Estimating the precipitable water vapour using Romania's permanent GPS network.

Olteanu, Vlad ; Badea, Alexandru ; Moise, Cristian 等

1. INTRODUCTION

The atmosphere influences the GPS signal when this passes through the ionosphere and the troposphere. The troposphere delay is divided into two parts based on the causes of the delay; therefore we have a wet delay caused by the presence of water vapour in the troposphere and a dry (or hydrostatic) part caused by the presence of other atmospheric constituents.

[delta][S.sup.i.sub.A[TROP]] - [delta][S.sup.i.sub.A[HYD]] + [delta][S.sup.i.sub.A[WET]] (1)

We should mention that a more complex approach exists, approach that presumes the dividing of the wet and dry delays once more by taking into consideration the azimuth symmetric and asymmetric delay components (Schuler, 2001).

The hydrostatic delay is known to have values of about 230 cm for a ground pressure of 1000 mb (Rocken, 1995). This part of the delay may be computed with sufficient precision, if meteorological observations are available at the GPS site as well. The wet part of the delay takes values between 0 and about 40 cm. This is very difficult to model due to the irregular distribution of water vapour in the atmosphere. To estimate this value, another parameter will be introduced in the mathematical model of GPS observation processing, parameter that will represent the Zenith Wet Delay. To determine the delay for a certain elevation angle, the obtained parameter is multiplied by a "mapping function". The total delay given by (1) may now be expressed as:

[delta][S.sup.i.sub.A[TROP]] = m[([[epsilon].sup.i.sub.A]).sub.[HYD]] x [ZHD.sub.A] + m[([[epsilon].sub.i.sub.A].sub.[WET]] x [ZWD.sub.A] (2)

Here, [epsilon] denotes the satellite elevation angle.

Thus for each GPS receiver of a network, the Zenith Wet Delay (ZWD) may be computed, ZWD that may be further transformed into an estimate of the PWV, by multiplying it with a value determined as a function of constants related to the refractivity of moist air and the temperature. The computation of the ZHD may be done with the aid of the models developed so far; we mention here the Hopfield model, the Saastamoinen model, etc. We present here the Saastamoinen function for computing the ZHD:

The pressure, the temperature and the relative humidity values may be taken from the RINEX meteorological files.

This paper outlines the basics steps in estimating the zenith wet delays and in transforming them into PWV values. It also presents the way in which this process was implemented in our developed "NWPBuilder" software.

2. ESTIMATION MODEL

The estimation model that we present here is the classic Least Squares method, with the mention that a Kalman filtering approach is more adequate. This kind of estimation is used by the IfEN's TropAC software and by the JPL's GIPSY (but this is based on un-differenced observations) as well. The MIT's processing software GAMIT uses the Least Squares estimation that we also used.

To eliminate or diminish some of the errors, the functional model is based on the double difference equation and on a linear combination between the observations on L1 and L2. The linear combination is chosen in such a way to reduce or eliminate the first ionospheric effect (Teunissen, 1998).

The functional un-linear model is therefore given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where v represents the residual, [DELTA][[phi].sub.AB.sup.ij] stands for the phase measurement (on the corresponding carrier frequency and expressed in cycles) between ground stations A and B and the satellites i and j, [a.sub.1] and [a.sub.2] are the linear combination's coefficients (in our case [a.sub.1] = 1 and [a.sub.2] = [f.sub.2]/[f.sub.1]).

Several parameters may be estimated depending on the user's necessities and preferences. In our case besides the zenith wet delays that we are interested in, the ground stations' coordinates and the ambiguities (actually the double differenced and linearly combined ambiguities) are estimated as well. For more complex approaches, but also more time and resource consuming, additional parameters may be estimated, such as orbit relaxations, azimuth dependent terms, etc (Hoffman--Wellenhof, 1997).

For the zenith wet delays, the corresponding coefficients that will be introduced in the design matrix may be computed with:

[[omega].sup.ij.sub.A] = [m[([[epsilon].sup.j.sub.A]).sub.[WET]] - m[([[epsilon].sup.i.sub.A]).sub.WET]] x ([a.sub.1]/[[lambda].sub.1] - [a.sub.2]/[[lambda].sub.2]) (5)

[[omega].sup.ij.sub.B] = [m[([[epsilon].sup.i.sub.B]).sub.WET]] - m[([[epsilon].sup.j.sub.B]).sub.[WET]] x ([a.sub.1]/[[lambda].sub.1] - [a.sub.2]/[[lambda].sub.2]) (6)

Transforming the obtained zenith wet delays into precipitable water vapour may now be done if the weighted mean temperature of the atmosphere is known ([T.sub.m]). This is done with the aid of the following relation:

PWV = [10.sup.6]/[rho] x [R.sub.v] x [[k.sub.3]/[T.sub.m] + [k'.sub.2]] x ZWD (7)

An important problem that may appear in the estimation is the so called "convergence problem" (Schuler, 2001). For short baselines the satellites are viewed under almost equal elevation angles. Taking into consideration the relations (5) and (6) the values of the coefficients will be rather small causing the impossibility of sensing absolute values of the zenith wet delays. Only relative delays will be sensed. These may be afterwards transformed into absolute delays if at least one absolute value is known. In this case the model of estimation should be changed. The problem may be easily remarked in:

[DELTA][delta][S.sub.[WET]] - m([[epsilon].sup.i.sub.B]) x [ZWD.sub.B] -m{[[epsilon].sup.i.sub.A]) x [ZWD.sub.A] [congruent to]

[congruent to] m([[epsilon].sup.t]). x ([ZWD.sub.B] - [ZWD.sub.A]) = m([[epsilon].sup.i]) x [DELTA][ZWD.sub.AB] (8)

Therefore, the implemented software will need to go through the following steps, as they are presented in the scheme below (Fig. 1).

3. CONCLUSIONS

Our software, "NWP Builder", is an application that was developed based on the steps presented above. However, we have to underline that this application was especially created especially for processing baselines from EUREF and IGS permanent stations network, stations that are also equipped with meteorological sensors.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Thus, the parameters resulted from the estimation process are the zenith wet delays and the ambiguities.

In the tests that we are presenting here we used observations from 3 permanent stations: BAIA (Romania), DEVA (Romania) and PDEL (Portugal). The observation files were downloaded from the GNSS Data Center site (http://igs.bkg.bund.de/). The observations were made on the 29th of May 2008.

The BAIA--DEVA baseline has a length of about 200 km. In this case, we could only determine the relative zenith wet delay. This is why we used a third site which is further away from the other two. The BAIA--PDEL baseline has a length of about 4000 km. This baseline was processed first in order to obtain the absolute values for the zenith delay for the two other sites. With their help we obtained the absolute values for the DEVA site. The results over the 12 hours of observations that were processed are shown in the next figure (Fig. 2).

Today from all the available stations within Romania's permanent GPS network only 9 are equipped with meteorological sensors. In time their number will increase and therefore it will be useful to determine the PWV at all these stations and use it in numerical weather and climate predictions.

We should mention that we intend to further develop our software and somehow combine it with the Virtual GPS Station Technique in order to improve the possibilities of positioning.

4. REFERENCES

Hoffman--Wellenhof, B. et al. (1997). GPS Theory and practice--fourth revised edition, Springer, ISBN 3-21182839-7, Vienna--New York

Rocken, C. et al (1995). GPS/STORM--GPS Sensing of Atmospheric Water Vapour for Meteorology, Journal of Atmospheric and Oceanic Technology, 12, 468-478

Schuler, T. et al. (2001)--GNSS Zenith Wet Delay Estimation Considering Their Stochastic Properties, Available from: http://forschung.unibw muenchen.de/papers/khtesmcixcat5b37vhrgygjtenr6tm.pdf, Accessed: 2007-10-01

Schuler, T. (2001) On Ground--Based GPS Tropospheric Delay Estimation, Available from: http://137.193.200.177/ediss/schueler-torben/inhalt.pdf Accessed: 2008-02-01

Teunissen, P.J.G., Kleusberg, A. (1998)--GPS for Geodesy, Springer Verlag, ISBN 3-540-63661-7, Germany