A PROPOSED GPS METHOD WITH MULTI-ANTENNAE AND SINGLE RECEIVER FOR THE IMPROVEMENT OF BASELINE HEIGHT COMPONENT
It is well known that GPS (ellipsoidal) height determination is weaker than that of horizontal coordinates. The weakness of GPS height determination can explained by the following facts: i).a high degree of correlation exists between the vertical coordinate and the clock parameters, as well as with the tropospheric delay parameters; ii) GPS satellites are only visible above the local horizon.
With the standard GPS field operation, clock parameters may be explicitly estimated in a GPS least squares adjustment with between-receiver single difference observations or they may be implicitly taken into account with a double difference observations processing scheme. The standard field operation consists of two or more receivers operating simultaneously without physical links between them. The proposed GPS method consists of two or more antennae connected to the same receiver for use in small monitoring networks (e.g., dam deformation monitoring) with baselines a few kilometres in length. In this scenario, the between-antenna single difference observations do not contain receiver clock errors, but careful calibration of the relative signal delay throughout the hardware (especially antenna cables) has to be performed. Because these single difference observations are free of clock parameters, more geometrical strength remains to determine the baseline components.
This paper is a theoretical study to demonstrate the feasibility of the proposed method and its usefulness to improve baseline height determination. In this study, the following topics are addressed. First, the observation equations are reviewed along with a discussion about error modelling with emphasis on intercable bias calibration. Simulation results for ambiguities-fixed solutions are presented and analysed for equatorial, mid-latitude and polar sites. For the proposed method, the impact of the propagation of intercable bias into station coordinates is studied. The propagation of systematic tropospheric errors and random measurement errors into GPS station coordinates for the proposed method is compared to standard data processing. The next paragraphs summarize the results of this study.
To determine height component to the mm-level, intercable biases must be calibrated at the same level. With the proposed GPS method, zero baseline tests (1 antenna, 2 cables and 1 receiver) are well suited to calibrate intercable biases. The change of cable length due to temperature variations can be taken into account with the knowledge of the thermal coefficient of delay of the cables and with the measurements of temperature at different points along the cable routes. For the proposed method, fiber optic cables would be preferred to coaxial cables because they have lower thermal coefficients and lower attenuation, they are unaffected by electromagnetic interference and they have excellent stability properties.
The tropospheric delay error is the main error source affecting baseline height components for small networks. For the standard method, the tropospheric zenith delay error is mainly magnified in the height component by a factor ranging between 2.6 and 6.5 (for elevation mask angles of 20° and 10° and for different latitude sites). For the proposed method, the magnification factor ranges between -1.9 to -4.6. If relative tropospheric zenith delay are suspected to remain after data modelling, a relative tropospheric zenith delay parameter must be estimated. This parameter will absorb the tropospheric error, but this has the consequence to amplify the propagation of measurement noise (random error) into the height component.
For standard data processing (without a tropospheric delay parameter), the height standard deviation is 2.3 to 2.7 times larger than the standard deviation of the horizontal coordinates, for equatorial sites and 2.6 to 3.0 times larger for mid-latitude sites. The ratio is 4.1 to 5.7, for polar sites. Moreover, for mid-latitude sites the standard deviation of the north component is 1.4 times larger than the standard deviation of the east component.
The height standard deviation varies quite significantly for different parameter combinations. If a tropospheric parameter is estimated with the standard data processing the height standard deviation becomes 2.4 to 3.7 times larger, for equatorial and mid-latitude sites and 3.0 to 5.5 times larger for polar sites. For the proposed method (without a tropospheric delay parameter) the height standard deviation is smaller (with respect to standard method without a tropospheric delay parameter) by a factor 2.7 to 3.6 for equatorial sites, 2.5 to 3.3 for mid-latitude sites and 3.1 to 5.1 for polar sites. Finally, even if a tropospheric parameter is estimated together with the proposed method, the height standard deviation is still 2 times smaller than the one associated to standard data processing (without a tropospheric delay parameter), for all sites and elevation mask angles. In other words, for the proposed method, the height standard deviations are comparable to those of the horizontal components, even if a tropospheric parameter is solved for (exception of polar sites).
It is shown that the proposed field operation and its associated data processing significantly improve GPS height determinations compared to standard GPS data processing schemes. The proposed field procedure is more cumbersome (long physical link between antennae and receiver) and requires careful relative cable calibration. However, for special precise applications (e.g., small networks with permanently installed cables) additional efforts can be justified. The benefit is the substantial improvement of GPS height determination. It is hoped that this work will stimulate further, hardware oriented, research along the same lines.