in the short baseline's position components, but on a smaller scale. As before, sidereal versus solar day processing did not show significant differences when viewing offsets in the north and vertical components of this baseline. The plot of vertical offsets has a larger scale than the other CAT1-PVE3 baseline plots, to prevent deemphasizing the significantly smaller error bars for all other position components.

The east component and the baseline length still show marked changes between sidereal and solar day processing, indicating that the east component was the major contributor to baseline length variablity. Although divergence of the best-fit baseline curves is apparent to the eye, the difference after 180 days is only 0.22 cm, or roughly 6.39 x 10-6 % of the total baseline length. Although this is a statistically small variation, millimeter precision is very important to the geophysical community and cannot be ignored here. As before, the different trends of the best-fit lines to both the sidereal and solar data fall outside the level of uncertainty.

When differencing the data points for the CAT1-PVE3 baseline, there is a very minute trend in the east component, but the trend seems to be linear instead of curving back to zero as in the longer baseline. No apparent trend appears in any of the other position components, as the second-degree polynomial fit remains close to zero.

The RMS and 'scatter about the line' measurements are very similar for all position components. Considering the reduction in scale from the long to short baseline (reduced by 98%), this is not unexpected.

To analyze absolute station positions after different types of GIPSY processing, I took the ECEF station positions estimated for each day and converted the (X,Y,Z) coordinates into latitude, longitude, and height. Since the scientific community uses geodetic coordinates more often than Earth-centered Earth-fixed coordinates, determining any differences in latitude, longitude, and height measurements will help determine any effect these results would have on the positioning community.

There is little difference between solar and sidereal day position estimates for any of the stations in this network. Station MDO1, shown below, provides a typical example of trends observed in the position data; other stations which are part of this network show the same overall trends. The data points for sidereal and solar day processing are nearly identical for the first 40-70 days, depending on the station and the coordinate (latitude, longitude, or height). After 40 to 70 days, the difference in solar and sidereal day satellite geometries has shifted in time by 2 2/3 to 4 2/3 hours, or 11-19% of a solar day; therefore, the data sets have 81-89% or more of data in common up to this point. After 40-70 days, correlations seem to break down, i.e. the solar and sidereal day data points become increasingly dissimilar but do not diverge severely.

Averages of the latitude, longitude, and height as calculated by the sidereal and solar day methods demonstrate that the mean station position is not dependent on the processing method (see chart below). Mean values are reported to the same precision as given by GIPSY final position solutions. To give perspective, the difference between methods yields a geometric difference roughly corresponding to the following errors in (X,Y,Z) components:

• X: 1/10 meter
• Y: millimeter
• Z: centimeter