On the relation between the wet delay and the integrated precipitable water vapour in the European atmosphere

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1 Meteorol. Appl. 7, (000) On the relation between the wet delay and the integrated precipitable water vapour in the European atmosphere T Ragne Emardson, Onsala Space Observatory, Chalmers University of Technology, SE Onsala, Sweden (now at Jet Propulsion Laboratory, California Institute of Technology, Mail Stop , 4800 Oak Grove Drive, Pasadena, CA 91109) Henrico J P Derks, Royal Netherlands Meteorological Institute, PO Box 01, 3730 AE De Bilt, Netherlands (now at Meteor Burst Communication Europe BV, Kerkenbos 10-47, 6546 BB Nijmegen, Netherlands) A ground-based network of Global Positioning System (GPS) receivers can be used to determine the excess propagation path of the GPS signals. Using surface pressure data it is possible to derive the excess path caused by water vapour. This quantity is often referred to as the wet delay. We have studied the relationship between the wet delay and the integrated precipitable water vapour (IPWV), based on more than radiosonde profiles from 38 sites in Europe. We have studied four different models based on the surface temperature, the latitude of the site, and the time of the year in different combinations. The accuracies of the models have been evaluated and also compared with the accuracy resulting from the use of published model parameters derived from radiosonde data acquired in the United States. The models using the surface temperature data yielded the smallest relative root mean square errors of 1.14% and 1.15% respectively. We find that the accuracy can sometimes be further improved, typically down to 1.0%, by adjusting the values of the model parameters to the area or site of interest. 1. Introduction The technique of using the Global Positioning System (GPS) to determine the amount of integrated precipitable water vapour (IPWV) has been demonstrated by, for example, Duan et al. (1996). The IPWV is the amount of water vapour represented by the height of the equivalent column of liquid water. This quantity is interesting for several meteorological applications. Information on the IPWV can be used as, for example, a constraint in numerical weather prediction (NWP) data assimilation or serve as an independent data source to validate NWP models. Furthermore, the IPWV is a useful tool when searching for trends in climate change. To determine the IPWV from GPS data, the total atmospheric zenith delay is estimated. Using surface pressure information, the total atmospheric zenith delay can be divided into a hydrostatic term and a term accounting for the wet delay. The hydrostatic delay, which is derived by applying the condition that hydrostatic equilibrium is satisfied, depends on the total weight of the atmosphere above. The wet delay is the propagation delay experienced by GPS signals due mainly to water vapour. The physical relation for the quotient Q, between the wet delay (l w ) and the IPWV (I) is given by Askne & Nordius (1987): l w v [ 3 m ] 8 Q = = 10 ρ R ( k / T ) + k ( 1) I where ρ is the density of liquid water, R v is the specific gas constant of water vapour, equal to J kg 1 K 1, and the atmospheric refractivity constants k 3 and k are approximately K mbar 1 and K mbar 1 respectively. T m is defined by v dz Tm = ρ ρ d z T where ρ v is the water vapour density, T is the temperature, and z is the vertical coordinate. To obtain the value of T m, water vapour pressure and temperature information along the vertical profile are required. However, Bevis et al. (199) derived a linear relation between the surface temperature T 0 and T m, based on almost 9000 radiosonde profiles from sites in the United States: T m = T 0 This relationship was then used to determine Q from equation (1). Emardson et al. (1998) derived a relation between the wet delay and the IPWV based on site latitude and the day of the year. In that study almost radiosonde profiles from the Scandinavian countries were used. v 61

2 T R Emardson and H J P Derks As the existing models are based on data from North America and Scandinavia, one can question whether these relations hold for all European sites. Hence the goal of our study was to investigate the validity of the existing models and to derive new models based upon European radiosonde data. We have evaluated the existing models as well as two new models. In the following section we describe the different models. We then present the results in sections 3 and 4, which are followed by conclusions in the last section.. Experimental methods We used radiosonde data from 38 different sites in Europe, spanning in time from 1989 to Figure 1 shows the location of the radiosonde launching sites. We calculated the wet delay and IPWV for each of the over profiles and archived the values together with the actual surface temperature. Of the total number of profiles, some hundred contained erroneous surface temperature measurements. These profiles were therefore removed from the data set before the analysis. With the remaining data we have estimated the values of different parameters in four models relating the wet delay to the IPWV. Two of these models use as input the surface temperature only, one model uses the site latitude and the time of year, and one uses both temperature and the time of year. (a) Simplified physical model Assuming a linear relation between T m and the surface temperature, T m = c 0 + c 1 (T a + T ) where T a is the mean surface temperature and T is the surface temperature minus the mean surface temperature for the area in kelvins, we can simplify the actual physical relation given in equation (1) to: a1 Qphysical = a0 + ( ) a + T where Q is in mm/mm. Here the coefficients a 0, a 1, and a, could correspond to: 8 8 a R k a R k 3 c0 0 = 10 ρ v 1 = 10 ρ v a T c = + c a (b) Polynomial model Using a Taylor series expansion on equation () we are able to write Q as power series of T : Q polynomial = a 0 + a 1 T + a T (3) Such a model is also easily motivated by plotting Q against the surface temperature. Figure shows this relation for the actual data set. (c) Annual model 1 1 The third relationship is based only on the annual change in Q due naturally to the seasonal variation in temperature. This model is the same as presented by Emardson et al. (1998) for a Swedish data set: t D Qannual = a + a + a θ sin π 365 t D a3 cos π ( 4) 365 where θ is the site latitude in degrees and t D is the decimal day of the year. Figure 1. Map showing the locations where radiosondes were launched. This map was produced with the GMT software (Wessel & Smith, 1995). 6 Figure. Q as a function of surface temperature. The solid curve represents the best fit to the data using the polynomial model.

3 Relation between wet delay and precipitable water (d) Hybrid model The last model is a combination of the polynomial and annual models and uses the fact that, after the temperature dependency has been removed from Q in equation (), there still exists an annual effect in the residuals: Q = a + at + a T + a + a hybrid The coefficients in these formulae have been estimated to minimise the residuals using the method of least squares. The area enclosed by the radiosonde data set includes different climatological regions. An effort was made to increase the accuracy of the new models, by dividing the radiosonde launch sites into four groups: a Mediterranean group, a Central European group, a Baltic Sea group, and an Atlantic group. The area can be divided in many different ways; the division in four regions made above is just one of many reasonable options. In addition to the regional optimisation, the coefficients of the new models were optimised by using the radiosonde data from each station separately. 3. Evaluation of models 0 1 3θ t D td a + sin π cos 365 π ( 5) Table 1 contains the values of the estimated coefficients for the four models given by equations () to (5), and the corresponding root mean square (RMS) error of the fit to the data set. We can see that the results for the three models using the surface temperature are very similar. For this data set, using the T m relation derived by Bevis et al (199) in equation (1): Q = T yields an RMS error of 1.47%. We believe that this 0 higher RMS value lies within the uncertainty of Q due to uncertainties in the parameters k and k 3. Figure 3 shows the Q residuals as a function of temperature after the four models have been applied. The effect of using temperature information directly, instead of using the Q annual dependency, is visible when the results of the models are compared. The annual model cannot track short time variations (days weeks) in the changes of warm and cold air masses. The explanation for this behaviour can be found in Figure 4, which shows Q, for the site Bordeaux, as a function of time. Figures 4(a) and 4(b) also show the annual and hybrid models respectively. As can be seen by comparing these two plots, both models follow the seasonal variations in Q. For the high and low values of Q, the annual model does not adapt as the hybrid model does. The advantage of having temperature measurements in the modelling of Q is, however, dependent on the accuracy of these measurements. A measurement error of 3 K in temperature yields an error in Q of 0.05 at a temperature of 73 K, which means that the error in the IPWV estimate would be 0.5 mm for a wet delay value of 0 cm. Adding this root-sum-squared to the error contributed by the polynomial model itself gives an IPWV RMS error of approximately the same size as the error from the annual model. Hence the temperature measurements at the GPS sites must have an RMS error better than 3 K in order to improve the results of the IPWV determination when using a temperature-dependent model instead of an annual model. The accuracy of operational meteorological sensors is in the order of 0.1 K. 4. Regional modelling For regional modelling a preliminary study of the results showed a poor agreement for some of the sites. We therefore moved the sites for which this occurred to adjacent regions and repeated the optimisation. For example, Hemsby was moved from the Atlantic to the Central European region. Tables and 3 show the estimated values for the coefficients in the polynomial and hybrid models using data from the four regional areas Table 1. Values of the coefficients used in the four models. The σ values are the formal errors, scaled so that χ per degree of freedom is one. The mean temperature is K. Model Coefficient Physical Polynomial Annual Hybrid Value σ Value σ Value σ Value σ a a a a a RMS (%)

4 T R Emardson and H J P Derks Figure 3. Q residuals as a function of temperature for (a) the simplified physical model, (b) the polynomial model, (c) the annual model, and (d) the hybrid model. Table. Values of the coefficients used in the polynomial model for four regions. T a is the mean surface temperature for the different data sets. Table 3. Values of the coefficients used in the hybrid model for four regions. For the mean surface temperature see Table. Region Number a 0 a 1 a T a of profiles (10 K 1 ) (10 5 K 1 ) (K) Baltic Central Atlantic Mediterranean Region a 0 a 1 a a 3 a 4 (10 K 1 ) (10 5 K 1 ) (10 ) (10 ) Baltic Central Atlantic Mediterranean defined. Table 4 gives an overview of the results for the polynomial model for all the sites used in the data set. This table shows the sites their region and number of profiles. The next three columns show RMS errors for the different sites using the polynomial model. In the first of these columns (labelled Europe ) the parameters in the model were estimated using the entire data set. The second column (labelled Region ) contains the RMS errors when the parameters were estimated by using data only from the region, and the third (labelled Site ) refers to when data were taken only from the actual station. It is clear that the Site RMS errors are lower than those for Region, which are in general lower than those for Europe. Table 4 also summarises all the optimised model parameters corresponding to Site results. The differences are not, however, very large compared with the improvement obtained by using the European instead of the North American models. This indicates that the model estimated for the whole of Europe is valid for most of the continent. For six of the sites, Madrid, Thessaloníki, Valentia, Visby, Leba, and Smolensk, the results are better when we use the European model compared with the regional one. In Leba, Thessaloníki, and Smolensk the explanation is a bias between the data and the regional model. If the model parameters for the Baltic area are used for the Smolensk data, this bias disappears and the RMS error is as small as 1.04%. The scatter increases, however, when using the Baltic model. The Madrid data fit poorly to the Mediterranean model. If we compare 64

5 Relation between wet delay and precipitable water Figure 4. Q time series for the site Bordeaux for (a) the annual model and (b) the hybrid model. them with the central region model, the RMS error is 1.04%. We presume this is because of the inland climate of Madrid, which differs significantly from that of the coastal Mediterranean sites. Table 5 summarises the results for the hybrid model. This table contains the same information as Table 4. As we stated before, the RMS errors of both the polynomial and hybrid models are approximately the same when using the entire data set. For the regional and the site-dependent modelling, however, the overall RMS errors are 1.06% and 1.0%, compared with 1.09% and 1.06% for the polynomial model. This shows that the hybrid model performs better than the polynomial model when smaller areas are concerned. This can be seen, for example, at the station Gibraltar. Using the polynomial model, the RMS error can be reduced from 1.36% for the European model to 1.11% for the site specific case. A further improvement to an RMS error of 1.01% can be obtained by using the hybrid model optimised for this site. The reason for the better results using the hybrid model, we presume, lies in the different height dependency of the temperature at different sites. Since we use the surface temperature to model the mean atmospheric temperature, we are sensitive to the different lapse rates at the sites. For a certain temperature at the surface, the mean atmospheric temperature will of course depend on how rapidly the temperature 65

6 T R Emardson and H J P Derks Table 4. Region and number of profiles plus the RMS errors using the polynomial model for the different sites, using coefficients based on data from the entire continent, the region, and just the actual site. Also shown are the values of the coefficients used in the model for all the sites. The parameters for the continent and region are given in Tables 1 and. T a is the mean surface temperature for the different data sets. The stations are allocated to the following regions: Mediterranean (M), Central European (C), Baltic Sea (B), and Atlantic (A). RMS error Coefficients of polynomial model Station* Region Number Europe Region Site a 0 a 1 a T a of profiles (10 K 1 ) (10 4 K ) (K) Athínai M Bordeaux C Brindisi M Bromma B Bucareşti C De Bilt C Ekofisk A Frösön B Gardermoen B Gibraltar M Hannover C Hemsby C Jokioinen B Jyväskylä B Keflavík A København B La Coruña M Landvetter B Leba C Lisboa M Luleå B Madrid M München C Nîmes C Ny Ålesund A Payerne C Riga B Roma M Smolensk C Sodankylä B Sankt-Peterburg B Sundsvall B Thessaloníki M Thule A Uccle C Valentia A Visby B Wien C Weighted means RMS * Station names are according to The Times Atlas of the World, 199 decreases with the height. Figure 5 shows, as an example, the lapse rates for the sites Athínai, Bordeaux, and Keflavík. The lapse rates shown are calculated as the mean temperature decrease with height from the ground to a height of 5 km, which is the height interval where most of the water vapour is found. To improve the clarity, the three curves in the figure are smoothed using a moving average with a window size of ten days. As can be seen from the figure, the lapse rates vary both seasonally and between the different sites. Using the polynomial model, the lapse rate sensitivity cannot be 66 accounted for, whereas the hybrid model is capable of modelling part of the seasonal dependency through its sine and cosine terms. The fact that geographical regions may contain sites with different climatological conditions should induce some caution in the creation of regional models. One could consider making new groups, using not only geographical selection criteria, but climatological criteria as well. On the other hand, such work would require every GPS site to be classified according to climato-

7 Relation between wet delay and precipitable water Table 5. RMS errors using the hybrid model for the different sites, using coefficients based on data from the entire continent, the region and just the actual site. Also shown are the values of the coefficients used in the model for all the sites. The parameters for the continent and region are given in Tables 1 and 3. The stations are allocated to the following regions: Mediterranean (M), Central European (C), Baltic Sea (B), and Atlantic (A). RMS error Coefficients used in the hybrid model Station* Region Number Europe Region Site a 0 a 1 a a 3 a 4 of profiles (10 K 1 ) (10 4 K ) (10 ) (10 ) Athínai M Bordeaux C Brindisi M Bromma B Bucareşti C De Bilt C Ekofisk A Frösön B Gardermoen B Gibraltar M Hannover C Hemsby C Jokioinen B Jyväskylä B Keflavík A København B La Coruña M Landvetter B Leba C Lisboa M Luleå B Madrid M München C Nîmes C Ny Ålesund A Payerne C Riga B Roma M Smolensk C Sodankylä B Sankt-Peterburg B Sundsvall B Thessaloníki M Thule A Uccle C Valentia A Visby B Wien C Weighted means RMS * Station names are according to The Times Atlas of the World, 199 logical group in order to use the proper coefficients. Therefore such groups are not studied further in this paper. 5. Conclusions In this study we have used profiles from sites in Europe to derive conversion formulae from the wet delay to the integrated precipitable water vapour (IPWV) based on surface temperature measurements, site latitude, and time of the year in different combinations, which we compare with the existing formulae. We show that the conversion factor Q can be determined with an RMS error of just above 1% for European stations. In view of these results, we recommend that European GPS users employ one of the models presented here. At most of the sites the estimated fit to the data was improved when we estimated the coefficients by using data from a region around the station. However, a disadvantage of such a division is the fact that stations enclosed by the same geographical region may differ climatologically. Hence it is possible that for any given site in Europe the European model gives bet- 67

8 T R Emardson and H J P Derks Figure 5. The atmospheric lapse rate as a function of day number for the sites Athínai (plus signs), Bordeaux (squares) and Keflavík (circles). ter results than do the regional models. To avoid this problem, a site-specific model can be used. One should notice, though, that the benefit of using smaller regions or site-specific models is in general small. A next step in the development of the relation between wet delay and IPWV would be the use of information from numerical weather prediction models. The mean atmospheric temperature T m could be generated by meteorological services, for example. The direct use of path delay within numerical weather prediction models would be an even greater step forward. Therefore, efforts should be made to assimilate the path delay into the weather prediction models. Even so, there will always be a need for conversion formulae from wet delay to IPWV when GPS estimates are to be compared with those from other instruments which measure the IPWV directly. Acknowledgments We would like to thank Bob Riddaway and two anonymous referees for their valuable comments. The data were supplied by the Deutscher Wetterdienst (DWD), UK Met. Office (UKMO), Swedish Meteorological and Hydrological Institute (SMHI), the Royal Netherlands Meteorological Institute (KNMI), and Fondazione Ugo Bordoni. The integration of the radiosonde profiles was performed by Gunnar Elgered. This work has been carried out as part of the WAVEFRONT Project, which is funded by the European Commission Environment and Climate Programme (EC Contract ENV4-CT ). The Project is a collaboration between the IESSG, University of Nottingham (UK), Onsala Space Observatory, CUT (Sweden), ETH (Swiss Federal Institute of Technology, Switzerland), and CSIC, Institut d Estudis Espacials de Catalunya (Spain), in association with the Astronomical Institute at the University of Berne (Switzerland), the Meteorological Office (UK) and the Danish Meteorological Institute (Denmark). Additional support was obtained from the Swedish National Space Board and Netherlands Remote Sensing Board. References Askne, J. & Nordius, H. (1987). Estimation of tropospheric delay for microwaves from surface weather data. Radio Sci., : Bevis, M., Businger, S., Herring, T. A., Rocken, C., Anthes, R. A. & Ware, R. H. (199). GPS meteorology: remote sensing of atmospheric water vapor using the Global Positioning System. J. Geophys. Res., 97: Duan, J., Bevis, M., Fang, P., Bock, Y., Chiswell, S., Businger, S., Rocken, C., Solheim, F., VanHove, T., Ware, R., McClusky, S., Herring, T. A. & King, R. W. (1996). GPS Meteorology: direct estimation of the absolute value of precipitable water. J. Appl. Meteorol., 35: Emardson, T. R., Elgered, G. & Johansson, J. M. (1998). Three months of continuous monitoring of atmospheric water vapor with a network of Global Positioning System receivers. J. Geophys. Res., 103: Wessel, P. & Smith, W. H. F. (1995). New version of the generic mapping tools released. EOS Trans., 76: