DESIGNATING THE ERROR OF VERTICAL COORDINATE OF AIRCRAFT POSITION IN THE GPS SYSTEM

Designating the error of vertical coordinate of Summary . This article presents the results of research concerning the determination of an error of the vertical position of an aircraft during a flight, taking into account, the flight altitude and the values of the ZTD tropospheric product. In particular, this work focuses on the impact of an error in designating the ZTD tropospheric product on the correct determination of a flight altitude. The authors of this work performed a scientific experiment for research data of two flights made by a Cessna 172 around the aerodrome in Dęblin. This article shows the findings of the tropospheric ZTD product specified for the single-frequency SPP positioning method and dual-frequency PPP positioning method. Based on the obtained results, it was discovered that the error of the ZTD tropospheric delay causes an altitude error from 0.08 m at the zenith angle equal to 80 o even to 0.79 m at the zenith angle equal to 85 o .


INTRODUCTION
The emergence of GNSS on-board receivers, on the market, has significantly facilitated the navigation exploitation of the aircraft. The use of GNSS on-board receivers outside the usual navigation functions provides a lot of additional information, for example, connected with specifying atmospheric parameters, that is, the ionospheric and tropospheric delays [5]. Thus, the ionosphere delay constitutes a dispersion part of the atmosphere, whereas the troposphere delay is a neutral part of the atmosphere for GNSS signals [7]. The impact of the ionospheric delay is particularly important in the GNSS differential measurements since it causes vector scaling between the reference station and the on-board GNSS receiver [3]. On the other hand, the tropospheric correction is reflected in determining the aircraft vertical coordinate. An incorrect determination of the aircraft altitude may consequently lead to an air incident or accident. It is, therefore, crucial to conduct scientific research with regard to the determination and monitoring the value of the tropospheric correction in the GNSS satellite measurements in aviation.
The basic equipment of an aircraft is a single frequency GNSS receiver. In this case, the impact of the tropospheric delay is designated using deterministic models [2]. For this reason, the tropospheric delay is added to the geometric distance between the satellite and the receiver in an observation equation in the Single Point Positioning (SPP) code method [6], similar to an observation equation in GNSS phase measurements. In the SPP method, the tropospheric correction is defined by means of the Slant Troposphere Delay (STD) values. The STD value comprises SHD (Slant Hydrostatic Delay) parameters and SWD (Slant Wet Delay) parameters [8]. The SHD and SWD components are calculated as follows: -SHD is determined on the basis of ZHD values (Zenith Hydrostatic Delay) and the mapping function, -SWD is determined on the basis of the ZWD value (Zenith Wet Delay) and the mapping function.
The sum of the ZHD and ZWD parameters in its final product is defined as ZTD (Zenith Troposphere Delay [4]. In case of on-board dual-frequency GNSS receivers, the scheme of designating the tropospheric correction varies. Namely, the ZHD parameter is still determined using the deterministic model, whereas, the ZWD parameter is designated in the stochastic process. To determine the troposphere correction employing the data from the dual-frequency GNSS receiver, the Precise Point Positioning (PPP) method is used [10]. Ultimately, the ZTD tropospheric product is defined as a sum of the ZHD component from the deterministic model and the ZWD component from the stochastic model.
In this paper, the authors present and describe research results concerning the determination of an aircraft vertical error coordinate in the aspect of applying the ZTD tropospheric correction. Furthermore, the authors designate the ZTD parameter based on the single-frequency SPP code method centred on GPS data from an on-board receiver. In addition, the authors of this work also determine the ZTD parameter, using the dual-frequency PPP method based on GPS data derived from an on-board receiver. The difference in the ZTD parameter calculated in the SPP and PPP method allows designating the aircraft vertical coordinate. The calculations are based on real GPS data derived from the on-board GNSS receiver mounted in the Cessna 172. The registered GPS data comes from two air tests performed on the same day (16 June) around the aerodrome in Dęblin.

RESEARCH METHODOLOGY
The lack of the aircraft vertical coordinate of the aircraft is determined using the following dependence [1]: where: dh -error of vertical coordinate of aircraft (expressed in metres), dZTD -error of Zenith Troposphere Delay (expressed in metres), z -zenith angle (expressed in degrees).
The parameter dZTD is determined from a relationship as below: where: SPP ZTD -Zenith Troposphere Delay calculated from single-frequency SPP method (expressed in metres),

PPP ZTD
-Zenith Troposphere Delay estimated from dual-frequency PPP method (expressed in metres).
Finally, after a transformation, we receive: Then, the parameter of the zenith angle z can be defined as the completion of the elevation of the angle as shown below: where: El -elevation angle (expressed in degrees).
The relationship from Equation 3 shows the error impact of designating the ZTD parameter upon the accuracy of determining a flight altitude for single-frequency GNSS observations. In is determined from the deterministic model in the SPP code method, whereas the parameter PPP ZTD constitutes the reference value of the ZTD parameter, computed in the PPP measurement technique.

THE RESEARCH EXPERIMENT
In the research experiment, the authors designated the aircraft vertical coordinate based on the received data about the ZTD parameter and with regard to the zenith angle. The experimental test was carried out for GPS data coming from two air tests, executed on the same day.  [9]. Furthermore, the ZTD parameter for the SPP code method was determined based on the troposphere model of Saastamoinen. In this case, the ZTD parameter was determined using the deterministic model of the tropospheric delay. In the second one, the authors designated the ZTD based on the PPP measurement technique. The calculations of the position of the Cessna 172, using a dual-frequency GPS code and phase measuring technique in the PPP measurement method was made in the CSRS-PPP v.2.26.0 programme [11]. Furthermore, the ZTD parameter for the PPP measurement technique was determined based on the troposphere VMF model. The ZHD parameter was determined from the deterministic model, whereas the ZWD parameter from the stochastic model of developing GPS code-phase observations. Thereafter, a comparison of the ZTD value was made in accordance with formula (2) to determine the tropospheric delay error. Finally, the parameter error of the vertical aircraft coordinate was determined in accordance with formula (3). It should be added that the calculation adopted boundary values of the zenith angle from 80 o , to 85 o , with, with a leap every 1 o . The numeric calculations were made in the Scilab v.6.0.0 [12].

RESULTS
The obtained results are depicted in Figs. 3 to 6. In the first place, the values of the parameter dZTD for flight no 1 were depicted in Fig. 3. The values dZTD range from 0.014 to 0.039 m. Furthermore, the average value of the parameter dZTD equals 0.020 m.  Fig. 4, it is worth stressing that the maximum values are noticeable when the parameter also takes the highest values. On the other hand, when the flight altitude increases, the parameter dh rises as well.

CONCLUSION
This article presents research findings describing the vertical coordinate error of an aircraft during the execution of experimental flights. This work shows the algorithm of designating the vertical component error of the aircraft position in the function of the ZTD troposphere product error as well as the zenith angle. The scientific experiment was supported by the conducted investigations, in which the research material from two air tests was used. The flights were executed by the Cessna 172 over the aerodrome in Dęblin. This article determines the ZTD tropospheric product on the basis of the deterministic model for the SPP code method. Moreover, the ZTD reference value was defined for the SPP dual-frequency measurement technology. The difference in parameter ZTD has allowed estimating the impact of the tropospheric delay on the designation of real aircraft altitude. Based on the obtained results of the investigations, it was discovered that in test no 1, the error of designating the altitude coordinate of the aircraft equals between 0.078 m for the zenith angle of 80 o to 0.443 m for the zenith angle of 85 o , -in test no 2, the error of designating the altitude coordinate of the aircraft equals between 0.293 m for the zenith angle of 80 o to 0.781 m for the zenith angle of 85 o , -an increase in the error parameter of the ZTD product results in an error increase of designating the altitude coordinate.
The test results can be used in practice for the application of the GPS navigation system in air transport. Moreover, the algorithm presented in this article can also be implemented in the GLONASS system in aviation. The presented work may have a significant impact on research related to the monitoring of the troposphere in air operations and in aircraft navigation.