Usage of a Correction Model to Enhance the Evaluation of the Zenith Tropospheric Delay
Meriem Jgouta, *, Benayad Nsiri and Rachid Marrakh.
GITIL Laboratory, University Hassan II,
Faculty of Sciences Ain Chock Casablanca, Km 8 Eljadidaroad, P.B 5366, Maarif 20100, MOROCCO Correspondence Author
Abstract
Tropospheric delay values are calculated only from global positioning system observations; however, if the global navigation satellite system data sites are unavailable or unrepresentative, the tropospheric delay is degraded resulting in an incorrect estimation. Currently in the GPS processing, tropospheric delay is estimated by the determination of priori values of total delay using a correction model, e.g. Global pressure and temperature model with global mapping function. This article proposes three mathematical models for the zenith tropospheric delay correction based on real GPS data and calculated by the Bernese software. The first model uses logarithmic function and the second uses rational function. The third has a high level of accuracy and shows a consistently smaller variation over the other two. The variation as represented by the standard deviation is larger for the second model. These models are evaluated and compared to the correction data given by the national institute of geographic information.
Keywords: GNSS; pseudo-range; tropospheric delay; MSE;
logarithmic regression.
INTRODUCTION
Surveillance of airspace using electromagnetic emitting sources such as signals knows a great success thanks to the development of space technology including GNSS (Global Navigation Satellite System). However, the initial role of a GNSS system is to provide a global coverage of geo- positioning for civilian use. There are several GNSS in the world such as GPS (Global Positioning System), the United States), Galileo (European Union), GLONASS (Russia), QZSS (Japan), Beidou (China) and IRNSS (India) (Tay, 2012). These systems competing and complementary to GPS will increase the availability; also improve the safety and the reliability thanks to the information on signal integrity (Viandier, 2011). GNSS performance is defined by the parameters of the system (position of the satellites, signal type); they also depend heavily on signal propagation environment. Air Navigation is achieved by a combination of a variety of systems, such as systems based on space e.g. GPS, GLONASS, Galileo, which consist of satellites and infrastructure to perform the navigation function. The aircraft- based systems as INS (Inertial Navigation System), radar, ADS (Air Data Systems)) work on board autonomously and acquire measurements to generate the information necessary
for air navigation, the ground based systems as ILS (Instrument Landing System), MLS (Microwave Landing System), VOR (Very High Frequency Omni-Ranging System), DME (Distance Measuring Equipment) are mainly located and maintained in the vicinity of the airport to guide aircraft in both en route (VOR, DME) and in the initial and final phases of flight (ILS, MLS)) (Bhatti, 2007).
The navigation message contains various information necessary for positioning. This message is transmitted at two carrier waves L1 and L2 with frequencies of 1575.4 MHz and 1227.6 MHz respectively (Viandier, 2011). Several books describe different aspects of transmitted satellite signals, their detection in receivers, and the geodesic methods of calculation (Hofmann et al., 2007) GPS positioning is carried out either by carrier phase measurement or by measuring of the pseudo- distance. Since the increase in the noise level reduces GPS positioning accuracy to a few meters in the pseudo-range measurements, these measurements are uninterrupted and non-ambiguous (Saha et al., 2007) which promote the measurement on the carrier phase. According to the literature and in a simplified way, the equations that describe the observations of code and phase for a satellite (s) and receiver (r) are (Misra et al., 2006):
1 2 The pseudo-distance measured,
The phase measured,
The geometric distance between the satellite (s) and receiver (r),
Wave length, Phase ambiguity,
and the ionospheric and tropospheric errors, : The error caused by any signal reflections, : Receiver noise,
: Offset of the receiver clock relative to the reference time, Offset of the satellite clock with respect to the reference time.
Atmospheric delays are among the sources of errors that affect the passage of the signal due to the refraction. Some applications in geodetic field and atmospheric studies require high-precision GPS positioning with location accuracy down to a millimeter or less. This action can be performed by post-
processing of GPS data with an accuracy of 5 to 10 mm (Gradinarsky et al., 2002) In meteorology, the use of GNSS signals delay for the remote sensing of atmospheric water vapor is essential. The ZTD (zenith tropospheric delay) consists of two parts: a predominant hydrostatic part ZHD and a wet part ZWD with ZHD representing 90% of ZTD. This term can be accurately determined by using the air density contrary to the wet part of the delay which is difficult to assess as a result of the temporal and spatial variation of atmospheric water vapor. The ZTD is about 2.3 m sea level and reaches values above 2.3m for an angle of elevation of 10
° (Hofmann et al., 2007). From the analysis of GNSS data, the ZTD can be obtained with sub-centimeter accuracy (Byunet et al., 2009) The tropospheric delay is the shortest in the zenith direction and will become larger with an increase of the zenith angle (Sakidinet et al., 2012) Several studies have shown that the ZTD may be significantly improved by using a suitable stochastic model that characterizes the accuracy of GNSS measurements (Luo et al., 2008; Jin et al., 2010).
Knowing the geometry of the satellites, the quality of the mapping function and the data availability we can accurately estimate the ZTD (Luo et al., 2013). Therefore, being the most crucial aspect is the precise assessment of ZWD is the quality of the ZHD; it will be greatly affected if there is no real and meteorological data near the site (Luo et al., 2013). We can at times deduce the meteorological parameters, (P, T, Hr) in a specific site from the standard atmosphere (Karabatic et al., 2011). This data will be used for the determination of ZHD, ZWD and ZTD.
Regardless of whether there are no perfect measurements, these can only be flawed depending on the protocol selected and the quality of the instruments taking the measurements.
The purpose of this document is to provide a priori ZTD correction models evaluated according to GPS data.
GPS ERROR SOURCES
There are numerous possible origins of faults which can deteriorate the position precision calculated and as a therefore negatively influence the working of a GPS receiver. These fault origins can be principally classified into two categories, system wide faults and specific operating environment or specific GPS receiver faults. System wide faults encompass selective availability (S/A), ephemeris fault, satellite clock fault, troposphere and ionosphere fault. We will only talk about the troposphere fault impact in the paper. System wide faults and their impact which can be lessened partly or completely removed by modes of differential correction technique are summed up in the next figure.
Figure 1: System errors end their impact
TROPOSPHERIC DELAY
The modeling of tropospheric effects began in the late of 1960s to correct distance measurements. At that time, using lasers was the only means of precise positioning. After the advent of a parametric model of the zenith tropospheric delay with a projection function that varies depending on the elevation of the satellite it is still used today, it is expressed respectively in two modes: observation on the code and phase and the mapping functions as follows:
1
2 : The total tropospheric delay.
and : hydrostatic and wet zenith delays respectively.
and : Mapping functions.
and : The northern and eastern parts of the tropospheric gradient.
: Azimuth.
Elevation angle of the satellite.
3
4
5 The quality of mapping functions is essential for a priori correction and precise parameter estimation. It's modeling has evolved considerably since the first models which were introduced by (Hopfield, 1969; Saastamoinen, 1972), although the principle is still based on the same parameters the changes
are mainly worn on the parameterization of functions to monitor the temporal and spatial variations. Once more, the analytical representations derived from climate data have been replaced by weather prediction models (Böhm et al., 2006, 2009). A comprehensive review and validations of some of these models can be found in (Tuka et al., 2013). According to the literature, Dalton's law and from the ground data, ZHD and ZWD used to correct raw observations after projection in the signal’s direction. They are expressed by the integral of the index of refractivity:
3 : Empirical coefficient.
: Air pressure.
: Temperature.
: Height.
Figure 2: Estimated values of tropospheric refractivity coefficients.
The ZHD is proportional to the pressure on the ground and comprises the component due to the dry air. It has long been calculated from an analytical model based on a standard pressure of 1013 hPa at sea level.The use of a global climatology of pressure and temperature or analysis derived from weather prediction models (Böhm et al., 2006) are now the most widespread. In the microwaves field, the refraction, significantly affected by the content of water vapor in the atmosphere, varies in space and time. Thus, the ZWD is the main source of errors for the GPS observations and the point of divergence between all the proposed models promulgated in recent decades, is used as an estimated parameter for higher accuracy (Xu et al., 2011), and can be transformed into precipitable water vapor (PWV) with the weighted mean temperature (Yao et al., 2014). The most popular models in the GPS processing software are the Hopfield and the Saastamoinen(Dodo et al., 2010). The parametric models of the ZHD and ZWD are respectively expressed as follows:
The Hopfield model:
3
4 The Saastamoinen model:
3
4 : The height of the tropopause (empirical model).
( ): Standard meteorological data.
: Orthometric height.
): Constants relating to the dry atmosphere and the water vapor.
: Latitude.
: Acceleration of gravity.
: Gas constant.
DATA PROCESSING
Since the interface of the IGN site, there is access to tropospheric data calculated by computing a first time in two different formats: The SINEX format and the Bernese format.The model used for these calculations is the Niell GMF (Global Mapping Function). During the transition to the latest version of the software in April 2014, the IGN has changed the model to comply with the recommendations of the EPN. GPT model (Global Pressure and Temperature Model) is now used with projection function as GMF.
The ZTD parameters are estimated at the time of calculations.
The calculations are performed with the Bernese GPS Software, version 5.0, GPSEST model, using parameters recommended by the European research program for hourly calculations.
Table 1: Options of calculation for tropospheric parameters («
Network map | RGP » available on: http://rgp.ign.fr/).
Stations Coordinats A priori ZTD
The mapping function used
Minimum elevation angle Standard values
(x, y, z) fixed for each station From Dry Niell model Wet Niell mapping function 10°
T=18°C, P=1013.25 hPa et H=50%
SIMULATION AND RESULTS
Currently, in the GPS processing, we estimate the tropospheric delay as follows: Usage of a model to determine the priori values of the total delay (ZTD) and estimate the correction using these values.
3 : The delays in the direction of a satellite.
In the Bernese software, the estimation is made by least squares. The priori delay is not calculated from meteorological data each time; the initial position is used with a standard pressure and temperature model, it has a constant value during the day. Thereafter, the Niell mapping function is applied to project a priori model in the zenith and estimate a correction value. The sum of ZTD and is the result of Bernese software ZTDB. The difference between the ZTDB and ZTD is used to build a linear correction model:
3 In order to accomplish the study, we were based on GPS observations with 211 stations over a week in January, 2015.
An extract of the database on the station ACOR is given as an example:
Table 2. Result of hourly calculation dated 3 and 4 January 2015, D: day, H: hour, M: minute, CORR: ZTD correction estimated by Bernese Software.
STAT D H M ZTD (m) CORR(m) SIGMA ZTDB (m) ACOR 3 23 0 2, 3204 0, 15012 0, 00174 2, 47055 ACOR 3 23 15 2, 3204 0, 15013 0, 00175 2, 47067 ACOR 3 23 30 2, 3204 0, 15014 0, 00176 2, 47083 ACOR 3 23 45 2, 3204 0, 15015 0, 00177 2, 47144 ACOR 4 0 0 2, 3204 0, 15016 0, 00178 2, 47246 ACOR 4 0 15 2, 3204 0, 15017 0, 00179 2, 47377 ACOR 4 0 30 2, 3204 0, 15018 0, 0018 2, 47463
An average value of ZTDB is calculated for each station defined by its height Hs and MSE (Mean Squared Error); we obtained a data table of which an extract is shown below:
Table 3: ZTDB average value is obtained by a calculation of the ZTDB’s mean value of each station, MSE, and Hs.
STAT ZTD (m) ZTDB (m) MSE Hs (m)
AGDS 2, 3194 2, 455745 0, 01858587 68, 164 AILT 2, 283 2, 37634 0, 00871609 186, 797 AJAC 2, 3092 2, 422258 0, 01345368 98, 791 ALBI 2, 2817 2, 42806 0, 01828174 206, 966 AMB2 2, 1713 2, 2525112 0, 00659506 617, 612 AMBL 2, 2954 2, 3631224 0, 00458871 129, 929
An accurate prediction of tropospheric delay requires an effective correction of the model which depends on the Hs.
We draw the scatter plot characterizing for each
station using the data in Table 6, then we determine the correction of tropospheric error model:
3 Where:
3 And
3 We get:
4
Figure 3: Comparison between the first proposed model and the IGN correction values.
After the superposition of with the correction values, we can clearly notice in the simulation that there is a coincidence in trend curves, by measuring their dispersion there is an absolute difference of 0.0006. There is also a large gap between the scatter plot of the correction values and , a standard deviation of 0.004. Thus we can see many outlier values.
The second model is:
3 In the second model, the trend curves do not coincide comparing with the first (see fig.2 and fig.3). By measuring their dispersion there is an absolute difference of 0.0027, we can also see a restriction of the gap from the to and both outlier values, the standard deviation between the scatter plots of the correction values and
is 0.0029. In order to have a model perfectly describing the distribution of data given by Bernese, we will use a combination of
and deduced from the previous analysis.
Figure 4: Simulation of the second proposed model compared with the correction values.
The third model is:
3
Figure 5: Simulation of the third model compared with Bernese corrections values.
For each model and there is an advantage and a disadvantage, according to the simulations, we have noted that the correction values are perfectly covered by the model. Thus, the standard deviation between the scatters plot is 0.0005 and the absolute difference is 0.001.
With higher temporal resolution testing data and yearly statistical results are shown in Table 4. had the smallest bias.
The position error during the trajectory before and after correction by the third tropospheric model is shown in Figure 6 and 7. The performance of the model is better. The position error reduced by 70%. The improvement is due to the fact that the measurements were made mainly in good GPS
conditions, for which the stand-alone GPS performs relatively well.
Table 4: Statistical results of ZTD Bias/RMS for each model (211 stations).
Model ZTD Bias (cm) ZTD RMS (cm)
Mean Mean
0.147 10-4 0.004
0.84 10-5 0.0029
0.53 10-5 0.005
Figure 6: Positioning error before using the tropospheric correction .
Figure 7: Positioning error after tropospheric correction.
CONCLUSIONS
The neutral atmosphere introduces a delay in the propagation of microwaves depending on the geographical location and
0 50 100 150 200 250 300 350 400 450 500
-50 -40 -30 -20 -10 0 10 20 30 40 50
iteration
meters
Positioning error and estimated error range
North direction East direction Down direction
0 50 100 150 200 250 300 350 400 450 500
-30 -20 -10 0 10 20 30
iteration
meters
Positioning error and estimated error range
North direction East direction Down direction
climatic conditions. The accuracy of the GPS ranging is critical for GPS-based aircraft navigation in particular during the landing and taking-off. The estimation of this tropospheric delay is possible only through modeling.
This document aims to demonstrate by different simulations three mathematical models from which we have chosen an appropriate model for a precise correction of a priori total zenithal tropospheric delay. The outcome of this work is:
Establishing a logarithmic relationship for the correction of ZTD in terms of the height of the station Hs through the data analysis of two hundred and eleven stations for a week.
Considering the models mentioned above, the third one reflects well the values that have already given by the IGN,
There is a decrease in outlier values.
The deviation between the values calculated from proposed , and the values given by IGN is 0.0005. Therefore, it is evident that there is a small
margin compared to and .
We still expect several possibilities to improve the correction of the tropospheric delay by:
Using other GPS data processing software, such as Gamit, Gipsy, Gins…
Testing other ZTD models, Exploring other study areas.
REFERENCES
[1] Tay, S., 2012, “Analysis and modeling of GNSS signals in marine environment, ” Theses., TELECOM Bretagne.
[2] Viandier, N., 2011, “Modeling and use of GNSS pseudorange errors in transport environment to enhance the localization performances”. Theses., Central graduate School of Lille.
[3] Bhatti, U. I., 2007, “Improved integrity algorithms for the integrated GPS/INS systems in the presence of slowly growing errors”, Imperial College London.
[4] Hofmann, B., Lichtenegger, H., and Wasle, E., 2007,
“GNSS-Global Navigation Satellite Systems: GPS, GLONASS, Galileo, and more”. Springer., pp. 518.
[5] Saha, K., Parameswaran, K., and Suresh Raju, C., 2007, “Tropospheric delay in microwave propagation for tropical atmosphere based on data from the Indian subcontinent”. Journal of Atmospheric and Solar- Terrestrial Physics., vol.69, pp. 875‑905.
[6] Misra, P., and Enge, P., 2006, “Global positioning system: signals, measurements and performance”, pp.
608.
[7] Gradinarsky, L. P., Johansson, J.M., Bouma, H.R., Scherneck, H.-G., and Elgered, G., 2002, “Climate monitoring using GPS”. Elsevier., vol. 27, pp. 335- 340.
[8] Byunet, S. H., andBar-Sever, Y. E., 2009, “A new type of troposphere zenith path delay product of the
international GNSS service”. J.Geod., vol. 83, pp. 1- 7.
[9] Sakidin, H., and chuanTay, C., 2012,
“Transformation of neill mapping function for GPS tropospheric delay”. Journal of Engineering and Technology., pp. 1-11.
[10] Luo, X., Mayer, M., and Heck, B., 2008, “Improving the stochastic model of GNSS observations by means of SNR-based weighting”. Springer., pp. 725-734.
[11] Jin, S.G., Luo, O., and Ren, C., 2010, “Effects of physical correlations on long-distance GPS positioning and zenith tropospheric delay estimates”.
Advances in Space Research., vol. 46, pp. 190-195.
[12] Luo, X., Heck, B., and Awange, J. L., 2013,
“Improving the estimation of zenith dry tropospheric delays using regional surface meteorological data”, Advances in Space Research., vol. 52, pp. 2204- 2214.
[13] Karabatic, A., Weber, R., and Haiden, T., 2011,
“Near real-time estimation of tropospheric water vapour content from ground based GNSS data and its potential contribution to weather now-casting in Austria”, Advances in Space Research., vol. 47, pp.
1691-1703.
[14] Hopfield, H. S., 1969, “Two-quartic tropospheric refractivity profile for correcting satellite data”.
Journal of Geophysical research., vol. 74, pp. 4487- 4499.
[15] Saastamoinen, J., 1972, “Atmospheric correction for the troposphere and stratosphere in radio ranging satellites”. The use of artificial satellites for geodesy., pp. 247-251.
[16] Ifadis, I., 2000, “A new approach to mapping the atmospheric effect for GPS observations”. Earth planets space., vol.52, pp. 703-708.
[17] Boehm, J., Niell, A., Tregoning, P., and Schuh, H., 2006, “Global Mapping Function (GMF): A new empirical mapping function based on numerical weather model data”, Geophysical Research Letters., vol. 33.
[18] Boehm, J., Kouba, J., and Schuh, H., 2009, “Forecast Vienna Mapping Functions for real-time analysis of space geodetic observations”, Journal of Geodesy., vol. 83, pp. 397-401.
[19] Tuka, A., and El-Mowafy, A., 2013, “Performance Evaluation of different Troposphere Delay Models and Mapping Functions”, Measurement 46., pp. 928- 937.
[20] Xu, C.Q., Shi, J.B., Guo, J.M., andXu, X.H., 2011,
“Analysis of Combining Ground-based GPS Network and Space-based COSMIC Occultation Observation for Precipitable Water Vapor Application Within China”, Geomatics Inf. Sci, Wuhan Univ., vol. 36, pp. 467-470.
[21] Yao, Y.B., Xu, C.Q., Zhang, B., and Cao, N., 2014,
“A new Global Empirical Model for Mapping Zenith Wet Delays onto Precipitable Water Vapor”, International journal of Geophysics., vol.197, pp.202-212.
[22] Dodo, JD., and Idowu, TO., 2010, “Regional Assessment of the GPS Tropospheric Delay Models on the African GNSS Network”, Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS)., pp. 113-121.
