Vector Closure Analysis
For determining the error of closure, simultaneous observations were carried out for three or four points. Out of these observations, triangles were formed by joining three points. Baseline vector has to be computed independently for each side of triangles by taking one station as the reference and others as rovers. Two different studies were carried out to evaluate the effect of duration of observation on vector closing error and geometry on vector closing error.
Effect of duration of observation on closing error
In order to analyse the effect of observation time on closing error; an experiment was carried out on two different triangles by using observations for 6 hrs. The common observation between the reference and rover points were divided into one hr, 2 hrs, 3 hrs and 4 hrs respectively. For each time interval, the baseline vector for all sides of the triangles were computed independently and closing error were computed. The results for two triangles were tabulated in tables 3.3 and 3.4. For tringle1, slope distances between the stations were 1.7 km, 0.5 km and 1.3 km respectively and for triangle 2; slope distances were 10.8 km, 8.3 km and 2.5 km.
| Triangle | Base Station Slope Distance |
| T3 | 11.6 km, 0.007 km, 11.6 km |
| T4 | 1.6 km, 0.022 km, 11.6 km |
| T5 | 11.6 km, 0.18 km, 11.8 km |
Effect of observation point geometry on closing error Error of closure for the present study was carried out for three triangles formed from simultaneous observations and the results are given in table 3.4. Three triangles were formed (T3, T4 and T5) and the slope distances between the base stations were as follows.
Results and Discussion
In order to evaluate the performance of various ionospheric models
on the accuracy of dual frequency observations, GPS data was
processed for five different ionospheric models. The analysis shows
that there is not much variation in the results of the various
ionospheric models for smaller observation time say 1/1.5 hrs. For
longer hours of observation computed model gives better accuracy.
Tables 2 gives the comparison of baseline differences, between
reference and rover points, computed using single and dual frequency
receivers. It has been observed that residual error in latitude, longitude
and height is around 0.50m, 0.25m and 1.0m respectively. The
difference of observations with in 1 m between single and dual
frequency receivers suggests that the baseline of 10 – 15 km hardly
makes any difference in positional accuracy (~ 1m), only
observation time matters.
Table 3.1 gives the comparison of baseline distances calculated
from the positional coordinates obtained using single and dual
frequency receivers with the actual ground distances. Table 3.1 shows
that variation is around ± 0.15 m.
Table. 3.1: Comparison of baseline distances with actual ground distances
| Stations | Distance Computation | Actual GroundDistance (m) | Residuals w.r.t. Actual Ground Distance |
| Single (m) | Dual (m) | Single (m) | Dual (m) |
| 01 – 02 | 768.067373 | 768.264363 | 768.2 | 0.132627 | -0.064363 |
| 01 – 03 | 3505.145652 | 3505.341638 | 3505.2 | 0.0543480 | -0.141638 |
Table 3.2 shows that one hour observation is sufficient for baselines upto two kms to achieve an accuracy of 1 ppm or better.
Table 3.2
| Time of observation | Closing error (cm) |
| dx | dy | dz |
| 1hour | - 0.03 | -0.07 | 0.00 |
| 2hours | -0.01 | -0.02 | 0.02 |
| 3hours | 0.00 | -0.02 | 0.01 |
| 4hours | 0.01 | -0.01 | 0.02 |
Table. 3.3
| Time of observation | Closing error (cm) |
| dx | dy | dz |
| 1 Hour | -0.18 | -0.51 | -0.44 |
| 2 Hours | -0.03 | -0.24 | -0.16 |
| 3 Hours | 0.05 | -0.06 | 0.01 |
| 3.83 Hours | 0.09 | 0.08 | 0.24 |
Table. 3.4: Effect of geometry on closing error
| Triangle | Closing error (cm) |
| dx | dy | dz |
| T3 | - 33.26 | - 15.52 | - 11.59 |
| T4 | 4.8 | - 10.63 | 3.09 |
| T5 | - 8.98 | - 12.56 | - 6.69 |
Table 3.3 shows that to achieve the above accuracy for a baseline of 10-15kms, two hours of observation is required.
Table 3.4 indicates that triangle T3 shows maximum closing error in all the three dimensions, which could be attributed to the poor geometry (very small base in comparison to other two sides) of the triangle. Moderate closure errors are seen in the case of T4 and T5 triangles. This could be because of small observation time and up to some extent geometry too.
Conclusions
The present experiment has given significant insight in the overall understanding of the various elements related to GPS measurements towards establishing the precise coordinates of the ground control points. The comparative evaluation of the performance of single frequency receivers vis-a vis dual frequency have been carried out. Also the impact of different ionospheric models on the accuracy of measurements have been studied. The baseline distances computed using single and dual frequency receivers show difference of less than a metre. The baseline distances as computed with differential GPS measurements showed a deviation of ± 0.15 m from the actual ground distance. Validation results also indicates that one hour observation is sufficient for baselines up to two km to achieve an accuracy of 1 ppm or better. However to achieve the same accuracy for a baseline of 10 to 15 km, two hours of observation is required.